Debugging Techniques
for Declarative Languages:
Profiling, Program Slicing, and Algorithmic Debugging
Josep Francesc Silva Galiana
Departamento de Sistemas Informáticos y Computación
Universidad Politécnica de Valencia
Memoria presentada para optar al tı́tulo de:
Doctor en Informática
Director:
Germán Vidal Oriola
Valencia, 2007
A Julio y Nina, que me dieron a Vanessa
A mi padre, mi guı́a
Prologue
I enjoy reading books about neutron stars and black holes, and one of my best friends
use to say that I am losing my time because I will never see a black hole, and it
is stupid to study something that will never influence my live... One hundred years
ago people used to read science fiction about the space and, nowadays, hundreds of
satellites are rounding the Earth: what is a dream today will be a fact tomorrow.
This is the way in which researchers must think, since much of their work is useless
today but, possibly, another researcher will make it useful tomorrow. In fact, this is
an annoying property of research: sometimes it is blind. When Fleming discovered
penicillin after the first world war, he did not know that he was going to save millions
of lives, and the scientific community did not realize about the importance of his
discovery until the second world war. Most of the results of this thesis has been
implemented and applied to a real language, thus producing useful technology; but
others simply no. Some results are theoretical results that I have not been able to
apply and I simply hope they will become useful in the future.
Some of the most important discoveries have been discovered accidentally or casually. When Archimedes discovered the Archimedes’ principle he was having a bath;
and Roentgen switched off the light and saw a glow from a drawer before he discovered X rays. Others, on the other hand, have been discovered after many years of
hard work. Einstein spent five years to generalize the relativity theory; and Mendel
had to plant and experiment for decades with thousands of pea before he arrived to
a definite conclusion and stated the principles of genetics.
Unfortunately (or maybe fortunately), this thesis belongs to the second group. I
started to study the principles of debugging techniques for declarative programming
in March of 2002. I got a grant in the Technical University of Valencia to participate
in the research of the ELP research group, and I went for the first time in the office
of my advisor German Vidal. He just asked me: “What do you like more, writing
programs or correcting them? ” In that moment I did not realize that my answer was
going to mark my future for at least the next four years...
v
vi
Debugging Techniques
for
Declarative Languages
vii
viii
Abstract
The task of debugging can be arduous. A bug can be evident with a single glance, or it
can be hidden in the deepest lurking place of our program. Nevertheless, surprisingly,
debugging is one of the software processes that has been mostly abandoned by the
scientific community, and the same debugging techniques used twenty years ago are
still being used today.
The situation is not different regarding declarative languages. Or it is indeed
worst; because declarative languages can pose additional difficulties due, for instance,
to the lazy evaluation mechanism.
In this thesis, we revise the current debugging methods for declarative languages
and we develop some new methods and techniques which are based on profiling,
program slicing, and algorithmic debugging. In short, the main contributions of the
thesis are:
• The definition of a profiling scheme for functional logic programs which is based
on the use of cost centers and that allows us to measure different kinds of
symbolic costs.
• The formulation of a new dynamic slicing technique based on redex trails, its
application to debugging and its adaptation for the specialization of modern
multi-paradigm declarative programs.
• The introduction of a new algorithmic debugging scheme which combines conventional algorithmic debugging with program slicing.
• The definition of three new strategies for algorithmic debugging.
• The development of a comparative study and a subsequent classification of program slicing techniques and algorithmic debugging strategies.
Part of the material included in this thesis has been previously presented in [Albert
et al., 2003; Braßel et al., 2004a; Ochoa et al., 2004a, 2005, 2006; Silva and Chitil,
2006; Silva, 2006].
ix
x
Resumen
La depuración de programas puede ser una tarea ardua. Esto se debe fundamentalmente a que los errores pueden ser evidentes al primer vistazo, o pueden estar escondidos en las zonas más profundas y ocultas de nuestros programas. Sin embargo,
sorprendentemente, la depuración es uno de los procesos software que ha sido menos
tratado por la comunidad cientı́fica; de hecho, las mismas técnicas de depuración que
se usaban hace veinte años, continúan utilizándose en la actualidad.
La situación no es diferente en el contexto de los lenguajes declarativos. O es
incluso peor, porque estos lenguajes suelen presentar dificultades adicionales a la
hora de depurarlos debido, por ejemplo, al mecanismo de evaluación perezosa.
En esta tesis se revisan los métodos actuales de depuración para lenguajes declarativos y se desarrollan nuevos métodos y técnicas basadas en el cómputo de costes, la
fragmentación de programas y la depuración algorı́tmica. Brevemente, las principales
contribuciones de la tesis son:
• La definición de un esquema de cómputo de costes para programas lógico funcionales que está basado en el uso de centros de coste y que permite medir
diferentes tipos de costes simbólicos.
• La formulación de una nueva técnica de fragmentación dinámica de programas
basada en redex trails, su aplicación a la depuración y su adaptación para la
especialización de programas declarativos multi-paradigma modernos.
• La introducción de un nuevo esquema de depuración algorı́tmica que combina
la depuración algorı́tmica convencional con la fragmentación de programas.
• La definición de tres nuevas estrategias para depuración algorı́tmica.
• El desarrollo de un estudio comparativo y una posterior clasificación de técnicas
de fragmentación de programas y estrategias de depuración algorı́tmica.
Parte del material de esta tesis ha sido presentado en [Albert et al., 2003; Braßel
et al., 2004a; Ochoa et al., 2004a, 2005, 2006; Silva and Chitil, 2006; Silva, 2006].
xi
xii
Resum
La depuració de programes pot ser una tasca àrdua. Açò es deu fundamentalment
a què els errors poden esdevindre evidents amb un sol colp d’ull o bé, poden estar
amagats en els fragments més profunds i ocults del nostre codi. En canvi, sorprenentment, la depuració és un dels procesos del programari que han estat menys tractats
per la comunitat cientı́fica; de fet, les mateixes tècniques de depuració que s’empraven
fa vint anys, encara continuen utilitzant-se en la actualitat.
La situació no és diferent en el context dels llenguatges declaratius. O tal volta
pitjor, perque aquestos llenguatges solen prenentar dificultats addicionals a l’hora de
depurar-los per culpa, per exemple, del mecanisme d’avalluació mandrós.
En aquesta tesi es revisen els mètodes actuals de depuració per a llenguatges
declaratius i es desenvolupen nous mètodes i noves tècniques basades amb el còmput
de costs, la fragmentació de programes i la depuració algorismica. Breument, les
principals contribucions de la tesis són:
• La definició d’un esquema de còmput per a programes lògic-funcionals que està
basat amb l’ús de centres de cost i permet mesurar els diferents tipus de costs
simbòlics.
• La formulació d’una nova tècnica de fragmentació dinàmica de programes basada
en “redex-trails”, la seua aplicació a la depuració declarativa i la seua adaptació
per a l’especialització de programes declaratius multi-paradigma moderns.
• La introducció d’un nou esquema de depuració algorı́smica que combina la depuració d’algorismes convencionals amb la fragmentació de programes.
• La definició de tres noves estratègies per a depuració algorı́smica.
• El desenvolupament d’un estudi comparatiu i una posterior classificació de
tècniques de fragmentació de programes i estratègies de depuració algorı́smica.
Part del material d’aquesta tesi ha estat presentat a [Albert et al., 2003; Braßel
et al., 2004a; Ochoa et al., 2004a, 2005, 2006; Silva and Chitil, 2006; Silva, 2006].
xiii
Contents
1 Introduction
1.1 Debugging Declarative Languages .
1.2 Debugging Techniques . . . . . . .
1.2.1 Traditional Debugging . . .
1.2.2 Profiling . . . . . . . . . . .
1.2.3 Algorithmic Debugging . .
1.2.4 Abstract Debugging . . . .
1.2.5 Tracing . . . . . . . . . . .
1.2.6 Program Slicing . . . . . .
1.3 Structure of the Thesis . . . . . . .
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Foundations
37
2 Preliminaries
2.1 Signature and Terms . . . . . . . . . .
2.2 Substitutions . . . . . . . . . . . . . .
2.3 Term Rewriting Systems . . . . . . . .
2.3.1 Narrowing . . . . . . . . . . . .
2.3.2 Needed Narrowing . . . . . . .
2.4 Declarative Multi-Paradigm Languages
II
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Profiling
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3 Time Equations for Lazy Functional (Logic) Languages
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Cost Augmented Semantics . . . . . . . . . . . . . . . . .
3.3 The Program Transformation . . . . . . . . . . . . . . . .
3.4 The Transformation in Practice . . . . . . . . . . . . . . .
xv
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3.5
3.4.1 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Other Applications and Extensions . . . . . . . . . . . . . . . .
Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Runtime Profiling of Functional Logic Programs
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
4.2 A Runtime Profiling Scheme . . . . . . . . . . . . .
4.3 Cost Semantics . . . . . . . . . . . . . . . . . . . .
4.4 Cost Instrumentation . . . . . . . . . . . . . . . . .
4.5 Implementation . . . . . . . . . . . . . . . . . . . .
4.5.1 Measuring Runtimes . . . . . . . . . . . . .
4.5.2 Extension for Symbolic Profiling . . . . . .
4.6 Related Work and Conclusions . . . . . . . . . . .
III
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Program Slicing
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5 Dynamic Slicing Based on Redex Trails
5.1 Introduction . . . . . . . . . . . . . . . . . .
5.2 Combining Tracing and Slicing . . . . . . .
5.3 Building the Extended Trail . . . . . . . . .
5.3.1 A Semantics for Tracing . . . . . . .
5.3.2 Including Program Positions . . . .
5.3.3 Logical Features . . . . . . . . . . .
5.4 Computing the Slice . . . . . . . . . . . . .
5.4.1 Using the Redex Trail . . . . . . . .
5.4.2 Logical Features . . . . . . . . . . .
5.5 Program Specialization Based on Dynamic
Slicing . . . . . . . . . . . . . . . . . . . . .
5.5.1 Specialization Based on Slicing . . .
5.6 Implementation Issues . . . . . . . . . . . .
5.6.1 Architecture . . . . . . . . . . . . .
5.6.2 The Slicer in Practice . . . . . . . .
5.6.3 Benchmarking the Slicer . . . . . . .
5.7 Related Work . . . . . . . . . . . . . . . . .
IV
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Algorithmic Debugging
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6 Combining Program Slicing and Algorithmic Debugging
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6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
xvi
6.2
6.3
6.4
6.5
6.6
6.7
6.8
The Idea: Improved Algorithmic Debugging .
Computation Graphs . . . . . . . . . . . . . .
The Augmented Redex Trail . . . . . . . . . .
Slicing the ART . . . . . . . . . . . . . . . . .
Combining Program Slicing and Algorithmic
Debugging . . . . . . . . . . . . . . . . . . . .
6.6.1 The Slicing Criterion . . . . . . . . . .
6.6.2 The Slicing Algorithm . . . . . . . . .
6.6.3 Completeness of the Slicing Algorithm
The Technique in Practice . . . . . . . . . . .
Related Work . . . . . . . . . . . . . . . . . .
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7 A Classification of Program Slicing and Algorithmic Debugging Techniques
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7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.2 Classifying Slicing Techniques . . . . . . . . . . . . . . . . . . . . . . . 180
7.3 Three New Algorithmic Debugging Strategies . . . . . . . . . . . . . . 183
7.3.1 Less YES First . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.3.2 Divide by YES & Query . . . . . . . . . . . . . . . . . . . . . . 187
7.3.3 Dynamic Weighting Search . . . . . . . . . . . . . . . . . . . . 188
7.4 Classifying Algorithmic Debugging Strategies . . . . . . . . . . . . . . 191
7.4.1 Combining Algorithmic Debugging Strategies . . . . . . . . . . 191
7.4.2 Comparing Algorithmic Debugging Strategies . . . . . . . . . . 193
8 Conclusions and Future Work
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8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9 Acknowledgements
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10 Agradecimientos
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xvii
xviii
Chapter 1
Introduction
In this chapter, we offer a general overview of the state of the art in our area of interest:
debugging of declarative languages. First, we briefly recall the main approaches and
results found in the literature. Then, we concisely present the main goals of this
thesis.
1.1
Debugging Declarative Languages
In 1998, Philip Wadler published in the Journal of Functional Programming a very
controversial article entitled “why no one uses functional languages” [Wadler, 1998].
He identified the difficulty of debugging such languages as one of the main causes.
In particular—he said—the main difficulty of debugging lazy languages is the lack of
proper debugging environments:
“Constructing debuggers and profilers for lazy languages is recognized as
difficult. Fortunately, there have been great strides in profiler research,
and most implementations of Haskell are now accompanied by usable time
and space profiling tools. But the slow rate of progress on debuggers for
lazy languages makes us researchers look, well, lazy.”
A good tracer, program slicer or algorithmic debugger able to help the programmer to
understand what could be wrong during the execution of a program can considerably
simplify the task of debugging; and, consequently, the time expended in this task.
Many efforts have been done to develop different environments for debugging
declarative programs (see, e.g., [Wallace et al., 2001; Caballero, 2005; MacLarty,
2005]) but the problem still remains. Unfortunately, none of these works have completely solved the problem. However, all these works are necessary to find a solution.
1
2
CHAPTER 1. INTRODUCTION
The techniques presented in this thesis constitute a step forward to solve some of the
problems related to the current state of the art in debugging.
Languages with a lazy evaluation strategy (in the following, lazy languages) impose additional difficulties over languages with an eager evaluation strategy (in the
following, eager languages); in general, these languages lack of suitable debugging
tools or, if they exist, they are mostly useless for programmers and, in practice, they
are hardly used due to their research-oriented nature and lack of stability [Wadler,
1998]. In the particular case of multi-paradigm languages, this lack is even worst
because these languages are relatively young, and, thus, the maturity level of the
debugging tools is sensibly lower.
Declarative multi-paradigm languages combine some of the main features from the
pure functional and logic programming paradigms, as well as from other paradigms
(e.g., concurrency). Integrated multi-paradigm languages allows us to see different
forms of computation as different aspects of a unique computation mechanism. This
versatility eases the task of programming allowing us to use in each aspect of a
problem the most natural solution, which could be based on functional programming,
logic programming, constraint programming, object-oriented programming, etc.
Among the most important multi-paradigm languages one can find Curry [Hanus,
1997, 2006a], Toy [López-Fraguas and Sánchez-Hernández, 1999], Mercury [MacLarty,
2005], and Oz [Muller et al., 1995]. The techniques proposed in this work could be
adapted to any functional logic language, as well as to pure logic and pure functional
languages; however, to be concrete, the developments and implementations described
have been based on the functional logic language Curry.
Curry is a language that combines elements from functional programming and
logic programming:
Functional Programming (e.g., Haskell [Peyton-Jones, 2003]) is based on the idea
that a computation consists on the evaluation (with referential transparency)
of expressions. The main contributions from this paradigm are higher-order
functions, function composition and the use of types.
Logic Programming (e.g., Prolog [Sterling and Shapiro, 1986]) defines computations as a series of deductions. The main contributions from this paradigm are
the logical variables and the search mechanisms.
In Chapter 2 a revision of this language is presented.
This thesis reviews some of the most important debugging methods for multiparadigm declarative languages; then it introduces new techniques to solve some of
the limitations of such languages. In particular, this thesis focuses on three well-known
techniques:
2
1.1. DEBUGGING DECLARATIVE LANGUAGES
3
Program Slicing: It is a technique to extract slices from a given program. A program slice is formed by all those program’s expressions that (potentially) affect
the values computed at some point of interest for the programmer. This point is
known as the slicing criterion. The task of computing program slices is known
as program slicing. The original definition of program slice—introduced by
Mark Weiser in 1979 [Weiser, 1979, 1984]—defined it as an executable program
extracted from another program by deleting sentences in such a way that the
slice reproduces part of the behavior of the original program. See Section 1.2.6
for a deeper explanation.
Profiling: Basically, it consists on the computation of the resources (time, memory,
IO operations, etc.) that programs use during their execution. Despite the
fact that this technique has classically used milliseconds to measure time and
bytes to measure memory, in Chapter 4 we will see a different approach based
on symbolic costs which is able to measure the number of elemental operations
that a program performs during its execution. See Section 1.2.2 for a deeper
explanation.
Algorithmic Debugging: This is a semi-automatic debugging technique which is
based on the answers of the programmer to a series of questions generated
automatically by the algorithmic debugger. After the programmer has answered
some questions, the debugger is able to automatically find the bug in the source
code. See Section 1.2.3 for a deeper explanation.
These techniques have been applied in many other contexts besides program debugging. In particular, program slicing has been applied, e.g., to program debugging
[Lyle, 1984; Lyle and Weiser, 1987; Agrawal, 1991; Shimomura, 1992; Kamkar, 1993],
testing [Laski, 1990; Bates and Horwitz, 1993; Kamkar et al., 1993; Rothermel and
Harrold, 1994; Binkley, 1995, 1998], parallelization [Choi et al., 1991; Weiser, 1983],
detection and removal of dead code [Jones and Métayer, 1989; Reps and Turnidge,
1996; Liu and Stoller, 2003], integration [Reps, 1991; Reps et al., 1994; Binkley et al.,
1995], software safety [Gallagher and Lyle, 1993], understanding [Cutillo et al., 1993;
Lanubile and Visaggio, 1993; Merlo et al., 1993] and software maintenance [Gallagher
and Lyle, 1991; Platoff et al., 1991; Ramalingam and Reps, 1991, 1992].
The techniques proposed in this thesis could also be used in the above mentioned
contexts. Firstly, the main objective was the development of a profiler for the PAKCS
programming environment [Hanus et al., 2006] of Curry. The techniques proposed
together with its implementation are described in Part II. Secondly, the purpose was
to complement the current Curry tracer with new debugging capabilities based on
slicing (the language Curry lacked of a program slicer). Part III describes how we
have extended the standard semantics of Curry to be able to slice programs, and how
3
4
CHAPTER 1. INTRODUCTION
we have implemented it in a single tool which integrates the Curry’s tracer, slicer and
program specializer. Thirdly, we focused on algorithmic debugging. In particular,
Part IV introduces some new algorithmic debugging strategies and compares them
with previous strategies in order to establish a classification of algorithmic debugging
strategies.
1.2
Debugging Techniques
In this section we review the state of the art in debugging techniques for declarative
languages. Most of the techniques discussed here have also been adapted to—or
come from—the imperative paradigm. We present here the most relevant techniques
that, in some sense, are related with the techniques later developed in the thesis. In
particular, those techniques that will be extended in the forthcoming chapters will
be explained in detail in their respective chapters. We omit here details about the
applications of these techniques to particular languages or implementations.
1.2.1
Traditional Debugging
One of the most common debugging practices still seem to be the use of print statements. This technique, also called “echo debugging” or “print debugging”, basically
consists in adding print statements in “key” parts of the program. With these print
statements, the programmer can check if some condition is satisfied, or he can also
observe the value of some variables of interest in some parts of the execution.
This kind of debugging can be done in every programming language without the
help of a debugging tool: it comes for free with the language. However, its main
drawback is that erroneous values are often observed long after the bug, and the
programmer has to iterate over the program locating the different print statements
over the source code which can be an arduous task. Moreover, in every iteration, the
programmer has to add the print statement, recompile the program and run it again
in order to see the value printed by the new print statement.
modify source code → compile → execute
Some debugging tools (see, e.g., [Somogyi and Henderson, 1999]), give the possibility
to inspect the state of the compiler at some point of the execution. The programmer
can insert breakpoints in the program—breakpoints allow us to see at different times
of a computation part of the computation state— and, during the execution, the
compiler shows the values of some specified variables at these points. In some tools,
it is also possible to specify some breakpoint conditions that the compiler evaluates.
If the conditions are satisfied, then the compiler stops at this breakpoint; otherwise,
the compiler ignores this breakpoint and the execution continues.
4
1.2. DEBUGGING TECHNIQUES
5
Some compilers incorporate mechanisms to statically identify potentially sources
of bugs. For instance, by identifying if a parameter is passed through a set of functions
without being used; or if an argument of a function never influences the result of the
function, etc. This kind of static analysis is known as Anomaly Detection [Osterweil
and Fosdick, 1976].
1.2.2
Profiling
The York functional programming group describes the necessity of profiling tools as
follows [York Functional Programming Group, 2006]:
“Functional programs often display elegance and simplicity; yet their computational behavior may be complex, depending on the subtle interplay of
several mechanisms in the process of evaluation. Solutions that are concise as defining formulae can give rise in their execution to surprisingly
large demands for machine resources.”
Profiling is a method to identify sections of code that consume large portions of
execution time. In a typical program, most execution time is spent in relatively few
sections of code. To improve performance, the greatest gains result from improving
coding efficiency in time-intensive sections. In particular, profilers can analyze code
and detect implementation inefficiencies, which are yet another form of a bug.
Profiling methods include:
Program counter (PC) sampling: This technique periodically interrupts the program being executed and logs the value of the PC. After execution, the logs can
be analyzed in order to determine which parts of the code consume the most
time.
Basic block counting: These techniques modify the program by inserting profiling
code at strategic locations. The augmented/instrumented code can calculate
the number of times it is executed.
Compiler profilers: This technique basically consist in the extension of a compiler
with profiling capabilities. While executing a program, the compiler logs the
number of resources (memory, time, disk space, etc.) needed by each portion of
code.
There are different kinds of profilers depending on the purpose of the programmer.
Firstly, the information provided by the profiler is rather different (for example, CPU
usage, call counts, call cost, memory usage, and I/O operations); and, moreover,
this information can be collected at different granularity levels (for example, at a
5
6
CHAPTER 1. INTRODUCTION
module level or at a function level). In addition, it is important to specify whether
the profiler must consider the shared libraries used by the application or not. And,
secondly, the method used to profile the programs determines whether the program
being profiled must be modified or not. For instance, compiler profilers can profile a
program without modifying it, while basic block counting profilers need to instrument
programs by some kind of program transformation.
One of the most important profilers for functional languages is the Haskell profiler
[Sansom and Peyton-Jones, 1997] which was the first to introduce the notion of cost
centers (a key piece in our work that we will deeply explain in Chapter 4). Roughly,
a cost center is a container that the programmer can set in the program by means
of a special operator, and which is used to accumulate the cost of executing one
part of the program (e.g., an expression). Another important characteristic of the
Haskell profiler is the possibility to measure symbolic costs. This means that this
profiler not only computes space and time costs, but it is also able to compute the
number of elemental operations performed during a computation. For instance, a
symbolic profiler can compute the number of function unfoldings performed during a
computation. The reader is referred to Chapter 4 for details about symbolic profiling.
Apart from the use of an augmented compiler to measure costs—this is the case,
e.g., of [Sansom and Peyton-Jones, 1997] and [Albert and Vidal, 2002]—another frequent approach to profiling is the definition of a program transformation. Approaches
based on a program transformation are, for instance, [Métayer, 1988; Sands, 1990;
Rosendahl, 1989]. We will discuss and compare them to our approach in Chapter 3.
1.2.3
Algorithmic Debugging
Algorithmic debugging is a debugging technique which relies on the programmer having an intended interpretation of the program. In other words, some computations
of the program are correct and others are wrong with respect to the programmer’s
intended semantics. Therefore, algorithmic debuggers compare the results of subcomputations with what the programmer intended. By asking the programmer questions or using a formal specification the system can identify precisely the location of
a program’s bug.
Essentially, algorithmic debugging is a two-phase process: An execution tree (see,
e.g., [Naish, 1997a]), ET for short, is built during the first phase. Each node in this
ET corresponds to an equation which consists of a function call with completely evaluated arguments and results1 . Roughly speaking, the ET is constructed as follows:
The root node is the main function of the program; for each node n with associated
function f , and for each function call in the right-hand side of the definition of f which
has been evaluated, a new node is recursively added to the ET as the child of n. This
1 Or
as much as needed if we consider a lazy language.
6
1.2. DEBUGGING TECHNIQUES
7
notion of ET is valid for functional languages but it is insufficient for other paradigms
as the imperative programming paradigm. In general, the information included in the
nodes of the ET includes all the data needed to answer the questions. For instance,
in the imperative programming paradigm, with the function (or procedure) of each
node it is included the value of all global variables when the function was called.
Similarly, in object-oriented languages, every node with a method invocation includes
the values of the attributes of the object owner of this method (see, e.g., [Caballero,
2006]). In the second phase, the debugger traverses the ET asking an oracle (typically
the programmer) whether each equation is correct or wrong. At the beginning, the
suspicious area which contains those nodes that can be buggy (a buggy node is associated with a buggy rule of the program) is empty; but, after every question, some
nodes of the ET leave the suspicious area. When all the children of a node with a
wrong equation (if any) are correct, the node becomes buggy and the debugger locates
the bug in the function definition of this node [Nilsson and Fritzson, 1994]. If a bug
symptom is detected then algorithmic debugging is complete [Shapiro, 1982]. It is
important to say that, once the execution tree is built, the problem of traversing it
and selecting a node is mostly independent of the language used (a clear exception
are those strategies which use information about the syntax of the program); hence
many algorithmic debugging strategies can theoretically work for any language.
Unfortunately, in practice—for real programs—algorithmic debugging can produce
long series of questions which are semantically unconnected (i.e., consecutive questions
which refer to different and independent parts of the computation) making the process
of debugging too complex.
Furthermore, questions can also be very complex. For instance, during a debugging
session with a compiler, the algorithmic debugger of the Mercury language [MacLarty,
2005] asked a question of more than 1400 lines.
Therefore, new techniques and strategies to reduce the number of questions, to
simplify them and to improve the order in which they are asked are a necessity to
make algorithmic debuggers usable in practice.
During the algorithmic debugging process, an oracle is prompted with equations
and asked about their correctness; it answers “YES” when the result is correct or
“NO” when the result is wrong. Some algorithmic debuggers also accept the answer “I
don’t know” when the programmer cannot give an answer (e.g., because the question
is too complex). After every question, some nodes of the ET leave the suspicious area.
When there is only one node in the suspicious area, the process finishes reporting this
node as buggy. It should be clear that algorithmic debugging finds one bug at a time.
In order to find different bugs, the process should be restarted again for each different
bug.
7
8
CHAPTER 1. INTRODUCTION
main = sqrtest [1,2]
sqrtest x = test (computs (listsum x))
test (x,y,z) = (x==y) && (y==z)
listsum [] = 0
listsum (x:xs) = x + (listsum xs)
computs x = ((comput1 x),(comput2 x),(comput3 x))
comput1 x = square x
square x = x*x
comput2 x = listsum (list x x)
list x y | y==0
= []
| otherwise = x:list x (y-1)
comput3 x = listsum (partialsums x)
partialsums x = [(sum1 x),(sum2 x)]
sum1 x = div (x * (incr x)) 2
sum2 x = div (x + (decr x)) 2
incr x = x + 1
decr x = x - 1
Figure 1.1: Example program
Let us illustrate the process with an example2 .
Example 1 Consider the buggy program in Figure 1.1 adapted to Haskell from [Fritzson et al., 1992]. This program sums a list of integers [1,2] and computes the square
of the result with three different methods. If the three methods compute the same result the program returns T rue; otherwise, it returns F alse. Here, one of the three
methods—the one adding the partial sums of its input number—contains a bug. From
2 While almost all the strategies presented here are independent of the programming paradigm
used, in order to be concrete and w.l.o.g. we will base our examples on the functional programming
paradigm.
8
1.2. DEBUGGING TECHNIQUES
9
Starting Debugging Session...
(1) main = False? NO
(2) sqrtest [1,2] = False? NO
(3) test [9,9,8] = False? YES
(4) computs 3 = [9,9,8]? NO
(5) comput1 3 = 9? YES
(7) comput2 3 = 9? YES
(16) comput3 3 = 8? NO
(17) listsum [6,2] = 8? YES
(20) partialsums 3 = [6,2]? NO
(21) sum1 3 = 6? YES
(23) sum2 3 = 2? NO
(24) decr 3 = 2? YES
Bug found in rule:
sum2 x = div (x + (decr x)) 2
Figure 1.2: Debugging session for the program in Figure 1.1
this program, an algorithmic debugger can automatically generate the ET of Figure 1.3
(for the time being, the reader can ignore the distinction between different shapes and
white and dark nodes) which, in turn, can be used to produce a debugging session as
depicted in Figure 1.2. During the debugging session, the system asks the oracle about
the correctness of some ET nodes w.r.t. the intended semantics. At the end of the
debugging session, the algorithmic debugger determines that the bug of the program is
located in function “sum2” (node 23). The definition of function “sum2” should be:
sum2 x = div (x*(decr x)) 2
Algorithmic debugging strategies are based on the fact that the ET can be pruned
using the information provided by the oracle. Given a question associated with a
node n of the ET, a NO answer prunes all the nodes of the ET except the subtree
rooted at n; and a YES answer prunes the subtree rooted at n. Each strategy takes
advantage of this property in a different manner.
A correct equation in the tree does not guarantee that the subtree rooted at this
equation is free of errors. It can be the case that two buggy nodes caused the correct
answer by fluke [Davie and Chitil, 2006]. In contrast, an incorrect equation does
guarantee that the subtree rooted at this equation does contain a buggy node [Naish,
1997a]. Therefore, if a program produced a wrong result, then the equation in the
root of the ET is wrong and thus there must be at least one buggy node in the ET.
We will assume in the following that the debugging session has been started after
discovering a bug symptom in the output of the program, and thus the root of the
9
10
(1) main -> False
(3) test (9,9,8) -> False
(5) comput1 3 -> 9
(6) square 3 -> 9
(4) computs 3 -> (9,9,8)
(25) listsum [1,2] -> 3
(7) comput2 3 -> 9
(16) comput3 3 -> 8
(26) listsum [2] -> 2
(8) listsum [3,3,3] -> 9
(12) list 3 3 -> [3,3,3]
(17) listsum [6,2] -> 8
(9) listsum [3,3] -> 6
(13) list 3 2 -> [3,3]
(18) listsum [2] -> 2
(21) sum1 3 -> 6
(23) sum2 3 -> 2
(10) listsum [3] -> 3
(14) list 3 1 -> [3]
(19) listsum [] -> 0
(22) incr 3 -> 4
(24) decr 3 -> 2
(11) listsum [] -> 0
(15) list 3 0 -> []
(20) partialsums 3 -> [6,2]
(27) listsum [] -> 0
CHAPTER 1. INTRODUCTION
Figure 1.3: Execution tree of the program in Figure 1.1
10
(2) sqrtest [1,2] -> False
1.2. DEBUGGING TECHNIQUES
11
tree contains a wrong equation. Hence, we know that there is at least one bug in
the program. We will also assume that the oracle is able to answer all the questions.
Then, all the strategies will find the bug.
Single Stepping (Shapiro, 1982)
The first algorithmic debugging strategy to be proposed was single stepping [Shapiro,
1982]. In essence, this strategy performs a bottom-up search because it proceeds by
doing a post-order traversal of the ET. It asks first about all the children of a given
node, and then (if they are correct) about the node itself. If the equation of this node
is wrong then this is the buggy node; if it is correct, then the post-order traversal
continues. Therefore, the first node answered NO is identified as buggy (because all
its children have already been answered YES).
For instance, the sequence of 19 questions asked for the ET in Figure 1.3 is:
Starting Debugging Session...
(3) test [9,9,8] = False? YES
(6) square 3 = 9? YES
(7) comput2 3 = 9? YES
(19) listsum [] = 0? YES
(5) comput1 3 = 9? YES
(11) listsum [] = 0? YES
(10) listsum [3] = 3? YES
(18) listsum [2] = 2? YES
(17) listsum [6,2] = 8? YES
(22) incr 3 = 4? YES
(9)
(8)
(21) sum1 3 = 6? YES
(24) decr 3 = 2? YES
listsum [3,3] = 6? YES
listsum [3,3,3] = 9? YES
(15) list 3 0 = []? YES
(14) list 3 1 = [3]? YES
(13) list 3 2 = [3,3]? YES
(23) sum2 3 = 2? NO
Bug found in rule:
(12) list 3 3 = [3,3,3]? YES
sum2 x = div (x + (decr x)) 2
Note that in this strategy questions are semantically unconnected.
Top-Down Search (Av-Ron, 1984)
Due to the fact that questions are asked in a logical order, top-down search [Av-Ron,
1984] is the strategy that has been traditionally used (see, e.g., [Caballero, 2005; Kokai
et al., 1999]) to measure the performance of different debugging tools and methods.
It basically consists in a top-down, left-to-right traversal of the ET and, thus, the
node asked is always a child or a sibling of the previous question node. When a node
is answered NO, one of its children is asked; if it is answered YES, one of its siblings
is. Therefore, the idea is to follow the path of wrong equations from the root of the
tree to the buggy node.
For instance, the sequence of 12 questions asked for the ET in Figure 1.3 is shown
in Figure 1.2.
11
12
CHAPTER 1. INTRODUCTION
This strategy significantly improves single stepping because it prunes a part of
the ET after every answer. However, it is still very naive, since it does not take into
account the structure of the tree (e.g., how balanced it is). For this reason, a number
of variants aiming at improving it can be found in the literature:
Top-Down Zooming (Maeji and Kanamori, 1987)
During the search of previous strategies, the rule or indeed the function definition
may change from one query to the next. If the oracle is human, this continuous
change of function definitions slows down the answers of the programmer because he
has to switch thinking once and again from one function definition to another. This
drawback can be partially overcome by changing the order of the questions: In this
strategy [Maeji and Kanamori, 1987], recursive child calls are preferred.
The sequence of questions asked for the ET in Figure 1.3 is exactly the same as
with top-down search (Figure 1.2) because no recursive calls are found.
Another variant of this strategy called exception zooming, introduced by Ian
MacLarty [MacLarty, 2005], selects first those nodes that produced an exception at
runtime.
Heaviest First (Binks, 1995)
Selecting always the left-most child does not take into account the size of the subtrees
that can be explored. Binks proposed in [Binks, 1995] a variant of top-down search
in order to consider this information when selecting a child. This variant is called
heaviest first because it always selects the child with a bigger subtree. The objective
is to avoid selecting small subtrees which have a lower probability of containing the
bug.
For instance, the sequence of 9 questions asked for the ET in Figure 1.3 would
3
be :
Starting Debugging Session...
(1)
main = False? NO
(2)
sqrtest [1,2] = False? NO
(4)
computs 3 = (9,9,8)? NO
(7)
comput2 3 = 9? YES
(16) comput3 3 = 8? NO
(20) partialsums 3 = [6,2]? NO
(21) sum1 3 = 6? YES
(23) sum2 3 = 2? NO
3 Here, and in the following, we will break the indeterminism by selecting the left-most node in
the figures. For instance, the fourth question could be either (7) or (16) because both have a weight
of 9. We selected (7) because it is on the left.
12
1.2. DEBUGGING TECHNIQUES
13
(24) decr 3 = 2? YES
Bug found in rule:
sum2 x = div (x + (decr x)) 2
Divide & Query (Shapiro, 1982)
In 1982, together with single stepping, Shapiro proposed another strategy: the socalled divide & query (D&Q) [Shapiro, 1982]. The idea of D&Q is to ask in every
step a question which divides the remaining nodes in the ET by two, or, if this is
not possible, into two parts with a weight as similar as possible. In particular, the
original algorithm by Shapiro always chooses the heaviest node whose weight is less
than or equal to w/2 where w is the weight of the suspicious area in the ET. This
strategy has a worst case query complexity of order b log2 n where b is the average
branching factor of the tree and n its number of nodes.
This strategy works well with a large search space—this is normally the case of
realistic programs—because its query complexity is proportional to the logarithm of
the number of nodes in the tree. If the ET is big and unbalanced this strategy is better
than top-down search [Caballero, 2005]; however, the main drawback of this strategy
is that successive questions may have no connection, from a semantic point of view,
with each other; requiring the programmer more time for answering the questions.
For instance, the sequence of 6 questions asked for the ET in Figure 1.3 is:
Starting Debugging Session...
(7)
comput2 3 = 9? YES
(16) comput3 3 = 8? NO
(17) listsum [6,2] = 8? YES
(21) sum1 3 = 6? YES
(24) decr 3 = 2? YES
(23) sum2 3 = 2? NO
Bug found in rule:
sum2 x = div (x + (decr x)) 2
Hirunkitti’s Divide & Query (Hirunkitti and Hogger, 1993)
In [Hirunkitti and Hogger, 1993], Hirunkitti and Hogger noted that Shapiro’s algorithm does not always choose the node closest to the halfway point in the tree
and addressed this problem slightly modifying the original divide & query algorithm.
13
14
CHAPTER 1. INTRODUCTION
Their version of divide & query is the same as the one of Shapiro except that their
version always chooses a node which produces a least difference between:
• w/2 and the heaviest node whose weight is less than or equal to w/2
• w/2 and the lightest node whose weight is greater than or equal to w/2
where w is the weight of the suspicious area in the computation tree.
For instance, the sequence of 6 questions asked for the ET in Figure 1.3 is:
Starting Debugging Session...
(7) comput2 3 = 9? YES
(16) comput3 3 = 8? NO
(17) listsum [6,2] = 8? YES
(21) sum1 3 = 6? YES
(24) decr 3 = 2? YES
(23) sum2 3 = 2? NO
Bug found in rule:
sum2 x = div (x + (decr x)) 2
Biased Weighting Divide & Query (MacLarty, 2005)
MacLarty proposed in his PhD thesis [MacLarty, 2005] that not all the nodes should
be considered equally while dividing the tree. His variant of D&Q divides the tree
by only considering some kinds of nodes and/or by associating a different weight to
every kind of node.
In particular, his algorithmic debugger was implemented for the functional logic
language Mercury [Conway et al., 1995] which distinguishes between 13 different node
types.
Hat-delta (Davie and Chitil, 2005)
Hat [Wallace et al., 2001] is a tracer for Haskell. Davie and Chitil introduced a
declarative debugger tool based on the Hat’s traces that includes a new strategy
called Hat-delta [Davie and Chitil, 2005]. Initially, Hat-delta is identical to top-down
search but it becomes different as the number of questions asked increases. The main
idea of this strategy is to use previous answers of the oracle in order to compute which
node has an associated rule that is more likely to be wrong (e.g., because it has been
answered NO more times than the others).
This strategy assumes that a rule answered NO n times out of m is more likely
to be wrong than a rule answered NO n′ times out of m if n′ < n 6 m. During
a debugging session, a sequence of questions, each of them related to a particular
14
1.2. DEBUGGING TECHNIQUES
15
rule, is asked. In general, after every question, it is possible to compute the total
number of questions asked for each rule, the total number of answers YES/NO, and
the total number of nodes associated with this rule. Moreover, when a node is set
correct or wrong, Hat-delta marks all the rules of its descendants as correctly or
incorrectly executed respectively. This strategy uses all this information to select the
next question. In particular, three different heuristics have been proposed based on
this idea [Davie and Chitil, 2006]:
• Counting the number of YES answers. If a rule has been executed correctly
before, then it will likely execute correctly again. The debugger associates to
each rule of the program the number of times it has been executed in correct
computations based on previous answers.
• Counting the number of NO answers. This is analogous to the previous heuristic
but collecting wrong computations.
• Calculating the proportion of NO answers. This is derived from the previous
two heuristics. For a node with associated rule r we have:
number of answers NO for r
number of answers NO/YES for r
If r has not been asked before a value of
1
2
is assigned.
Example 2 Consider this program:
4|0|0
8|4| 13
4|0|0
4|0|0
0|0| 21
sort [] = []
sort (x:xs) = insert x (sort xs)
insert x [] = [x]
insert x (y:ys)
| x<y = x:y:ys
| otherwise = insert x ys
where the left numbers indicate respectively the number of times each rule has been
executed correctly, the number of times each rule has failed and the proportion of NO
answers for this rule.
With this information, otherwise = insert x ys is more likely to be wrong.
Subterm Dependency Tracking (MacLarty et al., 2005)
In 1986, Pereira [Pereira, 1986] noted that the answers YES, NO and I don′ t know
were insufficient; and he pointed out another possible answer of the programmer:
Inadmissible (see also [Naish, 1997b]). An equation or, more precisely, some of its
15
16
CHAPTER 1. INTRODUCTION
arguments, are inadmissible if they violate the preconditions of its function definition. For instance, consider the equation insert ‘b’ “cc” = “bcc”, where function
insert inserts the first argument in a list of mutually different characters (the second
argument). This equation is not wrong but inadmissible, since the argument “cc” has
repeated characters. Hence, inadmissibility allows us to identify errors in left-hand
sides of equations.
However, with only these four possible answers the system fails to get fundamental
information from the programmer about why the equation is wrong or inadmissible.
In particular, the programmer could specify which exact (sub)term in the result or the
arguments is wrong or inadmissible respectively. This provides specific information
about why an equation is wrong (i.e., which part of the result is incorrect? is one
particular argument inadmissible?).
Consider again the equation insert ‘b’ “cc” = “bcc”. Here, the programmer
could detect that the second argument should not have been computed; he could
then mark the second argument (“cc”) as inadmissible. This information is essential
because it allows the debugger to avoid questions related to the correct parts of the
equation and concentrate on the wrong parts.
Based on this idea, MacLarty et al. [MacLarty, 2005] proposed a new strategy
called subterm dependency tracking. Essentially, once the programmer selects a particular wrong subterm, this strategy searches backwards in the computation for the
node that introduced the wrong subterm. All the nodes traversed during the search
define a dependency chain of nodes between the node that produced the wrong subterm and the node where the programmer identified it. The sequence of questions
defined in this strategy follows the dependency chain from the origin of the wrong
subterm.
For instance, if the programmer is asked question 3 from the ET in Figure 1.3,
his answer would be YES but he could also mark subexpression “8” as inadmissible.
Then, the system would compute the chain of nodes which passed this subexpression
from the node which computed it until question 3. This chain is formed by nodes
4, 16 and 17. The system would ask first 17, then 16, and finally 4 following the
computed chain.
In our example, the sequence of 8 questions asked for the ET in Figure 1.3, combining this strategy with top-down search, is:
Starting Debugging Session...
(1) main = False? NO
(2) sqrtest [1,2] = False? NO
(3) test (9,9,8) = False? YES
(17) listsum [6,2] = 8? YES
(16) comput3 3 = 8? NO
(20) partialsums 3 = [6,2]? NO
(the programmer marks ‘‘8")
(the programmer marks ‘‘2")
16
1.2. DEBUGGING TECHNIQUES
17
(23) sum2 3 = 2? NO
(24) decr 3 = 2? YES
Bug found in rule:
sum2 x = div (x + (decr x)) 2
1.2.4
Abstract Debugging
Abstract interpretation is a formal method introduced by Cousot and Cousot [1977]
to statically determine dynamic properties of programs. This method have also been
used for the debugging of programs in a technique known as abstract debugging.
Abstract debugging [Bourdoncle, 1993] was first introduced in the context of
higher-order imperative programs. In particular, its first implementation was the
Syntox system which was able to debug Pascal programs.
The technique starts with the programmer inserting a set of assertions in the source
code. This assertions can be invariant assertions, which are properties that must
always hold at the control point where they are placed; and intermittent assertions,
which are properties that must eventually hold at the control point where they are
placed. The assertions introduced by the programmer are interpreted as the intended
semantics of the program, thus their violation is considered a bug. Therefore, abstract
interpretation is used to find under which conditions does the assertions are violated.
Example 3 Consider the following Pascal program:
(1) program AbsDeb;
(2)
var i,n: integer;
(3)
T: array [1..100] of integer;
(4) begin
(5)
read(n);
(6)
for i:=0 to n do
*(7)
read(T[i]);
(8) end.
where the user introduced an assertion in the line marked with * specifying that, at this
line, 1 ≤ i ≤ 100 (i.e., the array’s index is not out of bounds). An abstract debugger
would propagate this condition backwards, and would determine that it is violated in
every execution of the program because the assignment i := 0. Then, the programmer
would correct this bug by replacing this assignment by i := 1. If the abstract debugger
is executed again with the correct program, it would determine that the assertion holds
whenever 1 ≤ n ≤ 100 at line (5). This is not a bug, but a precondition of the program
to ensure correctness.
17
18
CHAPTER 1. INTRODUCTION
Abstract debugging has been later applied in the logic paradigm by Comini et al.
[1994]. In particular, they have investigated the use of this technique in the context
of positive logic programs under the left-most selection rule.
1.2.5
Tracing
Tracing is a technique which relies in the construction of a data structure (a trace)
representing the execution of a program. This data structure is usually computed
during the execution of a program and can be later explored in order to get information
about the execution itself. The fact of being able to ‘record’ the execution of a program
in a data structure (the trace) and reproduce it forwards and/or backwards is also
known as execution backtracking [Agrawal et al., 1991].
A trace liberates from the time arrow of computation, thus it is possible to inspect a trace forwards or backwards looking how redexes were produced during the
computation.
A trace can contain information about the proper execution of the program being
debugged, but it can also contain other information (e.g., compiler information, source
code data, etc) so that many properties can be observed and improved. One of
the most appropriate data structures used for tracing in the context of declarative
languages are the redex trails, originally introduced by Sparud and Runciman [1997],
and its subsequent extension, the Augmented Redex Trails (ART) [Wallace et al.,
2001]. Redex trails will be presented and used in Chapter 5.
Tracing debuggers present to the user a view of the reductions that the compiler
did while evaluating a program. However, the subtle pragmatics of lazy evaluation
can lead to intricate program structures that produce very complex or illegible traces.
One of the solutions adopted is to provide the user with special tools that allow us
to browse traces at selected stages in the computation and inspect them by showing
the information in a more intuitive manner, for instance by showing the arguments
of functions in their most evaluated form.
Typically, two major technical problems that tracing debuggers need to overcome
are related to scaling up to large applications. The size of the traces can be huge
(indeed gigabytes). For nontrivial problems, either a stream approach is used (never
storing the whole trace to disk or memory), or only selected parts of the computation
are traced (e.g., by trusting some modules and/or functions). The structures that the
user can view and browse are also huge. Sometimes, the user can be prompted with
enormous data structures which are hardly understandable and, thus, useless. As an
example, the algorithmic debugger of Mercury [MacLarty, 2005] was used to debug
the own Mercury compiler—a program with more than 300.000 lines of code— and
during the debugging session the user was prompted with a question of more than
1400 lines.
18
1.2. DEBUGGING TECHNIQUES
1.2.6
19
Program Slicing
Program slicing was originally introduced in 1984 by Mark Weiser. Since then, many
researchers have extended it in many directions and for all programming paradigms.
The huge amount of program slicing-based techniques has lead to the publication of
different surveys [Tip, 1995; Binkley and Gallagher, 1996; Harman et al., 1996; Harman and Gallagher, 1998; De Lucia, 2001; Harman and Hierons, 2001; Binkley and
Harman, 2004; Xu et al., 2005] trying to clarify the differences between them. However, each survey presents the techniques from a different perspective. For instance,
[Tip, 1995; Binkley and Gallagher, 1996; Harman and Gallagher, 1998; De Lucia, 2001;
Harman and Hierons, 2001; Xu et al., 2005] mainly focus on the advances and applications of program slicing-based techniques; in contrast, [Binkley and Harman, 2004]
focuses on their implementation comparing empirical results; and [Harman et al.,
1996] tries to compare and classify them in order to predict future techniques and applications. In this section we describe all the techniques in a way similar to [De Lucia,
2001]. We propose an example program and we extract a slice from it by applying
every technique.
Program Slicing (Weiser, 1984)
The original ideas of program slicing come from the PhD thesis of Mark Weiser in
1979 [Weiser, 1979] that were presented in the International Conference on Software
Engineering in 1981 [Weiser, 1981] and finally published in Transactions on Software
Engineering in 1984 [Weiser, 1984].
Program slicing is a technique to decompose programs by analyzing their data and
control flow. Roughly speaking, a program slice consists of those program statements
which are (potentially) related to the values computed at some program point and/or
variable, referred to as a slicing criterion.
As it was originally defined:
“A slice is itself an executable program subset of the program whose behavior must be identical to the specified subset of the original program’s
behavior ”
However, the constraint of being an executable program has been sometimes relaxed.
Given a program p, slices are produced w.r.t a given slicing criterion hs, vi which specifies a sentence s and a set of variables v in p. Note that the variables in v not necessarily appear in s; consider for instance the slicing criterion h18, {lines, chars, subtext}i
for the program in Figure 1.4 (a). Observe that, in this case, not all the variables
appear in line 18.
As an example of slice, consider again the program in Figure 1.4 (a) (for the
time being the reader can ignore the breakpoints marked with *) where function
19
20
*(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
*(10)
(11)
(12)
*(13)
(14)
(15)
(16)
(17)
*(18)
CHAPTER 1. INTRODUCTION
read(text);
read(n);
lines = 1;
chars = 1;
subtext = "";
c = getChar(text);
while (c != ‘\eof’)
if (c == ‘\n’)
then lines = lines + 1;
chars = chars + 1;
else chars = chars + 1;
if (n != 0)
then subtext = subtext ++ c;
n = n - 1;
c = getChar(text);
write(lines);
write(chars);
write(subtext);
(1)
read(text);
(3)
lines = 1;
(6)
(7)
(8)
(9)
c = getChar(text);
while (c != ‘\eof’)
if (c == ‘\n’)
then lines = lines + 1;
(15)
c = getChar(text);
(16) write(lines);
(a) Example program
(b) Slice
Figure 1.4: Example of slice
getChar extracts the first character of a string. This program takes a text (i.e., a
string of characters including ‘\n’—carriage return—and ‘\eof’—end-of-file—) and a
number n, and it returns the number of characters and lines of the text and a subtext
composed of the first n characters excluding ‘\n’. A slice of this program w.r.t. the
slicing criterion h16, linesi is shown in Figure 1.4 (b).
Static Slicing (Weiser, 1984)
The original definition of program slicing was static [Weiser, 1984], in the sense that
it did not consider any particular input for the program being sliced. Particularly,
the slice shown in Figure 1.4 (b) is a static slice, because it does not consider any
particular execution (i.e., it works for any possible input data).
In order to extract a slice from a program, the dependences between its statements
must be computed first. The Control Flow Graph (CFG) is a data structure which
makes the control dependences for each operation in a program explicit. For instance,
the CFG of the program in Figure 1.4 (a) is depicted in Figure 1.5.
However, the CFG does not generally suffice for computing program slices because
it only stores control dependences and, for many applications (such as debugging),
20
1.2. DEBUGGING TECHNIQUES
21
(10)
(9)
Start
(1)
(2)
(13)
(14)
(3)
(8)
(11)
(12)
(16)
(17)
(18)
(15)
(4)
(5)
(6)
(7)
Stop
Figure 1.5: Control flow graph of Figure 1.4 (a)
Start
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(16)
(17)
(18)
(15)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
Figure 1.6: Program dependence graph of Figure 1.4 (a)
data dependences are also necessary. Ottenstein and Ottenstein [1984] noted that the
Program Dependence Graph (PDG) [Kuck et al., 1981; Ferrante et al., 1987] was the
“ideal” data structure for program slicing because it makes explicit both the data and
control dependences for each operation in a program. In essence, a PDG is an oriented
graph where the nodes represent statements in the source code and edges represent
control and data dependences between statements in such a way that they induce a
partial ordering in the nodes preserving the semantics of the program. As an example,
the PDG of the program in Figure 1.4 (a) is depicted in Figure 1.6 where solid arrows
represent control dependences and dotted arrows represent data dependences. Here,
nodes are labeled with the number of the sentence they represent; except node Start
which represents the start of the program. The solid arrow between nodes 7 and 8
indicates that the execution of sentence 8 depends on the execution of sentence 7. The
21
22
CHAPTER 1. INTRODUCTION
same happens with the solid arrow between sentences 8 and 12; thus, transitively, the
execution of sentence 12 also depends on the execution of sentence 7.
The slice shown in Figure 1.4 (b) is a static slice w.r.t. the slicing criterion
h18, linesi.
Dynamic Slicing (Korel and Laski, 1988)
One of the main applications of program slicing is debugging. Often, during debugging, the value of a variable v at some program statement s is observed to be incorrect.
A program slice w.r.t. hs, vi contains the cause of the error.
However, in such a case, we are interested in producing a slice formed by those
statements that could cause this particular error, i.e., we are only interested in one
specific execution of the program. Such slices are called dynamic slices [Korel and
Laski, 1988], since they use dynamic information during the process. In general,
dynamic slices are much smaller than static ones because they contain the statements
of the program that affect the slicing criterion for a particular execution (in contrast
to any execution as with static slicing).
During a program execution, the same statement can be executed several times
in different contexts (i.e., with different values of the variables); as a consequence,
pointing out a statement in the program is not enough in dynamic slicing. A dynamic
slicing criterion needs to specify which particular execution of the statement during
the computation is of interest; thus it is defined as hsi , v, {a1 . . . an }i, where i is the
number of occurrence of statement s in the execution history; v is the set of variables
we are interested in, and the set {a1 . . . an } are the initial values of the program’s
input. As an example, consider the input values {text = “hello world!\eof ”, n = 4}
for the program in Figure 1.4 (a). The execution history would be:
(1, 2, 3, 4, 5, 6, 7, (8, 11, 12, 13, 14, 15, 7)12 , 16, 17, 18)
where the superscript 12 indicates that the statements inside the parenthesis are
repeated twelve times in the execution history.
A dynamic slice of the program in Figure 1.4 (a) w.r.t. the slicing criterion
h1894 , {lines}, {text = “hello world!\eof ”, n = 4}i is shown in Figure 1.7. Note
that this slice is much smaller than its static counterpart because here the slicer can
compute the specific control and data-flow dependences produced by the provided
input data. However, it comes with a cost: the computation of such dependences
usually implies the computation of an expensive (measured in time and space) and
complex data structure (e.g., a trace [Sparud and Runciman, 1997]).
22
1.2. DEBUGGING TECHNIQUES
23
(3) lines = 1;
(16) write(lines);
Figure 1.7: Dynamic slice of Figure 1.4 (a) with respect to the slicing criterion
h1894 , {lines}, {text = “hello world!\eof ”, n = 4}i
Backward Slicing (Weiser, 1984)
A program can be traversed forwards or backwards from the slicing criterion. When
we traverse it backwards (the so called backward slicing), we are interested in all those
statements that could influence the slicing criterion. In contrast, when we traverse it
forwards (the so called forward slicing), we are interested in all those statements that
could be influenced by the slicing criterion.
The original method by Weiser—described above—was static backward slicing.
The main applications of backward slicing are debugging, program differencing and
testing. As an example, the slice shown in Figure 1.4 (b) is a backward slice of the
program in Figure 1.4 (a).
Forward Slicing (Bergeretti and Carré, 1985)
Forward slicing [Bergeretti and Carré, 1985] allows us to determine how a modification
in a part of the program will affect other parts of the program. As a consequence, it
has been used for dead code removal and for software maintenance. In this context,
Reps and Bricker were the first to use the notion of forward slice [Reps and Bricker,
1989]. However, despite backward slicing is the preferred method for debugging,
forward slicing has also been used for this purpose. In particular, forward slicing can
detect initialization errors [Gaucher, 2003].
A forward static slice of the program in Figure 1.4 (a) w.r.t. the slicing criterion
h3, {lines}i is shown in Figure 1.8.
(3) lines = 1;
(9)
lines = lines + 1;
(16) write(lines);
Figure 1.8: Forward static slice of Figure 1.4 (a) w.r.t. h3, {lines}i
Chopping (Jackson and Rollins, 1994)
In chopping [Jackson and Rollins, 1994], the slicing criterion selects two sets of variables, source and sink, and then it computes all the statements in the program that
23
24
CHAPTER 1. INTRODUCTION
being affected by source, they affect sink. Therefore, chopping is a generalization of
both forward and backward slicing where either source or sink is empty. As noted
by Reps and Rosay [Reps and Rosay, 1995], chopping is particularly useful to detect
those statements that “transmit effects” from one part of the program (source) to
another (sink).
For instance, a chop of the program in Figure 1.4 (a) w.r.t. the slicing criterion
hsource = {(3, {lines})}, sink = {(9, {lines})}i is shown in Figure 1.9.
(3)
(9)
lines = 1;
lines = lines + 1;
Figure 1.9: Chop of Figure 1.4 (a) with respect to the slicing criterion
hsource = {(3, {lines})}, sink = {(9, {lines})}i
Relevant Slicing (Agrawal, Horgan, Krauser and London, 1993)
A dynamic program slice only contains the program statements that actually affect the
slicing criterion. However, it is sometimes (e.g., in debugging) interesting to include
in the slice those statements that could have affected the slicing criterion. This is the
objective of relevant slicing [Agrawal et al., 1993], which computes all the statements
that potentially affect the slicing criterion.
A slicing criterion for relevant slicing is exactly the same as for dynamic slicing, but
if we compute a relevant slice and a dynamic slice w.r.t. the same slicing criterion, the
relevant slice is a superset of the dynamic slice because it contains all the statements
of the dynamic slice and it also contains those statements of the program that did
not affect the slicing criterion but could have affected it if they would have changed
(for instance because they were faulty).
Let us explain it with an example: consider the previously used slicing criterion
h1894 , {lines}, {text = “hello world!\eof ”, n = 4}i. The relevant slice computed is
shown in Figure 1.10. It contains all the statements of the dynamic slice (see Figure
1.7) and also it includes statements (6) and (8). Statement (6) could have influenced
the value of lines being redefined to “(6) c = ‘\n’;” and statement (8) could have
influenced the value of lines being redefined to “(8) if (c != ‘\n’)”.
It should be clear that the slices produced in relevant slicing can be non-executable.
The reason is that the main application of such a technique is debugging, where the
programmer is interested in those parts of the program that can contain the bug.
The statements included in the slice are those whose contamination could result in
the contamination of the variable of interest. In order to make the slice executable
preserving the behavior (including termination) of the original program it is necessary
24
1.2. DEBUGGING TECHNIQUES
25
to augment the slice with those statements that are required for the evaluation of all
the expressions included in the slice (even if this evaluation does not influence the
variable of interest).
The forward version of relevant slicing [Gyimóthy et al., 1999] can be useful in
debugging and in program maintenance. A forward relevant slice contains those
statements that could be affected by the slicing criterion (if it is redefined). Therefore,
it could be used to study module cohesion by determining what could be the impact
of a module modification over the rest of modules.
(3) lines = 1;
(6) c = getChar(text);
(8)
if (c == ‘\n’)
(16) write(lines);
Figure 1.10: Relevant slice of Figure 1.4 (a) with respect to the slicing criterion
h1894 , {lines}, {text = “hello world!\eof ”, n = 4}i
Hybrid Slicing (Gupta and Soffa, 1995)
When debugging, it is usually interesting to work with dynamic slices because they
focus on a particular execution (i.e., the one that showed a bug) of the program
being debugged; therefore, they are much more precise than static ones. However,
computing dynamic slices is very expensive in time and space due to the large data
structures (up to gigabytes) that need to be computed. In order to increase the precision of static slicing without incurring in the computation of these large structures
needed for dynamic slicing, a new technique called hybrid slicing [Gupta and Soffa,
1995] was proposed. A hybrid slice is more accurate—and consequently smaller—than
a static one, and less costly than a dynamic slice.
The key idea of hybrid slicing consists in integrating dynamic information into
the static analysis. In particular, the information provided to the slicer is a set
of breakpoints inserted into the source code that, when the program is executed,
can be activated thus providing information about which parts of the code have
been executed. This information allows the slicer to eliminate from the slice all the
statements which are in a non-executed possible path of the computation.
For instance, consider again the program in Figure 1.4 (a) which contains 4 breakpoints marked with ‘*’. A particular execution will activate some of the breakpoints;
this information can be used by the slicer. For instance, a possible execution could
be {1, 13, 13, 13, 10, 13, 18} which means that the loop has been entered 5 times, and
both branches of the outer if-then-else have been executed.
25
26
CHAPTER 1. INTRODUCTION
A hybrid slicing criterion is a triple hs, v, {b1 . . . bn }i, where s and v have the same
meaning than in static slicing and the set {b1 . . . bn } are the sequence of breakpoints
activated during a particular execution of the program.
Figure 1.11 shows an example of hybrid slice.
(1)
(4)
(6)
(7)
(8)
(11)
(15)
(17)
read(text);
chars = 1;
c = getChar(text);
while (c != ’\eof’)
if (c == ’\n’)
then chars = chars + 1;
c = getChar(text);
write(chars);
Figure 1.11: Hybrid slice of Figure 1.4 (a) with respect to the slicing criterion
h18, {chars}, {1, 13, 13, 13, 18}i
Intraprocedural Slicing (Weiser, 1984)
The original definition of program slicing has been later classified as intraprocedural
slicing (i.e., the slice in Figure 1.4 (b) is an intraprocedural slice), because the original
algorithm was defined for programs consisting of a single monolithic procedure; in
other words it did not take into account information related to the fact that slices
can cross the boundaries of procedure calls.
It does not mean that the original definition fails to slice multi-procedural programs; it means that it loses precision in such cases. As an example, consider the
program in Figure 1.12. A static backward slice of this program w.r.t. the slicing criterion h17, {a}i includes all the statements of the program except (12), (13) and (14).
However, it is clear that statements (3) and (8) included in the slice cannot affect the
slicing criterion. They are included in the slice because procedure sum influences the
slicing criterion, and statements (3) and (8) can influence procedure sum. However,
they cannot transitively affect the slicing criterion. This loss of precision has been
later solved by another technique called interprocedural slicing.
Interprocedural Slicing (Horwitz, Reps and Binkley, 1988)
In 1988, Horwitz et al. noted that the program dependence graph was not appropriate
for representing multi-procedural programs and they proposed a new dependence
graph representation of programs called system dependence graph [Horwitz et al.,
1988]. This new representation incorporates collections of procedures with procedure
26
1.2. DEBUGGING TECHNIQUES
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
27
program main
read(text);
lines = 1;
chars = 1;
c = getChar(text);
while (c != ‘\eof’)
if (c == ‘\n’)
then lines = sum(lines,1);
chars = sum(chars,1);
else chars = increment(chars);
c = getChar(text);
write(lines);
write(chars);
end
(15) procedure increment(a)
(16) sum(a,1);
(17) return
(18) procedure sum(a, b)
(19) a = a + b;
(20) return
Figure 1.12: Example of multiprocedural program
calls, and it allows us to produce more precise slices from multi-procedural programs
because it has information available about the actual procedures’ calling-context.
For instance, consider Program 1 in Figure 1.18 together with the slicing criterion
h3, {chars}i. An intraprocedural static slice contains exactly the same sentences
(Program 1). However, it is clear that procedure increment cannot affect the slicing
criterion (in fact it is dead code). Roughly speaking, this inaccuracy is due to the
fact that the call to procedure sum makes the slice include this procedure. Then, all
the calls to this procedure can influence it and thus, procedure increment is also
included.
In contrast, an interprocedural static slice (Program 2) uses information stored in
the system dependence graph about the calling context of procedures. Hence, it would
remove procedure increment thus increasing the precision w.r.t. the intraprocedural
algorithms.
Figure 1.13 shows the interprocedural slice of the program in Figure 1.12 w.r.t.
27
28
CHAPTER 1. INTRODUCTION
the slicing criterion h17, {a}i.
(1)
(2)
(4)
(5)
(6)
(7)
(9)
(10)
(11)
program main
read(text);
chars = 1;
c = getChar(text);
while (c != ‘\eof’)
if (c == ‘\n’)
then chars = sum(chars,1);
else chars = increment(chars);
c = getChar(text);
(15) procedure increment(a)
(16) sum(a,1);
(17) return
(18) procedure sum(a, b)
(19) a = a + b;
(20) return
Figure 1.13: Interprocedural slice of Figure 1.12 w.r.t. h17, {a}i
Quasi-Static Slicing (Venkatesh, 1991)
While static slicing computes slices w.r.t. any execution, dynamic slicing computes
slices w.r.t. a particular execution. However, it is sometimes (e.g., in program understanding) interesting to produce a slice w.r.t. a particular set of executions. Quasistatic slicing [Venkatesh, 1991] can be used in those applications in which a set of
the program inputs are fixed, and the rest of the inputs is unknown. This leads to a
potentially infinite set of considered executions.
A quasi-static slicing criterion is a tuple hs, v, ai where s and v have the same
meaning as in static slicing, and a is a mapping from (some of the) input variables to
values. For instance, a quasi-static slicing criterion for the program in Figure 1.4 (a)
could be h18, {subtext}, {n = 0}i. The slice computed w.r.t. this criterion is shown
in Figure 1.14.
Simultaneous Slicing (Hall, 1995)
Simultaneous slicing is a generalization of program slicing in which a set of slicing
criteria is considered. Hence, a set of points of interest is provided and thus the slice
28
1.2. DEBUGGING TECHNIQUES
29
(1) read(text);
(5) subtext = "";
(18) write(subtext);
Figure 1.14: Quasi-static slice of Figure 1.4 (a) w.r.t. h18, {subtext}, {n = 0}i
can be computed from the slices computed for each point. However, the construction
of such a slice does not simply reduces to the union of slices (this is not sound) and it
requires the use of more elaborated methods such as the SDS procedure [Hall, 1995].
Similarly to quasi-static slicing, simultaneous dynamic slicing computes a slice
w.r.t. a particular set of executions; however, while quasi-static slicing fixes a set of
the program inputs being the rest unknown, in simultaneous dynamic slicing [Hall,
1995] a set of “complete” inputs is known. Thus, a simultaneous dynamic slicing
criterion can be seen as a set of dynamic slicing criterions.
A simultaneous dynamic slicing criterion has the form: hsi , v, {I1 . . . In }i where i
is the number of occurrence of statement s in the execution history, v is the set of
variables of interest and {I1 . . . In } is a set of complete inputs for the program. For
instance, a slice of Figure 1.4 (a) w.r.t. the slicing criterion h1894 , {lines}, {I1 , I2 }i
where I1 = (text = “hello world!\eof ”, n = 4) and I2 = (text = “hello\nworld!\n
\eof ”, n = 0) is depicted in Figure 1.15.
Note that, in contrast to quasi-static slicing where the set of considered executions
can be infinite, in simultaneous dynamic slicing it is always finite.
(1)
(2)
(4)
(6)
(7)
(8)
(9)
(10)
(11)
(15)
(17)
read(text);
read(n);
chars = 1;
c = getChar(text);
while (c != ‘\eof’)
if (c == ‘\n’)
then lines = lines + 1;
chars = chars + 1;
else chars = chars + 1;
c = getChar(text);
write(chars);
Figure 1.15:
Simultaneous dynamic slice of Figure 1.4 (a) w.r.t.
h1894 , {lines}, {I1 , I2 }i, where I1 = (text = “hello world!\eof ”, n = 4) and
I2 = (text = “hello\nworld!\n\eof ”, n = 0)
29
30
CHAPTER 1. INTRODUCTION
In 1993, Lakhotia [1993] introduced a new kind of slicing criterion in order to compute module cohesion. The aim of his work was to collect the module’s components
which contribute to the final value of the output variables of this module. Therefore,
in this scheme, the slicing criterion is formed by a set of variables. It is called “end
slicing” because the slicing point is the end of the program. Then, an end slicing criterion is formed by a set of variables of the program, and thus it can be represented
as a set of slicing criteria (i.e., it is a particular case of simultaneous static slicing).
Program Dicing (Lile and Weiser, 1987)
Program dicing [Lyle and Weiser, 1987] was originally defined as: “remove those
statements of a static slice of a variable that appears to be correctly computed from
the static slice of an incorrectly valued variable”. From this definition it is easy to
deduce that program dicing was originally defined for debugging. In essence, a dice
is the set difference between at least two slices; typically, one of an incorrect value
and the other of a correct value. The main idea is that after some tests we can
find some correct and incorrect outputs. The statements in a slice of an incorrect
output that do not belong to the slice of a correct output are more likely to be wrong.
As a consequence, a program dicing criterion specifies n points in the program, one
for the correct computation and n − 1 for the wrong computations. For instance, a
backward static program dicing criterion for the program in Figure 1.4 (a) could be
h(18, {chars}), {(18, {lines})}i; its meaning is that variable chars at line 18 produced
a correct value (its slice contains the statements 1,4,6,7,8,10,11,15 and 17) whereas
variable lines at line 18 produced an incorrect value (its slice contains the statements
1,3,6,7,8,9,15 and 16). Note that only one point has been specified for the incorrect
computation, but a set is possible. The dice computed w.r.t. this criterion is shown
in Figure 1.8.
In contrast to program slicing where a dynamic slice is included in its corresponding static slice, a dynamic dice is not necessarily included in its corresponding static
dice. Chen and Cheung [1993] investigated under what conditions a dynamic dice is
more precise than a static dice.
Conditioned/Constraint Slicing (Ning, Engberts and Kozaczynski, 1994)
Although Ning et al. [1994] were the first to work with conditioned slices; this technique was formally defined for the first time by Canfora et al. [1994].
Similarly to simultaneous dynamic slicing and quasi-static slicing, conditioned
slicing [Canfora et al., 1994, 1998] computes slices w.r.t. a set of initial states of the
program. The original definition proposed the use of a condition (from the programming language notation) to specify the set of initial states. Posteriorly, Field et al.
[1995] proposed the same idea (known as parametric program slicing or constraint
30
1.2. DEBUGGING TECHNIQUES
31
slicing) based on constraints over the initial values. Finally, De Lucia et al. [1996]
proposed the condition to be a universally quantified formula of first-order predicate
logic. In this approach, which is a generalization of the previous works, a conditioned
slicing criterion is a quadruple hi, F, s, vi where i is a subset of the input variables of
the program, F is a logic formula on i, s is a statement of the program and v a subset
of the variables in the program.
The logic formula F identifies a set of the possible inputs of the program which
could be infinite. For instance, consider the conditioned slicing criterion h(text, n), F,
18, {subtext}i where F = (∀c ∈ text, c 6= ‘\n’, n > 0); the conditioned slice of the
program in Figure 1.4 (a) w.r.t. this criterion is shown in Figure 1.16.
It should be clear that conditioned slicing is a generalization of both simultaneous
dynamic slicing and quasi-static slicing because their respective slicing criteria are a
particular case of a conditioned slicing criterion. As an advantage, conditioned slicing
allows us to specify relations between input values. For instance, condition F above
specifies that if text is a single line, then n must be higher than 0.
(1)
(2)
(5)
(6)
(7)
(8)
(12)
(13)
(14)
(15)
(18)
read(text);
read(n);
subtext = "";
c = getChar(text);
while (c != ‘\eof’)
if (c == ‘\n’)
if (n != 0)
then subtext = subtext ++ c;
n = n - 1;
c = getChar(text);
write(subtext);
Figure 1.16: Conditioned slice of Figure 1.4 (a) with respect to the slicing criterion
h(text, n), F, 18, {subtext}i where F = (∀c ∈ text, c 6= ‘\n’, n > 0)
Abstract Slicing (Hong, Lee and Sokolsky, 2005)
Static slicing is able to determine for each statement in a program whether it affects
or is affected by the slicing criterion; however, it is not able to determine under which
variable values do the statements affect or are affected by the slicing criterion. This
problem is overcome by abstract slicing [Hong et al., 2005].
Essentially, abstract slicing extends static slicing with predicates and constraints
that are processed with an abstract interpretation and model checking based technique. Given a predicate of the program, every statement is labeled with the value
31
32
CHAPTER 1. INTRODUCTION
of this predicate under the statement affects (or is affected by) the slicing criterion.
Similarly, given a constraint on some variables of the program, every statement is
labeled indicating if they can or cannot satisfy the constraint. Clearly, static slicing
is a particular instance of abstract slicing where neither predicates nor constraints
are used. Moreover, in a way similar to conditioned slicing, abstract slicing is able
to produce slices w.r.t. a set of executions by restricting the allowed inputs of the
program using constraints.
In abstract slicing, a slicing criterion has the form hs, P, Ci where s is a statement
of the program and P and C are respectively a predicate and a constraint for some
statements defined over some variables of the program. For instance, C could be
((1), y > x), meaning that, at statement (1), the condition y > x holds. Bases on
previous slicing techniques, Hong et al. adapted this technique to forward/backward
slicing and chopping, giving rise to abstract forward/backward slicing and abstract
chopping.
As an example, consider the program in Figure 1.17 (a); the abstract slice of
this program w.r.t. the slicing criterion h(6), y > x, T ruei is depicted in Figure 1.17
(b). Here, we are interested in knowing the condition under each statement affects
the slicing criterion; we do not impose any condition. In the slice we see that the
statement labeled with ‘[true]’ will always affect the slicing criterion, ‘max = x’ will
only affect the slicing criterion if y ≤ x, and the ‘if-then’ statements will affect the
slicing criterion if y > x.
(1)
(2)
(3)
(4)
(5)
(6)
read(x);
read(y);
max = x;
if (y > x)
then max = y;
write(max);
(1)
(2)
(3)
(4)
(5)
(6)
(a) Example program
read(x);
read(y);
max = x;
if (y > x)
then max = y;
write(max);
[true]
[true]
[not(y>x)]
[y>x]
[y>x]
[true]
(b) Abstract slice
Figure 1.17: Abstract slicing: (b) is an abstract slice of (a) w.r.t. h(6), y > x, T ruei
Amorphous Slicing (Harman and Danicic, 1997)
All approaches to slicing discussed so far have been based on two assumptions: the
slice preserves (part of) the semantics of the program, and it is “syntax preserving”,
i.e., the slice is a subset of the original program statements. In contrast, amorphous
slices [Harman and Danicic, 1997] preserve the semantics restriction but they drop the
32
1.2. DEBUGGING TECHNIQUES
33
syntactic restriction: amorphous slices are constructed using some program transformation which simplifies the program and which preserves the semantics of the program
with respect to the slicing criterion.
This syntactic freedom allows amorphous slicing to perform greater simplifications, thus often being considerably smaller than conventional program slicing. This
simplifications are very convenient in the context of program comprehension where
the user needs to simplify the program as much as possible in order to understand a
part of the semantics of the program having the syntax less importance.
For instance, consider Program 1 in Figure 1.18 together with the slicing criterion
h3, {chars}i. An intraprocedural static slice of this program would contain exactly
the same sentences (Program 1). In contrast, an interprocedural static slice would
remove sentences 4, 5 and 6 (Program 2). Finally, its amorphous slice would be
Program 3. It should be clear that the three programs preserve the same semantics,
but Program 3 has been further simplified by partially evaluating [Jones, 1996] some
expressions (thus changing the syntax) of Program 1. Clearly, Program 3 is much
more understandable than Program 1 and Program 2.
(1) program main
(2) chars = sum(chars,1);
(3) end
(1) program main
(2) chars = sum(chars,1);
(3) end
(1) program main
(2) chars = chars + 1;
(3) end
(4) procedure increment(a)
(5) sum(a,1);
(6) return
(7) procedure sum(a,b)
(8) a = a + b;
(9) return
(7) procedure sum(a,b)
(8) a = a + b;
(9) return
Program 1
Program 2
Program 3
Figure 1.18: Example of amorphous slicing
It is known [Harman and Danicic, 1997] that amorphous static slicing subsumes
traditional static slicing, i.e., there will always be an amorphous static slice which is at
least as thin as the static slice constructed for the same slicing criterion. In addition,
there will usually be an amorphous slice which is thinner than the associated static
slice. This leads to the search of the minimal amorphous slice. However, as proved
by Harman and Danicic [Harman and Danicic, 1997], the computation of the minimal
amorphous static slice of an arbitrary program is, in general, undecidable.
33
34
CHAPTER 1. INTRODUCTION
Decomposition Slicing (Gallagher and Lyle, 1991)
Decomposition slicing [Gallagher and Lyle, 1991] was introduced in the context of
software maintenance to capture all computation on a given variable. The objective
of this technique is to extract those program statements which are needed to compute
the values of a given variable. Therefore, a decomposition slicing criterion is composed
of a single variable v. The slice is then built from the union of the static backward
slices constructed for the criteria {hn1 , vi, . . . , hnm , vi, hend, vi} where {n1 , . . . , nm } is
the set of lines in which v is output and end is the “End” of the program. It should
be clear that decomposition slicing is an instance of simultaneous slicing where the
set of slicing criteria is derived from the variable and the program.
A decomposition slice of the program in Example 1.4 (a) w.r.t. the slicing criterion
hlinesi is shown in Figure 1.4 (b).
1.3
Structure of the Thesis
The core of this document has been organized in four parts. The first part presents
some foundations. First, some preliminary definitions related to term rewriting systems and several basic concepts that will be used in the rest of the thesis are formally
described. Both the syntax and the semantics of the languages that will be later
used are introduced in Chapter 2. In particular, the multi-paradigm language Curry
is presented together with its internal representation, namely the flat programs and
their normalized version.
The second part contains two chapters which present two profiling techniques for
functional logic languages. First, in Chapter 3, we introduce a technique to formally
reason about program execution costs by computing precise time equations. This
technique computes the number of function unfoldings performed during the evaluation of a program. This chapter also includes a discussion about how the costs
computed for a particular execution could be used to reason about the general cost
of the program.
In Chapter 4 it is proposed a new technique for the development of profiling tools
and it presents its application to the Curry language. This new profiling scheme allows
us to measure different kinds of symbolic costs and attribute them to different cost
centers distributed over the program. The technique is based on the instrumentation
of a semantics with costs. However, in order to avoid the big overhead introduced
by this approach, a program transformation which is equivalent to the instrumented
semantics is also introduced.
The third part introduces a new dynamic program slicing technique based on
redex trails. This technique is introduced and discussed in Chapter 5. A redex trail
is a data structure which represents a computation. This technique is based on the
34
1.3. STRUCTURE OF THE THESIS
35
identification of control dependences in the redex trail in order to extract slices from
programs. The technique is defined for source Curry programs, but the redex trail
records the computation of equivalent FlatCurry programs (an internal representation
of Curry programs used by the compiler PAKCS. See Section 2.4 for details); therefore,
the technique needs a kind of translation between both languages. A formalization of
classical program slicing concepts is introduced in order to adapt these concepts to the
redex trails formalism. Moreover, it is shown that the same technique can be adapted
to perform program specialization. Finally, it is also described the implementation of
this technique in Curry.
The fourth part describes algorithmic debugging techniques, and it includes two
chapters. In Chapter 6, program slicing is combined with algorithmic debugging
in order to produce a more accurate debugging technique. The central idea is to
allow the user to provide the algorithmic debugger with more information. To do
so, the technique is described for higher-order term rewriting systems. In essence, it
consists on the use of program slicing to prune the execution trees used in algorithmic
debugging with the information provided by the user.
In Chapter 7, firstly, a classification of all program slicing techniques studied in
this chapter is introduced. This classification is then used in order to predict new
slicing techniques and to identify relationships between the techniques. Afterwards,
three new algorithmic debugging strategies are introduced. Then, they are compared
with the rest of strategies revised in this chapter and, as a result, a classification of
algorithmic debugging strategies is obtained.
Finally, the conclusions together with the future work are discussed in Chapter 8.
35
Part I
Foundations
37
Chapter 2
Preliminaries
We recall in this chapter some basic notions of term rewriting, functional programming, and functional logic programming that will be used in the rest of the thesis.
Firstly, we introduce some notations and concepts and provide a precise terminology.
Most of the definitions and notations in this chapter are taken from [Dershowitz and
Jouannaud, 1990; Klop, 1992; Antoy, 1992; Hanus, 1994; Baader and Nipkow, 1998;
Alpuente et al., 1999; Antoy et al., 2000; Albert, 2001]
2.1
Signature and Terms
Throughout the rest of the thesis, we consider a (many-sorted ) signature Σ partitioned
into a set C of constructors and a set F of (defined) functions or operations. We write
c/n ∈ C and f /n ∈ F for n-ary constructor and operation symbols, respectively.
There is at least one sort Bool containing the 0-ary Boolean constructors true and
f alse. The set of terms and constructor terms (possibly with variables from X ; e.g.,
x, y, . . .) are denoted by T (C ∪ F, X ) and T (C, X ), respectively. The set of variables
occurring in a term t is denoted by Var (t). A term t is ground if Var (t) = ∅. A term
is linear if it does not contain multiple occurrences of any variable. We write on for
the list of objects o1 , . . . , on .
A pattern is a term of the form f (dn ) where f /n ∈ F and d1 , . . . , dn ∈ T (C, X ).
A term is operation-rooted (constructor-rooted ) if it has an operation (constructor)
symbol at the root. root(t) denotes the symbol at the root of the term t.
2.2
Substitutions
We denote by {x1 7→ t1 , . . . , xn 7→ tn } the substitution σ with σ(xi ) = ti for i =
1, . . . , n (with xi =
6 xj if i 6= j), and σ(x) = x for all other variables x. The set
39
40
CHAPTER 2. PRELIMINARIES
Dom(σ) = {x ∈ X | σ(x) 6= x} is called the domain of σ. The codomain of a
substitution σ is the set Ran(σ) = {σ(x) | x ∈ Dom(σ)}. A substitution σ is
(ground) constructor, if σ(x) is (ground) constructor for all x ∈ Dom(σ). The identity
substitution is denoted by id. Substitutions are extended to morphisms on terms by
σ(f (tn )) = f (σ(tn )) for every term f (tn ). Given a substitution θ and a set of variables
V ⊆ X , we denote by θ|`V the substitution obtained from θ by restricting its domain
to V . We write θ = σ [V ] if θ|`V = σ |`V , and θ ≤ σ [V ] denotes the existence of a
substitution γ such that γ ◦ θ = σ [V ].
A term t′ is an instance of t if there is a substitution σ with t′ = σ(t). This implies
a subsumption ordering on terms which is defined by t ≤ t′ iff t′ is an instance of t.
A unifier of two terms s and t is a substitution σ with σ(s) = σ(t). A unifier σ is the
most general unifier (mgu) if σ ≤ σ ′ [X ] for each other unifier σ ′ . A mgu is unique
up to variable renaming.
2.3
Term Rewriting Systems
A set of rewrite rules l → r such that l 6∈ X , and Var (r) ⊆ Var (l) is called a term
rewriting system (TRS). The terms l and r are called the left-hand side and the righthand side of the rule, respectively. A TRS R is left-linear if l is linear for all l → r ∈ R.
A TRS is constructor-based (CB) if each left-hand side l is a pattern. Two (possibly
renamed) rules l → r and l′ → r′ overlap, if there is a non-variable position p in l and
a most general unifier σ such that σ(l|p ) = σ(l′ ). Extra variables are those variables in
a rule which do not occur in the left-hand side. The pair hσ(l)[σ(r′ )]p , σ(r)i is called
a critical pair (and is also called an overlay if p = Λ). A critical pair ht, si is trivial
if t = s (up to variable renaming). A left-linear TRS without critical pairs is called
orthogonal. It is called almost orthogonal if its critical pairs are trivial overlays. If it
only has trivial critical pairs it is called weakly orthogonal. Note that, in CB-TRSs,
almost orthogonality and weak orthogonality coincide.
A rewrite step is an application of a rewrite rule to a term, i.e., t →p,R s if there
exists a position p in t, a rewrite rule R = l → r and a substitution σ with t|p = σ(l)
and s = t[σ(r)]p (p and R will often be omitted in the notation of a computation
step). The instantiated left-hand side σ(l) is called a redex (reducible expression).
A term t is root-stable (often called a head-normal form) if it cannot be rewritten
to a redex. A constructor root-stable term is either a variable or a constructor-rooted
term, that is, a term rooted by a constructor symbol. By abuse, when it is clear from
the context, we use the notion “head-normal form” to refer to constructor root-stable
terms. A term t is called irreducible or in normal form if there is no term s with
t → s. Given a binary relation →, →+ denotes the transitive closure of → and →∗
denotes the reflexive and transitive closure of →.
40
2.3. TERM REWRITING SYSTEMS
41
Example 4 Consider the following TRS that defines the addition on natural numbers
represented by terms built from Zero and Succ:1
Zero + y = y
(Succ x) + y = Succ (x + y)
(R1 )
(R2 )
Given the term (Succ Zero) + (Succ Zero), we have the following sequence of rewrite
steps:
(Succ Zero) + (Succ Zero) →Λ,R2
→1,R1
2.3.1
Succ (Zero + (Succ Zero))
Succ (Succ Zero)
Narrowing
Functional logic programs mainly differ from purely functional programs in that function calls may contain free variables. In order to evaluate terms containing free variables, narrowing non-deterministically instantiates these variables so that a rewrite
step is possible. Formally,
Definition 1 (narrowing step) t ❀(p,R,σ) t′ is a narrowing step if p is a nonvariable position of t and σ(t) →p,R t′ .
We often write t ❀σ t′ when the position and the rule are clear from the context.
We denote by t0 ❀nσ tn a sequence of n narrowing steps t0 ❀σ1 . . . ❀σn tn with
σ = σn ◦ · · · ◦ σ1 (if n = 0 then σ = id), usually restricted to the variables of t0 .
Due to the presence of free variables, a term may be reduced to different values
after instantiating these variables to different terms. Given a narrowing derivation
t0 ❀∗σ tn , we say that tn is a computed value and σ is a computed answer for t0 .
Example 5 Consider again the definition of function “+” in Example 4. Given the
term x + (Succ Zero), narrowing non-deterministically performs the following derivations:
x + (Succ Zero) ❀Λ,R1 ,{x7→Zero}
Succ Zero
x + (Succ Zero) ❀Λ,R2 ,{x7→Succ y1 }
❀1,R1 ,{y1 7→Zero}
Succ (y1 + (Succ Zero))
Succ (Succ Zero)
x + (Succ Zero) ❀Λ,R2 ,{x7→Succ y1 }
❀1,R2 ,{y1 7→Succ y2 }
❀1.1,R1 ,{y2 7→Zero}
...
Succ (y1 + (Succ Zero))
Succ (Succ (y2 + (Succ Zero)))
Succ (Succ (Succ Zero))
Therefore, x + (Succ Zero) non-deterministically computes the values
1 In the examples, we write constructor symbols starting with upper case (except for the list
constructors, “[]” and “:”, which are a shorthand for Nil and Cons, respectively).
41
42
CHAPTER 2. PRELIMINARIES
• Succ(Zero) with answer {x 7→ Zero},
• Succ(Succ(Zero)) with answer {x 7→ Succ Zero},
• Succ(Succ(Succ(Zero))) with answer {x 7→ Succ (Succ (Succ Zero))}, etc.
As in logic programming, narrowing derivations can be represented by a (possibly
infinite) finitely branching tree. Formally, given a program R and an operation-rooted
term t, a narrowing tree for t in R is a tree satisfying the following conditions:
1. each node of the tree is a term,
2. the root node is t,
3. if s is a node of the tree then, for each narrowing step s ❀p,R,σ s′ , the node has
a child s′ and the corresponding arc is labeled with (p, R, σ), and
4. nodes which are constructor terms have no children.
In order to avoid unnecessary computations and to deal with infinite data structures, demand-driven generation of the search space has been advocated by a number
of lazy narrowing strategies Giovannetti et al. [1991]; Loogen et al. [1993]; MorenoNavarro and Rodrı́guez-Artalejo [1992]. Due to its optimality properties w.r.t. the
length of derivations and the number of computed solutions, needed narrowing Antoy
et al. [2000] is currently the best lazy narrowing strategy.
2.3.2
Needed Narrowing
Needed narrowing [Antoy et al., 2000] is defined on inductively sequential TRSs [Antoy, 1992], a subclass of left-linear constructor-based TRSs. Essentially, a TRS is
inductively sequential when all its operations are defined by rewrite rules that, recursively, make on their arguments a case distinction analogous to a data type (or
structural) induction. Inductive sequentiality is not a limiting condition for programming. In fact, the first-order components of many functional (logic) programs written
in, e.g., Haskell, ML or Curry, are inductively sequential.
These systems have the property that the rules defining any operation can be
organized in a hierarchical structure called definitional tree [Antoy, 1992], which is
used to implement needed narrowing.
Definitional Trees
A definitional tree can be seen as a partially ordered set of patterns with some additional constraints. The symbols branch and leaf , used in the next definition, are
uninterpreted functions used to classify the nodes of the tree.
42
2.3. TERM REWRITING SYSTEMS
43
Definition 2 (partial definitional tree [Antoy, 1992]) T is a partial definitional
tree, or pdt, with pattern π iff one of the following cases hold:
T = branch(π, o, T1 , . . . , Tk ) where π is a pattern, o is the occurrence of a variable of
π, the sort of π|o has constructors c1 , . . . , ck for some k > 0, and for all i in
{1, . . . , k}, Ti is a pdt with pattern π[ci (x1 , . . . , xn )]o , where n is the arity of ci
and x1 , . . . , xn are new distinct variables.
T = leaf (π) where π is a pattern.
We denote by P(Σ) the set of pdts over the signature Σ. Let R be a rewrite system.
T is a definitional tree of an operation f iff T is a pdt whose pattern argument is
f x1 . . . xn , where n is the arity of f and x1 , . . . , xn are new distinct variables, and for
every rule l → r of R with l = f x1 . . . xn , there exists a leaf leaf (π) of T such that
l is a variant (i.e., equal modulo variable renaming) of π, and we say that the node
leaf (π) represents the rule l → r. A definitional tree T of an operation f is called
minimal iff below any branch node of T there is a leaf representing a rule defining
f.
Definition 3 (inductively sequential operation) An operation f of a rewrite system R is called inductively sequential iff there exists a definitional tree T of f such
that each leaf node of T represents at most one rule of R and all rules are represented.
Definition 4 (inductively sequential rewrite system) A rewrite system R is called inductively sequential if all its defined functions are inductively sequential.
It is often convenient and simplifies understanding to provide a graphic representation of definitional trees, where each inner node is marked with a pattern, the
inductive position in branches is surrounded by a box, and the leaves contain the
corresponding rules.
Let us note that, even if we consider a first-order language, we use a curried
notation in the examples (as is usual in functional languages). Likewise, we use lower
case letter for variables and upper case letter for constructor symbols, following the
Curry syntax. Analogously, in our examples, rules have the form l = r instead of
l → r, as in our formal notation.
Example 6 Consider the rules defining the “6” function:
06y
(Succ x′ ) 6 0
(Succ x′ ) 6 (Succ y′ )
= True
= False
= x′ 6 y′
The definitional tree for the function “6” is illustrated in Figure 2.1. In the picture,
each branch node contains the pattern of the corresponding node of the definitional
43
44
CHAPTER 2. PRELIMINARIES
x 6y
✑◗
◗
✑
◗
◗
(Succ x′ ) 6 y
✑◗
✑
◗
✑
◗◗
✑
′
(Succ x ) 6 0 = False
(Succ x′ ) 6 (Succ y′ ) = x′ 6 y′
✑
✑
0 6 y = True
Figure 2.1: Definitional tree for the function “6”
tree. Every leaf node represents a rule. The inductive argument of a branch node is
pointed by surrounding the corresponding subterm within a box.
The needed narrowing strategy performs steps which are needed for computing solutions:
Definition 5 (needed narrowing step/derivation [Antoy et al., 2000])
1. A narrowing step t ❀p,R,σ t′ is called needed (or outermost-needed) iff, for
every η ≥ σ, p is the position of a needed (or outermost-needed) redex of η(t),
respectively.
2. A narrowing derivation is called needed (or outermost-needed) iff every step of
the derivation is needed (or outermost-needed, respectively).
In comparison to needed reduction, the definition of needed narrowing adds a new
level of difficulty for computing the needed steps. In addition to consider the normal
forms, one must take into account any possible instantiation of the input term. In
inductively sequential systems, the problem has an efficient solution which is achieved
by giving up the requirement that the unifier of a narrowing step be most general.
This strategy is basically equivalent to lazy narrowing [Moreno-Navarro and Rodrı́guezArtalejo, 1992] where narrowing steps are applied to the outermost function, if possible, and inner functions are only narrowed if their evaluation is demanded by a
constructor symbol in the left-hand side of some rule (i.e., a typical outermost strategy).
For instance, consider again Example 6. In a term like t1 6 t2 , it is always
necessary to evaluate t1 to some head normal form (i.e., a variable or a constructorrooted term) since all three rules defining “6” have a non-variable first argument. On
the other hand, the evaluation of t2 is only needed if t1 is of the form Succ t. Thus,
if t1 is a free variable, needed narrowing instantiates it to a constructor, here Zero or
Succ x. Depending on this instantiation, either the first rule is applied or the second
argument t2 is evaluated.
44
2.4. DECLARATIVE MULTI-PARADIGM LANGUAGES
2.4
45
Declarative Multi-Paradigm Languages
Functional logic languages have evolved to so called declarative multi-paradigm languages like, e.g., Curry [Hanus, 2006a], Toy [López-Fraguas and Sánchez-Hernández,
1999] and Escher [Lloyd, 1994]. In order to make things concrete, we consider in this
thesis the language Curry. Curry is an international initiative intended to provide
a common platform for the research, teaching and application of integrated functional logic languages. It is a programming language which combines the most important operational principles developed from the most relevant declarative programming
paradigms, namely functional programming and logic programming.
Curry combines features from functional programming (nested expressions, higherorder functions, lazy evaluation), logic programming (logical variables, partial data
structures, built-in search), and concurrent programming (concurrent evaluation of
expressions with synchronization on logical variables). This combination allows Curry
to provide additional features in comparison to the pure languages (compared to
functional programming: search, computing with partial information; compared to
logic programming: more efficient evaluation due to the deterministic and demanddriven evaluation of functions).
Curry follows a Haskell-like syntax, i.e., variables and function names start with
lowercase letters and data constructors start with an uppercase letter. The application
of function f to an argument e is denoted by juxtaposition, i.e., (f e).
The basic operational semantics of Curry is based on a combination of needed
narrowing and residuation [Hanus, 1997]. The residuation principle is based on the
idea of delaying function calls until they are ready for a deterministic evaluation.
Residuation preserves the deterministic nature of functions and naturally supports
concurrent computations. The precise mechanism—narrowing or residuation—for
each function is specified by evaluation annotations. The annotation of a function as
rigid forces the delayed evaluation by rewriting, while functions annotated as flexible
can be evaluated in a non-deterministic manner by narrowing.
In actual implementations, e.g., the PAKCS environment [Hanus et al., 2006] for
Curry, programs may also include a number of additional features: calls to external
(built-in) functions, concurrent constraints, higher-order functions, overlapping lefthand sides, guarded expressions, etc. In order to ease the compilation of programs as
well as to provide a common interface for connecting different tools working on source
programs, a flat representation for programs has been introduced. This representation
is based on the formulation of Hanus and Prehofer [1999] to express pattern-matching
by case expressions. The complete flat representation is called FlatCurry [Hanus
et al., 2006] and is used as an intermediate language during the compilation of source
programs.
45
46
CHAPTER 2. PRELIMINARIES
Flat Programs
In FlatCurry, all functions are defined at the top level (i.e., local function declarations in source programs are globalized by lambda lifting) and the pattern-matching
strategy is made explicit by the use of case expressions. Thus, a FlatCurry program
basically consists of a list of data type declarations (defining the data constructors
used in the program) and a list of function definitions. To support separate compilation, FlatCurry contains also information about modules (i.e., each module is
represented as a FlatCurry program). To support interactive tools (e.g., debuggers,
program verifiers, interactive back ends that allow the user to evaluate arbitrary expressions), the FlatCurry representation contains also some information about the
syntactic representation of source programs (list of infix operator declarations and a
translation table to map source program names to internal names and vice versa).
Altogether, a FlatCurry program is a tuple:
(module, imports, types, f unctions, operators, translation)
with the following components [Hanus, 2006b]:
module: The name of the module represented by this structure.
imports: The names of the modules imported by this program.
types: The list of data type declarations defined in this module. Each data type
declaration consists of a list of type variables as parameters (for polymorphic
data structures) and a list of constructor declarations where each constructor
declaration contains the name and arity of the data constructor and a list of
type expressions (the argument types of this data constructor).
functions: The functions defined in this module. A function declaration consists of
the name and arity of the function together with a type expression specifying
the function’s type and a single rule (where the right-hand side can contain
case expressions and disjunctions) specifying the function’s meaning. External
functions (i.e., primitive functions not defined in this or another module) contain
instead of the rule the external name of this function.
operators: The infix operators declared in this module. An operator declaration
contains the fixity (or associativity) and the precedence for each operator. Although an operator declaration contains no semantic information, it might be
used by interactive tools to print or read expressions.
translation: The translation table describes the mapping of identifiers (type constructors, data constructors, defined functions) used in the source program into
their corresponding internal names used in the intermediate FlatCurry program
46
2.4. DECLARATIVE MULTI-PARADIGM LANGUAGES
P
D
e
::=
::=
::=
|
|
D1 . . . Dm
f xn = e
t
case x of {pm → em }
f case x of {pm → em }
(program)
(rule)
(term)
(rigid case)
(flexible case)
t
p
::=
|
|
::=
x
c tn
f tn
c xn
47
(variable)
(constructor call)
(function call)
(flat pattern)
Figure 2.2: Syntax of flat programs
(since it is often necessary to rename identifiers to avoid name clashes, e.g., the
internal names are prefixed by the module name). Only the externally visible
names (i.e., names exported by this module) are listed here. This information
might be used by back ends for representing intermediate (e.g., during debugging) or final results or to allow the user to type in and evaluate arbitrary
expressions.
In order to simplify the presentation, in some chapters we will only consider the core of
the flat representation. Extending the developments to the remaining features is not
difficult and, indeed, the implementations reported for the debugging techniques in
this thesis cover many of these features. The syntax of flat programs is summarized in
Figure 2.2. Extra variables are intended to be instantiated by flexible case expressions.
We assume that each extra variable x is explicitly introduced in flat programs by a
direct circular let binding of the form “let x = x in e”. We also call such variables
which are bound to themselves logical variables.
We consider the following domains:
e1 , e2 , . . . ∈ E (expressions)
t1 , t2 , . . . ∈ T (terms)
v1 , v2 , . . . ∈ V (values)
The only difference between terms and expressions is that the latter may contain
case expressions. Values are terms in head normal form, i.e., variables or constructorrooted terms. A program P consists of a sequence of function definitions; each function is defined by a single rule whose left-hand side contains only different variables
as parameters. The right-hand side is an expression e composed by variables, constructors, function calls, and case expressions for pattern-matching. In general, a case
expression has the form:
(f )case x of {c1 xn1 → e1 ; . . . ; ck xnk → ek }
where (f )case stands for either fcase or case, x is a variable, c1 , . . . , ck are different
constructors, and e1 , . . . , ek are expressions (possibly containing nested (f )case’s).
The pattern variables xni are locally introduced and bind the corresponding vari47
48
CHAPTER 2. PRELIMINARIES
ables of the subexpression ei . The difference between case and fcase shows up when
the argument x evaluates to a free variable (within a particular computation): case
suspends—which corresponds to residuation, i.e., pure functional reduction—whereas
fcase non-deterministically binds this variable to the pattern in a branch of the case
expression—which corresponds to narrowing [Antoy et al., 2000]. Note that our functional logic language mainly differs from typical (lazy) functional languages in the
presence of flexible case expressions.
Example 7 Consider again the rules defining functions “+” (Example 4) and “6”
(Example 6). These functions can be defined in the flat representation as follows2 :
x + y = fcase x of { Zero
→ y;
Succ n → Succ (n + y) }
x 6 y = fcase x of { Zero
→ True;
Succ n → fcase y of { Zero
→ False;
Succ m → n 6 m } }
An automatic transformation from source (inductively sequential) programs to flat
programs has been introduced by Hanus and Prehofer [1999]. Translated programs
always fulfill the following restrictions: case expressions in the right-hand sides of
program rules appear always in the outermost positions (i.e., there is no case expression inside a function or constructor call) and all case arguments are variables, thus
the syntax of Figure 2.2 is general enough for our purposes. We shall assume these
restrictions on flat programs in the following.
The operational semantics of flat programs is shown in Figure 2.3. It is based
on the LNT—for Lazy Narrowing with definitional Trees—calculus of Hanus and
Prehofer [1999]. The one-step transition relation =⇒σ is labeled with the substitution
σ computed in the step. Let us briefly describe the LNT rules:
The select rule selects the appropriate branch of a case expression and continues
with the evaluation of this branch. This rule implements pattern matching.
The guess rule applies when the argument of a flexible case expression is a variable.
Then, this rule non-deterministically binds this variable to a pattern in a branch of
the case expression. The step is labeled with the computed binding. Observe that
there is no rule to evaluate a rigid case expression with a variable argument. This
situation produces a suspension of the evaluation.
The case eval rule can be applied when the argument of the case construct is not
in head normal form (i.e., it is either a function call or another case construct). Then,
it tries to evaluate this expression recursively.
2 Although we consider a first-order representation—the flat language—we use a curried notation
in concrete examples (as in Curry).
48
2.4. DECLARATIVE MULTI-PARADIGM LANGUAGES
(select)
(guess)
(case eval)
49
(f )case (c tn ) of {pm → em } =⇒id σ(ei )
if pi = c xn , c ∈ C, and σ = {xn 7→ tn }
f case x of {pm → em }
=⇒σ σ(ei )
if σ = {x 7→ pi } and i ∈ {1, . . . , m}
(f )case e of {pm → em } =⇒σ σ((f )case e′ of {pm → em })
if e is not in head normal form and e =⇒σ e′
(fun)
f tn
=⇒id σ(e)
if f xn = e ∈ P and σ = {xn 7→ tn }
Figure 2.3: Standard operational semantics (LNT calculus)
Finally, the fun rule performs the unfolding of a function call. As in proof procedures for logic programming, we assume that we take a program rule with fresh
variables in each such evaluation step.
Note that there is no rule to evaluate terms in head normal form; in this case, the
computation stops successfully. An LNT derivation is denoted by e0 =⇒∗σ en , which
is a shorthand for the sequence e0 =⇒σ1 . . . =⇒σn en with σ = σn ◦ · · · ◦ σ1 (if n = 0
then σ = id). An LNT derivation e =⇒∗σ e′ is successful when e′ is in head normal
form. Then, we say that e evaluates to e′ with computed answer σ.
Example 8 Consider the function “6” of Example 7 together with the initial call
“(Succ x) 6 y”. The LNT calculus computes, among others, the following successful derivation:
(Succ x) 6 y
=⇒id
fcase (Succ x) of
{Z
→ True;
(Succ n) → fcase y of {Z → False; (Succ m) → n 6 m}}
=⇒id
fcase y of {Z → False; (Succ m) → x 6 m}
=⇒{y7→Z} False
(fun)
(select)
(guess)
Therefore, (Succ x) 6 y evaluates to False with computed answer {y 7→ Z}.
In the remainder of this thesis, unless the contrary is stated, we will assume that computations always start from the distinguished function main which has no arguments.
Normalized Flat Programs
In some chapters we will assume that flat programs are normalized, i.e., let constructs
are used to ensure that the arguments of functions and constructors are always vari49
50
CHAPTER 2. PRELIMINARIES
P
D
e
::=
::=
::=
|
|
|
|
D1 . . . Dm
f xn = e
t
case x of {pm → em }
f case x of {pm → em }
e1 or e2
let x = e1 in e2
(program)
(rule)
(term)
(rigid case)
(flexible case)
(disjunction)
(let binding)
t
p
::=
|
|
::=
x
c xn
f xn
c xn
(variable)
(constructor call)
(function call)
(flat pattern)
Figure 2.4: Syntax of normalized flat programs
ables (not necessarily pairwise different). As in [Launchbury, 1993], this is essential
to express sharing without the use of complex graph structures.
Definition 6 (normalization [Albert et al., 2005]) The normalization of an expression e flattens all the arguments of function (or constructor) calls by means of
the mapping e∗ which is inductively defined as follows:
x∗
ϕ(xn )∗
ϕ(xi−1 , em )∗
(let xk = ek in e)∗
(e1 or e2 )∗
((f )case x of {pk → ek })∗
((f )case e of {pk → ek })∗
= x
= ϕ(xn )
= let xi = e∗1 in
ϕ(xi , e2 , . . . , em )∗
where e1 6∈ Var , xi is fresh
= let xk = ek ∗ in e∗
= e1 ∗ or e2 ∗
= (f )case x of {pk 7→ ek ∗ }
= let x = e∗ in
(f )case x of {pk 7→ ek ∗ }
where e 6∈ Var , x is fresh
Here, ϕ denotes a constructor or function symbol. The extension of normalization to
flat programs is straightforward, i.e., each rule f xn = e is transformed into f xn = e∗ .
For simplicity, the normalization mapping above introduces one new let construct
for each non-variable argument, e.g., f e is transformed into “let x = e in f x”.
Trivially, this could be modified in order to produce one single let with the bindings
for all non-variable arguments of a function (or constructor) call, which we assume
for the subsequent examples.
The syntax of normalized flat programs is shown in Figure 2.4. A program P consists of a sequence of function definitions D such that the left-hand side has pairwise
different variable arguments. The right-hand side is an expression e ∈ E composed
50
2.4. DECLARATIVE MULTI-PARADIGM LANGUAGES
51
(VarCons) Γ[x 7→ c(xn )] : x ⇓ Γ[x 7→ c(xn )] : c(xn )
(VarExp)
Γ:e ⇓ ∆:v
Γ[x 7→ e] : x ⇓ ∆[x 7→ v] : v
(Val)
Γ:v ⇓ Γ:v
(Fun)
Γ : ρ(e) ⇓ ∆ : v
Γ : f (xn ) ⇓ ∆ : v
(Let)
Γ[y 7→ ρ(e′ )] : ρ(e) ⇓ ∆ : v
Γ : let x = e′ in e ⇓ ∆ : v
(Or)
Γ : ei ⇓ ∆ : v
Γ : e1 or e2 ⇓ ∆ : v
(where i ∈ {1, 2})
(Select)
∆ : ρ(ei ) ⇓ Θ : v
Γ : x ⇓ ∆ : c(yn )
Γ : (f )case x of {pk → ek } ⇓ Θ : v
(where pi = c(xn )
(where e is not a value)
(where v is a value)
(where f (yn ) = e ∈ P and ρ = {yn 7→ xn })
(where ρ = {x 7→ y} and y is fresh)
and ρ = {xn 7→ yn })
(Guess)
Γ:x ⇓ ∆:y
∆[y →
7 ρ(pi ), yn 7→ yn ] : ρ(ei ) ⇓ Θ : v
Γ : fcase x of {pk → ek } ⇓ Θ : v
(where pi = c(xn ), ρ = {xn 7→ yn } and yn are fresh variables)
Figure 2.5: Rules of the natural semantics
by variables, data constructors, function calls, let bindings where the local variable x
is only visible in e1 and e2 , disjunctions (e.g., to represent set-valued functions), and
case expressions for pattern-matching.
The Natural Semantics
In this section, we describe a natural (big-step) semantics for functional logic languages (defined in [Albert et al., 2005]). Figure 2.5 shows the semantics. A heap,
denoted by Γ, ∆, or Θ, is a partial mapping from variables to expressions (the empty
heap is denoted by [ ]). The value associated to variable x in heap Γ is denoted by
Γ[x]. Γ[x 7→ e] denotes a heap with Γ[x] = e. A logical variable x is represented by
a circular binding of the form Γ[x] = x. A value v is a constructor-rooted term c en
(i.e., a term whose outermost function symbol is a constructor symbol) or a logical
variable (w.r.t. the associated heap). We use judgements of the form “Γ : e ⇓ ∆ : v”
which are interpreted as “in the context of heap Γ, the expression e evaluates to value
v, producing a new heap ∆”.
In order to evaluate a variable which is bound to a constructor-rooted term in the
51
52
CHAPTER 2. PRELIMINARIES
heap, rule VarCons reduces the variable to this term.
Rule VarExp achieves the effect of sharing. If the variable to be evaluated is
bound to some expression in the heap, then the expression is evaluated and the heap
is updated with the computed value; finally, we return this value as the result.
For the evaluation of a value, rule Val returns it without modifying the heap.
Rule Fun corresponds to the unfolding of a function call. The result is obtained by
reducing the right-hand side of the corresponding rule (we assume that the considered
program P is a global parameter of the calculus).
Rule Let adds its associated binding3 to the heap and proceeds with the evaluation
of its main argument. Note that we give the introduced variable a fresh name in order
to avoid variable name clashes.
Rule Or non-deterministically reduces an or expression to either the first or the
second argument.
Rule Select corresponds to the evaluation of a case expression whose argument
reduces to a constructor-rooted term. In this case, we select the appropriate branch
and, then, proceed with the evaluation of the expression in this branch by applying
the corresponding matching substitution.
Rule Guess applies when the argument of a flexible case expression reduces to
a logical variable. It binds this variable to one of the patterns and proceeds by
evaluating the corresponding branch. If there is more than one branch, one of them
is chosen non-deterministically. Renaming the pattern variables is necessary to avoid
name clashes. We also update the heap with the (renamed) logical variables of the
pattern.
A proof of a judgement corresponds to a derivation sequence using the rules of
Figure 2.5. Given a program P , the initial configuration has the form “[ ] : main”. If
the judgement
[ ] : main ⇓ Γ : v
holds, we say that main evaluates to value v. The computed answer can be extracted
from the final heap Γ by a simple process of dereferencing.
The Operational Semantics
We now present the (small-step) operational semantics of Albert et al. [2005]. Each
configuration of the small-step semantics is a triple, hΓ, e, Si, where Γ is a heap, e
is an expression (often called the control of the small-step semantics), and S is a
stack. A stack (a list of variable names and case alternatives where the empty stack
is denoted by [ ]) is used to represent the current context.
3 We
consider that Let only has one binding for simplicity.
52
2.4. DECLARATIVE MULTI-PARADIGM LANGUAGES
(VarCons) hΓ[x 7→ t], x, Si =⇒ hΓ[x 7→ t], t, Si
53
where t is constructor-rooted
where e is not constructorrooted and e 6= x
(VarExp)
hΓ[x 7→ e], x, Si =⇒ hΓ[x 7→ e], e, x : Si
(Val)
hΓ[x 7→ e], v, x : Si =⇒ hΓ[x 7→ v], v, Si where v is a value
(Fun)
hΓ, f (xn ), Si =⇒ hΓ, ρ(e), Si
where f (yn ) = e ∈ P and
ρ = {yn 7→ xn }
(Let)
hΓ, let x = e1 in e2 , Si
=⇒ hΓ[y 7→ ρ(e1 )], ρ(e2 ), Si
where ρ = {x 7→ y} and
y is a fresh variable
(Or)
hΓ, e1 or e2 , Si =⇒ hΓ, ei , Si
where i ∈ {1, 2}
(Case)
hΓ, (f )case x of {pk → ek }, Si
=⇒ hΓ, x, (f ){pk → ek } : Si
(Select)
hΓ, c(yn ), (f ){pk → ek } : Si
=⇒ hΓ, ρ(ei ), Si
where i ∈ {1, . . . k}, pi = c(xn ),
and ρ = {xn 7→ yn }
(Guess)
hΓ[y 7→(g,w) y], y, f {pk → ek } : Si
=⇒ hΓ[y 7→ ρ(pi ), yn 7→ yn ], ρ(ei ), Si
where i ∈ {1, . . . k}, pi = c(xn ),
ρ = {xn 7→ yn }, and yn are
fresh variables
Figure 2.6: Small-step operational semantics
The transition rules are shown in Figure 2.6. The meaning of the rules is the same
as in the natural semantics. Once a value is eventually computed, the computation
either terminates—when the stack is empty—or this value is used to update the
current heap (rule val).
The stack is used during the evaluation of case expressions. Rule Case initiates
the evaluation of a case expression by evaluating the case argument and pushing
the alternatives (f ){pk → ek } on top of the stack. If we reach a constructor-rooted
term, then rule Select is applied to select the appropriate branch and continue with
the evaluation of this branch. If we reach a logical variable and the case expression
on the stack is flexible (i.e., of the form f {pk → ek }), then rule Guess is used to
non-deterministically choose one alternative and continue with the evaluation of this
branch; moreover, the heap is updated with the binding of the logical variable to the
corresponding pattern.
Example 9 Consider function leq of Example 11 (where the bug is now corrected),
together with the initial call (leq Z (SZ)). The corresponding normalized flat program
53
54
CHAPTER 2. PRELIMINARIES
is the following:
main = let x3 = Z in
let x1 = Z in
let x2 = S x3 in leq x1 x2
leq x y = case x of
{ Z
→ True;
(S n) → case y of { Z
→ False;
(S m) → leq n m } }
The complete computation with the rules of Figure 2.6 is as follows:
h[ ], main, [ ]i
=⇒fun
=⇒let
=⇒let
=⇒let
=⇒fun
=⇒case
=⇒varcons
=⇒select
h[ ],
let x3 = Z in . . .,
[ ]i
h[x3 7→ Z],
let x1 = Z in . . .,
[ ]i
h[x1 7→ Z, . . .],
let x2 = S x3 in . . ., [ ]i
h[x2 7→ S x3, . . .], leq x1 x2,
[ ]i
h[. . .],
case x1 of . . .,
[ ]i
h[x1 7→ Z, . . .],
x1,
[{. . .}]i
h[x1 7→ Z, . . .],
Z,
[{. . .}]i
h[. . .],
True,
[ ]i
For clarity, each computation step is labeled with the applied rule. Therefore, main
computes the value True.
54
Part II
Profiling
55
Chapter 3
Time Equations for Lazy
Functional (Logic) Languages
There are very few approaches to measure the execution costs of lazy functional (logic)
programs. The use of a lazy execution mechanism implies that the complexity of an
evaluation depends on its context, i.e., the evaluation of the same expression may have
different costs depending on the degree of evaluation required by the different contexts
where it appears. In this chapter, we present a novel approach to complexity analysis
of functional (logic) programs. We focus on the construction of equations which
compute the time-complexity of expressions. In contrast to previous approaches, it is
simple, precise—i.e., it computes exact costs rather than upper/lower bounds—, and
fully automatic.
Part of the material of this chapter has been presented in the 2003 Joint Appia Gulp Prode Conference on Declarative Programming (AGP 2003), celebrated in
Reggio Calabria (Italy) in 2003 [Albert et al., 2003].
3.1
Introduction
There are very few approaches to formally reason about program execution costs in
the field of lazy functional (logic) languages. Most of these approaches analyze the
cost by first constructing (recursive) equations which describe the time-complexity
of a given program and then producing a simpler cost function (ideally as a closed
form) by some (semi-)automatic manipulation of these equations. In this chapter, we
concentrate on the first part of this process: the (automatic) construction of equations
which compute the time-complexity of a given lazy functional (logic) program.
Laziness introduces a considerable difficulty in the time-complexity analysis of
57
58
CHAPTER 3. TIME EQUATIONS FOR LAZY FUNCTIONAL...
functional (logic) programs. In eager (call-by-value) languages, the cost of nested
applications, e.g., f (g(v)), can be expressed as the sum of the costs of evaluating g(v)
and f (v ′ ), where v ′ is the result returned by the call g(v). In a lazy (call-by-name)
language, g(v) is only evaluated as much as needed by the outermost function f (in the
extreme, it could be no evaluated at all). Therefore, one cannot determine (statically)
the cost associated to the evaluation of the arguments of function calls. There exist
several methods which simplify this problem by transforming the original program
into an equivalent one under a call-by-value semantics and, then, by analyzing the
transformed program (e.g., [Métayer, 1988]). The translation, however, makes the
program significantly more complex and the number of evaluation steps is not usually
preserved through the transformation. A different approach is taken by [Sands, 1990;
Wadler, 1988], where the generated time-equations give upper (resp. lower) bounds
by using a strictness (resp. neededness) analysis. In particular, Wadler [1988] may
produce equations which are non-terminating even if the original program terminates;
in contrast, Sands [1990] guarantees that the time-equations terminate at least as
often as the original program. As an alternative approach, there are techniques (e.g.,
[Bjerner and Holmström, 1989]) which produce time equations which are precise, i.e.,
they give an exact time analysis. In this case, the drawback is that the time equations
cannot be solved mechanically.
In this chapter, we introduce a novel program transformation which produces (automatically) time equations which are precise—i.e., give an exact time—and, moreover, they can be executed in the same language of the source program—i.e., we define
a source-to-source transformation.
To be more precise, our aim is to transform a program so that it computes not
only values but their associated time-complexities. More formally, given a lazy (callby-name) semantics Sem and its cost-augmented version Semcost , we want to define
a program transformation trans(P ) = Pcost such that the execution of program P
with the cost-augmented semantics gives the same result as the execution of the transformed program Pcost with the standard semantics, i.e., Semcost (P ) = Sem(Pcost ).
Let us note that the considered problem is decidable. First, it is not difficult to construct a cost-augmented semantics which precisely computes the cost of a program’s
execution (see next section). Then the Futamura projections [Futamura, 1999] ensure
the existence of the desired transformed program Pcost : the partial evaluation [Jones
et al., 1993] of Semcost w.r.t. P gives a program P ′ such that P ′ produces the same
outputs as Semcost (P ), i.e.,
Sem(P, input) = output
Semcost (P, input) = (output, cost)
mix(Semcost , P ) = P ′ with P ′ (input) = (output, cost)
where mix denotes a partial evaluator. Therefore, Pcost (equivalently, P ′ in the above
58
3.2. COST AUGMENTED SEMANTICS
59
equations) exists and can be effectively constructed. However, by using a partial evaluator for the considered language (e.g., [Albert et al., 2002; Albert and Vidal, 2001]),
Pcost will be probably a cost-augmented interpreter slightly extended to only consider the rules of a particular program P . Therefore, similarly to the cost-augmented
interpreter, the program Pcost generated in this way will still use a sort of “ground
representation” to evaluate expressions. This makes the complexity analysis of the
program Pcost as difficult as the analysis of the original program (or even harder).
Furthermore, its execution will be almost as slow as running the cost-augmented
interpreter with the original program P (which makes it infeasible for profiling).
Our main concern in this chapter is to provide a more practical transformation.
In particular, our new approach produces simple equations, it gives exact timecomplexities, and it is fully automatic. For the sake of generality, our developments
are formalized in the context of a lazy functional logic language and, in particular, we
consider normalized flat programs in our formalizations. The use of logical variables
may be useful to analyze the complexity of the time-equations (as we will discuss in
Section 3.4.1). Nevertheless, our ideas can be directly applied to a pure (first-order)
lazy functional language.
The chapter is organized as follows. Section 3.2 introduces the cost-augmented
semantics of the considered lazy functional logic language. In Sect. 3.3, we present
our program transformation and state its correctness. The usefulness of the transformation is illustrated in Sect. 3.4 and several applications are outlined. Finally,
Section 3.5 reviews some related works.
3.2
Cost Augmented Semantics
In this section, we introduce a precise characterization of a cost semantics for our lazy
functional logic language.
We base our developments on the big-step semantics of Section 2.4. The only
difference is that our cost-augmented semantics does not model sharing, since then
the generated time-equations would be much harder to analyze (see Section 3.4.2).
In the following, we present an extension of the big-step semantics of Section 2.4 in
order to count the number of function unfoldings. The step-counting (state transition)
semantics for lazy functional logic programs is defined in Figure 3.1. We use judgments
of the form “Γ : e ⇓k ∆ : v” which are interpreted as “the expression e in the context
of the heap Γ evaluates to the value v with the (possibly modified) heap ∆ by unfolding
k function calls”. We briefly explain the rules of our semantics:
(VarCons) In order to evaluate a variable which is bound to a constructor-rooted
term in the heap, we simply reduce the variable to this term. The heap remains
unchanged and the associated cost is trivially zero.
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CHAPTER 3. TIME EQUATIONS FOR LAZY FUNCTIONAL...
(VarCons) Γ[x 7→ t] : x ⇓0 Γ[x 7→ t] : t
where t is constructor-rooted
(VarExp)
Γ[x 7→ e] : e ⇓k ∆ : v
Γ[x 7→ e] : x ⇓k ∆[x 7→ e] : v
(Val)
Γ : v ⇓0 Γ : v
(Fun)
where e is not constructor-rooted
and e 6= x
where v is constructor-rooted
or a variable with Γ[v] = v
Γ : ρ(e) ⇓k ∆ : v
7 xn }
where f (yn ) = e ∈ P and ρ = {yn →
Γ : f (xn ) ⇓k+1 ∆ : v
where ρ = {xk 7→ yk }
and yk are fresh variables
(Let)
Γ[yk 7→ ρ(ek )] : ρ(e) ⇓k ∆ : v
Γ : let {xk = ek } in e ⇓k ∆ : v
(Select)
∆ : ρ(ei ) ⇓k2 Θ : v
Γ : e ⇓k1 ∆ : c(yn )
Γ : (f )case e of {pk → ek } ⇓k1 +k2 Θ : v
(Guess)
∆[x 7→ ρ(pi ), yn →
Γ : e ⇓k1 ∆ : x
7 yn ] : ρ(ei ) ⇓k2 Θ : v
Γ : fcase e of {pk → ek } ⇓k1 +k2 Θ : v
where pi = c(xn )
and ρ = {xn 7→ yn }
where pi = c(xn ), ρ = {xn 7→ yn }, and yn are fresh variables
Figure 3.1: Step-counting semantics for lazy functional logic programs
(VarExp) If the variable to be evaluated is bound to some expression in the heap,
then this expression is evaluated and, then, this value is returned as the result.
The cost is not affected by the application of this rule.
It is important to notice that this rule is different from the one presented in
Section 2.4. In particular, in the original big-step semantics, it is defined as
follows:
Γ[x 7→ e] : e ⇓k ∆ : v
Γ[x 7→ e] : x ⇓k ∆[x 7→ v] : v
which means that the heap is updated with the computed value v for e. This is
essential to correctly model sharing (trivially, our modification does not affect
to the computed results, only to the number of function unfoldings). The generation of time-equations that correctly model sharing is a difficult topic which
is out of the scope of this thesis (see Section 3.4.2).
(Val) For the evaluation of a value, we return it without modifying the heap, with
associated cost zero.
(Fun) This rule corresponds to the unfolding of a function call. The result is obtained
by reducing the right-hand side of the corresponding rule. We assume that the
considered program P is a global parameter of the calculus. Trivially, if the
evaluation of the unfolded right-hand side requires k function unfoldings, the
original function call will require k + 1 function calls.
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3.3. THE PROGRAM TRANSFORMATION
61
(Let) In order to reduce a let construct, we add the bindings to the heap and proceed
with the evaluation of the main argument of let. Note that we rename the
variables introduced by the let construct with fresh names in order to avoid
variable name clashes. The number of function unfoldings is not affected by the
application of this rule.
(Select) This rule corresponds to the evaluation of a case expression whose argument
reduces to a constructor-rooted term. In this case, we select the appropriate
branch and, then, proceed with the evaluation of the expression in this branch
by applying the corresponding matching substitution. The number of function
unfoldings is obtained by adding the number of function unfoldings required to
evaluate the case argument with those required to evaluate the expression in
the selected branch.
(Guess) This rule considers the evaluation of a flexible case expression whose argument
reduces to a logical variable. It non-deterministically binds this variable to one
of the patterns and proceeds with the evaluation of the corresponding branch.
Renaming of pattern variables is also necessary to avoid variable name clashes.
Additionally, we update the heap with the (renamed) logical variables of the
pattern. Similarly to the previous rule, the number of function unfoldings is the
addition of the function unfoldings in the evaluation of the case argument and
the (non-deterministically) selected branch.
In the following, we denote by a judgment of the form Γ : e ⇓ ∆ : v a derivation with the standard semantics (without cost), i.e., the semantics of Figure 3.1 by
ignoring the cost annotations.
3.3
The Program Transformation
In this section, we present our program transformation to compute the time complexity of lazy functional logic programs. First, let us informally describe the approach of
Wadler [1988] in order to motivate our work and its advantages (the approach of Sands
[1990] is basically equivalent except for the use of a neededness analysis rather than
a strictness analysis). In this method, for each function definition f (x1 , . . . , xn ) = e,
a new rule of the form
cf (x1 , . . . , xn , α) = α ֒→ 1 + T [[e]]α
is produced, where cf is a time function and α denotes an evaluation context which
indicates whether the evaluation of a given expression is required. In this way,
transformed functions are parameterized by a given context. The right-hand side
is “guarded” by the context so that the corresponding cost is not added when its
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CHAPTER 3. TIME EQUATIONS FOR LAZY FUNCTIONAL...
evaluation is not required. A key element of this approach is the propagation of contexts through expressions. For instance, the definition of function T for function calls
is as follows:
T [[f (e1 , . . . , en )]]α = T [[e1 ]](f ♯1 α) + . . . + T [[en ]](f ♯n α) + cf (e1 , . . . , en , α)
The basic idea is that the cost of evaluating a function application can be obtained
from the cost of evaluating the strict arguments of the function plus the cost of the
outermost function call. This information is provided by a strictness analysis. Recall
that a function is strict in some argument i iff f (. . . , ⊥i , . . .) = ⊥, where ⊥ denotes
an undefined expression (i.e., the result of function f cannot be computed when the
argument i cannot be evaluated to a value).
Functions f ♯i are used to propagate strictness information through function arguments. To be more precise, they are useful to determine whether argument i of
function f is strict in the context α. Let us illustrate this approach with an example.
Consider the following function definitions:1
head xs =
foo xs
=
case xs of {(y : ys) → y}
let ys = head xs in head ys
According to Wadler [1988], these functions are transformed into
chead xs α
cfoo xs α
= α ֒→ 1
= α ֒→ 1 + chead xs (head♯1 α)
+ let ys = head xs in chead (ys α)
If the strictness analysis has a good precision, then whenever this function is called
within a strict context, (head♯1 α) should indicate that the head normal form of the
argument of head is also needed.
Unlike the previous transformation, we do not use the information of a strictness
(or neededness) analysis. A novel idea of our method is that we delay the computation
of the cost of a given function up to the point where its evaluation is actually required.
This contrasts to [Sands, 1990; Wadler, 1988] where the costs of the arguments of a
function call are computed a priori, hence the need of the strictness information. The
following definition formalizes our program transformation:
Definition 7 (time-equations) Let P be a program. For each program rule of the
form f (x1 , . . . , xn ) = e ∈ P , we produce a transformed rule:
fc (x1 , . . . , xn ) = let y = 1
in C[[e]] + y
1 Although we consider a first-order language, we use a curried notation in examples, as it is
common practice in functional programming.
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3.3. THE PROGRAM TRANSFORMATION
63
C[[x]] = x
C[[c(x1 , . . . , xn )]] = let y = c(x1 , . . . , xn )
in let z = 0
in (y, z)
C[[f (x1 , . . . , xn )]] = fc (x1 , . . . , xn )
C[[(f )case x of {pn → en }]] = (f )case x of {(pn , kn ) → C[[en ]] + kn }
C[[let yn = en in e]] = let yn = C[[en ]] in C[[e]]
Figure 3.2: Cost function C
where the definition of the cost function C is shown in Figure 3.2. Additionally, we
add the following function “+” to the transformed program:
(x, y) + z
=
case x of
{
c → let v = y + z
in (x, v)
c(xn ) → let v = y + z
in (x, v)}
for all constructor c/0 in P
for all constructor c/n in P
For clarity, we will use in the examples the non-normalized version of this function:
(c, k1 ) + k2
(c(xn ), k1 ) + k2
= (c, k1 + k2 )
= (c(xn ), k1 + k2 )
for all constructor c/0 in P
for all constructor c/n in P
Basically, for each constructor term c(xn ) computed in the original program, a pair
of the form (c(x′n ), k) is computed in the transformed program, where k is the number
of function unfoldings which are needed to produce the constructor “c”. For instance,
a constructor term—a list—like (x : y : []) will have the form (x : (y : ([], k3 ), k2 ), k1 ) in
the transformed program, where k1 is the cost of producing the outermost constructor
“:”, k2 the cost of producing the innermost constructor “:” and k3 the cost of producing
the empty list. The auxiliary function “+” is introduced to correctly increment the
current cost.
The definition of the cost function C distinguishes the following cases depending
on the structure of the expression in the right-hand side of the program rule, as it is
depicted in Figure 3.2:
• Variables are not modified. Nevertheless, observe that now their domain is
different, i.e., they denote pairs of the form (c(xn ), k).
• For constructor-rooted terms, their associated cost is always zero, since no function unfolding is needed to produce them.
• Each function call f (x1 , . . . , xk ) is replaced by its counterpart in the transformed
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CHAPTER 3. TIME EQUATIONS FOR LAZY FUNCTIONAL...
program, i.e., a call of the form fc (x1 , . . . , xn ).
• Case expressions are transformed by leaving the case structure and then replacing each pattern pi by the pair (pi , ki ), where ki denotes the cost of producing
the pattern pi (i.e., the cost of evaluating the case argument to some head
normal form). Then, this cost is added to the transformed expression in the
selected branch.
• Finally, let bindings are transformed by applying recursively function C to all
the expressions.
Theorem 8 states the correctness of our approach. In this result, we only consider
ground expressions—without free variables—as initial calls, i.e., we only consider
“functional” computations. The use of non-ground calls will be discussed in Section 3.4.1. Below, we denote by ∆c the heap obtained from ∆ by replacing every
binding x 7→ e in ∆ by x 7→ C[[e]].
Theorem 8 Let P be a program and let Pcost be its transformed version obtained by
applying Definition 7. Then, given a ground expression e, we have [ ] : e ⇓k ∆ : v
w.r.t. P iff [ ] : C[[e]] ⇓ ∆c : (C[[v]], k) w.r.t. Pcost .
Proof.
We prove a more general claim: for each cost-augmented derivation
Γ : e ⇓k ∆ : v
w.r.t. P , there exists a standard derivation
Γc : e′ ⇓ ∆c : (C[[v]], k)
w.r.t. Pcost and vice versa, where e′ is either a logical variable (according to Γc ) or
e′ = C[[e]].
We prove the claim by induction on the depth m of the derivation.
(Base case m = 0) In this case, only two rules can be applied:
• If e is a logical variable (i.e., a variable x such that Γ[x] = x) then, by rule Val
of the cost-augmented semantics, it is (Γ : x ⇓0 Γ : x) in the original program
P . Since e is a logical variable, e′ = (e, 0) and, by rule Val of the standard
semantics, we also have (Γc : (x, 0) ⇓ Γc : (x, 0)) in Pcost .
Similarly, if e = c(xn ), only rule Val can be applied: (Γ : c(xn ) ⇓0 Γ : c(xn )).
Thus, e′ = (c(xn ), 0) and, by rule Val of the standard semantics, we also have
(Γc : (c(xn ), 0) ⇓ Γc : (c(xn )), 0).
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3.3. THE PROGRAM TRANSFORMATION
65
• Otherwise, e = x is a variable with Γ = Γ[x 7→ c(xn ))] and rule VarCons of the
cost-augmented semantics is applied in program P :
Γ[x 7→ (c(xn )] : x ⇓0 Γ[x 7→ c(xn )] : c(xn )
Therefore, in the instrumented program, Pcost , it is Γc [x 7→ (c(xn ), k)] : x =
Γc [x 7→ (c(xn ), k)] : (c(xn ), k) and, thus, the cost k is not incremented.
(Inductive case m > 1) In this case, rules VarExp, Fun, Let, Or, Select, and Guess are
applicable. We consider each case separately.
(VarExp) Then, e is a variable with Γ[x] 6= x and we performed the following derivation step in P (with the cost-augmented semantics):
Γ : e1 ⇓k ∆ : v
Γ[x 7→ e1 ] : x ⇓k ∆[x 7→ v] : v
In the instrumented program, Pcost , it is e′ = C[[x]] = x. Therefore, the following
derivation holds in Pcost :
Γc : e1 ⇓ ∆c : (v, k)
Γc [x 7→ e1 ] : x ⇓ ∆c [x 7→ v] : (v, k)
Finally, the topmost judgement follows by the inductive hypothesis.
(Fun) Then, e is a function call and we performed the following derivation step in P
(with the cost-augmented semantics):
Γ : ρ(e1 ) ⇓k ∆ : v
Γ : f (xn ) ⇓k+1 ∆ : v
where f (yn ) = e1 ∈ P and ρ = {yn 7→ xn }. Therefore, e′ = C[[f (xn )]] =
fc (xn ) and fc (xn ) = e + 1 ∈ Pcost iff f (xn ) = e ∈ P . Hence, the following
subderivation holds in Pcost by applying rules Let and Fun of the standard
semantics:
Γc : ρ(e1 ) ⇓ ∆c : (v, k)
Γc [y 7→ 1] : ρ(e1 ) + y ⇓ ∆c : (v, k + 1)
c
Γ : let y = 1 in ρ(e1 ) + y ⇓ ∆c : (v, k + 1)
Γc : fc (xn ) ⇓ ∆c : (v, k + 1)
Finally, the claim follows by applying the inductive hypothesis to the topmost
judgement.
(Let) Then, e = (let x = e1 in e2 ) is a let expression and we performed the following
derivation step in P (with the cost-augmented semantics):
Γ[y 7→ ρ(e1 )] : ρ(e2 ) ⇓k ∆ : v
Γ : let x = e1 in e2 ⇓k ∆ : v
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CHAPTER 3. TIME EQUATIONS FOR LAZY FUNCTIONAL...
where ρ = {x 7→ y}. In the instrumented program, Pcost , it is e′ = C[[let x =
e1 in e2 ]] = let x = C[[e1 ]] in C[[e2 ]]. Therefore, the following derivation holds in
Pcost :
Γc [x 7→ ρ(C[[e1 ]])] : C[[e2 ]] ⇓ ∆c : (v, k)
Γc : let x = C[[e1 ]] in C[[e2 ]] ⇓ ∆c : (v, k)
Finally, the topmost judgement follows by the inductive hypothesis.
(Select) Then, e = (f )case x of {pk → ek } is a case expression and we performed the
following derivation step in P (with the cost-augmented semantics):
Γ : x ⇓k1 ∆ : c(yn )
∆ : ρ(ei ) ⇓k2 Θ : v
Γ : (f )case x of {pm → em } ⇓k1+k2 Θ : v
where pi = c(xn ), i ∈ {1, . . . , n}, and ρ = {xn 7→ yn }. Therefore, since x
evaluates to a non-variable expression, e′ = C[[(f )case x of {pm → em }]] =
(f )case x of {(pm , km ) → C[[em ]] + km }. Then, the following subderivation
holds in Pcost by applying rules Select of the standard semantics:
Γc : x ⇓ ∆c : (c(yn ), k1)
∆c : ρ(C[[ei ]]) ⇓ Θ : (v, k2)
Γc : (f )case x of {(pm , km ) → C[[em ]] + km } ⇓ Θc : (v, k1 + k2)
and the claim follows by applying the inductive hypothesis to the topmost judgements.
(Guess) Then, e = fcase x of {pk → ek } is a flexible case expression and we performed
the following derivation step in P (with the cost-augmented semantics):
Γ : x ⇓k1 ∆ : y
∆[y 7→ ρ(pi ), yn 7→ yn ] : ρ(ei ) ⇓k2 Θ : v
Γ : fcase x of {pk → ek } ⇓k1+k2 Θ : v
where pi = c(xn ), ρ = {xn 7→ yn }, and yn are fresh variables. Therefore, since
x evaluates to a logical variable, it is e′ = C[[(f )case x of {pm → em }]] =
(f )case x of {(pm , km ) → C[[em ]] + km }. Then, the following subderivation
holds in Pcost by applying rules Guess and Fun of the standard semantics:
Γc : x ⇓ ∆c : (c(yn ), k1)
∆c [y 7→ ρ(pi ), yn 7→ yn ] : ρ(C[[ei ]]) ⇓ Θ : (v, k2)
Γc : (f )case x of {(pm , km ) → C[[em ]] + km } ⇓ Θc : (v, k1 + k2)
and the claim follows by applying the inductive hypothesis to the topmost judgements.
✷
Roughly speaking, we obtain the same cost by evaluating an expression w.r.t.
the original program and the cost-augmented semantics defined in Figure 3.1 and by
evaluating it w.r.t. the transformed program and the standard semantics.
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3.4. THE TRANSFORMATION IN PRACTICE
3.4
67
The Transformation in Practice
In this section, we illustrate the usefulness of our approach by means of some selected
examples and point out possible applications of the new technique.
3.4.1
Complexity Analysis
The generated time-equations are simple and, thus, they are appropriate to reason
about the cost of evaluating expressions in our lazy language. We consider an example
which shows the behavior of lazy evaluation:
hd xs
from x
=
=
case xs of {(y : ys) → y}
let z = S x,
y = from z
in x : y
nth n xs
=
case n of
{
Z → hd xs;
(S m) → case xs of {(y : ys) → nth m ys}}
where natural numbers are built from Z and S. Function hd returns the first element
of a list, function from computes an infinite list of consecutive numbers (starting at
its argument), and function nth returns the element in the n-th position of a given
list. The transformed program is as follows:
hdc xs
fromc x
= case xs of {(y : ys, kxs ) → y + kxs } + 1
= let z = (S x, 0),
y = fromc z
in (x : y, 0) + 1
nthc n xs
= case n of
{ (Z, kn ) → hdc xs + kn ;
(S m, kn ) → case xs of
{(y : ys, kxs ) → nthc m ys + kxs } + kn } + 1
Consider, for instance, the function call2 “nth (S (S (S Z))) (from Z)”, which returns
the value (S (S (S Z))) with 9 function unfoldings. Consequently, the transformed call
nthc (S (S (S (Z, 0), 0), 0), 0) (fromc (Z, 0))
returns the pair ((S (S (S (Z, 0), 0), 0), 0), 9). Let us now reason about the complexity
of the original program. First, the equations of the transformed program can easily
be transformed as follows:3
2 Here,
we inline the let bindings to ease the reading of the function calls.
we only consider ground initial calls—and program rules have no extra-variables—we know
that case expressions will never suspend.
3 Since
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CHAPTER 3. TIME EQUATIONS FOR LAZY FUNCTIONAL...
hdc (y : ys, kxs )
fromc x
nthc (Z, kn ) xs
nthc (S m, kn ) (y : ys, kxs )
= y + (kxs + 1)
= (x : fromc (S x, 0), 1)
= (hdc xs) + (kn + 1)
= (nthc m ys) + (kxs + kn + 1)
Here, we have unnormalized and unflattened the program by using inlining (for let
expressions) and by moving the arguments of the case expressions to the left-hand
sides of the rules. This transformation is correct as proved by Hanus and Prehofer
(see Theorem 3.3 in [Hanus and Prehofer, 1999]. Now, we consider a generic call of
the form nthc v1 (from v2 ), where v1 and v2 are constructor terms (hence all the
associated costs are zero). By unfolding the call to fromc and replacing kn by 0 (since
v1 is a constructor term), we get:
nthc (Z, 0) (x : fromc (S x, 0), 1)
= (hdc (x : fromc (S x, 0), 1)) + 1
nthc (S m, 0) (x : fromc (S x, 0), 1) = (nthc m (fromc (S x, 0))) + 2
By unfolding the call to hdc in the first equation, we have:
nthc (Z, 0) (x : fromc (S x, 0), 1)
= x+3
nthc (S m, 0) (x : fromc (S x, 0), 1) = (nthc m (fromc (S x, 0))) + 2
Therefore, each recursive call to nthc requires 2 steps, while the base case requires 3
steps, i.e., the total cost of a call nth v1 (from v2 ) is n ∗ 2 + 3, where n is the natural
number represented by v1 .
A theorem prover may help to perform this task; and we think that logical variables
may also help to perform this task. For instance, if one executes the function call
nthc x (fromc (Z, 0)), where x is a logical variable, we get the following (infinite)
sequence (computed bindings for x are applied to the initial call):
nthc (Z, n1 ) (fromc (Z, 0))
=⇒ (Z, 3 + n1 )
nthc (S (Z, n1 ), n2 ) (fromc (Z, 0))
=⇒ (S (Z, 0), 5 + n1 + n2 )
nthc (S (S (Z, n1 ), n2 ), n3 ) (fromc (Z, 0)) =⇒ (S (S (Z, 0), 0), 7 + n1 + n2 + n3 )
...
If one considers that n1 = n2 = n3 = 0 (i.e., that the first input argument is a constructor term), then it is fairly easy to check that 2 steps are needed for each constructor
“S” of the first input argument of nthc , and 3 more steps for the final constructor
“Z”. Therefore, we have 2 ∗ n + 3, where n is the natural number represented by
the first input argument of nthc (i.e., we obtain the same result as with the previous
analysis).
On the other hand, there are already some practical systems, like ciaopp [Hermenegildo
et al., 2003], which are able to solve certain classes of equations. They basically try
to match the produced recurrence with respect to some predefined “patterns of equations” which represent classes of recurrences whose solution is known. These ideas
could be also adapted to our context.
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3.4. THE TRANSFORMATION IN PRACTICE
3.4.2
69
Other Applications and Extensions
Here we discuss some other practical applications which could be done by extending
the developments in this chapter.
There are several approaches in the literature to symbolic profiling (e.g., [Albert
and Vidal, 2002; Sansom and Peyton-Jones, 1997]). Here, the idea is to extend the
standard semantics with the computation of cost information (similarly to our stepcounting semantics, but including different cost criteria like case evaluations, function
applications, etc). A drawback of this approach is that one should modify the language
interpreter or compiler in order to implement it. Unfortunately, this possibility usually
implies a significant amount of work. An alternative could be the development of a
meta-interpreter extended with the computation of symbolic costs, but the execution
of such a meta-interpreter would involve a significant overhead at runtime. Therefore,
we have followed the first alternative. It is described in Chapter 4.
In this context, our program transformation could be seen as a preprocessing stage
(during compilation), which produces an instrumented program whose execution (in
a standard environment) returns not only the computed values but also the costs of
computing such values. For this approach to be feasible, an important property is the
executability of the transformed program. The following result states such a property.
Lemma 9 Let P be a normalized flat program (respecting the syntax of Figure 2.4)
and Pcost be the set of rules obtained according to Definition 7. Then Pcost is also a
valid normalized flat program.
Proof.
The claim follows easily from the following facts:
1. the original program is not modified; Only new expressions, and a new function
definition ( + )—which is flat and normalized—are added;
2. variables in left-hand sides are not modified; hence, the new time equations have
only different variables as parameters in the left-hand side;
3. cost function C do not introduce extra-variables in right-hand sides (apart from
those variables in let bindings, which are allowed); and
4. case structures are not modified.
✷
There are two possible extensions that we will develop in the next chapter. On
the one hand, we could adapt the use of cost centers in order to attribute costs
to concrete program functions or expressions, similarly to [Albert and Vidal, 2002;
Sansom and Peyton-Jones, 1997]. For instance, rather than using pairs (v, k) in the
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CHAPTER 3. TIME EQUATIONS FOR LAZY FUNCTIONAL...
transformed program, we could use triples of the form (v, k, cc) which indicate the
cost center cc to which the cost k should be attributed. On the other hand, counting
the number of steps is useful in general, but sometimes it is not very informative.
The transformation could also be extended to cover other different cost criteria, e.g.,
the number of case evaluations, the number of function applications, etc. (similarly
to symbolic profilers). In principle, this extension seems difficult since some of these
cost criteria are dynamic, i.e., they can only be measured at runtime. Therefore, it
is not easy to account for them statically in the transformed program. In the next
chapter, we first define an instrumented semantics able to compute different symbolic
costs; and then, we also define an equivalent program transformation which makes
practical the profiling of realistic programs.
Another possible application of our technique is the use of the computed cost
equations to produce standard recurrence equations which can be solved by existing
tools. For this purpose, a method to estimate the size of the terms would be needed.
For instance, one could consider the use of symbolic norms [Lindenstrauss and Sagiv,
1997]:
Definition 10 (symbolic norm) A symbolic norm is a function k·k : T (Σ, V) → IN
such that
Pn
ktk = m + i=0 ki kti k
where t = f (t1 , ..., tn ), n ≥ 0, and m and k1 , ..., kn are non-negative integer constants
depending only on f /n.
By using symbolic norms, we can associate sizes to terms and use them to produce
recurrence systems as it is shown in the next example.
Example 10 Consider again the unnormalized and unflattened versions of the functions hdc and nthc presented in Section 3.4.1 together with the symbolic norms:
( P
n
if t = f (t1 , . . . , tn ), n > 0
i=1 ||ti ||ts
||t||ts =
n
if t is a variable
1 + ||xs|| if t = (x : xs)
||t||ll =
n
if t is a variable
0
otherwise
which respectively correspond to the term-size and list-length symbolic norms. Cost
variables (i.e., those variables used in the rules to pass the cumulated cost from one
call to another) are initially bound to zero; and they take values during a computation. As we want to statically reason about the cost of the program, we can replace
that costs associated to arguments by zero. Hence, we have the program:
70
3.5. RELATED WORK
71
hdc (y : ys, 0)
= y + (0 + 1)
nthc (Z, 0) xs
= (hdc xs) + (0 + 1)
nthc (S m, 0) (y : ys, 0) = (nthc m ys) + (0 + 0 + 1)
After eliminating costs from the left-hand side, replacing + by +, and simplifying
the right-hand side, we can apply the norms to the arguments of function calls and
function definitions:
hdc ||y : ys||ll
nthc ||Z||ts ||xs||ll
nthc ||S m||ts ||(y : ys)||ll
=
=
=
y+1
hdc ||xs||ll + 1
nthc ||m||ts ||ys||ll + 1
Then, we delete all expressions in the right-hand side except costs and function calls,
because they do not contribute to the cost. The result is the following set of recurrence
rules:
hdc (n + 1)
nthc 0 n
nthc (n + 1) (m + 1)
= 1
= (hdc n) + 1
= (nthc n m) + 1
Therefore,
hd n
nth n m
=
=
1
n+2
where n > 0
where n, m ≥ 0
which is interpreted as: “ computing the head of a list of n elements implies one
function unfolding” and “ computing the nth element of a list of m elements implies
n + 2 function unfoldings”
Finally, the proposed transformation could also be useful in the context of program
debugging. In particular, some abstract interpretation based systems, like ciaopp
[Hermenegildo et al., 2003], allows the user to include annotations in the program
with the predicted cost. This cost will be then compared with the one automatically
generated by the compiler (here, our program transformation can be useful to compute
this cost). If both cost functions are not equivalent, this might indicate a programmer
error; for instance, the program may contain undesired loops which directly affect the
time complexity.
3.5
Related Work
There exist several approaches to automatic complexity analysis in the context of
eager functional languages (e.g., [Liu and Gómez, 1998]). Here, the construction
71
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CHAPTER 3. TIME EQUATIONS FOR LAZY FUNCTIONAL...
of cost equations is rather straightforward since the order of evaluation is statically
determined, i.e., given a nested function call f (g(t)), we first compute the cost of
evaluating g(t) to some value (say v) and, then, the cost of evaluating f (v). This
contrasts with lazy languages, where the order of evaluation depends dynamically on
the context of the given expression. This means that the above rule gives only a
rough upper bound. In general, if f only demands g to produce a certain number
of outermost constructors (to be consumed by f ), function g will not be further
evaluated. Consider, for instance, the following function definitions:
from x
head (x : xs)
= x : from (x + 1)
= x
where function from computes an infinite list of consecutive numbers starting at
x and function head returns the first element of a list. Given a call of the form
head (from 1), only two unfolding steps are necessary. In an eager language, this
evaluation will not terminate since function from is infinite. Note that determining
whether the evaluation of some subexpression is needed is in general an undecidable
problem.
Let us review some previous approaches to measuring the time-complexity of lazy
functional programs. There are some works which transform the original program into
an equivalent one under a call-by-value semantics and then analyze the transformed
program (e.g., [Métayer, 1988]). A drawback of this approach is that the translation
generally makes the program significantly more complex. Furthermore, the transformation does not ensure that the number of evaluation steps is preserved through the
transformation. An interesting approach is taken by Sands [1990] (which is based on
previous work by Wadler [1988]). His transformation produces time-equations which
approximate the time complexity. The approximation consists in giving lower or upper
bounds which are based on the results of a previous strictness analysis. An important
difference with our transformation is that we do not need such a strictness analysis.
Moreover, these approaches may produce equations which are non-terminating even
if the original program terminates. Our correctness results ensure that this cannot
happen in our transformation. There is also an interesting line of research aimed at
achieving exact time analyses. For instance, Bjerner and Holmström [1989] produce
time equations which are precise, i.e., they give the exact time complexity. The main
problem here is that the equations cannot be solved mechanically. An advantage of
our approach is that time equations are obtained as a source-to-source program transformation and they are amenable to be further manipulated and even executed in the
same language. Rosendahl [1989] defined a program transformation for first-order
functional programs whose transformed programs compute the number of call-byvalue reduction steps performed by the original program. Posteriorly, Sands [1996]
focussed on call-by-name reduction steps. His goal was to prove the correctness of
72
3.5. RELATED WORK
73
fold/unfold transformations, and his technique does it by annotating programs with
“ticks”, an identity function used to simulate a single tick of computation time. Later,
Voigtländer [2002] defined a program transformation to annotate left-linear rewrite
systems with a special identity function similar to the Sands’ ticks. In this case, the
function was used to measure the number of call-by-need steps in a computation. The
annotations were posteriorly used to prove syntactic conditions under which a transformed program is more efficient than the original program. This annotating scheme
used is similar to ours except that it does not consider the use of logical variables.
Our work is implicitly related to the profiling of lazy functional (logic) languages.
Practical approaches to profiling are based on cost-augmented semantics. The wellknown work of Sansom and Peyton-Jones [1997] formalizes a profiling scheme by
means of a small-step semantics enhanced with cost information. This semantics is
proved to be equivalent to the natural semantics of Launchbury [1993] and the cost
criteria are experimentally related to execution times. In the field of functional logic
programming, a similar approach has been taken in [Albert and Vidal, 2002]. The
previous section discussed how our program transformation could be used for the
same purpose.
Finally, in the field of partial evaluation, time equations have been used to assess
the effectiveness of the transformation process (see, e.g., [Albert et al., 2001; Vidal,
2002]). An essential difference w.r.t. our approach is the structure of the generated
cost equations. In [Albert et al., 2001; Vidal, 2002], they are generated throughout the
partial evaluation process in order to mimic the structure of the partially evaluated
program. Also, they are sometimes not executable (i.e., when the generated equations
are not “closed”). Our approach is thus more general, produces executable equations
and is not tied to a particular program transformation (as the aforementioned works).
73
Chapter 4
Runtime Profiling of
Functional Logic Programs
In this chapter, we introduce a profiling scheme for modern functional logic languages
covering notions like laziness, sharing, and non-determinism. It extends the profiling
scheme presented in the former chapter by computing different kinds of symbolic costs
and not only the number of function unfoldings.
Firstly, we instrument a natural (big-step) semantics in order to associate a symbolic cost to each basic operation (e.g., variable updates, function unfoldings, case
evaluations). While this cost semantics provides a formal basis to analyze the cost of
a computation, the implementation of a cost-augmented interpreter based on it would
introduce a huge overhead. Therefore, we also introduce a sound transformation that
instruments a program such that its execution—under the standard semantics—yields
not only the corresponding results but also the associated costs. Finally, we describe
a prototype implementation of a profiler based on the developments in this chapter.
Part of the material of this chapter has been presented in the 14th International
Symposium on Logic-based Program Synthesis and Transformation (LOPSTR’04) celebrated in Verona (Italy) in 2004 [Braßel et al., 2004a].
4.1
Introduction
Comparing always involves measuring. The only way to determine whether a version
of a program has improved the previous version, or whether two programs are equivalently efficient, is measuring their performance. The importance of profiling in improving the performance of programs is widely recognized. Profiling tools are essential
for the programmer to analyze the effects of different source-to-source program ma75
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CHAPTER 4. RUNTIME PROFILING OF FUNCTIONAL LOGIC...
nipulations (e.g., partial evaluation, specialization, optimization, etc). Despite this,
one can find very few profiling tools for modern declarative languages. This situation
is mainly explained by the difficulty to correctly map execution costs to source code,
which is much less obvious than for imperative languages. In this chapter, we tackle
the definition of a profiling scheme for modern functional logic languages covering
notions like laziness, sharing, and non-determinism (like Curry [Hanus, 2006a] and
Toy [López-Fraguas and Sánchez-Hernández, 1999]); currently, there is no profiling
tool practically applicable for such languages.
When profiling the runtime of a given program, the results highly depend on
the considered language implementation. Indeed, computing actual runtimes is not
always the most useful information for the programmer. Runtimes may help to detect
that some function is expensive but they do not explain why it is expensive (e.g., is
it called many times? Is it heavily non-deterministic?).
In order to overcome these drawbacks, we introduce a symbolic profiler which outputs the number of basic operations performed in a computation. For this purpose,
we start from the natural semantics for functional logic programs of Figure 2.5 and
instrument it with the computation of symbolic costs associated to the basic operations of the semantics: variable lookups, function unfoldings, case evaluations, etc.
These operations are performed, in one form or another, by likely implementations of
modern functional logic languages. Our cost semantics constitutes a formal model of
the attribution of costs in our setting. Therefore, it is useful not only as a basis to
develop profiling tools but also to analyze the costs of a program computation (e.g.,
to formally prove the effectiveness of some program transformation).
Trivially, one can develop a profiler by implementing an instrumented interpreter
which follows the previous cost semantics. However, this approach is not practicable
as it demands a huge overhead, making the profiling of realistic programs impossible.
Thus, in a second step, we design a source-to-source transformation that instruments a
program such that its execution—under the standard semantics—outputs not only the
corresponding results but also the associated costs. We formally prove the correctness
of our transformation (i.e., the costs computed in a source program w.r.t. the cost
semantics are equivalent to the costs computed in the transformed program w.r.t. the
standard semantics) and, finally, we discuss a prototype implementation of a profiler
based on the developments presented in this chapter.
The chapter is organized as follows. In the next section, we informally introduce
our model for profiling functional logic computations. Section 4.3 formalizes an instrumented semantics which also computes cost information. Section 4.4 introduces
a transformation instrumenting programs to compute symbolic costs. Section 4.5
describes an implementation of a profiler for Curry programs. Finally, Section 4.6
includes a comparison to related work and concludes.
76
4.2. A RUNTIME PROFILING SCHEME
4.2
77
A Runtime Profiling Scheme
Traditionally, profiling tools attribute execution costs to the functions or procedures
of the considered program. Following [Albert and Vidal, 2002; Sansom and PeytonJones, 1997], in this chapter we take a more flexible approach which allows us to
associate a cost center with any expression of interest. This allows the programmer to
choose an appropriate granularity for profiling, ranging from whole program phases to
single subexpressions in a function. Nevertheless, our approach can easily be adapted
to work with automatically instrumented cost centers; for instance, if one wants to
use the traditional approach in which all functions are profiled, each function can be
automatically annotated by introducing a cost center for the entire right-hand side.
Cost centers are marked with the (built-in) function scc (for set cost center).
Intuitively speaking, given an expression “scc(cc, e)”, the costs attributed to cost
center cc are the entire costs of evaluating e as far as the enclosing context demands
it, including the cost of
• evaluating any function called by the evaluation of the expression e,
but excluding the cost of
• evaluating the free variables of e (i.e., those variables which are not introduced
by a let binding in e) and
• evaluating any scc-expressions within e or within any function which is called
from e.
In this chapter we consider again the class of normalized flat programs to formalize
our profiling scheme. The following program contains two versions of a function to
compute the length of a list (for readability, we show the non-normalized version of
function main):
len(x) = fcase x of { []
→ 0;
(y:ys) → let z = 1,
w = len(ys)
in z + w }
len2s(x) = fcase x of { []
→ 0;
(y:ys) → fcase ys of
{ []
→ 1;
(z:zs) → let w = 2,
v = len2s(zs)
in w + v
} }
main = let list = scc("list",[1..5000])
in scc("len",len(list)) + scc("len2s",len2s(list))
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CHAPTER 4. RUNTIME PROFILING OF FUNCTIONAL LOGIC...
Cost criteria
Function unfolding
Binding a logical variable
Variable update
Variable lookup
Symbol
F
B
U
V
Cost criteria
Allocating a heap object
Case evaluation
Non-deterministic choice
Entering an scc
Symbol
H
C
N
E
Table 4.1: Basic costs
Here, main computes twice the length of the list [1..5000], which is a standard predefined way to define the list [1, 2, 3, . . . , 4999, 5000]. Each computation of the length
uses a different function, len and len2s. Intuitively, len2s is more efficient than len
because it performs half the number of function calls (indeed, len2s has been obtained
by unfolding function len). This is difficult to check with traditional profilers because
the overhead introduced to build the list hides the differences between len and len2s.
For instance, the computed runtimes in the PAKCS environment [Hanus et al., 2006]
for Curry are 9980 ms and 9990 ms for len([1..5000]) and len2s([1..5000]), respectively.1
Should one conclude that len and len2s are equally efficient? In order to answer
this question, a profiler based on cost centers can be very useful. In particular,
by including the three cost centers shown in the program above, the costs of len,
len2s, and the construction of the input list can be clearly distinguished. With our
execution profiler which distributes the execution time to different cost centers (its
implementation is discussed in Section 4.5.1), we have experimentally checked that
both functions are not equally efficient:
cost center
runtimes
main
17710
list
7668966
len
1110
len2s
790
Here, runtimes are expressed in a number of “ticks” (a basic time unit). Thanks to the
use of cost centers, we can easily check that len2s is an improvement of len. However,
what is the reason for such an improvement? We introduce symbolic costs—associated
with the basic operations of the language semantics—so that a deeper analysis can be
made. The considered kinds of costs are shown in Table 4.1. For the example above,
our symbolic profiler returns the following results (only the most relevant costs for
this example are shown):
1 The slow execution is due to the fact that experiments were performed with a version of the
above program that uses numbers built from Z and Succ (to avoid the use of built-in functions).
78
4.3. COST SEMANTICS
H
V
C
F
main
5000
5100
5100
5300
79
list
61700
280400
280400
168100
len
5100
5100
5100
5100
len2s
5100
5100
5100
2600
From this information, we observe that only function unfoldings (F ) are halved, while
the remaining costs are equal for both len and len2s. Therefore, we can conclude
that, in this example, unfolding function len with no input data only improves cost
F (which has a small impact on current compilers, as it has been shown before).
4.3
Cost Semantics
In this section, we instrument a natural (big-step) semantics for functional logic languages (defined in Figure 2.5) with the computation of symbolic costs. Figure 4.1
cc
shows the cost-augmented semantics. Here, Γ[x 7→ e] denotes a heap with Γ[x] = e
and associated cost center cc, i.e., we use this notation either as a condition on a heap
Γ or as a modification of Γ. We use judgements of the form “cc, Γ : e ⇓θ ∆ : v, ccv ”
which are interpreted as “in the context of heap Γ and cost center cc, the expression
e evaluates to value v with associated cost θ, producing a new heap ∆ and cost center
ccv ”.
In order to evaluate a variable which is bound to a constructor-rooted term in the
heap, rule VarCons reduces the variable to this term. Here, cost V is attributed to the
current cost center cc to account for the variable lookup (this attribution is denoted
by {cc ← V } and similarly for the other cost symbols).
Rule VarExp achieves the effect of sharing. If the variable to be evaluated is
bound to some expression in the heap, then the expression is evaluated and the heap
is updated with the computed value; finally, we return this value as the result. In
addition to counting the cost θ of evaluating expression e, both V and U are attributed
to cost centers cc and ccv , respectively.
For the evaluation of a value, rule Val returns it without modifying the heap. No
costs are attributed in this rule since actual implementations have no counterpart for
this action.
Rule Fun corresponds to the unfolding of a function call. The result is obtained by
reducing the right-hand side of the corresponding rule (we assume that the considered
program P is a global parameter of the calculus). Cost F is attributed to the current
cost center cc to account for the function unfolding.
Rule Let adds its associated binding to the heap and proceeds with the evaluation
of its main argument. Note that we give the introduced variable a fresh name in order
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CHAPTER 4. RUNTIME PROFILING OF FUNCTIONAL LOGIC...
cc
cc
c
c
(VarCons) cc, Γ[x 7−→
c(xn )] : x ⇓{cc←V } Γ[x 7−→
c(xn )] : c(xn ), ccc
(VarExp)
cce , Γ : e ⇓θ ∆ : v, ccv
cce
ccv
cc, Γ[x 7−→ e] : x ⇓{cc←V }+θ+{ccv ←U } ∆[x 7−→ v] : v, ccv
(Val)
cc, Γ : v ⇓{ } Γ : v, cc
(Fun)
cc, Γ : ρ(e) ⇓θ ∆ : v, ccv
cc, Γ : f (xn ) ⇓{cc←F }+θ ∆ : v, ccv
(Let)
cc, Γ[y 7−→ ρ(e′ )] : ρ(e) ⇓θ ∆ : v, ccv
cc, Γ : let x = e′ in e ⇓{cc←H}+θ ∆ : v, ccv
(Or)
cc, Γ : ei ⇓θ ∆ : v, ccv
cc, Γ : e1 or e2 ⇓{cc←N }+θ ∆ : v, ccv
(Select)
cc, ∆ : ρ(ei ) ⇓θ2 Θ : v, ccv
cc, Γ : x ⇓θ1 ∆ : c(yn ), ccc
cc, Γ : (f )case x of {pk → ek } ⇓θ1 +{cc←C}+θ2 Θ : v, ccv
(where e is
not a value)
(where v is a value)
(where f (yn ) = e ∈ P
and ρ = {yn 7→ xn })
cc
(where ρ = {x 7→ y}
and y is fresh)
(where i ∈ {1, 2})
(where pi = c(xn ) and ρ = {xn 7→ yn })
cc
cc
cc, ∆[y 7→ ρ(pi ), yn 7→ yn ] : ρ(ei ) ⇓θ2 Θ : v, ccv
cc, Γ : x ⇓θ1 ∆ : y, ccy
(Guess)
cc, Γ : fcase x of {pk → ek } ⇓θ1 +{cc←V,cc←U,cc←B,cc←n∗H}+θN +θ2 Θ : v, ccv
(where pi = c(xn ), ρ = {xn 7→ yn }, yn are fresh variables,
and θN = {cc ← N } if k > 1 and θN = { } if k = 1)
(SCC)
cc′ , Γ : e ⇓θ ∆ : v, ccv
cc, Γ : scc(cc′ , e) ⇓θ+{cc′ ←E} ∆ : v, ccv
Figure 4.1: Rules of the cost semantics
to avoid variable name clashes. In this case, cost H is added to the current cost center
cc.
Rule Or non-deterministically reduces an or expression to either the first or the
second argument. N is attributed to the current cost center to account for a nondeterministic step.
Rule Select corresponds to the evaluation of a case expression whose argument
reduces to a constructor-rooted term. In this case, we select the appropriate branch
and, then, proceed with the evaluation of the expression in this branch by applying
the corresponding matching substitution. In addition to the costs of evaluating the
case argument, θ1 , and the selected branch, θ2 , we add cost C to the current cost
center cc to account for the pattern matching.
Rule Guess applies when the argument of a flexible case expression reduces to
80
4.4. COST INSTRUMENTATION
81
a logical variable. It binds this variable to one of the patterns and proceeds by
evaluating the corresponding branch. If there is more than one branch, one of them
is chosen non-deterministically. Renaming the pattern variables is necessary to avoid
name clashes. We also update the heap with the (renamed) logical variables of the
pattern. In addition to counting the costs of evaluating the case argument, θ1 , and the
selected branch, θ2 , we attribute to the current cost center cc costs V (for determining
that y is a logical variable), U (for updating the heap from y 7→ y to y 7→ ρ(pi )), B
(for binding a logical variable), n ∗ H (for adding n new bindings into the heap) and,
if there is more than one branch, N (for performing a non-deterministic step). Note
that no cost C is attributed to cost center cc (indeed, cost B is alternative to cost C).
Finally, rule SCC evaluates an scc-expression by reducing the expression e in the
context of the new cost center cc′ . Accordingly, cost E is added to cost center cc′ .
A proof of a judgement corresponds to a derivation sequence using the rules of
Figure 4.1. Given a program P , the initial configuration has the form “ccmain , [ ] :
main”, where ccmain is a distinguished cost center. If the judgement
ccmain , [ ] : main ⇓θ Γ : v, ccv
holds, we say that main evaluates to value v with associated cost θ. The computed
answer can be extracted from the final heap Γ by a simple process of dereferencing.
Obviously, the cost semantics is a conservative extension of the original big-step
semantics of Figure 2.5, since the computation of cost information imposes no restriction on the application of the rules of the semantics.
4.4
Cost Instrumentation
As mentioned before, implementing an interpreter for the cost semantics of Figure 4.1
is impracticable. It would involve too much overhead to profile any realistic program.
Thus, we introduce a transformation to instrument programs in order to compute the
symbolic costs:
Definition 11 (cost transformation) Given a program P , its cost instrumented
version Pcost is obtained as follows: for each program rule
f (x1 , . . . , xn ) = e
Pcost includes, for each cost center cc in P , one rule of the form
fcc (x1 , . . . , xn ) = Fcc ([[e]]cc )
where Fcc (e) is the identity function on e. Counting the calls to Fcc in the proof tree
corresponds to the number of F ’s accumulated in cost center cc. Function [[ ]] (shown
in Figure 4.2) is used to instrument program expressions.
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CHAPTER 4. RUNTIME PROFILING OF FUNCTIONAL LOGIC...
[[x]]cc
= Vcc (x)
[[c(x1 , . . . , xn )]]cc
= c(cc, x1 , . . . , xn )
[[f (x1 , . . . , xn )]]cc
= fcc (x1 , . . . , xn )
let x = x in [[e]]cc
= Hcc let x = [[e′ ]]cc in [[e]]cc
let x = update([[e′ ]]cc ) in [[e]]cc
[[let x = e′ in e]]cc
[[e1 or e2 ]]cc
[[case x of {pk → ek }]]cc
= Ncc ([[e1 ]]cc or [[e2 ]]cc )
if e′ = x
if e′ = c(yn )
otherwise
= case [[x]]cc of {p′k → Ccc ([[ek ]]cc )}
where p′i = c(cc′ , yn ) for all pi = c(yn )
[[fcase x of {pk → ek }]]cc
= if isVar (x)
then Vcc (Ucc (Bcc (θN (fcase [[x]]cc of {p′k → |pk | ∗ Hcc ([[ek ]]cc )}))))
else fcase [[x]]cc of {p′k → Ccc ([[ek ]]cc )}
(
where p′i = c(cc, yn ) for all pi = c(yn ) and θN (e) =
[[scc(cc′ , e)]]cc
e
Ncc (e)
if k = 1
if k > 1
= Ecc′ ([[e]]cc′ )
Here, |p| denotes the arity of pattern p, i.e., |p| = n if p = c(xn ), and the auxiliary
function update is used to attribute cost U to the cost center of the computed value:
update(x) = case x of {ck (cck , xnk ) → Ucck (ck (cck , xnk ))}
where c1 , . . . , ck are the program constructors.
Figure 4.2: Cost transformation [[ ]]cc for instrumenting expressions
Observe that the transformed program contains as many variants of each function
of the original program as the number of different cost centers. Semantically, all
these variants are equivalent; the only difference is that we obtain the costs of the
computation by counting the calls to the different cost center identity functions (like
Fcc ).
Program instrumentation is mainly performed by function [[ ]]cc , where cc denotes
the current cost center. Let us informally explain how the transformation proceeds.
We distinguish the following cases depending on the expression, e, in a call of the
form [[e]]cc :
• If e is a variable, a call to function Vcc is added to attribute cost V to cost center
cc.
• If e = c(xn ) is a constructor-rooted term, we add a new argument to store the
current cost center. This is necessary to attribute cost U to the appropriate
82
4.4. COST INSTRUMENTATION
83
cost center (i.e., to the cost center of the computed value, see Figure 4.1).
• A call to a function f (xn ) is translated to a call to the function variant corresponding to cost center cc.
• If e = (let x = e1 in e2 ) is a let expression, a call to function Hcc is always added
to attribute cost H to cost center cc. Additionally, if the binding is neither a
logical variable nor a constructor-rooted term, the cost center cci , 1 ≤ i ≤ k, of
the computed value is determined (by means of an auxiliary function update,
see Figure 4.2) and a call to Ucci is added to attribute cost U to that cost center.
The auxiliary function update is needed when transforming Let expressions.
When a Let expression introduces a new binding into the heap that binds a
variable to an expression that can be further evaluated, it is unknown when
this binding will be updated. Therefore, the transformation introduces function
update into the binding. This ensures that a cost U will be attributed to the
cost center of the computed value only if (and when) it is computed.
• If e = (e1 or e2 ) is a disjunction, a call to Ncc is added to attribute N to cost
center cc.
• If e = case x of {pk → ek } is a rigid case expression, we recursively transform
both the case argument and the expression of each branch, where a call to Ccc
is added to attribute cost C to cost center cc. Observe that the cost center, cc′ ,
of the patterns is not used (it is only needed in the auxiliary function update).
• If e = fcase x of {pk → ek } is a flexible case expression, a runtime test (function
isVar ) is needed to determine whether the argument evaluates to a logical variable or not. This function can be found, e.g., in the library Unsafe of PAKCS.
If it does not evaluate to a logical variable, we proceed as in the previous case.
Otherwise, we add calls to functions Vcc , Ucc , Bcc , and Ncc (if k > 1). Also, in
each case branch, calls to Hcc are added to attribute the size of the pattern to
cost center cc. Here, we use n ∗ Hcc as a shorthand for writing n nested calls to
Hcc (in particular, 0 ∗ Hcc means that no call to Hcc is written).
• Finally, if e = scc(cc′ , e′ ) is an scc-expression, a call to function Ecc is added.
More importantly, we update the current cost center to cc′ in the recursive
transformation of e′ .
Derivations with the standard semantics (i.e., without cost centers) are denoted by
([ ] : main ⇓ Γc : v). Given a heap Γ, we denote by Γc the set of bindings x 7→ e′ such
cc
that x 7−→ e belongs to Γ, where e′ = e if e is a logical variable, e′ = [[e]]cc if e = c(xn ),
or e′ = update([[e]]cc ) otherwise. Also, in order to make explicit the output of the
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CHAPTER 4. RUNTIME PROFILING OF FUNCTIONAL LOGIC...
instrumented program with the standard semantics, we write ([ ] : main ⇓θ Γc : v),
where θ records the set of calls to cost functions (e.g., Hcc , Fcc ).
The correctness of our program instrumentation is stated as follows:
Theorem 12 (correctness) Let P be a program and Pcost be its cost instrumented
version. Then,
(ccmain , [ ] : main ⇓θ Γ : v, cc) in P iff ([ ] : mainccmain ⇓θ Γc : v ′ ) in Pcost
where v = v ′ (if they are variables) or v = c(xn ) and v ′ = c(cc, xn ).
Proof.
We prove a more general claim: for each cost-augmented derivation
cce , Γ : e ⇓θ ∆ : v, ccv
in P , there exists a standard derivation
Γc : e′ ⇓θ ∆c : [[v ′ ]]ccv
in Pcost and vice versa, where e′ is either a logical variable (according to Γc ) or
e′ = [[e]]cce , and v = v ′ (if they are variables) or v = c(xn ) and v ′ = c(cc, xn ).
We prove the claim by induction on the depth m of the derivation.
(Base case m = 0) In this case, only two rules can be applied:
• If e is a logical variable (i.e., a variable x such that Γ[x] = x) then, by rule Val
of the cost-augmented semantics, it is (cce , Γ : x ⇓{} Γ : x, cce ) in the original
program P . Since e is a logical variable, e′ = e and, by rule Val of the standard
semantics, we also have (Γc : x ⇓{} Γc : x) in Pcost .
Similarly, if e = c(xn ), only rule Val can be applied: (cce , Γ : c(xn ) ⇓{} Γ :
c(xn ), cce ). Thus, e′ = [[c(xn )]]cce = c(cce , xn ) and, by rule Val of the standard
semantics, we also have (Γc : c(cce , xn ) ⇓{} Γc : c(cce , xn )).
cc
c
• Otherwise, e = x is a variable with Γ = Γ[x 7−→
c(xn )] and rule VarCons of the
cost-augmented semantics is applied in program P :
cc
cc
c
c
c(xn )] : c(xn ), ccc
cce , Γ[x 7−→
c(xn )] : x ⇓{cce ←V } Γ[x 7−→
Therefore, in the instrumented program, Pcost , it is Γc [x 7→ [[c(xn )]]ccc ] : [[x]]cce =
Γc [x 7→ c(ccc , xn )] : Vcce (x) and, thus, the following derivation holds by applying
rules Fun (to unfold Vcce ) and VarCons:
Γc [x 7→ c(ccc , xn )] : x ⇓{ } Γc [x 7→ c(ccc , xn )] : c(ccc , xn )
Γc [x 7→ c(ccc , xn )] : Vcce (x) ⇓{cce ←V } Γc [x 7→ c(ccc , xn )] : c(ccc , xn )
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4.4. COST INSTRUMENTATION
85
(Inductive case m > 1) In this case, rules VarExp, Fun, Let, Or, Select, Guess, and SCC
are applicable. We consider each case separately.
(VarExp) Then, e is a variable with Γ[x] 6= x and we performed the following derivation step in P (with the cost-augmented semantics):
cce1 , Γ : e1 ⇓θ ∆ : v, ccv
cce1
cc
v
v] : v, ccv
cce , Γ[x 7−→ e1 ] : x ⇓{cce ←V }+θ+{ccv ←U } ∆[x 7−→
In the instrumented program, Pcost , it is e′ = [[x]]cce = Vcce (x) (since x is not
a logical variable) and [[e1 ]]cce = update([[e1 ]]cce ) (since e1 is neither x nor a
1
1
constructor-rooted term). Therefore, the following derivation holds in Pcost by
applying rules Fun and VarExp:
Γc [x 7→ update([[e1 ]]cce )] : [[e1 ]]cce
1
1
⇓θ ∆c [x 7→ [[v]]ccv ] : [[v]]ccv
Γc [x 7→ update([[e1 ]]cce )] : update([[e1 ]]cce ) ⇓θ+{ccv ←U } ∆c [x 7→ [[v]]ccv ] : [[v]]ccv
1
1
Γc [x 7→ update([[e1 ]]cce )] : x ⇓θ+{ccv ←U } ∆c [x 7→ [[v]]ccv ] : [[v]]ccv
1
Γc [x 7→ update([[e1 ]]cce )] : Vcce (x) ⇓{cce ←V }+θ+{ccv ←U } ∆c [x 7→ [[v]]ccv ] : [[v]]ccv
1
Finally, the topmost judgement follows by the inductive hypothesis.
(Fun) Then, e is a function call and we performed the following derivation step in P
(with the cost-augmented semantics):
cce , Γ : ρ(e1 ) ⇓θ ∆ : v, ccv
cce , Γ : f (xn ) ⇓{cce ←F }+θ ∆ : v, ccv
where f (yn ) = e1 ∈ P and ρ = {yn 7→ xn }. Therefore, e′ = [[f (xn )]]cce =
fcce (xn ) and the following subderivation holds in Pcost by applying twice rule
Fun of the standard semantics:
Γc : [[ρ(e1 )]]cce ⇓θ ∆c : [[v]]ccv
Γc : Fcce (ρ([[e1 ]]cce ) ⇓{cce ←F }+θ ∆c : [[v]]ccv
Γc : fcce (xn ) ⇓{cce ←F }+θ ∆c : [[v]]ccv
since, by definition of cost transformation, fcc (yn ) = Fcc ([[e1 ]]cc ) ∈ Pcost . Finally, the claim follows by applying the inductive hypothesis to the topmost
judgement.
(Let) Then, e = (let x = e1 in e2 ) is a let expression and we performed the following
derivation step in P (with the cost-augmented semantics):
cc
e
cce , Γ[y 7−→
ρ(e1 )] : ρ(e2 ) ⇓θ ∆ : v, ccv
cce , Γ : let x = e1 in e2 ⇓{cce ←H}+θ ∆ : v, ccv
where ρ = {x 7→ y}. Now, we consider three cases, depending on whether e1 = x
(a logical variable), e1 = c(xn ), or e1 is any other expression:
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CHAPTER 4. RUNTIME PROFILING OF FUNCTIONAL LOGIC...
• If e1 = x, then e′ = [[let x = e1 in e2 ]]cce = Hcce (let x = x in [[e2 ]]cce ) and
the following subderivation holds in Pcost by applying rules Fun and Let of
the standard semantics:
Γc [y 7→ y] : [[ρ(e2 )]]cce ⇓θ ∆c : [[v]]ccv
Γc : let x = x in [[e2 ]]cce ) ⇓θ ∆c : [[v]]ccv
Γc : Hcce (let x = x in [[e2 ]]cce ) ⇓{cce ←H}+θ ∆c : [[v]]ccv
and the claim follows by applying the inductive hypothesis to the topmost
judgement.
• If e1 = c(xn ), then
e′ = [[let x = e1 in e2 ]]cce = Hcce (let x = [[e1 ]]cce in [[e2 ]]cce )
and the following subderivation holds in Pcost by applying rules Fun and
Let of the standard semantics:
Γc [y 7→ ρ([[e1 ]]cce )] : [[ρ(e2 )]]cce ⇓θ ∆c : [[v]]ccv
Γc : let x = [[e1 ]]cce in [[e2 ]]cce ) ⇓θ ∆c : [[v]]ccv
Γc : Hcce (let x = [[e1 ]]cce in [[e2 ]]cce ) ⇓{cce ←H}+θ ∆c : [[v]]ccv
and the claim follows by applying the inductive hypothesis to the topmost
judgement.
• Finally, if e1 is any other expression, then e′ = [[let x = e1 in e2 ]]cce =
Hcce (let x = update([[e1 ]]cce ) in [[e2 ]]cce ) and the following subderivation
holds in Pcost by applying rules Fun and Let of the standard semantics:
Γc [y 7→ ρ(update([[e1 ]]cce ))] : [[ρ(e2 )]]cce ⇓θ ∆c : [[v]]ccv
Γc : let x = update([[e1 ]]cce ) in [[e2 ]]cce ) ⇓θ ∆c : [[v]]ccv
Γc : Hcce (let x = update([[e1 ]]cce ) in [[e2 ]]cce ) ⇓{cce ←H}+θ ∆c : [[v]]ccv
and the claim follows by applying the inductive hypothesis to the topmost
judgement.
(Or) Then, e is a disjunction and we performed the following derivation step in P
(with the cost-augmented semantics):
cce , Γ : ei ⇓θ ∆ : v, ccv
cce , Γ : e1 or e2 ⇓{cce ←N }+θ ∆ : v, ccv
where i ∈ {1, 2}. Therefore, e′ = [[e1 or e2 ]]cce = Ncce ([[e1 ]]cce or [[e2 ]]cce ) and
the following subderivation holds in Pcost by applying rules Fun and Or of the
standard semantics:
Γc : [[ei ]]cce ⇓θ ∆c : [[v]]ccv
Γc : [[e1 ]]cce or [[e2 ]]cce ⇓θ ∆c : [[v]]ccv
Γc : Ncce ([[e1 ]]cce or [[e2 ]]cce ) ⇓{cce ←N }+θ ∆c : [[v]]ccv
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4.4. COST INSTRUMENTATION
87
and the claim follows by applying the inductive hypothesis to the topmost judgement.
(Select) Then, e = (f )case x of {pk → ek } is a case expression and we performed the
following derivation step in P (with the cost-augmented semantics):
cce , ∆ : ρ(ei ) ⇓θ2 Θ : v, ccv
cce , Γ : x ⇓θ1 ∆ : c(yn ), ccc
cce , Γ : (f )case x of {pk → ek } ⇓θ1 +{cce ←C}+θ2 Θ : v, ccv
where pi = c(xn ), i ∈ {1, . . . , n}, and ρ = {xn 7→ yn }. Therefore, since x
evaluates to a non-variable expression, e′ = [[(f )case x of {pk → ek }]]cce =
(f )case [[x]]cce of {p′k → Ccce ([[ek ]]cce )} (we omit the evaluation of isV ar(x) for
clarity), where p′i = c(cc, zl ) for all pi = c(zl ), i = 1, . . . , k. Then, the following
subderivation holds in Pcost by applying rules Select and Fun of the standard
semantics:
Γc : [[x]]cce ⇓θ1 ∆c : [[c(yn )]]ccc
∆c : [[ρ(ei )]]cce ⇓θ2 Θ : [[v]]ccv
∆c : Ccce ([[ρ(ei )]]cce ) ⇓{cce ←C}+θ2 Θ : [[v]]ccv
Γc : (f )case [[x]]cce of {p′k → Ccce (ek )} ⇓θ1 +{cce ←C}+θ2 Θc : [[v]]ccv
and the claim follows by applying the inductive hypothesis to the topmost judgements.
(Guess) Then, e = fcase x of {pk → ek } is a flexible case expression and we performed
the following derivation step in P (with the cost-augmented semantics):
cc
cc
cce , ∆[y 7→e ρ(pi ), yn 7→e yn ] : ρ(ei ) ⇓θ2 Θ : v, ccv
cce , Γ : x ⇓θ1 ∆ : y, ccy
cce , Γ : fcase x of {pk → ek } ⇓θ1 +{cce ←V,cce ←U,cce ←B,cce ←n∗H}+θN +θ2 Θ : v, ccv
where pi = c(xn ), ρ = {xn 7→ yn }, yn are fresh variables, and θN = {cce ← N }
if k > 1 and θN = { } if k = 1. Therefore, since x evaluates to a logical variable,
it is e′ = [[fcase x of {pk → ek }]]cce =
Vcce (Ucce (Bcce (θNcce ((f )case [[x]]cce of {p′k → |pk | ∗ Hcce ([[ek ]]cce )}))))
(we omit the evaluation of isV ar(x) for clarity), where p′i = c(cc, zl ) for all
pi = c(zl ), i = 1, . . . , k. Then, the following subderivation holds in Pcost by
applying rules Guess and Fun of the standard semantics:
∆c : [[ρ(ei )]]cce ⇓θ2 Θ : [[v]]ccv
Γc : [[x]]cce ⇓{cce ←V }+θ1 ∆c : [[y]]ccy
∆c [y 7→ [[ρ(pi )]]cce , yn 7→ [[yn ]]cce ] :
n ∗ Hcce ([[ρ(ei )]]cce )
⇓{cce ←C,cce ←n∗H}+θ2 Θc : [[v]]ccv
Γc : Ucce (Bcce (θNcce ((f )case [[x]]cce of {p′k → |pk | ∗ Hcce ([[ek ]]cce )})))
⇓θ1 +{cce ←V,cce ←U,cce ←B,cce ←|pk |∗H}+θN +θ2 Θc : [[v]]ccv
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CHAPTER 4. RUNTIME PROFILING OF FUNCTIONAL LOGIC...
and the claim follows by applying the inductive hypothesis to the topmost judgements.
(SCC) Then, e = scc(cc, e1 ) and we performed the following derivation step in P
(with the cost-augmented semantics):
cc, Γ : e1 ⇓θ ∆ : v, ccv
cce , Γ : scc(cc, e1 ) ⇓{cc←E}+θ ∆ : v, ccv
Therefore, e′ = [[scc(cc, e1 )]]cce = Ecc ([[e1 ]]cc and the following subderivation
holds in Pcost by applying rules Fun and SCC of the standard semantics:
Γc : [[e1 ]]cc ⇓θ ∆c : [[v]]ccv
Γc : Ecc ([[e1 ]]cc ) ⇓{cc←E}+θ ∆c : [[v]]ccv
The claim follows by applying the inductive hypothesis to the topmost judgement.
✷
As an alternative to the transformation presented in this section, we could also instrument programs by partial evaluation [Jones et al., 1993], i.e., the partial evaluation
of the cost semantics of Section 4.3 w.r.t. a source program P should return an instrumented program which is equivalent to Pcost . However, this approach requires
both a implementation of the instrumented semantics as well as an optimal partial
evaluator for the considered language (in order to obtain a reasonable instrumented
program, rather than a slight specialization of the cost semantics). Thus, we preferred
to introduce a direct transformation.
Now, we illustrate our cost transformation by means of a simple example. Consider, for instance, the following definition for function len:
len(x) = fcase x of { Nil → Z;
Cons(y,ys) → let w = scc("b",len(ys))
in S(w) }
Then, the corresponding instrumented definition for the cost center "a" is the following:
lena (x) =
Fa (if isVar(x)
then Va (Ua (Ba (Na (fcase Va (x) of
{ Nil(cc) → Z(a);
Cons(cc,y,ys) → Ha (Ha (Ha (let w = update(lenb (ys))
in S(a,w))))} ))))
else fcase Va (x) of
88
4.5. IMPLEMENTATION
89
{ Nil(cc) → Z(a);
Cons(cc,y,ys) → Ca (Ha (let w = update(lenb (ys))
in S(a,w)))})
where the auxiliary function update is defined as follows:
update(x) = case x of { Z(cc) → Ucc (Z(cc));
S(cc,x) → Ucc (S(cc,x)) }
It is important to note that the transformation introduced in this chapter is different from the one presented in the previous chapter. Although, from a operational
point of view, the transformation of this chapter can be seen as a generalization of
the transformation in the previous chapter (i.e., The former extends the kind of symbolic costs computed, and introduces cost centers); the goal of the transformations
is essentially different. On the one hand, the objective of the transformation in the
previous chapter is to reason about costs of programs from a static point of view; that
is, to generalize the costs of programs for any input data. On the other hand, the
objective of the transformation in this chapter is to compute actual costs of programs
for a particular execution.
4.5
Implementation
The main purpose of profiling programs is to increase runtime efficiency. However, in
practice, it is important to obtain symbolic profiling information as well as measuring
runtimes. As discussed before, we want to provide cost centers for both kinds of profiling in order to be able to analyze arbitrary sub-computations independently of the
defined functions. For the formal introduction of costs and correctness proofs, symbolic costs are the appropriate means. Therefore, we introduced a program transformation dealing with symbolic costs. However, the presented program transformation
can easily be extended for measuring runtimes and distribute them through cost centers. In this section, we first present our approach to measure runtimes and function
calls (Section 4.5.1) and, then, describe the extensions to obtain symbolic profiling
(Section 4.5.2).
4.5.1
Measuring Runtimes
When trying to measure actual runtimes, the crucial point is to alter the runtime
behavior of the examined program as little as possible. If the program instrumented for
profiling runs 50% slower or worse, one profiles the process of profiling rather than the
program execution. Because of this, measuring actual runtimes is a matter of low-level
programming and, thus, highly depending on the actual language implementation.
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CHAPTER 4. RUNTIME PROFILING OF FUNCTIONAL LOGIC...
Our approach is specific to the Curry implementation PAKCS [Hanus et al., 2006].
In this programming environment, Curry programs are compiled by transforming flat
programs (cf. Figure 2.2) to SICStus Prolog (see [Antoy and Hanus, 2000] for details
about this transformation). In order to provide low-level profiling for PAKCS, we
instrument the program with the profiling mechanisms offered by SICStus Prolog.
Fortunately, SICStus Prolog features low-level profiling instruments which create an
overhead of approximately 22%. The Prolog tools provide precise measuring of the
number of predicate and clause calls. For measuring runtime, a number of synthetic
units is given which is computed according to Gorlick and Kesselman [1987].
The main challenge was to introduce the cost centers into Prolog profiling. Luckily, we found a way to do this without further slowing down the execution of the
program being profiled. The only overhead we introduce is code duplication, since we
introduce a different version of each function for each cost center, as in the program
transformation described above. Thus, for the program
main = SCC "len" (length (SCC "list" (enumFromTo 1 10)))
function main does not call a function length. Instead, it calls a variant with the
name “length{len}” and also a function named “enumFromTo{list}”. Gathering all
runtimes for functions with the attachment {cc}, one gets the runtime belonging to
that cost center. An obvious optimization is to eliminate unreachable functions like
length{list} in the example.
4.5.2
Extension for Symbolic Profiling
Our approach to symbolic profiling exactly represents the idea described in Section 4.4
above. For each cost, we introduce a new function, e.g., var lookup for cost V . There
are variations of these functions for the different cost centers, e.g., var lookup{list}
like in Section 4.5.1. After the execution of the transformed program, we simply count
each call to var lookup{list} to get the sum of costs V attributed to the cost center
list.
The advantage of this method is its simplicity. The demands to use our transformation for profiling with any implementation of Curry are not very high. The runtime
system must only be able to count the number of calls to a certain function which
is easy to implement. The disadvantage is the considerable (but still practicable)
slowdown as we are not only introducing new functions but also new function calls.
Nevertheless, this overhead does not influence the computation of symbolic costs.
It is worthwhile to note that, although the program transformation of Figure 4.2
is equivalent to the cost semantics of Figure 4.1 for particular computations (as stated
in Theorem 12), there is one important difference:
90
4.6. RELATED WORK AND CONCLUSIONS
91
While the cost semantics is don’t-care non-deterministic, the instrumented
programs accumulate all costs according to the search strategy.
For instance, the cost for a failing derivation is also accumulated in the cost of the
results computed afterwards. Furthermore, completely failing computations also have
an associated cost while no proof tree (and thus no costs) can be constructed in the
big-step semantics. From a practical point of view, this is an advantage of the program
transformation over the cost semantics, since the cost of failing derivations is relevant
in the presence of non-deterministic functions.
4.6
Related Work and Conclusions
The closest approaches to our work are [Sansom and Peyton-Jones, 1997] and [Albert
and Vidal, 2002]. On the one hand, [Sansom and Peyton-Jones, 1997] presents a formal specification of the attribution of execution costs to cost centers by means of an
appropriate cost-augmented semantics in the context of lazy functional programs. A
significant difference from our work is that our flat representation of programs allows
logical features (like non-determinism) and that we also present a formal transformation to instrument source programs. On the other hand, [Albert and Vidal, 2002]
introduces a symbolic profiling scheme for functional logic languages. However, the
approach of [Albert and Vidal, 2002] does not consider sharing (an essential component of lazy languages) and, thus, it is not an appropriate basis for the development
of profiling tools for current implementations of lazy functional logic languages. Also,
we introduced a program transformation that allows us to compute symbolic costs
with a reasonable overhead. Finally, in the context of the PAKCS environment for
Curry, we showed how actual runtimes can also be computed by reusing the SICStus
Prolog profiler.
91
Part III
Program Slicing
93
Chapter 5
Dynamic Slicing Based on
Redex Trails
Tracing computations is a widely used methodology for program debugging. Lazy
languages, however, pose new demands on tracing techniques because following the
actual trace of a computation is generally useless. Typically, tracers for lazy languages
rely on the construction of a redex trail, a graph that stores the reductions performed
in a computation. While tracing provides a significant help for locating bugs, the task
still remains complex. A well-known debugging technique for imperative programs is
based on dynamic slicing, a method for finding the program statements that influence
the computation of a value for a specific program input.
In this chapter, we introduce a novel technique for dynamic slicing in first-order
lazy functional languages. Rather than starting from scratch, our technique relies
on (a slight extension of) redex trails. We provide a notion of dynamic slice and
introduce a method to compute it from the redex trail of a computation. We also
sketch the extension of our technique to deal with a functional logic language. A
clear advantage of our proposal is that one can enhance existing tracers with slicing
capabilities with a modest implementation effort, since the same data structure (the
redex trail) can be used for both tracing and slicing.
In addition, we propose a lightweight approach to program specialization in the
context of lazy functional logic languages which is based on the dynamic slicing technique previously defined.
Part of the material of this chapter has been presented in the ACM SIGPLAN 2004
Symposium on Partial Evaluation and Program Manipulation (PEPM’04) celebrated
in Verona (Italy) in 2004 [Ochoa et al., 2004a]; in the First Workshop on Software
Analysis and Development for Pervasive Systems (SONDA’04) celebrated in Verona
(Italy) in 2004 [Ochoa et al., 2004b]; in the Workshop on Curry and Functional
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
Logic Programming (WCFLP 2005) celebrated in Tallinn (Estonia) in 2005 [Ochoa
et al., 2005]; and in the 10th European Conference on Logics in Artificial Intelligence
(JELIA’06) celebrated in Liverpool (England) in 2006 [Ochoa et al., 2006].
5.1
Introduction
In lazy functional programming languages, following the actual trace of a computation
is often useless due to the on demand style of evaluation. In the language Haskell
[Peyton-Jones, 2003], this drawback has been overcome by introducing higher-level
models that hide the details of lazy evaluation. This is the case of the following
systems: Hood [Gill, 2000], Freja [Nilsson and Sparud, 1997], and Hat [Sparud and
Runciman, 1997; Wallace et al., 2001]. Some of these approaches are based on the
construction of a redex trail [Sparud and Runciman, 1997], a directed graph which
records copies of all values and redexes (red ucible ex pressions) of a computation.
Basically, redex trails allow us to present different—more intuitive—views of a given
computation (e.g., by showing function calls with fully evaluated arguments). Wallace
et al. [2001] introduced an extended trail, the augmented redex trail (ART), in order
to cover all previous three approaches (i.e., those followed by Hood, Freja and Hat).
Braßel et al. [2004b] introduce an instrumented version of an operational semantics
for the kernel of a lazy functional logic language like Curry [Hanus, 2006a] that returns,
not only the computed values and bindings, but also a trail of the computation.
Similarly to the ART approach, this trail can also be used to perform tracing and
algorithmic debugging, as well as to observe data structures through a computation.
In contrast to previous approaches, the correctness of the computed trail is formally
proved, which amounts to say that it contains all (and only) the reductions performed
in a computation.
While tracing-based debugging provides a powerful tool for inspecting erroneous
computations, it should be clear that the task still remains complex. In this chapter
we use a well-known debugging technique for imperative programs known as program
slicing (see Chapter 1). When debugging functional programs, we usually start from
a particular computation that outputs an incorrect value. In this situation, dynamic
slicing may help the programmer in finding the location of the bug by extracting
a program slice which only contains the sentences whose execution influenced the
incorrect value. Therefore, we are mainly interested in dynamic slicing [Korel and
Laski, 1988], which only reflects the actual dependences of the erroneous execution,
thus producing smaller—more precise—slices than static slicing [Tip, 1995].
In this chapter, we present the first dynamic backward slicing technique for a
lazy functional logic language like Curry (though it could also be applied to a purely
functional language like Haskell). Rather than starting from scratch, our technique
96
5.1. INTRODUCTION
97
relies on a slight extension of redex trails. To be precise, we extend the redex trail
model of Braßel et al. [2004b] in order to also store the location, in the program, of
each reduced expression. A dynamic slice is then computed by
1. first gathering the set of nodes—in the redex trail—which are reachable from
the slicing criterion and, then,
2. by deleting (or hiding) those expressions of the original program whose location
does not appear in the set of collected nodes (see the next section for an informal
overview of our debugging technique).
From our point of view, dynamic slicing may provide a complementary—rather
than alternative—approach to tracing-based debugging. A clear advantage of our
approach is that one can enhance existing tracers with slicing capabilities with a
modest implementation effort, since the same data structure—the redex trail—is used
for both tracing and slicing.
The main contributions of this chapter can be summarized as follows:
• We introduce a flexible notion of slicing criterion for a first-order lazy functional
(logic) language. Previous definitions, e.g., [Biswas, 1997a; Reps and Turnidge,
1996], only considered a projection of the entire program as a slicing criterion.
• We extend (a simplified version of) the instrumented semantics of Braßel et al.
[2004b] in order to have an extended redex trail that also stores the positions,
in the source program, of each reduced expression, which is essential for slicing.
• We provide a notion of dynamic slice and introduce a method to compute it
from the extended redex trail of a computation.
• We show how tracing—based on redex trails—and slicing can be combined into
a single framework. Moreover, the viability of our approach is supported by a
prototype implementation of an integrated debugging tool.
This chapter is organized as follows. In the next section, we present an informal
overview of our approach to combine tracing and slicing into a single framework. Section 5.3 formalizes an instrumented semantics that builds an extended redex trail of
a computation which also includes the program positions of expressions. Section 5.4
defines the main concepts involved in dynamic slicing and introduces a method to
compute dynamic slices from the extended redex trail. Section 5.5 extends the slicing
technique to perform program specialization. Section 5.6 presents some implementation issues of a prototypical implementation which integrates both tracing and slicing.
Finally, Section 5.7 includes a comparison to related work.
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5.2
Combining Tracing and Slicing
In this section, we present an overview of a debugging tool for lazy functional languages that combines both tracing and dynamic slicing based on redex trails.
A redex trail —which was originally introduced by Sparud and Runciman [1997] in
the context of lazy functional programs—is defined as a directed graph which records
copies of all values and redexes (red ucible ex pressions) of a computation, with a
backward link from each reduct (and its proper subexpressions) to the parent redex
that created it. The ART (Augmented Redex Trail ) model of the Haskell tracer Hat
[Wallace et al., 2001] mainly extends redex trails by also including forward links from
every redex to its reduct. Braßel et al. [2004b] have introduced a data structure which
shares many similarities with the ART model but also includes a special treatment
to cope with non-determinism (i.e., disjunctions and flexible case structures) so that
it can be used for functional logic languages. Furthermore, in contrast to previous
approaches, the redex trails of Braßel et al. [2004b] are defined by instrumenting an
operational semantics for the language (rather than by a program transformation)
and their correctness is formally proved. Let us present this tracing technique by
means of an example. The technique presented in this chapter is based on normalized
flat programs. However, in this section, we consider that programs are not normalized
for clarity.
Example 11 Consider the program shown in Figure 5.1—inspired by a similar example by Liu and Stoller [2003]—where data structures are built from
data Nat
= Z | S Nat
data Pairs
= Pair Nat Nat
data ListNat = Nil | Cons Nat ListNat
The execution of the program in Figure 5.1 should compute the maximum of the
list [Z, S Z], i.e., S Z, and thus return 1; however, it returns 0.
In order to trace the execution of this program, the tracing tool of Braßel et al.
[2004b] builds the redex trail shown in Figure 5.2. In this redex trail, we can distinguish two kinds of arrows:1
Successor arrows: There is a successor arrow, denoted by a solid arrow, from each
redex to its reduct (e.g., from the node labeled with main to the node labeled
with printMax in Figure 5.2).
Argument arrows: Arguments of both function and constructor calls are denoted by a
pointer to the corresponding expression, which is denoted by a dashed arrow.
1 The redex trails of Braßel et al. [2004b] also contain a third kind of arrows—the parent arrows—that we ignore in this chapter because they are not necessary for computing program slices.
98
5.2. COMBINING TRACING AND SLICING
99
main = printMax (minmax [Z, S Z])
printMin t = case t of { Pair x y → printNat x }
printMax t = case t of { Pair x y → printNat y }
printNat n = case n of { Z → 0; (S m) → 1 + printNat m }
fst t = case t of { Pair x y → x }
snd t = case t of { Pair x y → y }
minmax xs = case xs of
{ y:ys → case ys of
{ []
→ Pair y y;
z:zs → let m = minmax (z:zs)
in Pair (min y (fst m))
(max y (snd m)) } }
min x y = ite (leq x y) x y
max x y = ite (leq x y) y x
ite x y z = case x of { True → y; False → z }
leq x y = case x of
{ Z
→ False;
(S n) → case y of { Z → False; (S m) → leq n m } }
Figure 5.1: Example program minmax
When the evaluation of an argument is not required in a computation, we have
a null pointer for that argument (e.g., the first argument of the node labeled
with Pair in Figure 5.2).
From the computed trail, the user can inspect the trace of a computation. In our
example, the tracing tool initially shows the following top-level computation for main:
0
0
0
0
=
=
=
=
main
printMax (Pair _ Z)
printNat Z
0
where “_” denotes an argument whose evaluation is not needed (i.e., a null pointer in
the redex trail). Each row shows a pair “val = exp” where exp is an expression and
val is its (possibly partial) value. Note that this top-level computation corresponds
to the six topmost nodes in Figure 5.2, where the two nodes labeled with a case
99
100
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
main
printMax
case
minmax
printNat
case
case
case
0
Pair
Cons
max
ite
Cons
case
leq
case
Z
False
Z
Figure 5.2: Redex trail for Example 11
structure are ignored to simplify the presentation. Note also that function arguments
appear as much evaluated as needed in the complete computation in order to ease the
understanding of the trace.
From the trace above, the user can easily see that the argument of printMax is
incorrect because the second argument of constructor Pair, i.e., the maximum of the
input list, should be (S Z) rather than Z. If the user selects the argument of printMax,
the expression (Pair Z), the following subtrace is shown:
0
= printMax (Pair _ Z)
Pair _ Z = minmax
(Z : _ : _)
Pair _ Z = Pair
_
Z
Here, the user can conclude that the evaluation of minmax (rather than its definition!)
contains a bug because it returns Z (the second element of Pair) as the maximum
of a list headed by Z, ignoring the remaining elements of the list (which were not
evaluated in the computation).
Now, the user can either try to figure out the location of the bug by computing
further subtraces (which ones?) or by isolating the code fragment which is responsible
of the wrong result by dynamic slicing and, then, inspecting the computed slice.
We propose the second approach and will show that incorporating dynamic slicing
into existing tracers (based on redex-trails) is simple and powerful. Essentially, our
approach allows the user to easily obtain the program slice associated with any entry
val = exp in the trace of a computation. Optionally, the user may provide a pattern π
that indicates which part of val is of interest and which part is not. Then, a dynamic
slice including all program constructs that are (potentially) needed to compute the
relevant part of val—according to π—from exp is shown. Therefore, in our context,
the slicing criterion is given by a tuple of the form hexp, πi.
We consider that patterns for slicing are built from data constructor symbols, the
100
5.2. COMBINING TRACING AND SLICING
main
[]
printMax
...
case
(printMax,/\)
minmax
...
101
printNat
(printMax,2.1)
case
(minmax,/\)
case
(minmax,2.1)
Cons
...
case
(printNat,/\)
0
(printNat,2.1)
Pair
...
max
...
ite
(max,2)
Cons
...
leq
(max,1)
case
(ite,/\)
case
(leq,/\)
Z
(main,1)
False
(leq,2.1)
Z
(main,1)
Figure 5.3: Extended redex trail for Example 11
special symbol ⊥ (which denotes a subexpression of the value whose computation is
not relevant), and ⊤ (a subexpression which is relevant). Consider, e.g., the following
slicing criterion:
hminmax (Z : : ), Pair ⊥ ⊤i
associated with the erroneous entry of the previous trace for minmax. Intuitively,
it determines the extraction of a slice with the expressions that are involved in the
computation of the second argument of Pair starting from the call to minmax. We
note, however, that the expressions involved in the evaluation of the arguments of the
call to minmax will not be part of the slice. This is reasonable since the user identified
a bug in the fact that minmax (Z : : ) evaluated to a wrong result, assuming that
the arguments of minmax were already evaluated.
Similarly to Reps and Turnidge [1996], we consider the computation of backward
slices where the information given by the slicing pattern π should be propagated
backwards through the computation to determine the program constructs that are
necessary to compute the relevant part of exp (according to π).
In order to compute the associated program slice, we extend the redex trail model
in order to also store the position, in the source program, of every expression in
the redex trail. Program positions are denoted by a pair (fun, pos) where fun is a
function symbol and pos is a sequence of natural numbers that precisely identify a
subexpression in the definition of fun (see Section 5.3.2 for a detailed definition).
For the example above, the extended redex trail including some of the program
positions—more details on this example can be found in Section 5.4—is shown in
Figure 5.3. The leftmost shadowed node stores the expression associated with the
slicing criterion, while the remaining shadowed nodes store the expressions which are
reachable from the slicing criterion (according to the slicing pattern).
The computed slice is thus the set of program positions in the shadowed nodes
101
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
main = printMax (minmax [Z, S Z])
printMin t = case t of { Pair x y → printNat x }
printMax t = case t of { Pair x y → printNat y }
printNat n = case n of { Z → 0; (S m) → 1 + printNat m }
fst t = case t of { Pair x y → x }
snd t = case t of { Pair x y → y }
minmax xs = case xs of
{ y:ys → case ys of
{ []
→ Pair y y;
z:zs → let m = minmax (z:zs)
in Pair (min y (fst m))
(max y (snd m)) } }
min x y = ite (leq x y) x y
max x y = ite (leq x y) y x
ite x y z = case x of { True → y; False → z }
leq x y = case x of
{ Z
→ False;
(S n) → case y of { Z → False; (S m) → leq n m } }
Figure 5.4: Slice of program minmax
of the extended redex trail. The corresponding program slice is shown in Figure 5.4,
where expressions that do not belong to the slice appear in gray. Some of these
expressions (e.g., let) have been deleted after a process of pretty printing discussed
in Section 5.5). By inspecting the slice, we can check that minmax only calls to
function max which, in turn, calls to functions ite, a standard conditional, and leq
(for “less than or equal to”). However, the only case which is not deleted from the
definition of leq can be read as “Z is not less than or equal to any natural number”,
which is clearly wrong.
Observe that the slice is particularly useful to understand the subtleties of lazy
evaluation. For instance, function minmax is only called once, with no recursive call
to inspect the tail of the input list (as a beginner to a lazy language would expect
despite the error). This motivates the need for powerful tools for debugging and
program understanding in lazy functional (logic) languages.
102
5.3. BUILDING THE EXTENDED TRAIL
5.3
103
Building the Extended Trail
In this section, we extend the instrumented semantics of Braßel et al. [2004b] so that
the computed redex trail also stores program positions; this additional information
will become useful to identify precisely the location, in the source program, of each
expression in the trail.
In order to keep the thesis self-contained and to ease the understanding, we introduce our instrumented semantics in a stepwise manner. Furthermore, we consider
neither disjunctions nor flexible case structures in Sections 5.3.1, and 5.3.2; the treatment of these language features is delayed to Section 5.3.3.
5.3.1
A Semantics for Tracing
First, we present the instrumented semantics of Braßel et al. [2004b] which extends
the operational semantics presented in Figure 2.6 to construct a redex trail of the
computation. However, in contrast to [Braßel et al., 2004b], we do not consider the
“parent” relation because it is not useful for computing program slices. Basically, the
trail is a directed graph with nodes identified by references2 that are labeled with
expressions. In the construction of this trail, the following conventions are adopted:
• r 7→ e denotes that the node with reference r is labeled with expression e.
Successor arrows are denoted by r 7→ which means that node q is the successor
q
of node r. The notation r 7→ e is used to denote both r 7→ e and r 7→.
q
q
• Argument arrows are denoted by x à r which means that variable x points to
node r. This relation suffices to store function arguments in our context because
only variable arguments are allowed in normalized programs. These arrows are
also called variable pointers.
Configurations in the instrumented semantics are tuples of the form hΓ, e, S, G, ri;
here, r denotes the current reference—where e will be eventually stored—and G is
the trail built so far. Analogously to the heap, G[r 7→ e] is used to extend the graph
G with a new node r labeled with the expression e.
The rules of the instrumented semantics are shown in Figure 5.5. Both rules,
VarCons and VarExp, add a new variable pointer for x to the current graph if it does
not yet contain such a pointer. This is used to account for the fact that the value
of x is needed in the computation. For this purpose, the auxiliary function “✶” is
introduced:
(
G[x à r] if there is no r′ such that (x à r′ ) ∈ G
G ✶ (x à r) =
G
otherwise
2 The domain for references is not fixed. One can use, e.g., natural numbers but more complex
domains are also possible.
103
104
(VarCons)
(VarExp)
(Val)
(Fun)
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
hΓ[x 7→ t], x, S, G, ri
=⇒ hΓ[x 7→ t], t, S, G ✶ (x à r), ri
where t is constructor-rooted
hΓ[x 7→ e], x, S, G, ri
=⇒ hΓ[x 7→ e], e, x : S, G ✶ (x à r), ri
where e is not constructor-rooted and e 6= x
hΓ[x 7→ e], v, x : S, G, ri
=⇒ hΓ[x 7→ v], v, S, G, ri
where v is a value
hΓ, f (xn ), S, G, ri
=⇒ hΓ, ρ(e), S, G[r 7→ f (xn )], qi
q
where f (yn ) = e ∈ R, ρ = {yn 7→ xn } and q is a fresh reference
(Let)
hΓ, let x = e1 in e2 , S, G, ri
=⇒ hΓ[y 7→ ρ(e1 )], ρ(e2 ), S, G[r 7→ ρ(let x = e1 in e2 )], qi
q
where ρ = {x 7→ y}, y is a fresh variable and q is a fresh ref.
(Case)
hΓ, case x of {pk → ek }, S, G, ri
=⇒ hΓ, x, ({pk → ek }, r) : S, G[r 7→ case x of {pk → ek }], qi
where q is a fresh reference
(Select)
hΓ, c(yn ), ({pk → ek }, r′ ) : S, G, ri
=⇒ hΓ, ρ(ei ), S, G[r 7→ c(yn ), r′ 7→], qi
q
where i ∈ {1, . . . k}, pi = c(xn ), ρ = {xn 7→ yn } and q is a fresh ref.
Figure 5.5: Instrumented semantics for tracing computations
Observe that this function is necessary to take care of variable sharing: if the value of
a variable was already demanded in the computation, no new variable pointer should
be added to the graph.
In rule Val, the current graph is not modified because computed values are only
stored when they are used (e.g., in rule Select).
In rules Fun and Let a new node r, labeled with the current expression in the
control, is added to the graph. The successor is set to q, a fresh reference, which is
now the current reference of the derived configuration.
104
5.3. BUILDING THE EXTENDED TRAIL
105
′
′
], qi
(Select-f) hΓ, c(yn ), ({pk → ek }, r ) : S, G, ri =⇒ hΓ, F ail, S, G[r 7→ c(yn ), r 7→
q
where 6 ∃i ∈ {1, . . . k} such that pi = c(xn )
(Success) hΓ, c(xn ), [], G, ri =⇒ hΓ, ⋄, [], G[r 7→ c(xn )], ✷i
Figure 5.6: Tracing semantics: failing and success rules
In rule Case we now push on the stack, not only the case alternatives, but also the
current reference. This reference will become useful later if rule Select is eventually
applied in order to set the right successor for the case expression (which is not yet
known when rule Case is applied). Note that no successor for node r is set in rule
Select because values are fully evaluated and, thus, have no successor.
When tracing computations, however, we are also interested in failing derivations.
For this purpose, rule Select-f of Figure 5.6 is added to the calculus. This rule applies
when there is no matching branch in a case expression. Furthermore, since expressions
are added to the graph a posteriori, it does not suffice to reach a “final” configuration
(i.e., a configuration with a value in the control and an empty stack). An additional
rule Success (shown in Figure 5.6) is needed to store the result of the computation.
In the derived configuration, we put the special symbol “⋄” to denote that no further
steps are possible.
In the instrumented semantics, a computation starts with a configuration of the
form h[], main, [], G∅ , ri, where G∅ denotes an empty graph and r an initial reference.
Example 12 Consider again the program of Example 9. By using the instrumented
semantics of Figures 5.5 and 5.6, the derivation shown in Figure 5.7 is computed
(here, natural numbers are used as references, where 0 is the initial reference). For
clarity, in each configuration, G denotes the graph of the previous configuration.
5.3.2
Including Program Positions
In this section we extend the previous semantics so that the computed redex trail also
includes program positions, which are used to uniquely determine the location—in the
source program—of each stored expression.
Definition 13 (position, program position) A position is denoted by a sequence
of natural numbers, where Λ is the empty sequence (i.e., the root position). They are
105
106
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
h[ ], main, [ ], G∅ , 0i
=⇒Fun
h[ ],
let x3 = Z in . . . ,
[ ],
G[0 7→ main], 1i
1
=⇒Let
h[x3 7→ Z],
let x1 = Z in . . . ,
[ ],
G[1 7→ let x3 = Z in . . . ], 2i
=⇒Let
h[x1 7→ Z, . . .],
let x2 = S x3 in . . . , [ ],
G[2 7→ let x1 = Z in . . . ], 3i
2
3
=⇒Let
h[x2 7→ S x3, . . .], leq x1 x2,
[ ],
G[3 7→ let x2 = S x3 in leq x1 x2], 4i
=⇒Fun
h[. . .],
case x1 of . . . ,
[ ],
G[4 7→ leq x1 x2], 5i
x1,
[({. . .}, 5)],
G[5 7→ case x1 of {. . . }}], 6i
G[x1 Ã 6], 6i
4
5
=⇒Case
h[x1 7→ Z, . . .],
=⇒VarCons
h[x1 7→ Z, . . .],
=⇒Select
h[. . .],
Z,
[({. . .}, 5)],
True,
[ ],
G[6 7→ Z, 5 7→], 7i
=⇒Success
h[. . .],
⋄,
[ ],
G[7 7→ True], ✷i
7
Figure 5.7: Derivation of Example 12
used to address subexpressions of an expression viewed as a tree, as follows:3
e|Λ = e
c(en )|i.w = ei |w
f (en )|i.w = ei |w
let x = e in e′ |1.w = e|w
let x = e in e′ |2.w = e′ |w
case e of {pn → en }|1.w = e|w
case e of {pn → en }|2.i.w = ei |w
for all expression e
if i ∈ {1, . . . , n}
if i ∈ {1, . . . , n}
if i ∈ {1, . . . , n}
A program position is a pair (g, w) that addresses the expression e|w in the right-hand
side of the definition, g(xn ) = e, of function g. Given a program P , Pos(P ) denotes
the set of all program positions in P .
Note that variables in a let construct or patterns in a case expression are not addressed
by program positions because they will not be considered in the slicing process. While
3 We
consider arbitrary (not necessarily normalized) expressions for generality.
106
5.3. BUILDING THE EXTENDED TRAIL
h[ ], main, [ ], G∅ , 0i
=⇒Fun
h[ ],
let x = Z in ...,
107
[ ], G[0 7→ main],
1i
1
=⇒Let
h[x 7→ Z],
let y = g x in f y, [ ], G[1 7→ let x = Z in . . . ],
=⇒Let
h[y 7→ g x, x 7→ Z], f y,
[ ], G[2 7→ let y = g x in f y], 3i
=⇒Fun
h[y 7→ g x, x 7→ Z], y,
[ ], G[3 7→ f y],
4i
=⇒VarExp
=⇒Fun
h[y 7→ g x, x 7→ Z], g x,
h[y 7→ g x, x 7→ Z], x,
[y], G[y à 4],
[y], G[4 →
7 g x],
4i
5i
[y], G[x à 5],
[ ], G,
[ ], G[5 →
7 Z],
5i
5i
✷i
2
2i
3
4
5
=⇒VarCons h[y 7→ g x, x 7→ Z], Z,
=⇒Val
h[y →
7 Z, x 7→ Z], Z,
=⇒Success h[y 7→ Z, x 7→ Z], ⋄,
Figure 5.8: Derivation of Example 13
other approaches to dynamic slicing, e.g., [Biswas, 1997b], consider some form of
explicit labeling for each program expression, our program positions can be seen as a
form of implicit labeling.
In order to add this additional information to the computed graph, two extensions
are necessary. The first extension is straightforward: we label each mapping x 7→ e in
the heap with the program position of expression e. Therefore, mappings in the heap
have now the form x 7→(g,w) e.
The second extension regards the addition of program positions to the nodes of
the graph. In principle, we could extend the redex trail model so that r 7→(g,w)
denotes that the expression which labels node r is located at program position (g, w).
Unfortunately, this simple solution is not appropriate because variables are not stored
in redex trails, but we still need to collect their program positions.
Example 13 Consider the following simple program:
main
= let x = Z in
let y = g x in f y
f x = x
g x = x
By using the instrumented semantics of Figures 5.5 and 5.6, the derivation shown in
Figure 5.8 is computed. The redex trail can be graphically depicted as follows:
main
let x = Z in ...
let y = g x in f y
f
g
Z
Observe that the values of variables have not been stored in the graph. Therefore, the
associated program slice containing the expressions in the redex trail would have the
following form:
107
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
main
= let x = Z in
let y = g x in f y
f x = x
g x = x
where functions f and g appear in gray because no expression in their right-hand sides
belong to the slice. This is clearly wrong because the right-hand sides of both functions,
f and g, have been evaluated.
In order to overcome this problem, we denote the mapping from nodes to program
positions by r 7→P , where P is a list of program positions. Now, the meaning of r 7→P
is the following:
• the program position of the expression labeling node r is the first element of list
P and
• if the tail of P is not empty, then it contains the program positions of the (chain
of) variables whose evaluation was required in order to reach the expression
labeling r.
Therefore, in the following, a configuration of the instrumented semantics is formally
defined as follows:
Definition 14 (configuration) A configuration of the semantics is represented with
a tuple hΓ, e, S, G, r, Pi where Γ ∈ Heap is the current heap, e ∈ Exp is the expression
to be evaluated (the control), S is the stack, G is a directed graph (the trail built so
far), r is the current reference, and P is the list of program positions associated with
e.
The new instrumented semantics, including the rules for failure and success, is shown
in Figure 5.9.4
In the first two rules, VarCons and VarExp, the evaluation of a variable x is demanded. Thus, the program position (g, w) of the expression to which x is bound
in the heap is added to the current list of program positions P. Here, P denotes
a (possibly empty) list of program positions for the chain of variables that finishes
in the current variable x (note that, in rule VarExp, e may be a variable). For this
purpose, the auxiliary function “✶” is redefined as follows:
(
G[x à r]
if there is no r′ such that (x à r′ ) ∈ G
G ✶P (x à r) =
G[r 7→P∪P ′ ] otherwise, with (r 7→P ′ ) ∈ G
The only difference with the original definition is that, when there is already a variable
pointer x à r in the graph, function ✶ should update the program positions associated
4 As
in Figure 5.5, we assume that q is always a fresh reference.
108
5.3. BUILDING THE EXTENDED TRAIL
(VarCons)
109
hΓ[x 7→(g,w) t], x, S, G, r, Pi
=⇒ hΓ[x 7→(g,w) t], t, S, G ✶ (x à r), r, (g, w) : Pi
where t is constructor-rooted
(VarExp)
hΓ[x 7→(g,w) e], x, S, G, r, Pi
=⇒ hΓ[x 7→(g,w) e], e, x : S, G ✶ (x à r), r, (g, w) : Pi
where e is not constructor-rooted and e 6= x
(Val)
hΓ[x 7→(g′ ,w′ ) e], v, x : S, G, r, (g, w) : Pi
=⇒ hΓ[x 7→(g,w) v], v, S, G, r, (g, w) : Pi
where v is a value
(Fun)
hΓ, f (xn ), S, G, r, Pi
=⇒ hΓ, ρ(e), S, G[r 7→P f (xn )], q, [(f, Λ)]i
q
where f (yn ) = e ∈ P and ρ = {yn 7→ xn }
(Let)
hΓ, let x = e1 in e2 , S, G, r, Pi
=⇒ hΓ[y 7→(g,w.1) ρ(e1 )], ρ(e2 ), S, G[r 7→P ρ(let x = e1 in e2 )], q, [(g, w.2)]i
q
where ρ = {x 7→ y}, y is a fresh variable and P = (g, w) : P ′ for some P ′
(Case)
hΓ, case x of {pk → ek }, S, G, r, Pi
=⇒ hΓ, x, ({pk → ek }, r) : S, G[r 7→P case x of {pk → ek }], q, [(g, w.1)]i
where P = (g, w) : P ′ for some P ′
(Select)
hΓ, c(yn ), ({pk → ek }, r′ ) : S, G, r, Pi
=⇒ hΓ, ρ(ei ), S, G[r 7→P c(yn ), r′ 7→(g,w):P ′ ], q, [(g, w.2.i)]i
q
where i ∈ {1, . . . k}, pi = c(xn ), ρ = {xn 7→ yn }
and (r′ 7→(g,w):P ′ ) ∈ G for some P ′
hΓ, c(yn ), ({pk → ek }, r′ ) : S, G, r, Pi
(Select-f) =⇒ hΓ, F ail, S, G[r 7→ c(y ), r′ 7→ ], q, [ ]i
n
P
[]
q
where 6 ∃i ∈ {1, . . . k} such that pi = c(xn ), with (r′ 7→[ ] ) ∈ G
(Success) hΓ, c(xn ), [], G, r, Pi =⇒ hΓ, ⋄, [], G[r 7→P c(xn )], ✷, [ ]i
Figure 5.9: Instrumented semantics for tracing and slicing
109
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
with reference r with the program positions associated with the current occurrence
of x.
In rule Val, the first element in the current list of program positions (i.e., the
program position of the expression in the control) is used to label the binding that
updates the current heap.
In rules Fun, Let and Case, the current list of program positions is used to label
the expression in the control which is stored in the graph. Note that, in rule Fun, the
list of program positions in the derived configuration is reset to [(f, Λ)] because the
expression in the control is the complete right-hand side of function f . Note also that,
in rule Let, the new binding y 7→(g,w.1) ρ(e1 ) which is added to the heap is labeled
with the program position of expression e1 . This information may be necessary in
rules VarCons and VarExp if the value of y is demanded.
In rule Select, observe that the reference r′ —which is stored in the stack by rule
Case—is used to determine the initial list of program positions in the derived configuration. Finally, in rules Select-f and Success an empty list of program positions
is set because the control of the derived configuration contains either the special
(constructor) symbol Fail or ⋄, none of which occur in the program.
In order to perform computations with the instrumented semantics, we construct
an initial configuration and apply the rules of Figure 5.9 until a final configuration is
reached:
Definition 15 (initial configuration) The initial configuration of computations has
the form h[], main, [], G∅ , r, [ ]i, where G∅ denotes an empty graph, r a reference,
and [ ] an empty list of program positions (since there is no call to main from the
right-hand side of any function definition).
Definition 16 (final configuration) The final configuration of computations has
the form hΓ, ⋄, [], G, ✷, [ ]i, where Γ is a heap containing the computed bindings and
G is the extended redex trail of the computation.
We denote by =⇒∗ the reflexive and transitive closure of =⇒. A derivation C =⇒∗ C ′
is complete if C is an initial configuration and C ′ is a final configuration.
Clearly, the instrumented semantics is a conservative extension of the original
small-step semantics of Figure 2.6, since the extensions impose no restriction on the
application of the standard part of the semantics (the first three components of a
configuration).
Example 14 Consider again the program of Example 13. By using the instrumented
semantics of Figure 5.9, the derivation shown in Figure 5.10 is now computed. The
redex trail can be graphically depicted as follows:
110
5.3. BUILDING THE EXTENDED TRAIL
111
h[ ], main, [ ], G∅ , 0, [ ]i
=⇒Fun h[ ], let x = Z in . . . [ ], G[0 7→[ ] main], 1, [(main, Λ)]i
1
=⇒Let h[x 7→(main,1) Z], let y = g x in f y, [ ], G[1 7→[(main,Λ)] let x = Z in . . .],
2
2, [(main, 2)]i
=⇒Let h[y 7→(main,2.1) g x, x 7→(main,1) Z], f y, [ ], G[2 7→[(main,2)] let y = g x in . . .],
3
3, [(main, 2.2)]i
=⇒Fun h[y 7→(main,2.1) g x, x 7→(main,1) Z], y, [ ], G[3 7→[(main,2.2)] f y], 4, [(f, Λ)]i
4
=⇒VarExp h[y 7→(main,2.1) g x, x 7→(main,1) Z], g x, [y], G[y à 4], 4,
[(main, 2.1), (f, Λ)]i
=⇒Fun h[y 7→(main,2.1) g x, x 7→(main,1) Z], x, [y], G[4 7→[(main,2.1),(f,Λ)] g x], 5, [(g, Λ)]i
5
=⇒VarCons h[y 7→(main,2.1) g x, x 7→(main,1) Z], Z, [y], G[x à 5], 5, [(main, 1), (g, Λ)]i
=⇒Val h[y 7→(main,1) Z, x 7→(main,1) Z], Z, [ ], G, 5, [(main, 1), (g, Λ)]i
=⇒Success h[y 7→(main,1) Z, x 7→(main,1) Z] ⋄, [ ], G[5 7→[(main,1),(g,Λ)] Z], ✷, [ ]i
Figure 5.10: Derivation of Example 14
main
[]
let x = Z in ...
[(main,/\)]
let y = g x in f y
[(main,2)]
f
[(main,2.2)]
g
[(main,2.1),
(f,/\)]
Z
[(main,1),
(g,/\)]
Therefore, in contrast to the situation in Example 13, the associated slice is now the
complete program.
The correctness of the redex trail has been proved by Braßel et al. [2004b]. Now, we
state the correctness of the computed program positions. In what follows, given a
configuration of the form C = hΓ, e, S, G, r, Pi, we let ctrl (C) = e and pos(C) = P.
Theorem 17 Let P be a program and (C =⇒∗ C ′ ) be a complete derivation in P ,
where G′ is the graph in the final configuration C ′ . If (r 7→P e) ∈ G′ , then either
P = [ ] (with e = main) or P = [(gk , wk ), . . . , (g1 , k1 )], k > 1, and the following
conditions hold:
1. there exists a subderivation C1 =⇒ . . . =⇒ Cj of C =⇒∗ C ′ , j > k, such that
pos(Cj ) = [(gk , wk ), . . . , (g1 , w1 )], and
2. there exists a rule gi (xni ) = ei ∈ P such that ei |wi = ctrl (Ci ), for all i =
1, . . . , j, with ej |wj = e.
In order to prove Theorem 17 we need first the following auxiliary lemma that proves
two useful invariants for the configurations in a derivation. In what follows, given
111
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
a configuration of the form C = hΓ, e, S, G, r, Pi, we let heap(C) = Γ, ctrl (C) = e,
stack (C) = S, graph(C) = G, ref (C) = r, and pos(C) = P.
Lemma 18 Let P be a program, C0 be an initial configuration, and (C0 =⇒∗ Cm ),
m > 0, be a derivation (not necessarily complete) in P . For each configuration
Ci = hΓ, e, S, G, r, Pi, 0 6 i 6 m, the following conditions hold:
(I1) either P = [ ] (and e = main, e = ⋄, or e = Fail ) or P = (g, w) : P ′ for some
P ′ and there exists a rule g(xn ) = e1 ∈ P such that e1 |w = e (up to variable
renaming), and
(I2) for all bindings x 7→(g′ ,w′ ) e′ in the heap Γ, there exists a rule g ′ (xk ) = e2 in P
such that e2 |w′ = e′ (up to variable renaming).
Proof. We prove the claim by induction on the length m of the considered computation.
Base case (m = 0) Then, the derivation contains only the initial configuration
h[], main, [], G∅ , r, [ ]i
and both invariants trivially hold.
Inductive case (m > 0) Let us consider an arbitrary derivation of the form
C0 =⇒ C1 =⇒ · · · =⇒ Cm−1 =⇒ Cm
By the inductive hypothesis, both (I1) and (I2) hold in Cm−1 . Now, we prove that
(I1) and (I2) also hold in Cm . If pos(Cm−1 ) = [ ], by definition of the instrumented
semantics, Cm−1 can only be an initial configuration (with ctrl (Cm−1 ) = main) or
a final one (with ctrl (Cm−1 ) = ⋄ or ctrl (Cm−1 ) = Fail ). Moreover, in our case,
Cm−1 can only be an initial configuration since we know that there exists a successor
configuration Cm . Therefore, only rule fun can be applied to Cm−1 . Here, invariant
(I1) trivially holds in Cm since pos(Cm ) = [(main, Λ)] clearly addresses the right-hand
side of function main. Invariant (I2) also holds because the heap is not modified.
Otherwise, assume that Cm−1 = hΓ, e, S, G, r, (g, w) : Pi. We consider the following cases depending on the applied rule:
(varcons) Then e = x is a variable. Consider that Γ = Γ[x 7→(g′ ,w′ ) t]. It is immediate that invariant (I1) holds in the derived configuration Cm = hΓ, t, S, G, r, (g ′ , w′ ) :
(g, w) : Pi because the invariant (I2) holds in Cm−1 . Also, invariant (I2) trivially
holds in Cm because the heap is not modified in the step.
(varexp) Then e = x is a variable and the proof is perfectly analogous to the
previous case.
(val) Then either e is a (logical) variable or a constructor-rooted term. In either
case, invariant (I1) trivially holds in Cm because both the control and the program
112
5.3. BUILDING THE EXTENDED TRAIL
113
position are not changed. As for invariant (I2), since invariant (I1) holds in Cm−1 ,
we know that program position (g, w) is a correct program position for v. Therefore,
the update of the heap is correct and, since invariant (I2) already holds in Cm−1 , it
still holds in Cm .
(fun) Then e = f (xn ) is a function call. Since the heap is not modified, invariant
(I2) trivially holds in Cm . Moreover, since the expression in the control is the renamed
(by ρ) right-hand side of function f , the program position (f, Λ) is obviously correct
(up to the variable renaming ρ) and invariant (I1) follows.
(let) Then e = (let x = e1 in e2 ) is a let expression. Since invariant (I1) holds
in Cm−1 , by definition of position, (g, w.2) correctly addresses the (renaming of)
expression e2 and, thus, invariant (I1) also holds in Cm . Similarly, (g, w.1) correctly
addresses the (renaming of) expression e1 and, since this is the only modification on
heap Γ and (I2) holds for Cm−1 , (I2) also holds in Cm .
(case) Then e = case x of {pk → ek }. Since the heap is not modified, invariant (I2)
trivially holds in Cm . Moreover, by definition of position, (g, w.1) correctly addresses
the case argument and, thus, invariant (I1) also holds in Cm .
(select) Then e = c(xn ). Since the heap is not modified in Cm , invariant (I2)
trivially holds. Moreover, it is easy to prove that the program position of the case expression that demanded the evaluation of c(xn ) is (g, w′ ) which is stored in the graph.
Therefore, by definition of position, (g, w′ .2.i) correctly addresses the expression ρ(ei )
in the i-th branch of the case expression (up to the variable renaming ρ).
(select-f) Then e = c(xn ). Since the heap is not modified in this step, invariant
(I2) trivially holds in Cm . Moreover, since the list of program positions in Cm is [ ],
invariant (I1) also holds.
(success) Then e = c(xn ). This case follows trivially because the heap is not
modified and the list of program positions in Cm is [ ].
✷
According to the instrumented semantics of Figure 5.9, not all expressions in the
control of a configuration are stored in the graph. The following definition, slightly
extended from Braßel et al. [2004b] to include program positions (and ignore parent
references), formalizes when a configuration contains an expression that will be stored
in the graph (in the next step of the computation).
Definition 19 A configuration hΓ, e, S, G, r, Pi is relevant iff one of the following
conditions hold:
1. e is a value and the stack S is not headed by a variable, or
2. e is a function call, a let expression, or a case expression.
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
The following lemma—a simplified version of the analogous lemma in Braßel et al.
[2004b] where logical features are not considered—states that, for each relevant configuration in a derivation, the associated information is stored in the graph.
Lemma 20 Let (C0 =⇒∗ Cn ), n > 0, be a derivation. For each i ∈ {0, . . . , n}, we
have that Ci = hΓ, e, S, G, r, Pi is a relevant configuration iff Gi+1 = G[r 7→P e],
where Gi+1 = graph(Ci+1 ).
Note that, in the above lemma, we denote by Gi+1 = G[r 7→P e] that the arrow
r 7→P e is added to G in the step Ci =⇒ Ci+1 .
Finally, we state and prove Theorem 17:
Theorem 17. Let P be a program and (C =⇒∗ C ′ ) be a complete derivation in
P , where G′ is the graph in the final configuration C ′ . If (r 7→P e) ∈ G′ , then either
P = [ ] (with e = main) or P = [(gk , wk ), . . . , (g1 , w1 )], k > 1, and the following
conditions hold:
1. there exists a subderivation C1 =⇒ . . . =⇒ Cj of C =⇒∗ C ′ , j > k, such that
pos(Cj ) = [(gk , wk ), . . . , (g1 , w1 )], and
2. there exists a rule gi′ (xni ) = ei ∈ P such that ei |wi′ = ctrl (Ci ), for all i = 1, . . . , j,
where gi′ = gi , wi′ = wi for all i = 1, . . . , k and gi′ = gk , wi′ = wk for all
i = k + 1, . . . , j, with ej |wj = e.
Proof. If P = [ ], then the proof follows trivially by Lemma 18 since (r 7→P main)
belongs to the final graph G′ by Lemma 20.
Now, consider P = [(gk , wk ), . . . , (g1 , k1 )], k > 1. Since we have (r 7→P e) ∈ G′ ,
by Lemma 20, there exists a relevant configuration Cj = hΓ, e, S, G, r, Pi. If P =
[(gk , wk )] contains only one program position, the proof follows straightforwardly by
Lemma 18 (with k = 1, j = 1). Otherwise, observe that only rules varcons, varexp
and val can be applied to a configuration so that either the list of program positions
is not modified or new program positions are added to it. Therefore, there must be
a subderivation of the form C1 =⇒ . . . =⇒ Cj of C =⇒∗ C ′ with j = k (if rule val is
never applied) or j > k (otherwise) such that pos(Cj ) = [(gk , wk ), . . . , (g1 , k1 )], k > 1.
Observe that the applications of rule val (if any) can only happen at the end of the
subderivation, since once a value is set in the control, rules varcons and varexp are no
longer applicable. By Lemma 18, we have that there exists a rule gi (xni ) = ei ∈ P
such that ei |wi = ctrl (Ci ), for all i = 1, . . . , k. Finally, since only rule val is applied
to Ck+1 , . . . , Cj , we have ek |wk = ctrl (Ci ), for all i = k + 1, . . . , j, with ej |wj = e, and
the claim follows.
✷
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5.4. COMPUTING THE SLICE
5.3.3
115
Logical Features
In this section, we sketch the extension of the instrumented semantics to deal with
the logical features of the language: disjunctions and flexible case structures. The
new rules are shown in Figure 5.11.5
Rule Or is used to non-deterministically choose one of the disjuncts. It is selfexplanatory and proceeds analogously to rules Fun, Let or Case.
Besides the original rule for case expressions (rule case), we add a new rule fcase
to deal with flexible case expressions. In this case, a symbol f is stored in the stack
preceding the case alternatives to indicate that the case expression is flexible. Note
also that, to be precise, this implies that rule select in Figure 5.9 should also be
slightly changed in order to accept a pair (f {pk → ek }, r′ ) on top of the stack.
In contrast to rule Select, rule Guess is used when we have a logical variable in
the control and the alternatives of a flexible case expression on the stack. Then, this
rule non-deterministically choose one alternative and continue with the evaluation of
this branch. Since the name of logical variables is not relevant, the new node r is
labeled with a special symbol LogVar . The successor of this node is then set to q
which is labeled with the selected binding for the logical variable. The list of program
positions for node q is empty because patterns—and their local variables—introduced
by instantiation do not have an associated program position. Finally, observe that
we use ( , ) to denote a null program position in the heap.
Rule Guess-f applies when we have a logical variable in the control and (the alternatives of) a rigid case expression on top of the stack, i.e., when the computation
suspends. Analogously to rule Select-f, an empty list of program positions is set in
the derived configuration.
Finally, rule Success-x is the counterpart of rule Success when the computed value
is a logical variable.
5.4
Computing the Slice
In this section, we formalize the concept of dynamic slice and introduce a method to
compute it from the extended redex trail.
As in the previous section, we consider a pure (first-order) functional language
without disjunctions nor flexible case structures; the treatment of these language
features is sketched in Section 5.4.2.
We define a slice as a set of program positions:
Definition 21 (slice) Let P be a program. A slice for P is any set W of program
positions with W ⊆ Pos(P ).
5 We
assume that q and s are always fresh references.
115
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
(Or)
hΓ, e1 or e2 , S, G, r, Pi =⇒ hΓ, ei , S, G[r 7→P e1 or e2 ], q, [(g, w.i)]i
q
′
where i ∈ {1, 2} and P = (g, w) : P for some P ′
(Fcase)
hΓ, fcase x of {pk → ek }, S, G, r, Pi
=⇒ hΓ, x, (f {pk → ek }, r) : S, G[r 7→P (f )case x of {pk → ek }],
q, [(g, w.1)]i
where P = (g, w) : P ′ for some P ′
(Guess)
hΓ[y 7→(h,w′ ) y], y, (f {pk → ek }, r′ ) : S, G, r, Pi
=⇒ hΓ[y 7→( , ) ρ(pi ), yn 7→( , ) yn ], ρ(ei ), S,
G[r 7→P LogVar , q 7→[ ] ρ(pi ), y à r, r′ 7→(g,w):P ′ ], s, [(g, w.2.i)]i
q
s
where i ∈ {1, . . . k}, pi = c(xn ), ρ = {xn 7→ yn }, (r′ 7→(g,w):P ′ ) ∈ G
for some P ′ , and yn are fresh variables
(Guess-f)
hΓ[y 7→(g,w) y], y, (f {pk → ek }, r′ ) : S, G, r, Pi
=⇒ hΓ[y 7→(g,w) y], F ail, S, G[r 7→P LogVar , y à r, r′ 7→[ ] ], s, [ ]i
s
where (r′ 7→) ∈ G
(Success-x)
hΓ[x 7→(g,w) x], x, [], G, r, Pi
p
=⇒ hΓ[x 7→(g,w) x], ⋄, [], G[r →
7 P LogVar , x à r], ✷, [ ]i
Figure 5.11: Instrumented semantics for tracing and slicing: logical features
Observe that we are not interested in producing executable slices but in computing
the program positions of the expressions that influence the slicing criterion. This is
enough for our purposes, i.e., for showing the original program with some distinguished
expressions.
In the literature of dynamic slicing for imperative programs, the slicing criterion
usually identifies a concrete point of the execution history, e.g., hn = 2, 81 , xi, where
n = 2 is the input for the program, 81 denotes the first occurrence of statement 8
in the execution history, and x is the variable we are interested in. In principle, it is
not difficult to extend this notion of slicing criterion to a strict functional language.
For instance, a slicing criterion could be given by a pair hf (vn ), πi, where f (vn ) is a
function call that occurs during the execution of the program—whose arguments are
fully evaluated because we consider a strict language—and π is a restriction on the
output of f (vn ).
Unfortunately, extending this notion to a lazy functional language is not trivial. For instance, identifying a function call f (en ) in the actual execution trace is
116
5.4. COMPUTING THE SLICE
117
impractical because en are usually rather complex expressions. Luckily, if a tracing
tool like the one presented in Section 5.2 is available—i.e., a tracing tool that shows
function calls with arguments as much evaluated as needed in the complete computation—then we can still consider a slicing criterion of the form hf (vn ), πi. In this
case, f (vn ) should be understood as “a function call f (en ) whose arguments en are
evaluated to vn in the complete computation”.
The connection with the tracing tool (as described in Section 5.2) should be clear:
given a (sub)trace of the form
val1
val2
...
valk
=
=
1
f1 (v11 , . . . , vm
)
2
2
f2 (v1 , . . . , vm )
=
k
fk (v1k , . . . , vm
)
i
a slicing criterion has the form hfi (v1i , . . . , vm
), πi, where π denotes a restriction on
vali . Therefore, the user can easily produce a valid slicing criterion from the trace
of a computation by only giving π (if any, since a default value ⊤ can also be used).
Traces usually contain a special symbol, “ ”, to denote that some subexpression is
missing because its evaluation was not required in the computation. Consequently,
we will also accept expressions containing this special symbol (see below).
Definition 22 (slicing criterion) Let P be a program and (C0 =⇒∗ Cm ) a complete derivation.6 A slicing criterion for P w.r.t. (C0 =⇒∗ Cm ) is a tuple hf (pvn ), πi
such that f ∈ F is a defined function symbol (arity n > 0), pv1 , . . . , pvn ∈ P V alue
are partial values, and π ∈ Pat is a slicing pattern. The domain PValue obeys the
following syntax:
pv ∈ PValue ::=
| x | c(pvk )
where c ∈ C is a constructor symbol (arity k > 0) and “ ” is a special symbol to denote
any non-evaluated expression.
The domain Pat of slicing patterns is defined as follows:
π ∈ Pat ::= ⊥ | ⊤ | c(πk )
where c ∈ C is a constructor symbol of arity k > 0, ⊥ denotes a subexpression of the
value whose computation is not relevant and ⊤ a subexpression which is relevant.7
Slicing patterns are particularly useful when we are only interested in part of the
result of a function call. For instance, if we know that a function call should return a
6 Here,
and in the following, we use the notation C0 =⇒∗ Cm as a shorthand to denote a derivation
of m steps of the form C0 =⇒ C1 =⇒ . . . =⇒ Cm .
7 Our patterns for slicing are similar to the liveness patterns used to perform dead code elimination
by Liu and Stoller [2003].
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
pair, and we are only interested in the first component of this pair, we could use the
slicing pattern (⊤, ⊥); if a function returns a list, and we are only interested in the
first two elements of this list, we could use the slicing pattern ⊤ : (⊤ : ⊥). Usually,
we are only interested in the computation of a value and, thus, the slicing pattern hnf
can be used. Of course, if we are interested in the full evaluation of an expression (i.e.,
as much as needed in the considered computation), we can use the slicing pattern ⊤.
In order to compute a slice, we first need to identify the configuration associated with a slicing criterion (scc) because only the program positions in the next
configurations are potential candidates for the slice:
candidates for the slice
z
}|
{
C1 =⇒ C2 =⇒ . . . =⇒ Ci =⇒ Ci+1 . . . =⇒ Cn
|{z}
scc
For this purpose, we need some auxiliary definitions. In what follows, given a configuration of the form C = hΓ, e, S, G, r, Pi, we let heap(C) = Γ, stack (C) = S, and
graph(C) = G.
First, we introduce the notion of lookup configuration [Braßel et al., 2004b], i.e.,
a configuration that demands the evaluation of a given variable for the first time in a
computation.
Definition 23 (lookup configuration) Let D = (C0 =⇒∗ Cm ), m > 0, be a
derivation. A configuration Ci , 0 6 i 6 m, is a lookup configuration of D iff
ctrl (Ci ) = x and there exists no configuration Cj , j < i, with ctrl (Cj ) = x.
When a lookup configuration hΓ, x, S, G, r, Pi appears in a computation, a variable
pointer x à r is added to the graph to point out that the value of x is demanded in
the computation:
Definition 24 (demanded variable) Let D = (C0 =⇒∗ Cm ), m > 0, be a derivation. We say that a variable x is demanded in D iff there exists a lookup configuration
Ci , 0 6 i 6 m, such that ctrl (Ci ) = x.
The following function is useful to extract the partial value denoted by an expression.
For this purpose, outer constructors are left untouched, non-demanded variables and
operation-rooted terms are replaced by the special symbol “ ” (i.e., “not a value”),
and demanded variables are dereferenced.
Definition 25 Let D = (C0 =⇒∗ Cm ), m > 0, be a complete derivation. Given an
expression e, we denote the partial value of e in D by val D (e), where function val is
defined as follows:
v
if e = x and x is demanded in D
valD (x)
val D (e) =
c(val D (xn )) if e = c(xn )
otherwise
118
5.4. COMPUTING THE SLICE
119
v
The auxiliary function valD
is used to dereference a variable w.r.t. (the heap computed
in) a derivation:
v
if Γ[x] = y and x 6= y
valD (y)
v
valD (x) =
c(val D (xn )) if Γ[x] = c(xn )
otherwise
where Γ = heap(Cm ) is the heap in the last configuration of D.
Let us illustrate this function with some examples. Consider three derivations, D1 ,
D2 and D3 , in which the computed heaps in the last configurations are as follows:8
Γ1
Γ2
Γ3
= [ x 7→ y,
= [ x 7→ y,
= [ m→
7 (f Z),
y 7→ (S n),
y 7→ (S z),
y 7→ (S n),
n 7→ Z
z 7→ Z,
n 7→ Z
]
n 7→ Z
]
]
Here Z, S ∈ C are constructors, f is a defined function symbol, and x, y, z, n, m are variables. Assuming that variables x, y, z are demanded in the considered computations
and m, n are not, we have:
val D1 (C x (f y)) = C (S )
= C (S Z)
val D2 (C x n)
= C
val D3 (C m y)
(S )
We can now formally identify the configuration associated with a slicing criterion.
Definition 26 (SC-configuration) Let D = (C0 =⇒∗ Cm ), m > 0, be a complete
derivation. Given a slicing criterion hf (pvn ), πi, the associated SC-configuration Ci ,
0 6 i 6 m, must fulfill the conditions9
• ctrl (Ci ) = f (xn ) and
• pvi = val D (xi ) for all i = 1, . . . , n.
Intuitively speaking, the SC-configuration associated with a slicing criterion hf (pvn ), πi
is a configuration, Ci , whose control has the form f (xn ) and the variables xn are eventually bound to the partial values pvn in the considered derivation.
Clearly, the notion of SC-configuration is only partially defined since it may happen
that the conditions of the above definition do not hold (e.g., if the slicing criterion
refers to a function that has not been evaluated in the considered computation).
8 Here
we ignore the program positions that label the bindings in the heap since they are not
relevant for computing the partial value.
9 If there are several configurations that fulfill these conditions (e.g., if the same function is called
several times in the computation with the same arguments), we choose the first of such configurations.
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
h[ ], main, [ ], G∅ , 0, [ ]i
=⇒Fun
h[ ],
=⇒Let
h[x 7→(main,1) Z],
=⇒Let
h[y 7→(main,2.1) g x, x 7→(main,1)
=⇒Fun
h[y 7→(main,2.1) g x, x 7→(main,1)
=⇒VarExp h[y 7→(main,2.1) g x, x 7→(main,1)
=⇒Fun
h[y 7→(main,2.1) g x, x 7→(main,1)
=⇒VarCons h[y 7→(main,2.1) g x, x 7→(main,1)
=⇒Val
h[y 7→(main,1) Z, x 7→(main,1) Z]
=⇒Success h[y 7→(main,1) Z, x 7→(main,1) Z]
Z]
Z]
Z]
Z]
Z]
let x = Z in . . . , [ ],
let y = g x in f y, [ ],
f y,
[ ],
y,
[ ],
g x,
[y],
x,
[y],
Z,
[y],
Z,
[ ],
⋄,
[ ],
G1 , 1,
G2 , 2,
G3 , 3,
G4 , 4,
G5 , 4,
G6 , 5,
G7 , 5,
G8 , 5,
G9 ,✷,
P1 i
P2 i
P3 i
P4 i
P5 i
P6 i
P7 i
P8 i
P9 i
Figure 5.12: A complete subderivation
For simplicity, in the following we assume that there is always an SC-configuration
associated to any slicing criterion.
For instance, given the program and derivation of Example 14, the SC-configuration
associated with the slicing criterion hf Z, ⊤i is the configuration
h [y 7→(main,2.1) g x, x 7→(main,1) Z],
G[2 7→[(main,2)] let y = g x in . . .],
3
f y, [ ],
3,
[(main, 2.2)] i
since its control is (f y) and the variable y is bound to Z in the heap [y 7→(main,1)
Z, x 7→(main,1) Z] of the final configuration of the derivation.
We also need the notion of complete subderivation:
Definition 27 (complete subderivation) Let P be a program and D = (C0 =⇒∗
Cm ) a complete derivation in P . Let Ci , 0 > i > m, be a configuration. The
subderivation Ci =⇒∗ Cj , i > j > m, is the complete subderivation for Ci in D iff
the following conditions hold:
1. ctrl (Cj ) ∈ Value,
2. stack (Ci ) = stack (Cj ), and
3. 6 ∃k. i < k < j, stack (Ci ) = stack (Ck ) and ctrl (Ck ) ∈ Value.
Consider, for instance, the derivation of Example 14 which is shown in Figure 5.12.
Here, the complete subderivation for the configuration
h[y 7→(main,2.1) g x, x 7→(main,1) Z], g x, [y], G[y à 4], 4, [(main, 2.1), (f, Λ)]i
is shown in a box (where graphs and program positions are ignored for conciseness).
Now, we can already introduce our notion of dynamic slice (in the following, the
auxiliary function hd returns the first element of a list):
120
5.4. COMPUTING THE SLICE
121
Definition 28 (dynamic slice, ds) Let P be a program and let D = (C0 =⇒∗ Cn )
be a complete derivation in P . Let hf (pvl ), πi be a slicing criterion and Ci its associated SC-configuration. The dynamic slice for P w.r.t. the slicing criterion hf (pvl ), πi
is given by
{hd (pos(Ci ))} ∪ ds([Ci+1 , . . . , Cj ], π, { }, D)
where Ci =⇒∗ Cj is the complete subderivation for Ci in D and the definition of the
auxiliary function ds is shown in Figure 5.13.
We note that function ds is not applied to the SC-configuration Ci since it may
store, not only the program position of f (xn ), but also the positions of some variables
whose evaluation introduced this function call (see the discussion in Section 5.3.2).
Therefore, in order to avoid gathering these undesired positions, we only collect the
position in the head of the list of positions in Ci and, then, apply function ds to the
subderivation starting in the next configuration.
Note also that the third parameter of function ds is used to store the set of variables
whose evaluation is relevant for the slice. Essentially, we consider that every variable
introduced in a let expression before the SC-configuration is not relevant, according to
the assumption that the evaluation of the arguments in the call of the slicing criterion
should not be taken into account. This set, initially empty, is then updated whenever
a let expression is found.
Let us now informally explain how a dynamic slice is computed through a simple
example.
Example 15 Consider the program shown in Figure 5.14. Given the slicing criterion hg Z, C ⊥ ⊤i, the computation described in Figure 5.15 illustrates the way in
which function ds proceeds. For simplicity, we only show the control of each configuration. Here, the computation for function main is shown in the leftmost column;
then, each time a case expression demands the evaluation of its argument, the computation moves one level to the right (and, analogously, when the demanded value is
computed, it moves back one level to the left). In this computation, the expressions of
those configurations whose program positions are collected by ds are shown in a box.
5.4.1
Using the Redex Trail
According to Definition 28, one could develop a slicing tool by implementing a metainterpreter for the instrumented operational semantics (Figure 5.9) and, then, extracting the program positions from these configurations using function ds (Figure 5.13).
Our aim in this chapter, however, is to compute the slice from the extended redex
trail, since it is already available in existing tracing tools.
121
122
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
ds([Cj ], π, V, D) = pos(Cj ) ∪ ds aux (Cj , π, V, D)
ds([Ci , . . . , Cj ], π, V, D) =
pos(Ci ) ∪ ds([Ci+1 , . . . , Cj ], π, V ′ , D)
pos(Ci ) ∪ ds([Ci+1 , . . . , Cj ], π, V, D)
pos(Ci ) ∪ ds([Ck+1 , . . . , Cj ], π, V, D)
pos(Ci ) ∪ ds([Ci+1 , . . . , Ck ], hnf, V, D)
∪ ds([Ck+1 , . . . , Cj ], π, V, D)
ds([Ci+1 , . . . , Cj ], π, V, D)
ds aux (Cj , π, V, D) =
S
(x,π ′ )∈ctrl(Cj )⊓V π
[i < j]
′
if ctrl (Ci ) = (let x = e in e )
where V ′ = V ∪ {x}
if ctrl (Ci ) is a function call
if ctrl (Ci ) = (case x of . . .) and
x 6∈ V where Ci+1 =⇒∗ Ck is the
complete subderivation
for Ci+1 in D
if ctrl (Ci ) = case x of . . . and
x ∈ V where Ci+1 =⇒∗ Ck is the
complete subderivation
for Ci+1 in D
otherwise
ds([Cx , . . . , Cx′ ], π ′ , V, D)
where Cj =⇒∗ Cx in D, ctrl (Cx ) = x,
Cx is a lookup configuration,
and Cx =⇒∗ Cx′ is the complete
subderivation for Cx in D
{(xi , πi ) | xi ∈ V, πi 6= ⊥, and i ∈ {1, . . . , m}} if v = c(xm )
and π = c(πm )
v ⊓V π =
{(xi , ⊤) | xi ∈ V and i ∈ {1, . . . , m}}
if e = c(xm )
and
π=⊤
{}
otherwise
Figure 5.13: Extracting a dynamic slice from a derivation (function ds)
From the extended redex trail, the computation of a dynamic slice proceeds basically as follows: first, the node associated with the slicing criterion is identified (the
SC-node, the counterpart of the SC-configuration); then, the set of program positions
of the nodes which are reachable from the SC-node (taking into account the slicing
pattern) are collected.
In order to identify the SC-node, we first need to define the counterpart of function
val D :
122
5.4. COMPUTING THE SLICE
123
main = let x = pair in f x
f x = case x of
{ C w1 w2 → case w1 of { Z → case w2 of { Z → Z }}}
pair = let z = zero in g z
g z = let y = one in case z of
{ Z → case y of
{ S w → case w of
{ Z → let v1 = Z in
let v2 = Z in C v1 v2 }}}
zero = Z
one = let v = Z in S v
Figure 5.14: Program of Example 15
Definition 29 Let D = (C0 =⇒∗ Cm ), m > 0, be a complete derivation and G =
graph(Cm ). The partial value associated with variable x in G is given by ValD (x),
where function ValD is defined as follows:
valvG (q)
if (x à r) ∈ G and (r 7→) ∈ G
q
ValD (x) =
c(ValD (xn )) if (x à r) ∈ G and (r 7→ c(xn )) ∈ G
_
otherwise
The auxiliary function valvG is used to follow the successor chain in a graph:
v
if (r 7→) ∈ G
valG (q)
q
v
valG (r) =
(x
))
if
(r
→
7
c(xn )) ∈ G
c(Val
D
n
otherwise
Observe the similarities between functions val D and ValD . The main difference is
that, in order to get the final value of a variable, val D looks at the bindings in the
heap of the final configuration of a computation, while ValD follows a path in the
graph of the final configuration of this computation.
Now, the node associated with a slicing criterion in the extended redex trail can
be determined as follows:
Definition 30 (SC-node) Let P be a program and (C0 =⇒∗ Cm ) a complete derivation, where G = graph(Cm ). Given a slicing criterion hf (pvn ), πi, the associated
123
124
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
main
↓
let x = pair
↓
fx
↓
case x of
→x
↓
pair
↓
let z = zero
↓
gz
// SC-configuration
↓
let y = one
// variable y added to V
↓
case z of
→z
↓
zero
↓
Z
ւ
case y of → y
↓
one
↓
let v = Z
↓
Sv
case v of
let v1 = Z
↓
let v2 = Z
↓
C v1 v2
ւ
case v1 of →v1
↓
Z
ւ
case v2 of → v2
↓
Z
ւ
Z
ւ
→ v
↓
Z
ւ
// subcomputation ignored:
// variable z not in V
// subcomputation considered:
// variable y in V
// variable v added to V
// subcomputation considered:
// variable v in V
// variable v1 added to V
// variable v2 added to V
//
//
//
//
C v1 v2 ⊓V C ⊥ ⊤ returns
{ (v1,⊥), (v2,⊤) }
subcomputation ignored:
slicing pattern is ⊥
// subcomputation considered:
// slicing pattern is ⊤
// and variable v2 is in V
Figure 5.15: Computation of a dynamic slice (Example 15)
124
5.4. COMPUTING THE SLICE
DS (G, r, π, V ) =
P ∪ DS AUX (G, c(xn ), r, π, V )
P ∪ DS (G, q, π, V ′ )
P ∪ DS (G, q, π, V )
P ∪ DS (G, q ′ , hnf , V )
∪ DS (G, q, π, V )
P ∪ DS (G, q, π, V )
DS AUX (G, c(xn ), r, π, V ) =
125
if (r 7→P c(xn )) ∈ G
if (r 7→P let x = e in e′ ) ∈ G,
q
where V ′ = V ∪ {x}
if (r 7→P case x of {pk → ek }) ∈ G
q
and x 6∈ V
if (r 7→P case x of {pk → ek }) ∈ G
q
and x ∈ V , where (x à q ′ ) ∈ G
otherwise, where (r 7→P ) ∈ G
q
S
(x,π ′ )∈c(xm )⊓V π
DS (G, rx , π ′ , V )
where (x à rx ) ∈ G
Figure 5.16: Extracting a dynamic slice from a redex trail (function DS )
SC-node must fulfill the conditions:10
• (r 7→ f (xn )) ∈ G and
• pvi = ValD (xi ) for all i = 1, . . . , n.
Analogously to the notion of SC-configuration, the SC-node is only partially defined
since it may happen that the conditions of the above definition do not hold. For
simplicity, in the following we assume that there is always an SC-node associated to
any slicing criterion.
We now introduce a function to compute a dynamic slice from the program positions of the nodes that are reachable from the SC-node (according to the slicing
pattern).
10 As
in Definition 26, if there are several nodes that fulfill these conditions, we choose the first one
following a traversal of the graph that is defined as follows:
r; traversal(G, q ′ ); traversal(G, q) if (r 7→ case x of . . .) ∈ G
q
and (x à q ′ ) ∈ G
traversal(G, r) =
r; traversal(G, q)
otherwise, with (r 7→) ∈ G
q
i.e., whenever we find a case expression, we first follow the evaluation of the case argument and,
then, the selected branch. The initial call would be traversal(G, r) where G is the computed graph
and r is the reference of the initial configuration.
125
126
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
Definition 31 (function DS ) Let G be an extended redex trail. Let hf (pvn ), πi be
a slicing criterion and r the reference of its associated SC-node. A slice of hf (pvn ), πi
in G is given by {hd (P)} ∪ DS (G, q, π, { }), where (r 7→P ) ∈ G and function DS is
q
defined in Figure 5.16.
The parallelisms between functions ds and DS should be clear. While traversing the
complete subcomputation for the slicing criterion, both functions
• consider the subcomputations for those variables introduced in a let expression
after the slicing criterion and
• ignore the subcomputations for those variables introduced in a let expression
before the slicing criterion.
Once the value of the slicing criterion is reached, both functions use the slicing pattern
to determine which further subcomputations should be considered.
Now, we prove that slices computed by function DS are indeed dynamic slices
according to Definition 28:
Theorem 32 (completeness and minimality) Let P be a program and (C0 =⇒∗
Cm ), m > 0, a complete derivation, with G = graph(Cm ). Let hf (pvn ), πi be a slicing
criterion and W be the dynamic slice computed according to Definition 28. Then, we
have DS (G, r, π, {}) = W, where r is the SC-node associated with the slicing criterion.
First, we need some results from Braßel et al. [2004b] that we slightly modify to
include program positions and ignore parent arrows: the notion of successor derivation
and Propositions 34 and 35, which correspond, respectively, to Propositions 5.5 and
5.8 in Braßel et al. [2004b]. First, we introduce the notion of successor derivation.
Definition 33 (successor derivation) Let D = (C0 =⇒∗ Cm ), m > 0, be a derivation. Let Ci = hΓ, e, S, G, r, Pi, 0 ≤ i < m, be a relevant configuration such that e is a
function call, a let expression, or a case expression. Then, Ci =⇒∗ Cj , i < j ≤ m, is
a successor subderivation of D iff (i) Cj is relevant, (ii) stack(Ci ) = stack(Cj ), and
(iii) there is no relevant configuration Ck , i < k < j, such that stack(Ci ) = stack(Ck ).
Intuitively, a successor derivation represents a single reduction step including the associated subcomputations (if any). The following result states that, for each successor
subderivation in a computation, the instrumented semantics adds a corresponding
successor arrow to the graph, and vice versa.
Proposition 34 Let D = (C0 =⇒∗ Cm ), m > 0, be a derivation. For all i, j (0 ≤
i < j < m), there exists a successor subderivation
Ci = hΓ, e, S, G, r, Pi =⇒∗ hΓ′ , e′ , S ′ , G′ , r′ , P ′ i = Cj
in D iff (r 7→′ P e) ∈ graph(Cj+1 ) and (r′ 7→P ′ e′ ) ∈ graph(Cj+1 ).
r
q
126
5.4. COMPUTING THE SLICE
127
As mentioned before, when a lookup configuration hΓ, x, S, G, r, p, Pi appears in a
computation, a variable pointer x à r should be added to the graph. Moreover,
reference r stores the dereferenced value of heap variable x. This is expressed by
means of function Γ∗ which is defined as follows:
(
Γ∗ (y) if Γ[x] = y and x 6= y
∗
Γ (x) =
Γ[x]
otherwise
The next proposition formally states that, for each variable whose value is demanded
in a computation, we have an associated variable pointer in the graph (it is a slight
extension of a similar result in Braßel et al. [2004b] in order to consider program
positions).
Proposition 35 Let D = (C0 =⇒∗ Cm ), m > 0, be a derivation. For all i (0 <
i < m), there exists a lookup configuration Ci = hΓ, x, S, G, r, Pi in D iff (x Ã
r) ∈ graph(Cj+1 ) and r 7→P ′ e′ ∈ graph(Cj+1 ), where Cj = hΓ′ , e′ , S ′ , G′ , r, P ′ i,
i ≤ j < m, is a relevant configuration and there is no relevant configuration Ck with
i < k < j.
Observe that all the configurations in the subderivation Ci =⇒∗ Cj−1 have a variable
in the control. Therefore, only rules varcons and varexp can be applied, which implies
that their program positions are accumulated to the list of program positions in Cj .
In order to prove the completeness of the computed slice, we first need to prove
that, given a slicing criterion, the associated SC-node correctly points to the node
that stores the control of the associated SC-configuration. For this purpose, we need
some preliminary results.
The next lemma shows that reductions in a derivation can also be obtained from
the extended redex trail by following the successor relation.
Lemma 36 Let D = (C0 =⇒∗ Cm ), m > 0, be a complete derivation and Ci =
hΓ, e, S, G, r, Pi, 0 < i < m, be a relevant configuration. There exists a complete
subderivation Ci =⇒∗ Cj for Ci in D iff there exists a successor chain (r 7→P e), (q1 7→
q1
P1
q2
e1 ), . . . , (qn 7→Pn en ) ∈ graph(Cm ) with en = ctrl (Cj ).
Proof. Since Ci is a relevant configuration, we have that Ci =⇒∗ Cj can be decomposed into a sequence of successor subderivations
Ci = hΓ, e, S, G, r, Pi
=⇒∗
=⇒∗
...
=⇒∗
hΓ1 , e1 , S1 , G1 , q1 , P1 i
hΓ2 , e2 , S2 , G2 , q2 , P2 i
hΓn , en , Sn , Gn , qn , Pn i
Therefore, by Proposition 34, the claim follows.
127
✷
128
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
The next lemma shows that, for each variable x demanded in a derivation D, we
have val D (x) = ValD (x).
Lemma 37 Let D = (C0 =⇒∗ Cm ), m > 0, be a complete derivation and Ci =
hΓ, x, S, G, r, Pi, 0 < i < m, be a lookup configuration. Then, val D (x) = ValD (x).
Proof. We prove this claim by structural induction on the computed partial value
pv.
(Base case) Then, pv is a constructor constant c. By definition of function val , we
v
v
v
(yn ) = c. Hence, we have Γ∗ (x) = c.
(y1 ) = . . . = valD
(x) = valD
have: val D (x) = valD
Moreover, by Proposition 35, we have that (x à r) ∈ G and (r 7→ Γ∗ (x)) ∈ G, where
G = graph(Cm ). Thus, val D (x) = ValD (x) = c.
(Inductive case) We now consider that the partial value pv is a constructor call
v
c(xn ). Then, Γ[x] = c(xn ) and val D (x) = valD
(x) = c(val D (xn )). By Proposition 35,
∗
we have that (x à r) ∈ G and (r 7→ Γ (x)) ∈ G, where G = graph(Cm ). Therefore,
the claim follows by the inductive hypothesis since val D (xi ) = ValD (xi ) for all i =
1, . . . , n.
✷
We now establish the equivalence between the notions of SC-node and SC-configuration
for a given slicing criterion.
Theorem 38 Let D = (C0 =⇒∗ Cm ) be a complete derivation. Given a slicing
criterion hf (pvn ), πi, the associated SC-configuration is Ci = hΓ, f (xn ), S, G, r, Pi iff
the associated SC-node is r = ref (Ci ).
Proof. Both the definition of SC-configuration and SC-node require two conditions.
We consider each of them separately:
(1) Trivially, Ci = hΓ, f (xn ), S, G, r, Pi is a relevant configuration. Then, by Lemma 20,
we have that Gi+1 = G[r 7→P f (xn )] and, thus, (r 7→P f (xn )) ∈ graph(Cm ).
(2) This case follows by Lemma 37, since it implies that val D (xi ) = ValD (xi ), with
(xi à ri ) ∈ graph(Cm ), for all i = 1, . . . , n.
✷
Finally, we state and prove Theorem 32. In the following, we say that a call,
ds([Ci , . . . , Cj ], π, V, D), is relevant if the configuration Ci is relevant.
Theorem 32. (completeness and minimality) Let P be a program and (C0 =⇒∗
Cm ), m > 0, a complete derivation, with G = graph(Cm ). Let hf (pvn ), πi be a slicing
criterion and W be the dynamic slice computed according to Definition 28. Then,
we have DS (G, r, π, { }) = W, where r is the SC-node associated with the slicing
criterion.
128
5.4. COMPUTING THE SLICE
129
Proof.
(Completeness) We prove a more general claim: we prove that for any,
possibly incomplete, computation for ds, the same positions are also gathered by
some, possible incomplete, computation for DS .
Formally, let us denote a (possibly incomplete) evaluation of the dynamic slice
using function ds as follows:
P0 ∪ ds 0 = P1 ∪ ds 1 = . . . = Pz ∪ ds z
where
• each new equality is obtained from the previous one by unfolding a single call
to ds one or more times until it reduces to a (possibly empty) union of relevant
calls to ds,
• Pk is the already computed set of positions, and
• ds k is the (possibly empty) union of relevant calls that have not been reduced
yet
for all k = 0, . . . , z.
Now, we prove that, for any z ≥ 0, we have a corresponding (possibly incomplete)
sequence of reductions of calls to DS :
P0′ ∪ DS 0 = P1′ ∪ DS 1 = . . . = Pz′ ∪ DS z
such that Pz′ ⊇ Pz and, for every call ds(Cz′ : CSz , πz , Vz , D) in ds z (if any), we have
that there exists a call DS (G, rz , πz′ , Vz′ ) in DS z with rz = ref (Cz′ ), (rz 7→P z ez ) ∈ G,
P z = pos(Cz′ ), ez = ctrl (Cz′ ), πz′ = πz , and Vz′ = Vz .
We prove the claim by induction on the number z of relevant calls to ds reduced
in the considered evaluation.
(Base case z = 0) Then, we have P0 = {hd (pos(Ci ))} and
ds 0 = ds([Ci+1 , . . . , Cj ], π, { }, D)
where Ci is the SC-configuration associated to the slicing criterion hf (pvn ), πi and
Ci =⇒∗ Cj is the complete subderivation for Ci in the considered computation. For
simplicity, we assume that Ci+1 is relevant (otherwise, we would proceed as in the
inductive case, see below). By Theorem 38, the SC-node associated to hf (pvn ), πi is
ref (Ci ). Moreover, since Ci is relevant, by Lemma 20, we have (r 7→P ctrl (Ci )) ∈ G,
q
with r = ref (Ci ), q = ref (Ci+1 ), and P = pos(Ci ). Hence, we construct the following
initial expression for DS :
P0′
DS 0
}|
{ z
}|
{
z
{hd (pos(Ci ))} ∪ DS (G, ref (Ci+1 ), π, { })
129
130
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
and the claim follows.
(Inductive case z > 0) Let us consider a possibly incomplete evaluation in which
z − 1 relevant calls to ds have been reduced. Then, by the inductive hypothesis, we
have the following (possibly incomplete) sequence of reductions of calls to DS :
′
P0′ ∪ DS 0 = P1′ ∪ DS 1 = . . . = Pz−1
∪ DS z−1
′
such that Pz−1
⊇ Pz−1 and, for every call
′
ds(Cz−1
: CSz−1 , πz−1 , Vz−1 , D)
in ds z−1 (if any), there exists a call
′
′
DS (G, rz−1 , πz−1
, Vz−1
)
′
′
in DS z−1 such that rz−1 = ref (Cz−1
), (rz−1 7→P z−1 ez−1 ) ∈ G, P z−1 = pos(Cz−1
),
′
′
′
ez−1 = ctrl (Cz−1 ), πz−1 = πz−1 , and Vz−1 = Vz−1 .
If there is no such a call in ds z−1 , the proof is done. Otherwise, let us consider a
′
′
: CSz−1 , πz−1 , Vz−1 , D). We denote by ds z−1 the union of
call of the form ds(Cz−1
calls to ds in ds z−1 but the considered one.
According to the definition of ds, we distinguish two cases, depending on whether
CSz−1 is empty or not. Let us consider first that CSz−1 is not empty. Then, we
consider the following cases:
′
• ctrl (Cz−1
) = (let x = e in e′ ). Then, according to the definition of ds, this call
reduces to
′
pos(Cz−1
) ∪ ds(CSz−1 , πz−1 , Vz , D)
where Vz = Vz−1 ∪ {x}.
p
p
1
Assume CSz−1 = Cz−1
: . . . : Cz−1
: CSz , where Cz−1
, p ≥ 1, is the first
relevant configuration in CSz−1 (observe that, by construction, there must be
p−1
1
at least one relevant configuration). Since Cz−1
are not relevant con, . . . , Cz−1
figurations, by definition of ds, we have that
p
ds(CSz−1 , πz−1 , Vz , D) = ds(Cz−1
: CSz , πz−1 , Vz , D)
by repeatedly applying the last case of ds. This is a consequence of the fact
that, if a configuration is not relevant, its control can only contain a variable or
a constructor call. Therefore, the next expression of the sequence is
ds z
Pz
}|
{
}|
{ z
z
′
p
′
Pz−1 ∪ pos(Cz−1
) ∪ ds(Cz−1
: CSz , πz−1 , Vz , D) ∪ ds z−1
130
5.4. COMPUTING THE SLICE
131
′
′
′
Now, we consider the call DS (G, rz−1 , πz−1
, Vz−1
),
) in DS z−1 rz−1 = ref (Cz−1
z−1
′
′
′
= pos(Cz−1 ), ez−1 = ctrl (Cz−1 ), πz−1 = πz−1 ,
(rz−1 7→P z−1 ez−1 ) ∈ G, P
′
′
and Vz−1
= Vz−1 . We denote by DS z−1 the union of calls to DS in DS z−1 but
the above one.
p
′
First, since Cz−1
=⇒∗ Cz−1
is a successor subderivation, by Proposition 34, we
have
′
(ra 7→Pa ctrl (Cz−1
)) ∈ G
rb
and
p
(rb 7→′ Pb ctrl (Cz−1
)) ∈ G
r
with
p
p
′
′
), rb = ref (Cz−1
), Pb = pos(Cz−1
)
), Pa = pos(Cz−1
ra = ref (Cz−1
and r′ a fresh reference. Hence, according to the definition of function DS , we
have that
p
′
′
′
DS (G, rz−1 , πz−1
, Vz−1
) = pos(Cz−1
) ∪ DS (G, ref (Cz−1
), πz−1 , Vz )
Therefore, the next expression of the sequence is now
DS z
P′
z
}|
{
z
}|
{ z
′
p
′
′
Pz−1 ∪ pos(Cz−1 ) ∪ DS (G, ref (Cz−1 ), πz−1 , Vz ) ∪ DS z−1
′
and the claim follows since Pz−1
⊇ Pz−1 .
′
• ctrl (Cz−1
) is a function call. This case is perfectly analogous to the previous
one with the only difference that Vz = Vz−1 .
′
• ctrl (Cz−1
) = case x of . . . and x 6∈ Vz−1 . Then, assume
p
1
CSz−1 = Cz−1
: . . . : Cz−1
: Cz′ : CSz
p
1
1
where Cz−1
=⇒∗ Cz−1
, p > 1, is the complete subderivation for Cz−1
in the
considered computation (i.e., the subderivation in which the variable x is evaluated) and that Cz′ is relevant (otherwise, we would proceed as in the first case).
By definition of ds, we have
′
ds(Cz−1
: CSz−1 , πz−1 , Vz−1 , D)
′
= Pos(Cz−1 ) ∪ ds(Cz′ : CSz , πz−1 , Vz−1 , D)
Therefore, the next expression of the sequence is
ds z
Pz
}|
{
z
}|
{ z
′
′
Pz−1 ∪ pos(Cz−1
) ∪ ds(Cz′ : CSz , πz−1 , Vz−1 , D) ∪ ds z−1
131
132
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
′
′
Now, we consider the call DS (G, rz−1 , πz−1
, Vz−1
) in DS z−1 such that rz−1 =
′
z−1
′
′
ref (Cz−1 ), (rz−1 7→P z−1 ez−1 ) ∈ G, P
= pos(Cz−1
), ez−1 = ctrl (Cz−1
),
′
′
′
πz−1 = πz−1 , and Vz−1 = Vz−1 . We denote by DS z−1 the union of calls to DS
in DS z−1 but the above one.
′
First, since Cz−1
=⇒∗ Cz′ is a successor subderivation, by Proposition 34, we
have
′
)) ∈ G
(ra 7→Pa ctrl (Cz−1
rb
and
(rb 7→′ Pb ctrl (Cz′ )) ∈ G
r
with
′
′
ra = ref (Cz−1
), Pa = pos(Cz−1
), rb = ref (Cz′ ), Pb = pos(Cz′ )
and r′ a fresh reference. Hence, according to the definition of function DS , we
have that
′
′
′
DS (G, rz−1 , πz−1
, Vz−1
) = pos(Cz−1
) ∪ DS (G, ref (Cz′ ), πz−1 , Vz−1 )
Therefore, the next expression of the sequence is now
DS z
P′
z
}|
{
z
}|
{ z
′
′
′
′
Pz−1 ∪ pos(Cz−1 ) ∪ DS (G, ref (Cz ), πz−1 , Vz−1 ) ∪ DS z−1
′
and the claim follows since Pz−1
⊇ Pz−1 .
′
• ctrl (Cz−1
) = case x of . . . and x ∈ Vz−1 . Assume now that we have CSz−1 =
p
p
1
1
Cz−1 : . . . : Cz−1
: Cz′ : CSz where Cz−1
=⇒∗ Cz−1
, p > 1, is the complete
1
subderivation for Cz−1 in the considered computation (i.e., the subderivation in
which the variable x is evaluated) and that Cz′ is relevant (otherwise, we would
proceed as in the first case). By definition of ds, we have
′
ds(Cz−1
: CSz−1 , πz−1 , Vz−1 , D)
p
′
1
= Pos(Cz−1
) ∪ ds([Cz−1
, . . . , Cz−1
], hnf , Vz−1 , D)
′
∪ ds(Cz : CSz , πz−1 , Vz−1 , D)
Moreover, for simplicity, we assume that, in the sequence of configurations
p
1
2
Cz−1
, . . . , Cz−1
, the configuration Cz−1
is the first relevant configuration. Hence,
we have
p
1
ds([Cz−1
, . . . , Cz−1
], hnf , Vz−1 , D)
p
2
= ds([Cz−1
, . . . , Cz−1
], hnf , Vz−1 , D)
Therefore, the next expression of the sequence is
Pz
}|
{
z
′
Pz−1 ∪ pos(Cz−1
)
p
2
∪ ds([Cz−1
, . . . , Cz−1
], hnf , Vz−1 , D)
′
′
∪ ds(Cz : CSz , πz−1 , Vz−1 , D) ∪ ds z−1
{z
}
|
ds z
132
5.4. COMPUTING THE SLICE
133
′
′
Now, we consider the call DS (G, rz−1 , πz−1
, Vz−1
) in DS z−1 such that rz−1 =
′
z−1
′
′
ref (Cz−1 ), (rz−1 7→P z−1 ez−1 ) ∈ G, P
= pos(Cz−1
), ez−1 = ctrl (Cz−1
),
′
′
′
πz−1 = πz−1 , and Vz−1 = Vz−1 . We denote by DS z−1 the union of calls to DS
in DS z−1 but the above one.
′
On the one hand, since Cz−1
=⇒∗ Cz′ is a successor subderivation, by Proposition 34, we have
′
)) ∈ G
(ra 7→Pa ctrl (Cz−1
rb
and
(rb 7→′ Pb ctrl (Cz′ )) ∈ G
r
with
′
′
ra = ref (Cz−1
), Pa = pos(Cz−1
), rb = ref (Cz′ ), Pb = pos(Cz′ )
1
and r′ a fresh reference. On the other hand, since ctrl (Cz−1
) = x, we have two
possibilities:
1
– If Cz−1
is a lookup configuration (i.e., this is the first time that the value
2
of x is demanded in the computation), since Cz−1
is relevant, by Proposition 35, there exists
(x à rc ) ∈ G
and
2
(rc 7→
P ctrl (Cz−1 )) ∈ G
′′ c
r
with
1
2
2
rc = ref (Cz−1
) = ref (Cz−1
), Pc = pos(Cz−1
), r′′ fresh
Hence, by definition of function DS , we have
′
′
DS (G, rz−1 , πz−1
, Vz−1
) =
′
pos(Cz−1
)
2
∪ DS (G, ref (Cz−1
), hnf , Vz−1 )
′
∪ DS (G, ref (Cz ), πz−1 , Vz−1 )
Therefore, the next expression of the sequence is now:
Pz′
z
}|
{
′
′
Pz−1
∪ pos(Cz−1
)
2
∪ DS (G, ref (Cz−1
), hnf , Vz−1 )
′
∪ DS (G, ref (Cz′ ), πz−1 , Vz−1 ) ∪ DS z−1
{z
}
|
DS z
′
and the claim follows since Pz−1
⊇ Pz−1 .
1
– Consider now that Cz−1
is not a lookup configuration, i.e., the value of x
has already been demanded in the computation. Hence, the heap contains
133
134
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
a binding that maps x to a value. Therefore, in this case, we have that
2
p = 2, ctrl (Cz−1
) is the (already computed) value of x, and
Pz
z
}|
{
′
Pz−1 ∪ pos(Cz−1
)
2
∪ ds([Cz−1
], hnf , Vz−1 , D)
′
′
∪ ds(Cz : CSz , πz−1 , Vz−1 , D) ∪ ds z−1
{z
}
|
ds z
is the next expression of the sequence. In the graph, however, we have
(x à rd ) ∈ G
2
such that rd is not labeled with ctrl (Cz−1
) but points to a path in the
2
graph that ends in a reference labeled with ctrl (Cz−1
). This is due to the
fact that all occurrences of x in the graph are shared (see rules varcons
and varexp in Figure 5.9) and, thus, x points to the full path representing
the original evaluation of x (associated to the first time it was demanded).
Hence, there must be a successor chain
n−1
1
1
2
(rd 7→
7→Pdn end ) ∈ G
P 1 ed ), (qd 7→
P 2 ed ), . . . , (qd
1 d
2 d
qd
qd
2
2
such that ctrl (Cz−1
) = end and pos(Cz−1
) = Pdn (i.e., the computed value
of x must be the same every time its value is demanded). Therefore, in
this case, we construct the following expression:
Pz′
z
}|
{
′
′
Pz−1
∪ pos(Cz−1
) ∪ Pd1 ∪ . . . ∪ Pdn−1
∪ DS (G, qdn−1 , hnf , Vz−1 )
′
∪ DS (G, ref (Cz′ ), πz−1 , Vz−1 ) ∪ DS z−1
|
{z
}
DS z
′
and the claim follows since Pz−1
⊇ Pz−1 , (qdn−1 7→Pdn end ) ∈ G, Pdn =
2
2
pos(Cz−1
), and end = ctrl (Cz−1
).
′
• ctrl (Cz−1
) is a variable or a constructor call. This case is not possible since,
′
then, Cz−1 would not be relevant.
Let us now consider that CSz−1 is an empty list. In this case, we have:
′
ds([Cz−1
], πz−1 , Vz−1 , D)
′
′
, πz−1 , Vz−1 , D)
= pos(Cz−1
) ∪ ds aux (Cz−1
134
5.4. COMPUTING THE SLICE
135
By definition, this expression is reduced as follows:
′
ds([Cz−1
], πz−1 , Vz−1 , D)
1
′
) ∪ ds(C x : CSx1 , πx1 , Vz−1 , D)
= pos(Cz−1
∪ ...
v
∪ ds(C x : CSxv , πxv , Vz−1 , D)
′
) lookup configuration that fulfills the following
so that Cxi is the first (from Cz−1
conditions: its control stores a variable xi in Vz−1 and there exists a pair (xi , πxi )
i
′
in ctrl (Cz−1
) ⊓ πz−1 for i = 1, . . . , v. Here, we consider that C x denotes the first
i
relevant configuration from Cxi and C x : CSxv contains the list of configurations in the
i
complete subderivation for C x in D, for all i = 1, . . . , v.
By the inductive hypothesis, there exists a call to function DS which is of the
′
′
′
form DS (G, rz−1 , πz−1
, Vz−1
) such that rz−1 = ref (Cz−1
), (rz−1 7→P z−1 ez−1 ) ∈ G,
′
′
′
′
P z−1 = pos(Cz−1
), ez−1 = ctrl (Cz−1
), πz−1
= πz−1 , and Vz−1
= Vz−1 . By definition
of function DS , we have:
′
′
DS (G, rz−1 , πz−1
, Vz−1
) =
′
pos(Cz−1
)
′
), rz−1 , πz−1 , Vz−1 )
∪ DS AUX (G, ctrl (Cz−1
Now, by Proposition 35, we have
′
DS AUX (G, ctrl (Cz−1
), rz−1 , πz−1 , Vz−1 )
1
⊆ DS (G, ref (C x ), πx1 , Vz−1 )
∪ ...
v
∪ DS (G, ref (C x ), πxv , Vz−1 )
and the claim follows. Observe that DS AUX may in principle compute additional
positions since function ds aux only considers lookup configurations and this restriction has no counterpart in function DS AUX .
(Minimality) Analogously to the previous case, we prove a more general claim. First,
we say that a call, DS (G, r, π, V ), is collecting if it is the first call to DS with reference r in the considered computation.11 Let us consider a (possibly incomplete)
computation for DS as follows:
P0′ ∪ DS 0 = P1′ ∪ DS 1 = . . . = Pz′ ∪ DS z
where
• each new equality is obtained from the previous one by reducing a single call to
DS one or more times until it reduces to a (possibly empty) union of collecting
calls,
11 The notion of collecting call in the evaluation of DS is essentially the counterpart of the notion
of relevant call in the evaluation of ds.
135
136
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
• Pk′ is the already computed set of positions, and
• DS k is the (possibly empty) union of collecting calls that have not been evaluated yet
for all k = 0, . . . , z.
Now, we prove that, for any z ≥ 0, there is a (possibly incomplete) sequence of
calls to ds:
P0 ∪ ds 0 = P1 ∪ ds 1 = . . . = Pz ∪ ds z
such that Pz ⊇ Pz′ and, for every call DS (G, rz , πz′ , Vz′ ) in DS z with (rz 7→P z ez ) ∈
G, there exists a call ds([Cz′ ], πz , Vz , D) in ds z with ref (Cz′ ) = rz , pos(Cz′ ) = P z ,
ctrl (Cz′ ) = ez , πz = πz′ , and Vz = Vz′ .
Now, the proof would proceed similarly to the proof of completeness since the
needed results (Propositions 34 and 35, Lemma 20, and Theorem 38) hold in both
directions. Actually, the positions gathered by DS and ds are always the same but in
one case: when considering that the reference rz−1 is labeled with a case expression
case x of . . . and x ∈ Vz−1 . In the following, we only show the proof of this case.
′
′
In this case, we consider a collecting call DS (G, rz−1 , πz−1
, Vz−1
) with (rz−1 7→
x of . . .) ∈ G and x ∈
in DS z−1 but the above one.
P z−1 case
′
Vz−1
.
We denote by
′
DS z−1
rz
the union of calls to DS
′
By the inductive hypothesis, we have that Pz−1 ⊇ Pz−1
and there exists a call
′
′
′
ds(Cz−1 : CSz−1 , πz−1 , Cz−1 , D) in ds z−1 with ref (Cz−1 ) = rz−1 , ctrl (Cz−1
) =
′
z−1
′
′
case x of . . ., pos(Cz−1 ) = P
, πz−1 = πz−1 , and Vz−1 = Vz−1 . By definition
of function DS , we have
′
′
DS (G, rz−1 , πz−1
, Vz−1
) = P z−1
′
∪ DS (G, rx , hnf , Vz−1
)
′
′
∪ DS (G, rz , πz−1 , Vz−1 )
where (x à rx ) ∈ G. Therefore, the next expression in the reduction sequence is
P′
DS z
z
}|
{
}|
{ z
z
′
′
′
′
′
) ∪ DS (G, rz , πz−1
, Vz−1
) ∪ DS z−1
Pz−1 ∪ P z−1 ∪ DS (G, rx , hnf , Vz−1
Since rz is the successor of rz−1 , we have by Proposition 34 that there exists a successor
′
derivation of the form Cz−1
=⇒ Cx =⇒∗ Cx′ =⇒ Cz′ in D where Cx =⇒∗ Cx′ is the
complete subderivation for Cx .
Let (rx 7→Px ex ) ∈ G. By Proposition 35, there exists a lookup configuration C;
assume that C ′ is the first relevant configuration from C, where ctrl (C ′ ) = ex , and
pos(C ′ ) = Px . Now, we distinguish two cases:
136
5.4. COMPUTING THE SLICE
137
′
• Assume first that Cz−1
=⇒ C (i.e., the lookup configuration C is Cx ). In this
case, by definition of function ds, we construct the expression
Pz
}|
{
z
′
Pz−1 ∪ pos(Cz−1
) ∪ ds([C ′ , . . . , Cx′ ], hnf , Vz−1 , D)
′
∪ ds(Cz′ : CSz , πz−1 , Vz−1 , D) ∪ ds z−1
|
{z
}
ds z
and the claim follows by the induction hypothesis analogously to the corresponding case in the proof of completeness.
• Assume now that the next configuration, Cx , is not a lookup configuration in
D. In this case, we have Cx =⇒ Cx′ , Cx′ stores in the control the value of x,
′
and the call DS (G, rx , hnf , Vz−1
) is not a collecting call. Hence, there must be
a successor chain in the graph:
1
1
2
n−1
(rx 7→
7→Pxn enx ) ∈ G
P 1 ex ), (qx 7→
P 2 ex ), . . . , (qx
1 x
2 x
qx
qx
such that ctrl (Cx′ ) = enx and pos(Cx′ ) = Pxn (i.e., the computed value of x is the
same in both cases). Then, we have
′
DS (G, rx , hnf , Vz−1
)
′
= Px1 ∪ DS (G, qx1 , hnf , Vz−1
)
= ...
= Px1 ∪ . . . ∪ Pxn
such that Px1 ∪ . . . ∪ Pxn−1 ⊆ Pz−1 because the associated calls are not collecting.
Therefore, by definition of function ds, we construct the expression
Pz
}|
{
z
′
Pz−1 ∪ pos(Cz−1
) ∪ Px1 ∪ . . . ∪ Pxn−1
∪ ds([Cx′ ], hnf , Vz−1 , D)
′
∪ ds(Cz′ : CSz , πz−1 , Vz−1 , D) ∪ ds z−1
|
{z
}
ds z
and the claim follows by the inductive hypothesis.
✷
Example 16 Consider the (normalized version) of the program in Example 11 and
the slicing criterion hminmax (Z : : ), Pair ⊥ ⊤i. As we have already seen in
Section 5.2, the extended redex trail of Figure 5.3 is computed. Figure 5.17 shows
the SC-node together with the nodes which are reachable from the SC-node according
to the slicing pattern (Pair ⊥ ⊤), including the associated lists of program positions.
The computed slice is the one already shown in Figure 5.4.
137
138
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
minmax
case
case
Pair
[(minmax,/\)]
[(minmax,2.1)]
[(minmax,2.1.2.2.2.2.2.2.2)]
[(main,2.2.2.2.2.2.1),
(printMax,1)]
max
Z
ite
case
[(max,2)]
[(ite,/\)]
[(minmax,2.1.2.2.2.2.2.2.1),
[(main,1),
(printNat,1)]
(ite,2.2)]
leq
case
False
[(leq,/\)]
[(leq,2.1)]
[(max,1),
(ite,1)]
Figure 5.17: Reachable nodes w.r.t. hminmax (Z : : ), Pair
5.4.2
Z, 1, Pair ⊥ ⊤i
Logical Features
In this section, we sketch the extension of the previous developments in this section
in order to cope with the logical features of the considered functional logic language.
The first notion that should be extended is that of slicing criterion. Definition 22
is not appropriate in a lazy functional logic language because of non-determinism,
since the same function call may be reduced to different values within the same
computation. Consider, e.g., the following function definition:
coin = Z or (S Z)
Here, coin can be non-deterministically reduced to Z and (S Z). In order to identify a
concrete occurrence of the given call, we add another element to the slicing criterion:
the computed value.12 Therefore, a slicing criterion is defined in the functional logic
setting as a tuple hf (vn ), v, πi, where f (vn ) is a function call whose arguments appear
as much evaluated as needed in the complete computation, v is the computed value,
and π is a pattern that determines the interesting part of the computed value, e.g.,
hcoin, Z, ⊤i.
Regarding the notions of SC-configuration and SC-node, only functions val D and
ValD should be slightly extended in order to also consider a logical variable as a value.
v
For this purpose, auxiliary function valD
is defined as follows:
v
valD
(y)
c(val (x ))
D n
v
valD
(x) =
LogVar
if Γ[x] = y and x 6= y
if Γ[x] = c(xn )
if Γ[x] = x
otherwise
12 Observe
that we can still have an expression like coin + coin that evaluates to the same value
(S Z) in different ways. Here, we implicitly consider the first of such computations. This is somehow
a limitation, but it is not realistic to expect that users are able to distinguish among different
computations which share both the same initial function call and the final value.
138
5.5.
PROGRAM SPECIALIZATION BASED ON DYNAMIC SLICING
139
Correspondingly, function valvG is defined as follows:
valvG (q)
c(ValD (xn ))
valvG (r) =
LogVar
if (r 7→) ∈ G
q
if (r 7→ c(xn )) ∈ G
if (r 7→ LogVar ) ∈ G
otherwise
Finally, functions ds and DS require no significant changes, i.e., the only difference
is that the underlying calculus also includes the rules of Figure 5.11.
Analogously to the development in the previous section, the extension of our results and, particularly, of Theorem 32 to cope with the logical features of the language
does not involve significant difficulties, because the results from Braßel et al. [2004b]
required for proving this theorem have been already proved for a language which
includes these logical features.
5.5
Program Specialization Based on Dynamic
Slicing
This section extends the slicing technique introduced in the former sections. Our aim
is similar to that of Reps and Turnidge [1996], who designed a program specialization
method for strict functional programs based on static slicing. However, in contrast to
Reps and Turnidge [1996], we consider lazy functional logic programs and our technique is based on dynamic slicing. We consider dynamic slicing since it is simpler
and more accurate than static slicing. Informally speaking, specialization proceeds as
follows: first, the user identifies a representative set of slicing criteria; then, dynamic
slicing is used to compute a program slice for each slicing criterion; finally, the specialized program is built by conveniently unioning the computed slices (e.g., with the
SDS procedure by Hall [1995]). This is a lightweight approach since dynamic slicing
is simpler than static slicing (as in [Reps and Turnidge, 1996]). Unfortunately, in our
previous formalization, we defined a program slice as a set of program positions rather
than as an executable program (See Definition 21), since this is sufficient for debugging. Therefore, we also introduce an appropriate technique to extract executable
slices from a set of program positions.
5.5.1
Specialization Based on Slicing
Consider again function minmax shown (now corrected) in Figure 5.18.
Our aim now is the specialization of this program in order to extract those statements which are needed to compute the minimum of a list. For this, we first consider
139
140
CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
minmax xs = case xs of
{ y:ys → case ys of
{ []
→ Pair y y;
z:zs → let m = minmax (z:zs)
in Pair (min y (fst m))
(max y (snd m)) } }
min x y = ite (leq x y) x y
max x y = ite (leq x y) y x
fst t = case t of { Pair x y → x }
snd t = case t of { Pair x y → y }
ite x y z = case x of { True → y; False → z }
leq x y = case x of
{ Z
→ True;
(S n) → case y of { Z → False; (S m) → leq n m } }
Figure 5.18: Example program minmax
the selection of appropriate slicing criteria and, then, the extraction of the corresponding slice (the specialized program).
Figure 5.19 shows the slice computed for minmax w.r.t. the slicing criterion
hminmax [Z, S Z], Pair ⊤ ⊥i. This notion of slice is sufficient for debugging, where
we are interested in highlighting those expressions which are related to the slicing
criterion. In order to specialize programs, however, we should produce executable
program slices. For this purpose, we need to introduce an algorithm that extracts an
executable program slice from the original program and the computed set of program
positions. For instance, an executable program slice associated to minmax is shown
in Figure 5.20, where the distinguished symbol “?” is used to point out that some
subexpression is missing due to the slicing process.
Informally, the specialized program is obtained as follows: those functions that
were not used in any computation are completely deleted; for the remaining expressions, we follow a simple rule: if their location belongs to the computed set, it appears
in the slice; otherwise, it is replaced by the special symbol “?”.
Observe that some further simplifications are obvious, as for instance deleting the
unused branches of case expressions, or replacing the expression
let ? in Pair (min y ?) ?
140
5.5.
PROGRAM SPECIALIZATION BASED ON DYNAMIC SLICING
141
minmax xs = case xs of
{ y:ys → case ys of
{ []
→ Pair y y;
z:zs → let m = minmax (z:zs)
in Pair (min y (fst m))
(max y (snd m)) } }
min x y = ite (leq x y) x y
max x y = ite (leq x y) y x
fst t = case t of { Pair x y → x }
snd t = case t of { Pair x y → y }
ite x y z = case x of { True → y; False → z }
leq x y = case x of
{ Z
→ True;
(S n) → case y of { Z → False; (S m) → leq n m } }
Figure 5.19: Slice of program minmax w.r.t. hminmax [Z, S Z], Pair ⊤ ⊥i
by Pair (min y ?) ?, since the first argument of the let expression is not used.
Therefore, after the specialization, we use a simplification process in order to further
simplify programs. For instance, the simplified version of the program in Figure 5.20
is shown in Figure 5.21.
Clearly, producing a specialized program for a concrete computation is not always
appropriate. For instance, in the slice above, function minmax does not work with lists
one element. To overcome this problem, [Reps and Turnidge, 1996] considers static
slicing where no input data are provided and, thus, the extracted slices preserve the
semantics of the original program for any input data. Our lightweight approach is
based on dynamic slicing and, thus, a representative set of slicing criteria—covering
the interesting computations—should be determined in order to obtain program slices
which are general enough.
Now, we formalize our method to extract an executable program slice from the
computed program positions.
Definition 39 (executable slice) Let P be a program and W an associated dynamic slice. The corresponding executable slice, PW , is obtained as follows:
Λ
PW = {f (xn ) = [[e]]Q | (f, Λ) ∈ W and Q = {p | (f, p) ∈ W} }
141
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
minmax xs = case xs of
{ y:ys → case ys of
{ []
→ ?;
z:zs → let ?
in Pair (min y ?) ? } }
min x y = ite (leq x ?) x ?
ite x y z = case x of { True → y; False → ? }
leq x y = case x of
{ Z
→ True;
(S n) → ? } }
Figure 5.20: Executable slice of program minmax w.r.t. hminmax [Z, S Z], Pair ⊤ ⊥i
minmax xs = case xs of
{ y:ys → case ys of
z:zs → Pair (min y ?) ? } }
min x y = ite (leq x ?) x ?
ite x y z = case x of { True → y }
leq x y = case x of { Z → True } }
Figure 5.21:
Simplified
hminmax [Z, S Z], Pair ⊤ ⊥i
executable
142
slice
of
program
minmax
w.r.t.
5.6. IMPLEMENTATION ISSUES
143
p
where [[e]]Q = ? if p 6∈ Q and, otherwise (i.e., p ∈ Q), it is defined inductively as
follows:
p
[[x]]Q = x
p
[[ϕ(x1 , . . . , xn )]]Q = ϕ(x1 , . . . , xn )
p
p.1
p.2
[[let x = e′ in e]]Q = let x = [[e′ ]]Q in [[e]]Q
p
p.1
p.2
[[e1 or e2 ]]Q = [[e1 ]]Q or [[e2 ]]Q
p
p.2.k
[[(f )case x of {pk → ek }]]Q = (f )case x of {pk → [[ek ]]Q
}
where ϕ ∈ (C ∪ F) is either a constructor or a defined function symbol.
The slice computed in this way would be trivially executable by considering “?” as a
fresh 0-ary constructor symbol (since it is not used in the considered computation).
However, from an implementation point of view, this is not a good decision since we
should also extend all program types in order to include “?”. Fortunately, in our
functional logic setting, there is a simpler solution: we can replace “?” by a fresh
constructor failed (which represents ⊥).
Now, the specialization process should be clear. First, we compute dynamic
slices—sets of program positions—for the selected slicing criteria. Then, an executable slice is built for the combination of the computed dynamic slices. Moreover,
there are some post-processing simplifications that can be added in order to reduce
the program size and improve its readability:
let x = ? in e =⇒ ρ(e)
if ρ = {x 7→ ?}
e or ? =⇒ e
(f )case x of {pk → ek } =⇒ (f )case x of {p′m → e′m }
if {p′m → e′m } = {pi → ei | ei 6= ?, i = 1, . . . , k}
More powerful post-processing simplifications could be added but they require a form
of amorphous slicing (see Chapter 1).
5.6
Implementation Issues
In order to check the usefulness and practicality of the ideas presented in this chapter,
we have developed a prototype implementation of the debugging tool for the multiparadigm language Curry [Hanus, 2006a] that has been sketched in Section 5.2. It
combines both tracing and slicing by extending a previous tracer for Curry programs
[Braßel et al., 2004b]. Three main extensions were necessary:
• First, we modified the instrumented interpreter of Braßel et al. [2004b] that
generates the redex trail in order to also include program positions. Program
positions are added in such a way that they do not interfere with the tracing
143
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
(|e|)Λ = Λ
(|c(en )|)i.w = i.(|ei |)w
(|f (en )|)i.w = i.(|ei |)w
(|let x = e in e′ |)1.w = px .(|e|)w
(|let x = e in e′ |)2.w = (|σ(e′ )|)w
(|(f )case e of {pn → en }|)1.w = 1.(|e|)w
(|(f )case e of {pn → en }|)2.i.w = 2.i.(|ei |)w
for all expression e
if i ∈ {1, . . . , n}
if i ∈ {1, . . . , n}
where px is the position of x in e′
where σ = {x 7→ e}
if i ∈ {1, . . . , n}
Figure 5.22: Conversion of positions from normalized flat programs to flat programs
[|e|]nΛ = (n, Λ)
[|c(ek )|]ni.w = (m, i.v)
[|f (ek )|]ni.w = (m, i.v)
[|(f )case e of {pk → ek }|]n2.i.w = (n + m, v)
for all expression e
if i ∈ {1, . . . , k} and [|ei |]nw = (m, v)
if i ∈ {1, . . . , k} and [|ei |]nw = (m, v)
if i ∈ {1, . . . , k} and [|ei |]lw = (m, v),
where l = brs(pi−1 → ei−1 )
Figure 5.23: Conversion of positions from flat programs to source programs
process.
• Then, we defined algorithms to convert program positions for normalized flat
programs—the programs considered in the instrumented semantics—to source
Curry programs. This is necessary to be able to show program slices at an adequate level of abstraction. The kernel of the implemented conversion algorithms
are shown in Figures 5.22 and 5.23. For simplicity, here we consider that let
expressions are not allowed in source Curry programs, i.e., they are only used
to take care of sharing.
Given an expression e in a normalized flat program and its position w, we have
that (|e|)w returns the position of e in the corresponding flat (non-normalized)
program. Basically, transforming a non-normalized flat program into a flat
program consists in applying inlining to every let expression. Consequently,
function (| |) only changes those positions inside a let expression in order to
reflect these inlining steps. For instance, the position 2 addresses the call f x
in the right-hand side of the following normalized rule:
main = let x = Z in f x
The associated position in the non-normalized rule
main = f Z
is (|let x = Z in f x|)2 = (|f Z|)Λ = Λ.
144
5.6. IMPLEMENTATION ISSUES
145
Then, function [| |] takes an expression e in a flat program and its position w,
so that [|e|]1w returns the position of e in the corresponding source program.
Transforming a flat program into a source Curry program mainly consists in replacing case expressions by pattern matching in the left-hand sides of different
rules. Therefore, function [| |] only changes positions inside a case expression
so that prefixes which are used to identify a particular case branch are deleted
and the rule number within the function definition is determined instead. The
superscript in function [| |] is used to calculate the rule number within a definition (so it is initialized to 1). Auxiliary function brs is used to count the
number of case branches within a list of expressions. Here, we assume that
source Curry programs have no case expressions and that case expressions have
always a variable argument.13
For instance, consider the following flat rule:
g x = case x of {Z → Z; S y → f y}
The position 2.2 addresses the call to f y in the right-hand side of the second
branch of the case expression. Given the associated source Curry rule:
gZ
=
g (S y) =
Z
fy
the position associated with f y is given by [|case x of {Z → Z; S y → f y}|]12.2 =
[|case x of {Z → Z; S y → f y}|]12.2 = (1 + 1, Λ) = (2, Λ) (i.e., the right-hand
side of the second rule), since [|f y|]1Λ = (1, Λ).
We note, however, that there exist some features of Curry that are not covered
in our current prototype, like higher-order functions and local definitions.
• Finally, we have implemented a viewer that takes a slicing criterion and an
extended redex trail and shows the associated program slice. This viewer first
determines the SC-node (according to Definition 30) and then implements function ds to collect the set of program positions in the slice. Now, program
positions for the considered normalized flat program are first converted to their
associated positions in the source Curry program which is then shown to the
user by highlighting the expressions whose position belong to the slice.
Let us note that the complexity of the slicing algorithm is not a main concern in
this research. Our slicing technique is built on top of an existing tracer. Therefore,
if an efficient implementation of the tracer is available, then also slicing would run
efficiently because only a slight extension is necessary (adding program positions to
13 This is reasonable because flat programs obtained by translating source Curry programs without
case expressions always fulfill it.
145
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
Figure 5.24: Opening a file for debugging
redex trails). Only the viewer, which implements the computation of the SC-node
and the set of program positions which are reachable from this node (clearly a linear
function), requires some additional cost.
The user interface of our slicing tool has been completely developed using the GUI
library for Curry [Hanus, 2000]. In particular, we have developed a new library which
implements several widgets for Tcl/Tk (dialog boxes, popup menus, etc).
5.6.1
Architecture
In this section, we describe the structure of our slicing tool. It can be used both
for debugging—by automatically extracting the program fragment which contains an
error—and for program specialization—by generating executable slices w.r.t. a given
slicing criterion.
Tracer
It is introduced in Braßel et al. [2004b]. The tracer executes a program using an
instrumented interpreter. As a side effect of the execution, the redex trail of the
computation is stored in a file. The tracer is implemented in Haskell and accepts
first-order lazy functional logic programs that can be traced either backwards or
146
5.6. IMPLEMENTATION ISSUES
147
forwards. In our slicer, we only slightly extended the original tracer in order to also
store in the redex trail the location (the so called program position), in the source
program, of every reduced expression.
Viewer
Once the computation terminates (or it is aborted by the user in case of a looping
computation), the viewer reads the file with the redex trail and allows the user to
navigate through the entire computation. The viewer, also introduced in Braßel et al.
[2004b], is implemented in Curry, a conservative extension of Haskell with features
from logic programming including logical variables and non-determinism. The viewer
is useful in our approach to help the user to identify the slicing criterion.
Slicer
Given a redex trail and a slicing criterion, the slicer outputs a set of program positions
that uniquely identify the associated program slice. The slicing tool is implemented
in Curry too, and includes an editor that shows the original program and, when a
slice is computed, it also highlights the expressions that belong to the computed slice.
Specializer
Similarly to the slicer, the specializer also computes a set of program positions—a
slice—w.r.t. a slicing criterion. However, rather than using this information to highlight a fragment of the original program, it is used to extract an executable slice
(possibly simplified) that can be seen as a specialization of the original program for
the given slicing criterion.
5.6.2
The Slicer in Practice
The debugging tool starts by allowing the user to select either a source file (.curry)
or its flat representation (.fcy), see Figure 5.24. After a file has been loaded in,
its flat representation is shown—with some pretty printing—in the main screen, see
Figure 5.25.
Then, the user can start tracing the program by using the tool implemented by
Braßel et al. [2004b] (as in Section 5.2). For this purpose, the viewer of Braßel et al.
[2004b] is loaded in and the extended redex trail generated by the meta-interpreter
is passed as an argument to it. We have implemented a new viewer that allows the
user to see the computed slice.
The slicing process can be initiated by providing a slicing criterion associated to
an entry in a given trace. In particular, we have implemented the functions that,
given a slicing criterion, compute the associated SC-node as well as the reachable
147
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
Figure 5.25: Viewing the source program
nodes from the SC-node (function ds), according to the slicing pattern, and then
return their program positions (i.e., the computed slice). The main extensions to
this library allow to change the foreground colors of widgets, and provide several new
dialog boxes, among others.
Once a point in the trace of the program is selected for slicing, our tool takes on
and performs the slicing of the program by first collecting all the reachable nodes in
the graph from the selected slicing node (SC-node).
Finally, the source program is shown in the main screen, coloring in gray those
parts of the program that do not belong to the slice, as shown in Figure 5.26.
The debugging tool has several other useful features. For instance, it generates
both a latex (.tex) and a postscript (.ps) version of the computed slice, which is useful
for printing and discussing errors. Also, it allows the user to set some configuration
parameters, such as the preferred postscript viewer, the colors for the displayed slice,
etc.
An Example of Debugging Session
In order to show the usefulness of the slicer, we now present a debugging session that
combines tracing and slicing.
We consider the program shown in Figure 5.27 (for the time being, the reader
148
5.6. IMPLEMENTATION ISSUES
149
Figure 5.26: Viewing the slice of the source program
can safely ignore the distinction between gray and black text). In this program, the
function main returns the maximum number of hits of a given web page in a span
of time. Function main simply calls minMaxHits which traverses a list containing
the daily hits of a given page and returns a data structure with the minimum and
maximum of such a list.
The execution of the program above should return 65, since this is the maximum
number of hits in the given span of time. However, the code is faulty and prints 0
instead. We can trace this computation in order to find the source of the error. The
tracer initially shows the following top-level trace:
0 = main
0 = printMax
(Hits
0 = prettyPrint
0
0)
0 = if then else True 0 0
0 = 0
Each row in the trace has the form val = exp, where exp is an expression and val is
the computed value for this expression.
By inspecting this trace, it should be clear that the argument of printMax is
erroneous, since it contains 0 as the maximum number of hits. Note that the minimum
(represented by “_” in the trace) has not been computed due to the laziness of the
149
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
data T = Hits Int Int
main = printMax (minMaxHits webSiteHits)
webSiteHits = [0, 21, 23, 45, 16, 65, 17]
printMin t = case t of (Hits x ) -> show x
printMax t = case t of (Hits y) -> show y
fst t = case t of (Hits x ) -> x
snd t = case t of (Hits y) -> y
minMaxHits xs = case xs of
(y:ys) -> case ys of
[]
-> (Hits y y);
(z:zs) -> let m = minMaxHits (z:zs)
in (Hits (min y (fst m))
(max y (snd m)))
min x y = if (leq x y) then x else y
max x y = if (leq x y) then y else x
leq x y = if x==0 then False
else if y==0 then False else leq (x-1) (y-1)
Figure 5.27: Example program minMaxHits
considered language. Now, if the user selects the argument of printMax, the following
subtrace is shown:
0
(Hits
(Hits
= printMax
(Hits
0) = minMaxHits (0: )
0) = Hits
0
0)
From these subtraces, the user can easily conclude that the evaluation of function
minMaxHits (rather than its definition) contains a bug since it returns 0 as the maximum of a list without evaluating the rest of the list.
At this point, the tracer cannot provide any further information about the location
of the bug. This is where slicing comes into play: the programmer can use the slicer in
order to isolate the slice which is responsible of the wrong result; in general, it would
be much easier to locate the bug in the slice than in the complete source program. For
instance, in this example, the slice would contain the black text in Figure 5.27. Indeed,
this slice contains a bug: the first occurrence of False in function leq (less than or
equal to) should be True. Note that, even though the evaluation of minMaxHits was
150
5.7. RELATED WORK
benchmark
time
minmax
horseman
lcc
colormap
familyc on
familyn d
Average
11 ms.
19 ms.
33 ms.
3 ms.
4 ms.
2 ms.
12 ms.
151
Table 5.1: Benchmark results
orig size
slice size
1.035
625
784
587
1453
731
bytes
bytes
bytes
bytes
bytes
bytes
724
246
613
219
262
289
bytes
bytes
bytes
bytes
bytes
bytes
reduction (%)
69.95
39.36
78.19
37.31
18.03
29.53
43.73
erroneous, its definition was correct.
5.6.3
Benchmarking the Slicer
In order to measure the specialization capabilities of our tool, we conducted some
experiments over a subset of the examples listed in
http://www.informatik.uni-kiel.de/~curry/examples.
Some of the benchmarks are purely functional programs (horseman, family nd), some
of them are purely logic (colormap, family con), and the rest are functional logic
programs.
Results are summarized in Table 5.1. For each benchmark, we show the time
spent to slice it, the sizes of both the benchmark and its slice, and the percentage of
source code reduction after slicing. As shown in the table, an average code reduction
of more than 40% is reached.
5.7
Related Work
Despite the fact that the usefulness of slicing techniques has been known for a long
time in the imperative paradigm, there are very few approaches to program slicing
in the context of declarative languages. In the following, we review the closest to
our work. Reps and Turnidge [1996] presented a backward slicing algorithm for firstorder functional programs. This work is mainly oriented to perform specialization
by computing slices for part of a function’s output, which is specified by means of a
projection. In contrast to our approach, they consider a strict language and static
slicing, which is less useful for debugging (since it is generally less precise). Moreover,
only a fixed slicing criterion is considered: the output of the whole program. [Biswas,
1997a,b] considered a higher-order strict functional language and develops a technique for dynamic backward slicing based on labeling program expressions and, then,
151
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CHAPTER 5. DYNAMIC SLICING BASED ON REDEX TRAILS
constructing an execution trace. Again, only a fixed slicing criterion is considered
(although the extension to more flexible slicing criteria is discussed).
Liu and Stoller [2003] defined an algorithm for static backward slicing in a firstorder strict functional language, where their main interest is the removal of dead code.
Although the underlying semantics is different (lazy vs. strict), our slicing patterns
are inspired by their liveness patterns.
Hallgren [2003] presented (as a result of the Programatica project) the development of a slicing tool for Haskell. Although there is little information available, it
seems to rely on the construction of a graph of functional dependences (i.e., only function names and their dependences are considered). Therefore, in principle, it should
produce bigger—less precise—slices than our technique.
Rodrigues and Barbosa [2005] introduced a novel approach to slicing functional
programs which is based on the use of projection (for backward slicing) and hiding
(for forward slicing) functions as slicing criteria. When these functions are composed
with the original program, they lead to the identification of the intended slice. In
contrast to our approach, no data structure storing program dependences is built.
The closest approach is the forward slicing technique for lazy functional logic
programs introduced by Vidal [2003]. In this approach, a slight extension of a partial
evaluation procedure for functional logic programs is used to compute program slices.
Since partial evaluation naturally supports partial data structures, the distinction
between static and dynamic slicing is not necessary (the same algorithm can be applied
in both cases). This generality is a clear advantage of [Vidal, 2003], but it often implies
a loss of precision.
Our slicing technique is based on an extension of the redex trail introduced by
Braßel et al. [2004b]. This work introduces an instrumented operational semantics
for lazy functional logic programs that generates a redex trail of a computation. This
redex trail shares many similarities with the ART model of Wallace et al. [2001] but
there is an important difference:
• The ART model of Wallace et al. [2001] is defined in terms of a program transformation. In contrast, Braßel et al. [2004b] present a direct, semantics-based
definition of redex trails. This improves the understanding of the tracing process
and allows one to prove properties like “every reduction occurring in a computation can be observed in the trail”. Chitil [2001] presents a first approach to
such a direct definition by introducing an augmented small-step operational semantics for a core of Haskell that generate ARTs. However, this work does not
consider correctness issues.
Chitil [2005] has proposed a source-based trace exploration for the higher-order lazy
functional language Haskell which combines ideas from algorithmic debugging, traditional stepping debuggers, and dynamic slicing. Although this proposal shares many
152
5.7. RELATED WORK
153
similarities with our approach, no correctness results are provided (since no semanticsbased definition of redex trails is available).
153
Part IV
Algorithmic Debugging
155
Chapter 6
Combining Program Slicing
and Algorithmic Debugging
As discussed in Chapter 1, program slicing and algorithmic debugging are two of the
most relevant debugging techniques for declarative languages. In this chapter, we
show for functional languages how the combination of both techniques produces a
more powerful debugging scheme that reduces the number of questions that programmers must answer to locate a bug.
Part of the material of this chapter has been presented in the 8th ACM-SIGPLAN
International Symposium on Principles and Practice of Declarative Programming
(PPDP’06) celebrated in Venice (Italy) in 2006 [Silva and Chitil, 2006].
6.1
Introduction
In this chapter we combine algorithmic debugging [Shapiro, 1982] with program slicing [Weiser, 1984]. By combining algorithmic debugging and program slicing the
debugging process becomes more interactive, allowing the programmer to provide
more information to the debugger. This information reduces the number of questions
and guides the process, thus making the sequence of questions semantically related.
The main contributions of this chapter can be summarised as follows:
1. We combine both program slicing and algorithmic debugging in a single, more
powerful framework in the context of functional programming.
2. We adapt the well-known program slicing concepts of slice, slicing criterion,
complete slice, and so on to the augmented redex trail, a trace structure that
can be the basis for both algorithmic debugging and program slicing, and we
define and prove properties for this formalism.
157
158
CHAPTER 6. COMBINING PROGRAM SLICING AND...
3. We introduce an algorithm for slicing execution trees which is proven complete,
in the sense that every slice produced contains all the nodes of the tree that are
needed to locate a bug.
The rest of the chapter is organised as follows. In the next section we outline how
algorithmic debugging can be improved by letting the programmer provide more information. In Section 6.4 we define the augmented redex trail and in Section 6.5 we
define which kind of slices of an augmented redex trail we are interested in. In Section 6.6 we combine these slicing concepts with algorithmic debugging and introduce
an algorithm to compute the slices we need. Then, we give an example of a debugging session that uses our technique in Section 6.7. In Section 6.8 we discuss related
debugging techniques.
6.2
The Idea: Improved Algorithmic Debugging
As shown in Example 1, a conventional algorithmic debugger will ask the programmer
whether an equation
f oo a1 . . . an = e
is correct or wrong; the programmer will answer “yes”, “no” or “maybe” depending
on whether he thinks that it is respectively correct, wrong or he just does not know.
Our main objective in this research is to be able to get more information from
the programmer and consequently reduce the number of questions needed to locate a
bug. For this purpose, we consider the following cases:
1. The programmer answers “maybe”. This is the case when the programmer
cannot provide any information about the correctness of the equation.
2. The programmer answers “yes”. This is the case when the programmer
agrees with both the complete result and all the expressions in the left-hand
side of the equation.
3. The programmer answers “no”. The programmer can disagree with an
equation and provide the system with additional information related to the
wrong parts of the equation. This happens either because the result is wrong,
or because some precondition of the equation has been violated (i.e., some parts
of the left-hand side were not expected).
• When the result is wrong, the programmer could say just “no”, but he
could also be more specific. He could exactly point to the parts of the
result which are wrong. For instance, in the equation above, he could
specify that only a subexpression of e is wrong, and the rest of the result
158
6.3. COMPUTATION GRAPHS
159
is correct. This information is useful because it could help the system to
avoid questions about correct parts of the computation.
• The programmer could detect that some expression inside the arguments
of the left-hand side of the equation should not have been computed. For
instance, consider the equation insert ′ b′ “cc” = “bcc”, where function
insert inserts the first argument in a list of mutually different characters
(the second argument). Clearly the result of the equation is correct; but
the programmer could detect here that the second argument should not
have been computed. This means that even for a correct equation the
programmer could provide the system with information about bugs. In this
case, the programmer could mark the second argument (“cc”) as wrong
(i.e., inadmissible [Naish, 1997b] for this function because it violates a
precondition).
• Finally, both, the result and the arguments, could be correct, but the
function name could be wrong1 , in the sense that it should have never
been called. In this case, the programmer could mark the function name
as wrong.
The information provided by the programmer allows the system to slice the execution
tree, thus reducing the number of questions. The more information he provides, the
smaller the execution tree becomes. For instance, the slice computed if the programmer answers “no” is usually larger than the one computed if he answers “no, and this
function should not have been computed ”. Of course, as in conventional algorithmic
debugging, the programmer may only give information about the wrong expressions
he is able to detect. In the worst case, he can always answer “yes”, “no” or “maybe”.
In summary, in order to provide information about errors, the programmer can
specify that some part of the result or the arguments, or the function name is wrong.
The slicing process needed for each case is different.
In this chapter we consider higher-order term rewriting systems that we introduce
in the next section.
6.3
Computation Graphs
In this chapter, we represent terms and expressions as graphs; and we represent computations with graphs rewriting. Therefore, we need to formally define this formalism.
Some of the definitions in this section are taken from [Chitil and Luo, 2006].
1 As we consider a higher-order language, this is a special case of a wrong argument, the function
name being the first (wrong) part of the function application.
159
160
CHAPTER 6. COMBINING PROGRAM SLICING AND...
Since we consider higher-order TRSs, we only need to define the domains of variables X and constructor terms C (we do not distinguish between operation and constructor symbols). Except the opposite is stated, we keep the same definitions and
notation of Chapter 2.
A term is a variable, a constructor, or an application of two terms. A rewrite rule
is a pair of terms, written by l → r, such that root(l) 6∈ X and Var (r) ⊆ Var (l). A
finite set of rewrite rules is a higher-order term rewriting system.
6.4
The Augmented Redex Trail
In the first phase of algorithmic debugging the program is executed and the execution
tree is built. However, the execution tree is not actually built directly. Direct construction of the execution tree during the computation would not be very efficient;
in particular, the nodes of the execution tree are labeled with equations that contain
many repeated subexpressions and, for a lazy functional language, the tree nodes
are constructed in a rather complex order. So instead of the final execution tree a
more elaborated trace structure is constructed that avoids duplication of expressions
through sharing.
The augmented redex trail (ART) [Sparud and Runciman, 1997; Wallace et al.,
2001] is such a trace structure. The ART is used in the Haskell tracer Hat [Sparud
and Runciman, 1997; Wallace et al., 2001] and has similarities with the trace structure
constructed by the earlier algorithmic debugger Freja [Nilsson, 1998]. The ART has
been designed so that it can be constructed efficiently and after its complete construction the execution tree needed for algorithmic debugging can be constructed from it
easily. In addition, an ART contains more information than an execution tree. An
execution tree can be constructed from an ART but not vice versa. The ART supports several other views of a computation besides the execution tree. Nonetheless an
ART is not larger than an execution tree with sharing. An ART is a complex graph
of expression components that can represent both eager or lazy computations. In this
chapter we will use the following definition of ART, which is different from the one
introduced by Braßel et al. [2004b] and used in Chapter 5.
Definition 40 (Augmented redex trail) A node n is a sequence of the letters l,
r and t; which stands respectively for left, right and top. There is also a special node
⊥ (bottom).
A node expression T is a function symbol, a data constructor, a node or an
application of two nodes.
A trace graph is a partial function G : n 7→ T such that its domain is prefixclosed (i.e., if ni ∈ dom(G), then n ∈ dom(G)), but ⊥ ∈
/ Dom(G). We often regard a
trace graph as a set of tuples {(n, G(n)) | n ∈ dom(G)}
160
6.4. THE AUGMENTED REDEX TRAIL
161
tt
t
ttt
2
tl
tr
sum2
ttl
ttr
3
2
ttll
ttlrt
ttlr
5
div
ttlrl
ttlrrt
ttlrr
ttlrrtt
2
ttlrll
ttlrrl
+
ttlrrtl
ttlrrtr
1
decr
ttlrrtll
−
Figure 6.1: Example of ART
A node can represent several terms, because any nodes n and nt are considered
equal. A node approximates a term if it represents this term, except that some
subterms may be replaced by ⊥.
A node n plus a node nt represent a reduction step of the program, if node n
represents the redex and node nt approximates the reduct, and for each variable of the
corresponding program rule there is one shared node.
A trace graph G where node ǫ (empty sequence) approximates the start term M and
every pair of nodes n and nt represents a reduction step with respect to the program
P , is an augmented redex trail (ART) for start term M and program P .
The sequence of the letters l, r and t of every node is used as identifier, thus every
node has a unique sequence of these letters. Any application of two terms t1 and t2
represented with the sequence x, has an arc to these terms whose nodes are identified
respectively with the sequences xl and xr; except if the terms were already in the
EDT. When a term t1 represented by a node x is reduced to another term t2 through
a single reduction step, t2 is identified with the sequence xt. We often call this kind
of nodes ended with t as reductions. Finally, a node labeled with ⊥ represents a
non-evaluated term.
For example, consider the graph depicted in Figure 6.1. This graph is part of
the ART generated from the program in Figure 1.1. In particular, it corresponds to
the function call “sum2 3” producing the result “2”. As specified in Definition 40,
161
162
CHAPTER 6. COMBINING PROGRAM SLICING AND...
each node is identified by a sequence of letters ‘t’, ‘l’ and ‘r’2 . Every node contains
the expression it represents, except for a node which represents an application of
two expressions connected to it by simple lines. A double line represents a single
reduction step. For instance, node ttlrrt represents the function call “3 − 1” that has
been reduced to “2”.
In the following, a (atom) denotes a function symbol or constructor, and n, m
and o denote nodes. We usually write n ∈ G instead of n ∈ Dom(G) when it is clear
from the context. We also say that a node n ∈ G is reduced iff nt ∈ G. Given a node
n ∈ G where G(n) is an application of nodes m · o, we respectively refer to m and o
as left(n) and right(n).
The parent of a node is the node which created it, that is, given a reduction step
t → s, and a node n representing a subterm of s, then, the parent of n is the node
which represents t. Formally:
parent(nl) = parent(n)
parent(nr) = parent(n)
parent(nt) = n
The recursive application of function parent produces the ancestors of a node:
ancestors(n)
=
(
{m} ∪ ancestors(m)
∅
if m = parent(n) defined
otherwise
Finally, to extract the most evaluated form represented by a node, function mef is
defined by:
a
mef (G, m)
mef (G, n) =
mef (G, m) · mef (G, o)
mef (G, nt)
if
if
if
if
G(n) = a
G(n) = m
G(n) = m · o and nt 6∈ G
nt ∈ G
Intuitively, function mef traverses the graph G following reduction edges (n to nt)
as far as possible. Note that this function is similar to function val introduced in
Definition 25, but mef is defined for a higher-order language.
Now we see how easily the execution tree is obtained from the ART. We first
formally define execution tree.
Definition 41 (execution tree) Let G be an ART. The execution tree E associated
to G is a tree whose nodes are labeled with equations constructed with the auxiliary
2 For easier reading we here call the first node t; within the ART of the complete computation the
node is actually trtrtrtrlr.
162
6.4. THE AUGMENTED REDEX TRAIL
163
function. equ is used to extract expressions from the nodes of the ET, and it is defined
as follows:
(
a
if G(n) = a
equ(G, n) =
mef (G, m) · mef (G, o) if G(n) = m · o
The equations have the form l = r such that:
• the root node is equ(G, ǫ) = mef (G, t), if t ∈ G,
• for each reduction step n to nt in G there is a node equ(G, n) = mef (G, nt) in
E, and
• for each pair of nodes n′1 , n′2 ∈ E, with n′1 labeled equ(G, n1 ) = r1 and n′2 labeled
equ(G, n2 ) = r2 ; n′1 is the parent of n′2 iff n2 = n1 to where o ∈ {r, l}∗ .
The reduced nodes of the ART become the nodes of the execution tree. The
function parent expresses the child–parent relationship between these nodes in the
execution tree. The equation of a node n can be reconstructed by applying the
function mef to left(n) and right(n) to form the left-hand side, and to nt to form the
right-hand side. For instance, for the ART in Figure 6.1 the algorithmic debugger
produces the execution tree in Figure 6.2. It contains five questions, one for each
reduction step (reduced nodes have been shaded). For example, the question asked
for node ttlr is “3 + 2 = 5?”.
sum2 3 = 2
div 5 2 = 2
3+2=5
decr 3 = 2
3-1=2
Figure 6.2: Execution tree of the ART in Figure 6.1
Definition 42 (nodes associated to expressions) Let G be an ART and E the
execution tree associated to G. A node n ∈ G is associated to an expression e ∈ E iff
n represents e and 6 ∃ n′ ∈ G | n′ = no, o ∈ {r, l, t}∗ and n′ represents e.
The actual implementation of ARTs includes more detailed information than described here; however, this extra information is not relevant for the purposes of this
research. As an example of a complete ART, the execution tree in Figure 1.3 was
generated from the graph shown in Figure 6.3.
163
164
CHAPTER 6. COMBINING PROGRAM SLICING AND...
Constant Def
main
Application
($)
Application
putStrLn
Hidden
($)
Application
show
Hidden
{IO}
sqrtest
Application
(:)
test
Application
’F’
(:)
(&&)
Hidden
’a’
(:)
Application
Projection
’l’
(:)
computs
Application
Application
(,,)
’s’
(==)
(:)
(==)
Application
Hidden
Hidden
True
False
’e’
Projection
comput2
Application
[]
arrsum
Application
Application
list
comput3
Application
Guard
comput1
Application
Application
Guard
(==)
(+)
Application
Application
Hidden
(:)
Hidden
fromInteger
arrsum
Application
0
Hidden
False
9
list
Constant Use
0
Guard
otherwise
Application
Application
Guard
(==)
(+)
Application
Application
Hidden
(:)
Hidden
fromInteger
arrsum
Application
Hidden
0
False
6
list
Constant Use
Application
0
Guard
otherwise
arrsum
Application
Application
Application
(+)
partialsums
Application
Guard
(==)
Constant Use
Application
otherwise
fromInteger
0
Hidden
(:)
Hidden
arrsum
arrsum
Application
Application
list
fromInteger
Guard
0
Application
False
3
otherwise
Hidden
Application
(:)
Constant Def
Hidden
(+)
Application
Hidden
0
(:)
Application
Application
sum1
sum2
[]
Hidden
Application
div
div
0
Application
Application
(-)
fromInteger
square
(-)
Application
Application
Application
Application
(*)
fromInteger
(*)
(-)
Application
Application
0
Hidden
True
Hidden
1
Hidden
0
0
fromInteger
2
Hidden
6
Hidden
incr
Application
2
5
(+)
Hidden
1
1
(-)
Application
Hidden
Hidden
fromInteger
fromInteger
Application
(:)
9
1
Hidden
2
1
1
Hidden
4
1
Hidden
(+)
Application
Application
fromInteger
arrsum
Hidden
1
1
Application
1
Hidden
(:)
3
(+)
Application
Application
fromInteger
arrsum
1
Hidden
Application
2
Hidden
[]
2
fromInteger
2
0
Hidden
0
Figure 6.3: ART of the program in Figure 1.1
164
Hidden
Application
12
fromInteger
2
decr
Application
Application
1
fromInteger
Application
Hidden
fromInteger
Hidden
Application
(+)
Application
Hidden
Application
Hidden
arrsum
fromInteger
Hidden
Application
Hidden
Application
[]
Hidden
Application
arrsum
Application
(==)
Application
Application
Hidden
Application
True
8
(+)
1
2
2
6.5. SLICING THE ART
6.5
165
Slicing the ART
Program slicing techniques have been based on tree-like data structures such as the
program dependence graphs (see Chapter 1). In the context of execution trees we want
to define the slicing mechanism over their underlying ART. This provides us with two
advantages:
1. At this level we have much more information about the computation which
allows the production of more precise slices. It is straightforward to extract an
execution tree from the sliced ART [Chitil, 2005].
2. When we define the slicing technique at the ART level, it can be used for any
other purpose (e.g., for proper program slicing which has not been defined for
these graphs, for program specialisation, etc.).
Now, we define the notion of slice and some properties in the context of ARTs.
Definition 43 (slice) Let G be an ART. A slice of G is a set of nodes {n1 , . . . , nk } ⊆
Dom(G).
To compute slices w.r.t. a particular node of interest we need to identify which nodes
in the computation have influenced this node of interest. We introduce the following
definition:
Definition 44 (Influence between nodes) Let G be an ART for start term M
and program P . Node n1 influences node n2 , written n1 ❀
❀ n2 iff for all ARTs G ′ for
′
start term M and program P with n2 ∈ G we have n1 ∈ G ′ .
Note that we have not fixed any evaluation order for constructing ARTs, but functional programs are deterministic. Hence different ARTs differ in which parts of the
computation are how much evaluated.
Influence is transitive and reflexive.
❀) Let G be an ART for start term M
Lemma 45 (Transitivity and reflexivity ❀
and program P . For nodes n1 , n2 and n3 in G:
• n1 ❀
❀ n2 and n2 ❀
❀ n3 implies n1 ❀
❀ n3
• n1 ❀
❀ n1
Proof.
We consider influence for a graph G with start term M and program P .
′
Let G be any ART for the same start term and program such that n3 ∈ G ′ . With
❀ n3 follows that node n2 is in G ′ . Then with n1 ❀
n2 ❀
❀ n2 follows that node n1 is in
′
❀
G . So according to Definition 44, n1 ❀ n3 . Reflexivity trivially holds by definition
of influence.
165
166
CHAPTER 6. COMBINING PROGRAM SLICING AND...
✷
We perform program slicing on the ART G without knowing the program P whose
computation is described by G. So we cannot compute other ARTs G ′ for start term
M and program P ; furthermore, there is a large, possibly infinite number of such G ′ s.
Our aim is to find an algorithm that obtains a slice from the given ART G ensuring
completeness (i.e., all nodes that influence the node of interest belong to the slice).
Therefore we have to approximate the slice that influences a given node:
Definition 46 (Weak influence between nodes) Given an ART G. Reduction
n1 weakly influences node n2 , written n1 ❀ n2 iff either
1. n2 = n1 o for some o ∈ {l, r, t}∗ , or
2. there exist m, o ∈ {l, r, t}∗ such that m ∈ ancestors(n2 ), n1 = mo and o does
not start with t.
The idea is to include all prefixes of the given node (case 1) and all nodes that may
be needed for the reduction step of an ancestor. Therefore, if an ancestor of n2 is
an application, then, all the nodes involved in the arguments of this application do
weakly influence n2 (case 2).
Lemma 47 (Transitivity and reflexivity of ❀) Let G be an ART for start term
M and program P . For nodes n1 , n2 and n3 in G:
• n1 ❀ n2 and n2 ❀ n3 implies n1 ❀ n3
• n1 ❀ n1
Proof. Reflexivity is trivial. For transitivity let n1 ❀ n2 and n2 ❀ n3 . We perform
case analysis according to the two conditions of Definition 46:
• n2 = n1 o1 and n3 = n2 o2 . Then n3 = n1 o1 o2 , so n1 ❀ n3 .
• n2 = n1 o1 and n2 = mo2 with m ∈ ancestors(n3 ). If m = n1 p, then n1 pp′ = n3
because m is an ancestor of n3 and thus n1 ❀ n3 . Otherwise, n1 = mp. Then
n2 = mpo1 . Since o2 does not start with t, p cannot start with t. So n1 ❀ n3
following the second part of Definition 46.
• n1 = mo1 where m ∈ ancestors(n2 ) and n3 = n2 o2 . Since ancestors(n2 ) ⊆
ancestors(n3 ), we have m ∈ ancestors(n3 ). So n1 ❀ n3 .
• n1 = m1 o1 where m1 ∈ ancestors(n2 ) and n2 = m2 o2 where m2 ∈ ancestors(n3 ).
So m1 o′ = n2 = m2 o2 . If m2 = m1 o, then m1 ∈ ancestors(n3 ), so n1 ❀ n3 .
Otherwise, m1 = m2 o. Since o2 does not start with t, also o does not start with
t. Furthermore, n1 = m2 oo1 and m2 ∈ ancestors(n3 ). So n1 ❀ n3 .
166
6.5. SLICING THE ART
167
✷
Definition 48 (Slice of influence) Given an ART G and a node n ∈ G the slice of
G w.r.t. node n is slice(G, n) = {m ∈ G | m ❀ n}.
Proposition 49 (Slice of influence)
{(m, G(m)) | m ∈ slice(G, n)} is an ART for the same start term and program as G.
In order to prove this proposition we need first the following auxiliary lemmas:
Lemma 50 (Prefix-closed) Let G be an ART with n ∈ G. Then slice(G, n) is
prefix-closed.
Proof. Let us assume that mo ∈ slice(G, n), i.e., mo ❀ n. Because m is a prefix
of mo, it follows with the first condition in Definition 46 that m ❀ mo. Then, by
Lemma 47, m ❀ n. So m ∈ slice(G, n).
✷
Lemma 51 (Reductions in slice) Let G be an ART with n ∈ G. For any reduction
step m to mt in {(m, G(m)) | m ∈ slice(G, n)}, node m represents the same redex and
node mt approximates the same reduct as in G.
Proof.
We show that all relevant nodes exist unchanged in {(m, G(m)) | m ∈
slice(G, n)}: From the second part of Definition 46, it follows that all nodes reachable
from m in G are also in slice(G, n). Moreover, by Definition 48 {(m, G(m)) | m ∈
slice(G, n)} is a subset of G. Therefore, m represents the same redex as in G. The fact
that mt ∈ slice(G, n) suffices already for mt to approximate the reduct of m; since
{(m, G(m)) | m ∈ slice(G, n)} is a subset of G it must be the same reduct as in G.
✷
Now, we can prove Proposition 49:
Proof. According to lemma 50 slice(G, n) is prefix-closed; and according to lemma 51
for every node nt ∈ Dom({(m, G(m)) | m ∈ slice(G, n)}), the node n represents the
redex and node nt approximates the reduct of the reduction step n to nt.
✷
Weak influence approximates influence:
167
168
CHAPTER 6. COMBINING PROGRAM SLICING AND...
Proposition 52 (Influence between nodes)
Let G be an ART with n ∈ G. Then slice(G, n) ⊇ {m ∈ G | m ❀
❀ n}.
Proof. We prove that m 6❀ n implies m ❀
6 n. According to Proposition 49 slice(G, n)
❀
is an ART for the same start term and program as G. Moreover, we know that
n ∈ slice(G, n) because the reflexivity property of Lemma 47. From m 6❀ n, it follows
that m ∈
/ slice(G, n). Hence, according to Definition 44, m 6❀
❀ n because there exists
an ART to which n belongs and m does not.
✷
So we will compute slices of an ART G that weakly influence a node n.
6.6
Combining Program Slicing and Algorithmic
Debugging
The combination of algorithmic debugging and program slicing can lead to a much
more accurate debugging session. An algorithmic debugger can interpret information
from the programmer about the correctness of different parts of equations, and, by
means of a program slicer, use this information to slice execution trees, thus reducing
the number of questions needed to find a bug. In this section, we show how program
slicing can be combined with algorithmic debugging.
6.6.1
The Slicing Criterion
To define the slicing technique we need to define first a slicing criterion in this context
which specifies the information provided by the programmer. During algorithmic
debugging, the programmer is prompted and asked about the correctness of some
series of equations such as the following:
f1 v11 . . . vi1
f2 v12 . . . vj2
...
fn v1n . . . vkn
= val1
= val2
= valn
Then, the programmer determines for each equation whether it is correct or wrong. In
particular, when the programmer identifies a wrong (sub)expression inside an equation, he may produce a slice of the execution tree w.r.t. this wrong expression.
Therefore, from the point of view of the programmer, a slicing criterion is a wrong
(sub)expression: he should be able to specify (i.e., mark) which parts of the equations
shown during algorithmic debugging are wrong. However, from the point of view of
168
6.6. COMBINING PROGRAM SLICING AND...
169
the debugger, the internal representation of the execution tree is an ART and thus a
slicing criterion should be a pair of nodes hm, ni in the ART denoting where the slice
starts (m) and where it ends (n).
To automatically produce a slicing criterion from the expression marked by the
programmer, we use slicing patterns. Wee need to redefine the domain Pat of slicing
patterns of Chapter 5 so that higher-order is considered. It is defined as follows:
π ∈ Pat ::= ⊥ | ⊤ | π1 π2
where ⊥ denotes a subexpression that is irrelevant and ⊤ a relevant subexpression.
Note that a (partially or completely evaluated) value is an instance of a slicing
pattern where atoms have been replaced by ⊥ and ⊤. Patterns are used to distinguish
between correct and wrong parts of equations, and hence to find the nodes of the
ART which could be the cause of the bug. For instance, once the programmer has
selected some (wrong) subexpression, the system automatically produces a pattern
from the whole expression where the subexpression marked by the programmer have
been replaced by ⊤, and the rest of subexpressions have been replaced by ⊥. Because
algorithmic debugging finds a single bug at a time, we only consider a single expression
marked in π; different wrong expressions could be produced by different bugs.
Example 17 Consider the following equation from the execution tree of Figure 1.3:
computs 3 = (9,9,8)
If the programmer marks 8 as wrong, the pattern generated for the expression
(9,9,8), internally represented as (((T·9)·9)·8), would be (⊥· ⊤), where T is the
n-ary tuple constructor.
As shown in the previous example, the pattern allows us to discard those subexpressions which are correct. With the information provided by the pattern it is easy to
find the nodes in the ART which are associated with wrong expressions. The following definition introduces function patNodes which identifies the node in the ART
associated with the wrong expression marked by the programmer.
Definition 53 (patNodes) Given an ART G, a pattern π for an expression M and
a node n such that mef (G, n) = M , patNodes(G, π, n) finds in the ART the node
associated with the wrong expression specified in π. patNodes is defined as follows:
{}
patNodes(G, π, nt)
patNodes(G, π, m)
patNodes(G, π, n) =
{n}
patNodes(G, π1 , n1 )
∪ patNodes(G, π2 , n2 )
169
if π = ⊥
if π 6= ⊥ and nt ∈ G
if π 6= ⊥, nt 6∈ G and G(n) = m
if π = ⊤, nt 6∈ G and G(n) =
6 m
if π = π1 · π2 , nt 6∈ G
and G(n) = n1 · n2
170
CHAPTER 6. COMBINING PROGRAM SLICING AND...
After patNodes(G, π, n) has found in the ART the node associated with the wrong
expression specified in π, the slicer can produce a slice for this node which contains
all those nodes in G that could influence the wrong expression.
For instance, given the equation f oo M = N , the programmer should be able to
specify that he wants to produce a slice from either f oo, some subexpression of M or
some subexpression of N . Therefore, we distinguish between three possible ways of
defining a slicing criterion:
• The programmer marks a subexpression of N and the associated pattern π
is generated; then, the algorithmic debugger computes the tuple hm, ni where
{n} = patNodes(G, π, m) such that mef (G, left(m)) mef (G, right(m)) = f oo M
and, thus, m must be a reduced node.
• The programmer marks a subexpression of M and the associated pattern π
is generated; then, the algorithmic debugger computes the tuple hǫ, ni where
{n} = patNodes(G, π, m) such that mef (G, m) = M is the second argument of
the application f oo M .
• The programmer marks the function name f oo, and the algorithmic debugger
produces a tuple hǫ, ni where mef (G, n) = f oo is the first argument of the
application f oo M .
We need to distinguish between errors in the right-hand side of an equation and errors
in the left-hand side because the slices for them are essentially different. In the former
case, we only care about new nodes, because we trust the arguments of the function
call in the left-hand side of the equation and, thus, all the nodes which were computed
before this function call could not cause the bug; however, in the later case, the error
could be in any previous node that could have influenced (rather than had influenced )
the value of the incorrect argument or function name.
Definition 54 (Slicing criterion)
Let G be an ART. A slicing criterion for G is a tuple hm, ni; such that m, n ∈ G and
n = mo, where o ∈ {l, r, t}∗ .
A slicing criterion points out two nodes in the graph. In contrast, slicing patterns
can select a set of nodes of interest. Therefore, from a tuple hm, si generated from a
slicing pattern, we produce a set of slicing criteria {hm, ni} such that n ∈ s. Later,
we compute the slice associated to hm, si from the union of the slices of {hm, ni}.
The two nodes specified in a slicing criterion delimit the area of the graph where
the bug must be. The objective of the slicer is to find these nodes and collect all the
nodes which are weakly influenced by the first node and that weakly influence the
second node. This kind of slice is known in the literature as a chop (see Section 1.2.6):
170
6.6. COMBINING PROGRAM SLICING AND...
path(G, m, n)
collect(G, m, n) =
171
if parent(m) = parent(n)
or if parent(m) and parent(n)
are undefined
{p} ∪ collect(G, m, p)
otherwise, where parent(n) = p
∪ subnodes(G, p) ∪ path(G, pt, n)
subnodes(G, n) =
subn(G, n) =
′
(
path(G, n, n ) =
subn(G, m) ∪ subn(G, o)
subn(G, m)
subn(G, o)
{}
if G(n) = m · o, m = nl and o = nr
if G(n) = m · o, m = nl and o 6= nr
if G(n) = m · o, m 6= nl and o = nr
otherwise
{n} ∪ subnodes(G, n) ∪ subn(G, nt)
if nt ∈ G
{n} ∪ subnodes(G, n)
if nt 6∈ G
(
{n}
if n = n′
{n′ } ∪ path(G, n, m)
if n′ = mi, m ∈ {l, r, t}∗ and i ∈ {l, r, t}
Figure 6.4: Function collect to slice ARTs
Definition 55 (Chop of influence)
Let G be an ART and hm, ni a slicing criterion for G. The chop S of G w.r.t. hm, ni
is chop(G, m, n) = {n′ ∈ G | m ❀ n′ and n′ ❀ n}.
Obviously slice(G, n) = chop(G, ǫ, n).
6.6.2
The Slicing Algorithm
Figure 6.4 shows the formal definition of the slicing algorithm for ARTs. For the
sake of convenience, we give in this figure two versions of the algorithm that can
be differentiated with the underlining. On the one hand, this algorithm is able to
compute a slice from an ART. On the other hand, we present a conveniently modified
version which only computes those nodes that are useful during algorithmic debugging.
We first describe the algorithm to slice ARTs (for the time being the reader can ignore
the underlining):
Function collect computes chop(G, m, n). It collects all the nodes for which m is
a prefix that belong to one of the following sets:
• The nodes which are in the path from m to n,
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CHAPTER 6. COMBINING PROGRAM SLICING AND...
• The ancestors of n, and
• All the nodes which participated in the evaluation of the arguments of the
ancestors of n (we call such a set of nodes the subnodes).
Function collect proceeds by identifying the parent of the incorrect node (i.e., the
function that introduced it) and its subnodes; then, it recursively collects all the
nodes which could influence the parent (all its ancestors). Functions subnodes and
subn are used to compute all those nodes which participated in the evaluation of the
arguments of a given application node n. Finally, function path computes the path
between two given nodes.
The algorithm presented in Figure 6.4 is a program slicing technique for ARTs.
However, the slices it produces contain nodes which are useless for algorithmic debugging. In algorithmic debugging we are only interested in reductions, i.e., reduced
nodes, because they are the only nodes which yield questions during the algorithmic
debugging process [Chitil, 2005]. For this reason, we also introduce a slightly modified
version which only computes those nodes from the ART which are needed to produce
the execution trees used during algorithmic debugging.
Firstly, only reduced nodes must be computed, thus the underlined expressions in
Figure 6.4 should be omitted from the algorithm. And, secondly, we must ensure that
the produced slice is not empty. This is ensured by the fact that the slicing criterion
defined for algorithmic debugging always takes a reduced node as the starting point.
Therefore, the slice is never empty because at least the (reduced) node associated
with the left-hand side of the wrong equation is included in the slice.
6.6.3
Completeness of the Slicing Algorithm
The following result states the completeness of the slicing algorithm:
Theorem 56 (Completeness) Given an ART G and a slicing criterion hm, ni for
G, collect(G, m, n) = chop(G, m, n).
Proof.
We prove the theorem by showing that:
1. chop(G, m, n) ⊆ collect(G, m, n) and
2. collect(G, m, n) ⊆ chop(G, m, n)
We prove first that chop(G, m, n) ⊆ collect(G, m, n):
We consider two possible cases according to collect’s definition. Let us assume
first that either parent(m) = parent(n) or parent(m) and parent(n) are undefined. In
this case, collect(G, m, n) = path(G, m, n), thus it only collects (first rule of function
collect) the nodes which are in the path from m to n.
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6.6. COMBINING PROGRAM SLICING AND...
173
Now, we prove that chop(G, m, n) = path(G, m, n). Because m and n have the same
parent, and m is a prefix of n (by Definition 54), we know that n = mo where o ∈
{l, r}∗ . Then, by Definitions 46 and 55 only the prefixes of n with m as prefix belong to
chop(G, m, n). These nodes form the path between m and n. Then, chop(G, m, n) =
path(G, m, n) = collect(G, m, n). Therefore, the claim holds.
If, on the contrary, parent(m) 6= parent(n) = p or parent(m) is undefined and
parent(n) = p, rule 2 from collect is applied. In this case, all the nodes that belong
to the set S = {p} ∪ collect(G, m, p) ∪ subnodes(G, p) ∪ path(G, pt, n) are collected.
We prove the first claim by contradiction assuming that ∃n′ ∈ chop(G, m, n) and
n 6∈ S. Because n′ 6∈ S, we know that n′ is not an ancestor of n nor a subnode of an
ancestor of n. Thus, by Definition 46, n′ must be a prefix of n. Moreover, we know
that n′ is not in the path between m and n, thus it must be a prefix of both n and
m. But this is a contradiction w.r.t. the definition of chop (Definition 55), because
m ❀ n′ and thus n′ cannot be a prefix of m. Then, chop(G, m, n) ⊆ collect(G, m, n)
′
Now, we prove that collect(G, m, n) ⊆ chop(G, m, n):
We consider again two possible cases according to collect’s definition. Let us
assume first that either parent(m) = parent(n) or parent(m) and parent(n) are undefined. In this case, collect(G, m, n) = path(G, m, n), thus it only collects (first rule
of function collect) the nodes which are in the path from m to n.
Now, we prove that chop(G, m, n) = path(G, m, n). Because m and n have the same
parent, and m is a prefix of n (by Definition 54), we know that n = mo where o ∈
{l, r}∗ . Then, by Definitions 46 and 55 only the prefixes of n with m as prefix belong to
chop(G, m, n). These nodes form the path between m and n. Then, chop(G, m, n) =
path(G, m, n) = collect(G, m, n). Therefore, the claim holds.
If, on the contrary, parent(m) 6= parent(n) = p or parent(m) is undefined and
parent(n) = p, rule 2 from collect is applied. In this case, all the nodes that belong
to the set S = {p} ∪ collect(G, m, p) ∪ subnodes(G, p) ∪ path(G, pt, n) are collected.
We assume now that ∃n′ ∈ S and n′ 6∈ chop(G, m, n). Then, either p or a node
in collect(G, m, p), subnodes(G, p) or path(G, pt, n) does not belong to chop(G, m, n).
First, we know that m is a prefix of n in the initial call (by Definition 54) and in all
the recursive calls by rule 2 of collect(G, m, n)); and parent(m) 6= parent(n) = p thus
n = mo being o a sequence of letters containing at least one t. Then, we know that
m = p or m is a prefix of p. Therefore p ∈ chop(G, m, n) and n′ 6= p. In addition,
by rule 2 of Definition 46, we know that the subnodes of p weakly influence n, and
they are weakly influenced by m because m is their prefix. Hence subnodes(G, p) ⊆
chop(G, m, n) and thus n′ 6∈ subnodes(G, p). Moreover, because m is a prefix of n
and by rule 1 of Definition 46 we have that ∀m′ ∈ path(G, m, n), m ❀ m′ ❀ n and
consequently path(G, m, n) ⊆ chop(G, m, n), thus n′ 6= m′ . Then, n′ does not belong
to collect(G, m, p), and thus, ∀n′ ∈ S, n′ ∈ chop(G, m, n) and the claim holds.
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CHAPTER 6. COMBINING PROGRAM SLICING AND...
✷
As discussed before, we cannot guarantee minimality (i.e., ensure that only nodes
that influence the slicing criterion are collected) w.r.t. Definition 44. This loss of
precision is unavoidable because during algorithmic debugging we have available only
one ART (the wrong computation). However, the algorithm in Figure 6.4 is minimal
w.r.t. Definition 46 (i.e., it ensures that only nodes that weakly influence the slicing
criterion are collected). Nevertheless, even if the underlying ART corresponds to
a lazy evaluation language (this is the case, for instance, of a debugging session in
Haskell) and, thus, not needed terms are represented by ⊥ in the ART, the slice
produced is not minimal w.r.t. Definition 44. This phenomenon is due to sharing: A
previously computed reduced node can appear as argument in a non-strict position
of a function call. Then, it is not possible to know which part of this reduction was
done before and which part was done after the start of the slice. Therefore, only when
the slicing criterion starts at the beginning of the computation (classical backward
slicing) the slice produced is minimal, and thus it contains all and only the nodes that
could influence the slicing criterion.
Theorem 57 (Completeness of execution trees)
Given an execution tree E for an ART G, an equation l = r ∈ E and an expression
e that is a subexpression of l or r, marked as wrong during an algorithmic debugging session, then collect(G, epsilon, n), where n ∈ G is associated to e, contains the
(buggy) nodes which caused e.
Proof.
By Theorem 56 we have that collect(G, epsilon, n) = chop(G, epsilon, n),
and hence, by Definition 55, collect(G, epsilon, n) = {n′ ∈ G | epsilon ❀ n′ and
n′ ❀ n}. With the first rule of Definition 46 holds that ∀m ∈ G, epsilon ❀ m and,
thus, collect(G, epsilon, n) = {n′ ∈ G | n′ ❀ n}. Moreover, by Proposition 52 we
have that collect(G, epsilon, n) ⊇ {n′ ∈ G | n′ ❀
❀ n} and hence, collect(G, epsilon, n)
contains the (buggy) nodes which caused e.
✷
6.7
The Technique in Practice
In this section we show an example of how to integrate our technique with an algorithmic debugger. Consider again the example program in Figure 1.1 and its associated
execution tree in Figure 1.3. The debugging session using our technique is depicted
in Figure 6.5.
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6.7. THE TECHNIQUE IN PRACTICE
175
Starting Debugging Session...
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
main = False? NO
sqrtest [1,2] = False? NO
test (9,9,8) = False? YES (The programmer selects "8")
computs 3 = (9,9,8)? NO
comput3 3 = 8? NO
listsum [6,2] = 8? YES
partialsums 3 = [6,2]? NO (The programmer selects "2")
sum2 3 = 2? NO
decr 3 = 2? YES
Bug found in rule:
sum2 x = div (x + (decr x)) 2
Figure 6.5: Debugging session
This debugging session is very similar to the one in Figure 1.2. There are only two
differences: The programmer is allowed to provide the system with more information
and he does in questions ‘3’ and ‘7’; and there are less questions to answer, the latter
being the consequence of the former.
Internally, when the programmer selects an expression, he is indirectly defining
a slicing criterion. Then, the system uses this slicing criterion to slice the execution
tree, thus reducing the number of questions. In the example, when the programmer
selects the expression “8”3 , the system automatically removes from the execution
tree eleven possible questions: all the subtrees of the equations “comput1 3 = 9”
and “comput2 3 = 9” (see dark nodes on the left of Figure 1.3). Similarly, when
the programmer selects the expression “2”, the system automatically removes from
the execution tree two possible questions: the subtree of the equation “sum1 3 = 6”
(questions 21 and 22 in Figure 1.3). Note that the slicing technique is completely
transparent for the programmer.
In general, the amount of information deleted from the tree is directly related to
the size of the data structures in the equations, because when the programmer selects
one subexpression, the computation of the other subexpressions can be avoided. Another important advantage of combining program slicing and algorithmic debugging
is that it allows the programmer to guide the questions of the algorithmic debug3 Note that, even when the equation is correct, the programmer can mark an expression as inadmissible, providing the system with useful information about the bug.
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CHAPTER 6. COMBINING PROGRAM SLICING AND...
ger. Traditionally, the programmer had to answer the questions without taking part
in the process. Thanks to the slicing mechanism, the programmer can select which
(sub)expressions he wants to inspect (i.e., because they are wrong or just suspicious),
thus guiding the algorithmic debugger and producing a sequence of questions semantically related for the programmer.
For large computations a trace is huge. Hat stores its ART on disk, so that the
size of the ART is not limited by the size of the main memory. In contrast, the
Mercury debugger materialises parts of the execution tree at a time, by rerunning the
computation. Because a slice can be a large part of the ART, collecting all its nodes
could be very expensive for both implementation choices. However, for directing the
algorithmic debugging process, we do not need all nodes at once. We only have to
determine them one at a time to determine the next question. We are in a different
situation when we want to highlight in the program source the program slice associated
with the trace slice. Such an operation can take time linear in the size of the trace and
is thus unsuitable for an interactive tool, as the source-based algorithmic debugger of
Hat demonstrates (see related work).
6.8
Related Work
The idea of improving algorithmic debugging by allowing the programmer to provide
specific information about the location of errors is not new. In fact, it was introduced
by Pereira almost two decades ago [Pereira, 1986]. The original work of Pereira
defined—in the context of Prolog—how an algorithmic debugger could take advantage
of additional information from the programmer in order to reduce the number of
questions. Surprisingly, later research in this subject practically ignored this work The
first attempt to adapt the ideas of Pereira to the imperative programming paradigm
was [Fritzson et al., 1992]. In this work, the authors propose the use of program
slicing in order to reduce execution trees, thus reducing the number of questions
in algorithmic debugging. Dynamic program slicing was the technique that Pereira
needed in his development, but his work was done in 1986. Even though program
slicing was published in 1984 [Weiser, 1984], dynamic program slicing was introduced
only in 1988 [Korel and Laski, 1988].
Although the use of program slicing in algorithmic debugging was shown to be
very useful, the technique by Fritzson et al. is still limited, because the programmer
is only allowed to specify which variables have a wrong value, but he is not allowed
to specify which part of the value is wrong. This information could notably improve
the effectiveness of the technique as it was shown in Example 6.5. In addition, the
authors do not give details about the dynamic slicing algorithm used, so it is not
clear whether the variables selected by the programmer must be output variables,
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6.8. RELATED WORK
177
because they only considered the use of program slicing in wrong equations (the concept of inadmissibility was lost). Later the ideas introduced by Fritzson et al. have
been applied to the logic language paradigm in [Paakki et al., 1994] and extended in
different ways. In particular, Kokai et al. [1997] have defined the IDTS system which
integrates algorithmic debugging, testing and program slicing techniques in a single
framework. Our work can be seen as an adaptation of these ideas to the functional
paradigm. Functional programs impose different technical challenges (i.e., generalised
use of higher-order functions, absence of nondeterminism and unification) which implies the use of different formalisms compared to logic programs. For instance, while
ARTs directly represent the sequence of function unfoldings, which allows us to trace
data flows, the slicing of Prolog proof trees requires explicit annotation of the tree
with information about the relations between terms. These differences produce an
essentially different slicing processes.
MacLarty [2005] have implemented an algorithmic debugger for the functional
logic language Mercury which allows the programmer to specify which parts of an
equation are wrong (including subterms). In particular, they have defined a strategy
called “subterm dependency tracking” which, given a wrong term, finds the ‘origin’
of this subterm in the execution tree. This information is used by the tool for determining which questions are more likely to be wrong, thus improving the algorithmic
debugging process (see Section 1.2.3). Although this process can be considered as a
kind of program slicing, they have not defined it formally, and, for the case of wrong
input arguments, the method has only been proposed but not implemented nor proved
correct.
The source-based trace explorer of Hat [Chitil, 2005] implements algorithmic debugging and can display a simple program slice that contains the bug location and
that shrinks when more questions are answered. However, the programmer cannot
indicate wrong subexpressions and the slice does not direct the algorithmic debugging
process.
To the best of our knowledge, our technique is the first formal approach that 1)
is able to slice execution trees for either input or output expressions, 2) copes with
complete expressions and subexpressions and 3) is proven complete. We consider functional programs, a paradigm that completely lacked debugging techniques combining
algorithmic debugging and program slicing.
177
Chapter 7
A Classification of Program
Slicing and Algorithmic
Debugging Techniques
This chapter introduces two classifications. Firstly, the program slicing techniques
presented in Section 1.2.6 are compared and classified in order to identify relationships
between them. Secondly, three new algorithmic debugging strategies are introduced,
and they are compared and classified together with those presented in Section 1.2.3.
Part of the material of this chapter has been presented in the International Symposium on Logic-based Program Synthesis and Transformation (LOPSTR 2006) celebrated in Venice (Italy) in 2006 [Silva, 2006].
7.1
Introduction
In this chapter we discuss and compare the program slicing techniques and algorithmic
debugging strategies of Sections 1.2.6 and 1.2.3.
Firstly, following the approach of Harman et al. [1996], the techniques in Section 1.2.6 are compared in order to clarify and establish the relations between them.
This comparison gives rise to a classification of techniques which can help to guide
future research directions in this field. In particular, we compare and classify the
techniques aiming at identifying relations between them in order to answer questions
like Is one technique a particular case of another? Is one technique more general
or more expressive than another? Are they equivalent but expressed with different
formalisms?
Secondly, we introduce three new algorithmic debugging strategies that can reduce
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CHAPTER 7. A CLASSIFICATION OF PROGRAM SLICING...
the number of questions asked during an algorithmic debugging session.
Finally, we classify and study the costs of the new algorithmic debugging strategies
and those of Section 1.2.3. Based on this study, we introduce a parameterizable
algorithm which allows us to dynamically combine different algorithmic debugging
strategies in a single debugging session.
The chapter has been structured as follows: The next section revisits the classification introduced by Harman et al. [1996] and we extend it with new slicing techniques
and dimensions. The analysis provides useful information that allows us to classify
the slicing techniques and establish relations between them. In Section 7.3 three new
algorithmic debugging strategies are introduced. Finally, in Section 7.4 we present a
comparison of algorithmic debugging strategies and we study their costs.
7.2
Classifying Slicing Techniques
In this section we use a classification by Harman et al. [1996] in order to compare and
classify all slicing techniques presented in Section 1.2.6. Table 7.1 extends Harman et
al’s classification with new dimensions in order to be able to classify new techniques.
In the table, we have omitted those slicing criteria which have been identified as a
particular case of another technique in Section 1.2.6 and thus, they do not impose
additional restrictions or dimensions in the table. For instance, we omit in the table
“End Slicing” because it is a particular case of simultaneous slicing. In addition,
we only include in the table the most general form of slicing for each technique.
For instance, in the case of quasi-static slicing, the original definition by Vengatesh
considered slicing only “at the end of the program”; here, in contrast, we consider a
generalization (see Section 1.2.6) in which any point of the program can be selected for
slicing. Similarly, we consider simultaneous slicing as a set of any slicing criteria (in
contrast to simultaneous dynamic slicing where all the criteria are dynamic); and we
consider conditioned amorphous slicing [Harman and Danicic, 1997] which generalizes
static amorphous slicing.
In the table, following Harman et al’s terminology, the sets I, S and N refer
respectively to the set of all program variables, possible initial states and iterations.
For each slicing technique (first column),
• column Var. (Variables) specifies the number of variables participating in a
slicing criterion C, where P(I) indicates that a subset of I participates in C,
• column Initial States shows the number of initial states considered by the corresponding slicing criterion,
• column Slice Points describes the number of program points included in C, and
hence the number of slices actually produced (and combined) to extract the
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7.2. CLASSIFYING SLICING TECHNIQUES
181
final slice. Here, n derived means that n different points are implicitly specified
(see Section 1.2.6); and 1+n means that 1 program point is specified together
with n breakpoints,
• column I.C. (Iteration Counts) indicates, for the specified slice points, which
iterations are of interest,
• column Dir. (Direction) states the direction of the traversal to produce the slice
which can be backwards (B), forwards (F) or a combination of them. Here,
despite some techniques accept both directions, we assign to every technique
the direction specified in its original definition (see Section 1.2.6),
• column P. (Precision) is marked if the slice is precise, in the sense that it
does not contain statements which cannot influence (or, in forward slicing, be
influenced by) the slicing criterion. Finally,
• column Sy/Se Pres. (Syntax/Semantics Preserving) contains boolean parameters that indicate respectively whether the slice produced is a projection of the
program syntax or it preserves a projection of the original semantics. Here, (e)
means that the slice produced is executable.
The comparison of slicing techniques presented in Table 7.1 is a powerful tool for
the study of current slicing techniques and the prediction of new ones. For instance,
the last five rows in Table 7.1 correspond to new slicing techniques predicted by
analyzing the information of the table.
The first new technique is point slicing. In all the slicing techniques presented so
far one or more variables of the program are part of the slicing criterion. In contrast,
point slicing does not consider any variable in the slicing criterion. Point slicing
selects a sentence of the program and computes backwards (resp. forwards) all the
sentences that could have been executed before (respectively after) it. This is useful
in debugging. For instance, during print debugging [Agrawal, 1991] the programmer
places print statements in some strategic points of the program, and then executes it
in order to see if some of them have been reached. Usually, a print statement is placed
in a part of the code in order to check if it is executed or not. A point slice w.r.t.
this statement can be useful in order to know which possible paths of the program
could reach the execution of this sentence. Note that print slicing does not follow
control or data dependences, but control flows, and thus it is not subsumed by any
of the other slicing techniques in the table. As an example, the statement “print
(‘Executed’);” does not influence any other statement, and hence, it will not belong
to any slice taken w.r.t. a subset of the variables of the program. The conditioned
version of point slicing restricts the possible paths by limiting the initial states of the
execution.
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CHAPTER 7. A CLASSIFICATION OF PROGRAM SLICING...
Slicing
Technique
Var.
Initial
States
Slice
Points
I.C.
Dir.
P.
Sy/Se
Pres.
Static Slicing
Dynamic Slicing
Forward Slicing
Quasi Static Slicing
Conditioned Slicing
Decomposition Slicing
Chopping
Relevant Slicing
Hibrid Slicing
Intraprocedural Slicing
Interprocedural Slicing
Simultaneous Slicing
Dicing
Abstract Slicing
Amorphous Slicing
P(I)
P(I)
P(I)
P(I)
P(I)
1
P(I)
P(I)
P(I)
P(I)
P(I)
P(I)
P(I)
P(I)
P(I)
{S}
1
{S}
P(S)⋆
P(S)⋆⋆
{S}
{S}
1
P(S)⋆ ⋆ ⋆
{S}
{S}
P(S)
{S}
P(S)⋆⋆
P(S)⋆⋆
1
1
1
1
1
n derived
pair
1
1+n
1
1
n
n
1
1
{N }
1
{N }
{N }
{N }
{N }
{N }
1
{N }
{N }
{N }
{N }
{N }
{N }
{N }
B
B
F
B
B
B
B∧F
B
B
B
B
B∨F
B
B
B
y
y
y
y
y
y
y
n
y
n
y
y
y
y
y
y/y(e)
y/y(e)
y/y
y/y(e)
y/y(e)
y/y
y/n
y/n
y/y(e)
y/y(e)
y/y(e)
y/y(e)
y/n
y/y(e)
n/y(e)
∅
∅
P(I)
P(I)
∅
{S}
P(S)⋆⋆
P(S)⋆⋆
P(S)⋆⋆
{S}
1
1
pair
1
1
{N }
{N }
{N }
{N }
{N }
B
B
B∧F
F
F
n
n
y
y
n
y/y(e)
y/y(e)
y/n
n/y(e)
y/y(e)
New Techniques
Point Slicing
Conditioned Point
Conditioned Chopping
Forward Amorphous
Forward Point
⋆ all agree on input prefix
⋆⋆ all agree on condition
⋆ ⋆ ⋆ all agree on breakpoints
Table 7.1: Classification of program slicing techniques
Conditioned chopping is a very general form of slicing which subsumes both conditioned slicing and chopping, and hence, dynamic chopping (neither defined yet).
Dynamic chopping can be useful, for instance, in debugging. When tracing a computation, a programmer usually proceeds (big) step by step until a wrong subcomputation is found (e.g., a procedure call that returned a wrong result). Dynamic chopping
comes in handy at this point because it can compute the sentences that influenced
the wrong result from the procedure call. This dynamic chop is, in general, much
smaller than the static chop because it only considers a particular execution.
Finally, the last two rows of the table, forward amorphous and forward point
slicing correspond to the forward versions of these techniques.
The information in the table can also be used to identify relations between slicing
techniques. We have identified some relations and represented them in the graph of
182
7.3. THREE NEW ALGORITHMIC DEBUGGING STRATEGIES
Amorphous
Abstract
Conditioned
Program Slicing
n
Constrained Quasi-Static
n
Dynamic
Generalization
n
Decomposition
Simultaneous
InterProcedural
Static
183
2
Dicing
Chopping
IntraProcedural
Forward
Relevant
Superset
Backward
Composed of
Figure 7.1: Relationships between program slicing techniques
Figure 7.1. There are three kinds of arrows in the graph:
Generalization (S1 −→ S2 ): A slicing technique S2 generalizes another technique
S1 iff all the slicing criteria that can be specified with S1 can also be specified
with S2 .
Superset (S1 99K S2 ): The slice produced by a slicing technique S2 is a superset
of the slice produced by another technique S1 iff all the statements in S1 also
belong to S2 .
Composed of (S1 · · n· · ⊲ S2 ): A slicing technique S1 is composed of n other techniques S2 iff the slicing criterion of S1 contains n slicing criteria of S2 . In the
graph, for clarity, “Program Slicing” represents different slicing techniques. For
instance, chopping is composed of two slicing techniques (forward slicing and
backward slicing) and dicing is composed of n slicing techniques (one backward
slice of an incorrect value, and n − 1 backward slices of a correct value).
The classification points out abstract slicing as one of the most general slicing
techniques thanks to its use of predicates and conditions. While chopping generalizes
forward and backward slicing, an abstract chop [Hong et al., 2005] generalizes forward
and backward abstract slicing. Hence abstract chopping subsumes static, dynamic,
backward and forward slicing.
7.3
7.3.1
Three New Algorithmic Debugging Strategies
Less YES First
This section introduces a new variant of top-down search which further improves
heaviest first. It is based on the fact that every equation in the ET is associated with
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CHAPTER 7. A CLASSIFICATION OF PROGRAM SLICING...
a rule of the source code (i.e., the rule that the debugger identifies as buggy when
it finds a buggy node in the ET). Taking into account that the final objective of the
process is to find the program’s rule which contains the bug—rather than a node in
the ET—and considering that there is not a relation one-to-one between nodes and
rules because several nodes can refer to the same rule, it is important to also consider
the node’s rules during the search. A first idea could be to explore first those subtrees
with a higher number of associated rules (instead of exploring those subtrees with a
higher number of nodes).
Example 18 Consider the following ET:
1
2
3
4
5
6
8
7
8
8
7
where each node is labeled with its associated rule and where the oracle answered NO
to the question in the root of the tree. While heaviest first selects the right-most child
because this subtree has four nodes instead of three, less YES first selects the left-most
child because this subtree contains three different rules instead of two.
Clearly, this approach relies on the idea that all the rules have the same probability
of containing the bug (rather than all the nodes). Another possibility could be to
associate a different probability of containing the bug to each rule, e.g., depending on
its structure: Is it recursive? Does it contain higher-order calls?.
The probability of a node to be buggy is q · p where q is the probability that the
rule associated to this node is wrong, and p is the probability of this rule to execute
incorrectly. Therefore, under the assumption that all the rules have the same probability of being wrong, the probability P of a branch b to contain the bug is:
P =
Pn
i=1
pi
R
where n is the number of nodes in b, R is the number of rules in the program, and pi is
the probability of the rule in node i to produce a wrong result if it is incorrect. Clearly,
if we assume that a wrong rule always produces a wrong result1 (i.e., ∀i.pi = 1), then
1 This assumption is valid for instance in those flattened functional languages where all the conditions in the right-hand side of function definitions have been distributed between its rules. This is
relatively frequent in internal languages of compilers, but not in source languages.
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7.3. THREE NEW ALGORITHMIC DEBUGGING STRATEGIES
185
r
the probability is
where r is the number of rules in b, and thus, this strategy is (on
R
average) better than heaviest first. For instance, in Example 18 the left-most branch
has a probability of 38 to contain a buggy node, while the right-most branch has a
probability of 82 despite it has more nodes.
However, in general, a wrong rule can produce a correct result, and thus we need
to consider the probability of a wrong rule to return a wrong answer. This probability
has been approximated by the debugger Hat-delta (see Section 1.2.3) by using previous
answers of the oracle. The main idea is that a rule answered NO n times out of m is
more likely to be wrong than a rule answered NO n′ times out of m if n′ < n 6 m.
Here, we use this idea in order to compute the probability of a branch to contain a
buggy node. Hence, this strategy is a combination of the ideas from both heaviest first
and Hat-delta. However, while heaviest first considers the structure of the tree and
does not take into account previous answers of the user, Hat-delta does the opposite;
thus, the advantage of less YES first over them is the use of more information (both
the structure of the tree and previous answers of the user).
A direct generalization of Hat-delta for branches would result in counting the
number of YES answers of a given branch; but this approach would not take into
account the number of rules in the branch. In contrast, we proceed as follows:
When a node is set correct, we mark its associated rule and all the rules of its
descendants as correctly executed. If a rule has been executed correctly before, then
it will likely execute correctly again. The debugger associates to each rule of the
program the number of times it has been executed in correct computations based on
previous answers. Then, when we have to select a child to ask, we can compute the
total number of rules in the subtrees rooted at the children, and the total number of
answers YES for every rule.
This strategy selects the child whose subtree is less likely to be correct (and thus
more likely to be wrong). To compute this probability we calculate for every branch
b a weight wb with the following equation:
wb =
n
X
1
(Y ES)
i=1 ri
(Y ES)
where n is the number of nodes in b and ri
the rule r of the node i.
is the number of answers YES for
As with heaviest first, we select the branch with the biggest weight, the difference
is that this equation to compute the weight takes into account previous answers of
the user. Moreover, we assume that initially all the rules have been answered YES
once, and thus, at the beginning, this strategy asks those branches with more nodes,
but it becomes different as the number of questions asked increases.
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CHAPTER 7. A CLASSIFICATION OF PROGRAM SLICING...
As pointed out by Davie and Chitil [2006], independently of the considered strategy, taking into account the rule associated with each node can help to avoid many
unnecessary questions. Consider the following example:
Example 19 The ET associated with function append:
(1)
(2)
append [ ] y = y
append (x:xs) y = x:append xs y
w.r.t. the call “append [1,2,3,4] [5,6]” is the following:
2
2
2
2
1
where each node is labeled with its associated rule. Here, the bug can be found with
only one question to the node labeled “1”. If the answer is NO the bug is in rule 1,
if the answer is YES then we can stop the search because the rest of the nodes have
the same associated rule (2) and, thus, the bug must necessarily be in rule 2. Note
that the use of this information can significantly improve the search. With a list of n
numbers as the first argument, other strategies could need n+1 questions to reach the
bug.
Davie and Chitil have generalized this idea and introduced the tree compression technique [Davie and Chitil, 2006]. This technique proceeds as follows: For any node n1
with associated rule r1 and child n2 with associated rule r1 , it replaces node n2 with
its children (i.e., the children of n2 become the children of n1 ). Therefore, the tree
compression technique prunes ETs even before starting to ask any question.
With this strategy, the sequence of 9 questions asked for the ET in Figure 1.3 is:
Starting Debugging Session...
(1) main = False? NO
(2) sqrtest [1,2] = False? NO
(4) computs 3 = (9,9,8)? NO
(7) comput2 3 = 9? YES
(16) comput3 3 = 8? NO
(20) partialsums 3 = [6,2]? NO
(21) sum1 3 = 6? YES
(23) sum2 3 = 2? NO
(24) decr 3 = 2? YES
Bug found in rule:
sum2 x = div (x + (decr x)) 2
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7.3. THREE NEW ALGORITHMIC DEBUGGING STRATEGIES
7.3.2
187
Divide by YES & Query
The same idea used in less YES first can be applied in order to improve divide & query.
Instead of dividing the ET into two subtrees with a similar number of nodes, we can
divide it into two subtrees with a similar weight. The problem that this strategy tries
to address is the D&Q’s assumption that all the nodes have the same probability of
containing the bug. In contrast, this strategy tries to compute this probability.
By using the equation to compute the weight of a branch, this strategy computes
the weight associated to the subtree rooted at each node. Then, the node which divides
the tree into two subtrees with a more similar weight is selected. In particular, the
node selected is the node which produces a least difference between:
• w/2 and the heaviest node whose weight is less than or equal to w/2
• w/2 and the lightest node whose weight is greater than or equal to w/2
where w is the weight of the suspicious area in the ET.
As with D&Q, different nodes could divide the ET into two subtrees with a similar
weights; in this case, we could follow another strategy (e.g., Hirunkitti) in order to
select one of them.
We assume again that initially all the rules have been answered YES once. Therefore, at the beginning this strategy is similar to D&Q, but the differences appear as
the number of answers increases.
Example 20 Consider again the ET in Example 18. Similarly to D&Q, the first
node selected is the top-most “8” because only structural information is available.
Let us assume that the answer is YES. Then, we mark all the nodes in this branch
as correctly executed. Therefore, the next node selected is “2”; because, despite the
subtrees rooted at “2” and “5” have the same number of nodes and rules, we now have
more information which allows us to know that the subtree rooted at “5” is more likely
to be correct since node “7” has been correctly executed before.
The main difference with respect to D&Q is that divide by YES & query not only
takes into account the structure of the tree (i.e., the distribution of the program rules
between its nodes), but also previous answers of the user.
With this strategy, the sequence of 5 questions asked for the ET in Figure 1.3 is:
Starting Debugging Session...
(7) comput2 3 = 9? YES
(16) comput3 3 = 8? NO
(21) sum1 3 = 6? YES
(23) sum2 3 = 2? NO
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CHAPTER 7. A CLASSIFICATION OF PROGRAM SLICING...
(24) decr 3 = 2? YES
Bug found in rule: sum2 x = div (x + (decr x)) 2
7.3.3
Dynamic Weighting Search
Subterm dependency tracking (see Section 1.2.3) relies on the idea that if a subterm
is marked, then the error will likely be in the sequence of functions that produced
and passed the incorrect subterm until the function where the programmer found it.
However, the error could also be in any other equation previous to the origin of the
dependency chain.
Here, we propose a new strategy which is a generalization of subterm dependency
tracking and which can integrate the knowledge acquired by other strategies in order
to formulate the next question.
The main idea is that every node in the ET has an associated weight (representing
the probability of being buggy). After every question, the debugger gets information
that changes the weights and it asks for the node with a higher weight. When the
associated weight of a node is 0, then this node leaves the suspicious area of the ET.
Weights are modified based on the assumption that those nodes of the tree which
produced or manipulated a wrong (sub)term, are more likely to be wrong than those
that did not. Here, we compute weights instead of probabilities w.l.o.g. because this
will ease a later combination of this strategy with previous ones. We assume initially
that all the nodes have a weight 1 and that a weight 0 means “out of the suspicious
area”.
Computing Weights from Subterms
Firstly, as with subterm dependency tracking, we allow the oracle to mark a subterm
from an equation as wrong (instead of the whole equation). Let us assume that
the programmer is being asked about the correctness of the equation in a node n1 ,
and he marks a subterm s as wrong (or inadmissible). Then, the suspicious area is
automatically divided into four sets. The first set contains the node, say n2 , that
introduced s into the computation and all the nodes needed to execute the equation
in node n2 . The second set contains the nodes that, during the computation, passed
the wrong subterm from equation to equation until node n1 . The third set contains all
the nodes which could have influenced the expression s in node n2 from the beginning
of the computation. Finally, the rest of the nodes form the fourth set. Since these
nodes could not produce the wrong subterm (because they could not have influenced
it), the nodes in the fourth set are extracted from the suspicious area and, thus, the
new suspicious area is formed by the sets 1, 2 and 3.
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7.3. THREE NEW ALGORITHMIC DEBUGGING STRATEGIES
189
Each subset can be assigned a different probability of containing the bug. Let us
show it with an example.
Example 21 Consider the ET in Figure 1.3, where the oracle was asked about the
correctness of equation 3 and he pointed out the computed subterm “8” as inadmissible.
Then, the four sets are denoted in the figure by using different shapes and colors:
• Set 1: those nodes which evaluated the equation 20 to produce the wrong subterm
are denoted by an inverted trapezium.
• Set 2: those nodes that passed the wrong subterm until the programmer detected
it in the equation 3 are denoted by an ellipse.
• Set 3: those nodes that could influence the wrong subterm are denoted by a
trapezium.
• Set 4: the rest of nodes are denoted by a grey rectangle.
The source of a wrong subterm is the equation which computed it. From our experience, all the nodes involved in the evaluation of this equation are more likely to
contain the bug. However, it is also possible that the functions that passed this wrong
term during the computation should have modified it and they did not. Therefore,
they could also contain the bug. Finally, it is also possible (but indeed less likely)
that the equation that computed the wrong subterm had a wrong argument and this
was the reason why it produced a wrong subterm. In this case, this inadmissible argument should be further inspected. In the example, the wrong term “8” was computed
because equation 20 had a wrong argument “[6,2]” which should be “[6,3]”; the
nodes which computed this wrong argument have a trapezium shape.
Consequently, in the previous example, after the oracle marked “8” as wrong in
equation 3, we could increase the weight of the nodes in the first subset with 3, the
nodes in the second subset with 2, and the nodes in the third subset with 1. The
nodes in the fourth subset can be extracted from the suspicious area because they
could not influence the value of the wrong subterm and, consequently, their probability
of containing the bug is zero (see Section 6.6).
These subsets of the ET are in fact slices of different parts of the computation. In
order to extract these slices, we introduce some expressions which use the definitions
introduced in Section 6.4.
Lemma 58 (Origin of an expression in ETs) Given an ET E for an ART G and
an expression e ∈ E, e 6= main associated to n ∈ G marked as wrong or inadmissible
during an algorithmic debugging session, then, the equation in E that introduced e
into the computation is:
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CHAPTER 7. A CLASSIFICATION OF PROGRAM SLICING...
equ(G, m) = mef (G, mt)
where m = parent(n)
We often call this equation “ the origin of e”.
Proof. Firstly, e cannot be further evaluated because it appears in an equation of
E and from Definition 42 holds that n represents e in G. Let us assume that l = r
is the equation in E that introduced e into the computation (the origin of e). Then,
either e ∈ l or e ∈ r.
Since we assume that the computation starts in the distinguished function main
which have no arguments, and, because new expressions are only introduced in righthand sides, if e ∈ l then e = main. But one of the premises of the lemma is that
e 6= main, and thus, e ∈ r.
Let n′ be the node associated to r. Since e cannot be further evaluated then
parent(n) = parent(n′ ). Therefore, m = parent(n) is the node associated to l, and
mt = n′ is the node associated to r; then, following Definition 41, equ(G, m) =
mef (G, mt) is l = r.
✷
Note that e does not necessarily appear in the equation l = r ∈ E which is the
origin of e, because e could be a subexpression of l′ in the equation l′ = r′ ∈ E where
the oracle identified e. For instance, consider the question “3 − 1 = 2?” produced
from the ART in Figure 6.3. If the oracle marks 1 as inadmissible, then the equation
which introduced 1 into the computation is “decr 3 = 2?” which not includes 1 due
to the normalization process.
Each of the four subsets is computed as follows:
• The equations in E which belong to the Set 1 are those which are in the subtree
rooted at the origin of e.
• Given an ET E and two equations eq, eq ′ ∈ E where the expression e appears in
eq and the equation eq ′ is the origin of e, then, if the oracle marks e as wrong or
inadmissible during an algorithmic debugging session, the equations in E which
belong to the Set 2 are all the equations which are in the path between eq and
eq ′ except eq and eq ′ .
• The Set 4 contains all the nodes that could not influence the expression e, and
thus, it can be computed following the approach of Section 6.6.1. In particular,
by using the algorithm defined in Figure 6.4 a slice w.r.t. the expression e
can be computed. All the equations equ(G, n) = mef (G, nt) in E associated to
reductions n to nt in G that do not belong to the slice form Set 4.
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7.4. CLASSIFYING ALGORITHMIC DEBUGGING STRATEGIES
191
• Finally, Set 3 contains all the equations in E that do not belong to the Sets 1,
2 and 4.
This strategy can integrate information used by other strategies (e.g., previous
answers of the oracle) in order to modify nodes’ weights. In the next section we
introduce an algorithm to combine information from different strategies.
7.4
Classifying Algorithmic Debugging Strategies
7.4.1
Combining Algorithmic Debugging Strategies
In Figure 7.3 we can see that each strategy uses different information during the
search. Hence a combination of strategies can produce more efficient new strategies.
The algorithm in Figure 7.2 is a simple generic procedure for algorithmic debugging.
For simplicity, it assumes that the information provided by the oracle is stored in a
data structure ‘info’ which is combined by means of the operator ∪. Essentially, info
stores all the answers and expressions marked by the oracle, the information about
which nodes belong to the suspicious area and, in general, all the information needed
for the strategies being used as, for instance, the number of YES answers for every
node. It uses the following three functions:
bugFound(info,ET): It is a simple test to check if the bug can be found with the
information provided by the oracle so far.
askQuestion(node): It prompts the oracle with the question of a node and gets
new information.
computeWeights(info,ET): This function determines the new weights of the nodes
and selects the heaviest node w.r.t. a particular strategy.
Therefore, the strategy used in the algorithm is only implemented by computeWeights.
Now, we propose a combination of strategies to implement this function.
Function computeWeights proceeds by computing the weight of every node, and
then returning the node with the maximum weight. We compute the weight of a node
from the sum a + b + c + d + e + f where each component is the result of applying a
different strategy plus an addend. For instance:
a
b
c
d
e
f
=
=
=
=
=
=
a′ +
b′ +
c′ +
d′ +
e′ +
f′ +
Optimizations
Computing Weights from Subterms
Divide by YES & Query
Less YES First
Hirunkitti
Heaviest First
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CHAPTER 7. A CLASSIFICATION OF PROGRAM SLICING...
Input: an execution tree (ET )
Output: a node
Initialization: Nodes’ weight set to “1”; info = ∅
Repeat
node = computeWeights(info, ET );
info = info ∪ askQuestion(node);
Until bugF ound(info, ET )
Return: buggy node
Figure 7.2: Algorithmic debugging procedure
where a′ . . . f ′ are the addends and every strategy produces a weight between 0 and
a bound C. It is not difficult to modify a given strategy to produce a weight for
each node. For instance, heaviest first can be modified in order to assign a higher
weight to the nodes with more descendants. Hence, in Example 18, nodes labeled
5 and 2 would be assigned a weight of three, whilst nodes 3, 4, 6 and 7 would be
assigned a weight of two. In addition, we must ensure that weights are not greater
than the bound C, thus, in order to assign a weight to node n, we can use the equation:
C · weight(n)
weight(r)
where r is the root node of the ET. Other strategies can be modified in a similar way,
e.g., Less YES first assigns a higher weight to the nodes whose subtree is more likely
to contain the buggy node, and D&Q assigns a higher weight to the nodes that are
closer to the center of the ET.
Here, we consider Optimizations as another strategy because, independently of the
strategy used, it is always possible to apply some optimizations during the debugging
process. Some of them are obvious, as for instance excluding the first question (main)
from the suspicious area (because its answer is always NO), or propagating answers
to duplicate questions. For instance, in the ET of Figure 1.3 questions 11, 19 and 27
are identical and, thus, if one of them is answered with YES or NO, so are the others.
Other optimizations, on the other hand, are less obvious and they can influence the
order of the questions as it was shown in Example 19. Optimizations should be used
to select a particular node by assigning to it a weight C; and to eliminate nodes
which cannot be buggy (i.e., nodes equal to a node answered YES) by assigning them
a weight zero. Therefore, the addend of optimizations should be the greatest addend.
For instance, the previous combination of strategies with the addends a′ = 47C, b′ =
23C, c′ = 11C, d′ = 5C, e′ = 2C and f ′ = 0 would apply the optimizations first (if
192
7.4. CLASSIFYING ALGORITHMIC DEBUGGING STRATEGIES
193
possible), then, whenever two nodes have the same weight, computing weights from
subterms would be applied to break the tie, and so on. This is due to the fact that
these addends guarantee the following relations:
a > b + c + d + e + f,
b > c + d + e + f,
c > d + e + f,
d > e + f,
e>f
Note the importance of C—but not of the value of C—to ensure the previous relations.
In contrast, with the addends a′ = b′ = ... = f ′ all the strategies would be applied at
the same time.
The best combination of strategies depends in each case on the current context
(i.e., branching factor of the tree, previous answers of the oracle, etc) thus the combination of strategies used by the debugger should also be dynamically modified during
the algorithmic debugging process. Therefore, the debugger itself should state the
addends dynamically after every question. In the next section we compare the cost
of a set of strategies in order to theoretically determine their addends.
7.4.2
Comparing Algorithmic Debugging Strategies
A summary of the information used by every strategy is shown in Figure 7.3. The
meaning of each column is the following:
• ‘(Str)ucture’ is marked if the strategy takes into account the distribution of
nodes (or rules) in the tree;
• ‘(Rul)es’ is marked if the strategy considers the rules associated with nodes;
• ‘(Sem)antics’ is marked if the strategy follows an order of semantically related
questions, the more marks the more strong relation between questions;
• ‘(Ina)dmissibility’ is marked if the strategy accepts “inadmissible” answers;
• ‘(His)tory’ is marked if the strategy considers previous answers in order to select
the next node to ask (besides cutting the tree);
• ‘(Div)isible’ is marked if the strategy can work with a subset of the whole ET.
ETs can be huge and thus, it is desirable not to explore the whole tree after
every question. Some strategies allow us to only load a part of the tree at a
time, thus significatively speeding up the internal processing of the ET; and
hence, being much more scalable than other strategies that need to explore the
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CHAPTER 7. A CLASSIFICATION OF PROGRAM SLICING...
Strategy
Single Stepping
Top-Down Search
Top-Down Zooming
Heaviest First
Less YES First
Divide & Query
Biased Weighting D&Q
Hirunkitti’s D&Q
Divide by YES & Query
Hat-delta
Subterm Dependency Track.
Dynamic Weighting Search
Str.
X
X
X
X
X
X
X
Rul.
X
X
X
X
Sem.
X
XX
X
X
XXX
-
Ina.
X
X
His.
X
X
X
X
Div.
X
X
X
-
Cost
n
b·d
b·d
b·d
b·d
b · log2 n
b · log2 n
b · log2 n
n
n
n
Figure 7.3: Comparing algorithmic debugging strategies
whole tree before every question. For instance, top-down can load the nodes
whose depth is less than d, and ask d questions before loading another part of
the tree. Note, however, that some of the non-marked strategies could work
with a subset of the whole ET if they where restricted. For instance, heaviest
first could be restricted by simply limiting the search for the heaviest branch
to the loaded nodes of the ET. Other strategies need more simplifications: less
YES first or Hat-delta could be restricted by only marking as correctly executed
the first d levels of descendants of a node answered YES; and then restricting
the search for the heaviest branch (respectively node) to the loaded nodes of
the ET. Finally,
• ‘Cost’ represents the worst case query complexity of the strategy. Here, n
represents the number of nodes in the ET, d its maximum depth and b its
branching factor.
The cost of single stepping is too expensive. Its worst case query complexity is
order n, and its average cost is n/2.
Top-down and its variants have a cost of b · d which is significantly lower than the
one of single stepping. The improvement of top-down zooming over top-down is based
on the time needed by the programmer to answer the questions; their query complexity
is the same. Note that, after the tree compression described in Section 7.3.1, no zoom
is possible and, thus, both strategies are identical.
In contrast, while in the worst case the costs of top-down and heaviest first are
equal, in the mean case heaviest first performs an improvement over top-down. In
particular, on average, for each wrong node with b children si , 1 ≤ i ≤ b:
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7.4. CLASSIFYING ALGORITHMIC DEBUGGING STRATEGIES
195
b+1
of the children.
2
Pb
i=1 weight(si ) · pos(si )
of the children.
• Heaviest first asks
Pb
i=1 weight(si )
• Top-down asks
where function weight computes the weight of a node and function pos computes
the position of a node in a list containing it and all its brothers which is ordered
by their weights.
In the case of less YES first, the improvement is based on the fact that the heaviest
branch is not always the branch with a higher probability of containing the buggy
node. While heaviest first and less YES first have the same worst case query complexity, their average cost must be compared empirically.
D&Q is optimal in the worst case, with a cost order of (b · (log2 n)).
The cost of the rest of strategies is highly influenced by the answers of the user.
The worst case of Hat-delta happens when the branching factor is 1 and the buggy
node is in the leaf of the ET. In this case the cost is n. However, in normal situations,
when the ET is wide, the worst case is still close to n; and it occurs when the first
branch explored is answered YES, and the information of YES answers obtained
makes the algorithmic debugger explore the rest of the ET bottom up. It can be seen
in Example 22.
Example 22 Consider the following ET:
1
2
3
4 4 4
3
2
3
3
2
3
...
4 4 4
3
3
3
4 4 4
If we use the version of Hat-delta which counts correct computations, then the first
node asked would be the left-most 2. If it is answered YES, then all the nodes in this
branch are marked as correctly executed, and thus, rule 2 has been correctly executed
once, rule 3 has been correctly executed twice and rule 4 has been correctly executed
three times. Therefore, the rest of the nodes are asked bottom up. In this case, the
.
cost is n − (d−1)d
2
Despite subterm dependency tracking is a top-down version enriched with additional
information provided by the oracle, this information (that we assume correct here to
195
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CHAPTER 7. A CLASSIFICATION OF PROGRAM SLICING...
compute the costs) could make the algorithmic debugger ask more questions than with
the standard top-down. In fact, this strategy—and also dynamic weighting search if
we assume that top-down is used by default—has a worst case query complexity
of n because the expressions marked by the programmer can make the algorithmic
debugger explore the whole ET. The next example shows the path of nodes asked by
the algorithmic debugger in the worst case.
Example 23 Consider the following program:
f
g
h
i
j
k
x
x
x
x
x
x
=
=
=
=
=
=
g (k (g x))
j (i (h x))
x+1
x+1
x+1
x+1
whose ET w.r.t. the call f 1 is:
f1=8
g5=8
j7=8
i6=7
k4=5
h5=6
j3=4
g1=4
i2=3
h1=2
Here, the sequence of nodes asked with subterm dependency tracking is depicted with
dotted arrows and the expressions marked by the programmer as wrong or inadmissible
are underlined. Then, the debugging session is:
Starting Debugging Session...
f 1 = 8? NO (The user selects
j 7 = 8? YES (The user selects
i 6 = 7? YES (The user selects
h 5 = 6? YES
g 5 = 8? YES (The user selects
k 4 = 5? YES (The user selects
j 3 = 4? YES (The user selects
i 2 = 3? YES (The user selects
h 1 = 2? YES
g 1 = 4? NO
Bug found in rule:
g x = j (i (h x))
"8")
"7")
"6")
"5")
"4")
"3")
"2")
196
7.4. CLASSIFYING ALGORITHMIC DEBUGGING STRATEGIES
197
And, thus, all the ET’s nodes are asked.
As a summary, the following diagram shows the relation between the costs of the
strategies discussed so far:
> >
Top-Down Search = Top-Down Zooming > Heaviest First
Less YES First
> >
Divide & Query > Hirunkitti
>
Single Stepping > Divide by YES & Query
>
Hat Delta
Dynamic Weighting Search
Subterm Dependency Tracking
This diagram together with the costs associated to every strategy in Figure 7.3
allows algorithmic debuggers to automatically determine which strategy to use depending on the structure of the ET. This means that the algorithmic debugger can
dynamically change the strategy used during a debugging session. For instance, after
every question the new maximum depth (d) of the ET and its remaining number of
nodes (n) is computed; if d < log2 n then heaviest first is used in order to select a new
node, else Hirunkitti is applied. If we use this result in the combination of strategies
proposed in Section 7.4.1 we can conclude that if d < log2 n then f ′ < e′ .
197
Chapter 8
Conclusions and Future Work
We conclude by summarizing the main contributions of this thesis and pointing out
several directions for further research.
8.1
Conclusions
Bugs are a human symptom. No matter which language or methodology we use, we
will involuntarily introduce bugs in our specifications. It is not due to our lack of
knowledge or undiscipline; the reason relies on the human incapacity to predict all
the future possible consequences of our actions in the present. The automatization
of software development tasks is gradually decreasing the number of initial bugs in
programs, nevertheless, until the day that all the software development process is fully
automated—while humans participate in the process—programs will contain bugs.
This pessimism is not unfounded (some discussions can be found, e.g., in [Agrawal,
1991]), and shows the intrinsically necessity of debugging tools. Software can be a
very sophisticated and complex beast, what justifies the more than thirty years of
research and efforts to produce usable debugging tools. Every new programming
language or technique opens a number of doors for new research: program slicing
produced static slicing, dynamic slicing, quasi-static slicing, simultaneous dynamic
slicing, conditioned slicing and many other approaches; profiling produced program
counter sampling, block counting and compiler profilers, including techniques to measure CPU usage, call counts, call cost, memory usage, I/O operations, etc; tracing
produced breakpoints, execution backtracking, and has been combined with a number
of techniques such as program slicing and algorithmic debugging; all of them towards
the search of the same thing: finding and eliminating bugs from programs.
After all, the task of debugging still remains complex. The problem is not solved
by far, and it calls for further investigation. One of the main problems of current
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debugging tools is that after generating and using all the data structures (e.g., redex trails, instrumented programs, etc.) needed for debugging, they become mostly
useless, and they must be generated and analyzed again for a different program or
computation. This is one of the causes that many debugging tools can be directly
classified as abandonware. Some bugs can be easily found, others, on the contrary,
can be extremely difficult. If the debugging tool is not close to the user, or it requires
much knowledge or work to be used, it is not used in practice. In general, many
programmers are too lazy to learn and use complex debugging tools that hopefully
will help them to find a bug that maybe they can find with a print statement.
Learning to use a debugging tool is a long term investment; it becomes interesting
while you use them more and more, because both your knowledge about the debugger
and the knowledge of the debugger1 about your programs increase. Unfortunately,
there is currently a strong tendency to reject this investment. This situation could
change with the improvement of debugging tools rather than its intrinsic techniques.
As stated earlier [Wadler, 1998], there is a wide gap between researchers and programmers: The formers put much efforts in defining debugging techniques and little
efforts in producing prototypes that, at the end, are far from the programmers needs.
Debugging tools help the user to solve problems; if they introduce new problems
they become useless. For instance, if the debugger needs a complex specification
(e.g., about the intended semantics of the program) it can enter into a “buggy search
of bugs” where the programmer has accidentally introduced a bug in the specification
for the debugger. Therefore, one strong requirement of debugging tools is the usability.
They must be easy to use and provide the user with concrete information. Modern
graphical interfaces can play a crucial role in order to increment the usage rate of
debugging tools.
In this thesis, we have developed some debugging techniques that (hopefully)
constitute a step forward in the search for the free-bugs software. In short, the main
contributions of the thesis are:
1. Profiling Techniques.
We have introduced a novel program transformation to measure the cost of
lazy functional (logic) computations. This is a difficult task mainly due to the
on-demand nature of lazy evaluation. The approach is simple, precise—i.e., it
computes exact costs rather than upper/lower bounds—, and fully automatic.
However, the symbolic costs computed are only based on function unfoldings.
Therefore, we introduced another profiling scheme for modern functional logic
languages which is more advanced and is able to analyze a set of different symbolic costs.
1 This is the case, e.g., of modern declarative debuggers, which maintain a database of incremental
acquired knowledge about programs.
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8.1. CONCLUSIONS
201
Firstly, we instrument a natural (big-step) semantics for functional logic programs which covers laziness, sharing and non-determinism. This contrasts with
[Sansom and Jones, 1997], where logical features are not considered, and [Albert and Vidal, 2002], where sharing is not covered (which drastically reduces its
applicability). The instrumented semantics associates a symbolic cost to each
basic operation (e.g., variable updates, function unfoldings, case evaluations).
While this cost semantics provides a formal basis to analyze the cost of a computation, the implementation of a cost-augmented interpreter based on it would
introduce a huge overhead. Therefore, we also introduced a sound transformation that instruments a program such that its execution—under the standard
semantics—yields not only the corresponding results but also the associated
costs.
2. Program Slicing Techniques.
We have presented a new scheme to perform dynamic slicing in lazy functional
logic programs which is based on redex trails. For this purpose, we have introduced a non-trivial notion of slicing criterion which is adequate in the context
of lazy evaluation. We have also formalized the concept of minimal dynamic
backward slicing in this context. We have defined the first (complete) dynamic
slicing technique for the Curry language, we have shown how tracing (based
on redex trails) and slicing can be combined into a single framework, and, we
have presented a lightweight approach to program specialization based on the
dynamic slicing technique.
Finally, we have compared a wide variety of program slicing based techniques in
order to classify them according to their properties and in order to establish a
hierarchy of slicing techniques. In particular, we have extended a comparative
table of slicing techniques by Harman et al. with new dimensions and techniques.
The classification of techniques presented not only shows the differences between
the techniques, but it also allows us to predict future techniques that will fit a
combination of parameters not used yet.
With the information provided in this classification we have studied and identified three kinds of relations between the techniques: composition, generalization
and set relations. This study allows us to answer questions like: Is one technique
more general than another technique? Is the slice produced by one technique
included in the slice of another technique? Is one technique composed of other
techniques?
3. Algorithmic Debugging Techniques.
We have presented a technique to improve algorithmic debugging by combining
it with program slicing. Firstly, we have adapted some program slicing concepts
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CHAPTER 8. CONCLUSIONS AND FUTURE WORK
(and proven some of their properties) to the context of augmented redex trails.
Based on this adaptation, we have introduced an algorithm to slice ARTs and
proven its completeness. We have slightly modified the slicing algorithm for the
case of algorithmic debugging in order to optimise the number of nodes in slices
so that only profitable nodes to build execution trees are included in the slice.
With the information provided by the user the introduced algorithm produces a
slice which only contains the relevant parts of the execution tree, thus reducing
the number of questions.
We have introduced three new strategies and some optimizations for algorithmic
debugging. Less YES first tries to improve heaviest first and divide by YES &
query tries to improve D&Q by considering previous answers of the oracle during
the search. Dynamic weighting search allows the user to specify the exact part
of an equation which is wrong. This extra information can produce a much
more accurate debugging session.
We have introduced an algorithm to combine different strategies, and we have
classified and studied their costs in order to determine how they should be
combined in different contexts. The study has determined which strategies are
(on average) better. In particular, the result of the study points out heaviest
first, less YES first, Hirunkitti and divide by YES & query and Hat Delta as
the most efficient. The theoretical comparative has produced objective criteria
to dynamically change the strategy used during a debugging session depending
on the ET’ structure.
8.2
Future Work
There are several interesting directions for continuing the research presented in this
thesis. Let us briefly enumerate some of them:
1. Profiling Techniques.
In Chapter 3, we only sketched the use of logical variables for reasoning about
the complexity of a program. This idea can be further developed and, perhaps,
combined with partial evaluation so that we get a simpler representation of the
time-equations for a given expression. The extension of our developments to
cope with higher-order functions seems essential to be able to analyze typical
lazy functional (logic) programs. For instance, this can be done by defunctionalization, i.e., by including an explicit (first-order) application operator.
2. Program Slicing.
There are several interesting topics for future work concerning the slicing technique presented in Chapter 5. On the one hand, we plan to define appropriate
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8.2. FUTURE WORK
203
extensions in order to cover the additional features of Curry programs (higherorder, constraints, built-in functions, etc). On the other hand, we plan to investigate the definition of appropriate methods for the static slicing of functional
logic programs. In this case, we think that our approach could still be used but
a sort of abstract redex trail becomes necessary for guaranteeing its finiteness
even when no input data are provided.
Another interesting extension of this work consists in adapting our developments
to pure functional languages. For instance, Haskell tracers (e.g., Sparud and
Runciman [1997]; Wallace et al. [2001]) usually rely on some program transformation to instrument a source program so that its execution—under the
standard semantics—generates the associated redex trail. The inclusion of program positions within these approaches can also be done in a similar way as in
our approach. Then, the implementation of a dynamic slicer mainly amounts
to implement an algorithm to compute reachable nodes in a graph.
For the specialization technique we plan to extend the current method in order
to perform more aggressive simplifications (i.e., a form of amorphous slicing).
This extended form of slicing can be very useful in the context of pervasive
systems where resources are limited and, thus, code size should be reduced as
much as possible. Again, another promising topic is the definition of program
specialization based on static slicing rather than on dynamic slicing. In particular, it would be interesting to establish the strengths and weaknesses of each
approach.
3. Algorithmic Debugging.
The classification and comparison of algorithmic debugging strategies presented
in Chapter 7 must be further investigated through empirical experiments with
real programs. An empirical comparison of all the strategies would allow us
to determine a weighting for their combination. With the knowledge acquired
from this experiment we will be able to approximate the strategies’ weights and
to determine how they should change and on which factors this change depends.
This thesis has been written in English, which is not the author’s native language.
The author apologizes for any mistakes which may be due to this circumstance.
203
Acknowledgements
205
Chapter 9
Acknowledgements
When I started to work in this thesis, my motivation lied me making me think that it
was going to be easy. Some time later, I was completely lost in the middle of nothing:
without any idea of how to step forward or if there was any sense in my work. This
horrible feeling has invaded me many times, but I have always found a hand that hold
me and encouraged me to go on.
These hands are the real authors of this work.
The unconditional and continuous support of my parents has always hold me and
I am completely sure that it will continue holding me for the rest of my life. This is
one of these debts that you know that you will never be able to paid.
Probably none of my family will never understand a word of this thesis, but they
have always being asking me about its problems and my solutions. This love is one
of the hands that I use to remember that I cannot defraud them.
Also my friends, Amparo, Carlos, Lolita, Vivi and, fortunately, many others—who
often know what I think much better than I do—are able to transform my biggest
problems into tiny sand grains with their friendly advices.
But all these hands together were not enough!
Sometimes I had to fight with technical problems that were much bigger than I
was. In these situations I only had to knock on one of the front red doors of knowledge
and get an incredible precise answer. Yes, I never was alone; I had a suit of brains,
each of them an expert in its field, helping me to find the path to hide solutions. By
door order: Cesar Ferri, José Orallo, Santiago Escobar, Salvador Lucas, Marı́a José
Ramı́rez, Marı́a Alpuente, Javier Oliver, Marisa Llorens, Carlos Herrero, Pepe Iborra,
Raúl Gutierrez, Antonio Bella, Beatriz Alarcón, Salvador Tamarit, Gustavo Arroyo
and Diego Cheda.
All of them are my colleagues who always help me when I need it and who I admire
for their work. Thanks indeed to all of you.
207
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CHAPTER 9. ACKNOWLEDGEMENTS
I have a special thank for three people that have been very close to me during the
thesis. They had the ability to make my work easier.
First, I want to thank my formerly officemate Alicia Villanueva for her friendly
support and patience. Secondly, I want to thank my colleague Guadalupe Ramos who
has provide me many ideas in our discussions, and who always listened to me with
incredible patience, indeed when he was sure that I was wrong and he was right. I am
sure you will succeed with your projects. I will miss you a lot. Finally, I am greatly
thankful to my friend Vicent Estruch who is a model of person that I will always
admire. I feel very lucky to be his friend.
All of them constitute the ELP research group that I am proud to belong to.
I also want to thank my coauthors and colleagues for their valuable contributions
to this thesis; specially to Claudio Ochoa and Demis Ballis for being as they are.
During the first year they gave me power for the rest of the thesis. Also to Michael
Hanus, Frank Huch, Bernd Braßel and Sebastian Fischer for giving me the first push
in Kiel. Finally, to Olaf Chitil, Thomas Davie and Yong Luo for kindly inviting me
to their university and make my research period at Canterbury a pleasure.
When something finishes, you always think in how it started. I will always thank
the patience of Isidro Ramos and Pepe Carsı́ who introduced me in the research world
for the first time.
Finally, there has been a hand who was the most strong holding me. It is the
infinite tender love that I continuously get from Vanessa. She made me ignore the
problems because I always knew that it really did not matter if I was going to succeed
on my work. After all, she was going to be there. Proud of me...
Valencia, 2007
Josep Silva
208
Agradecimientos
209
Chapter 10
Agradecimientos
Cuando comencé a trabajar en esta tesis, mi motivación me mintió haciéndome creer
que iba a ser una tarea sencilla. Algún tiempo después estaba completamente perdido
en medio de la nada: sin saber cómo seguir adelante o si habı́a algún sentido en mi
trabajo. Este horrible sentimiento me ha invadido muchas veces, pero siempre he
encontrado una mano que me ha levantado y me ha animado a seguir.
Estas manos son los verdaderos autores de este trabajo.
El continuo e incondicional apoyo de mis padres siempre me ha ayudado a continuar y estoy totalmente seguro de que continuará haciéndolo durante el resto de mi
vida. Ésta es una de esas deudas que sabes que nunca podrás pagar.
Probablemente nadie en mi familia entenderá nunca una palabra de esta tesis; sin
embargo, siempre me han estado preguntando acerca de los problemas que encontraba
y las soluciones que iba planteando. Este amor es una de las manos que utilizo para
recordar que no puedo defraudarles.
También mis amigos, Amparo, Carlos, Lolita, Vivi y, afortunadamente, muchos
otros—que a menudo saben lo que pienso mucho mejor que yo mismo—son capaces
de transformar mis mayores problemas en insignificantes granos de arena con sus
cariñosos consejos.
Pero todas estas manos juntas no fueron suficientes!
Algunas veces tuve que enfrentarme con problemas técnicos que eran mucho más
grandes que yo. En esos momentos, sólamente tuve que llamar a una de las puertas rojas del conocimiento que tengo enfrente y obtener una increı́blemente precisa
respuesta. Sı́, nunca estuve sólo, tenı́a una baterı́a de cerebros, cada uno de ellos experto en una materia, ayudándome a encontrar el camino a las respuestas ocultas. Por
orden de puerta: Cesar Ferri, José Orallo, Santiago Escobar, Salvador Lucas, Marı́a
José Ramı́rez, Marı́a Alpuente, Javier Oliver, Marisa Llorens, Carlos Herrero, Pepe
Iborra, Raúl Gutierrez, Antonio Bella, Beatriz Alarcón, Salvador Tamarit, Gustavo
211
212
CHAPTER 10. AGRADECIMIENTOS
Arroyo y Diego Cheda.
Todos ellos son mis compañeros, que siempre me han ayudado cuando lo he necesitado y a los cuales admiro por su trabajo. Gracias de verdad a todos vosotros.
Tengo un agradecimiento especial para tres personas que han estado muy cerca
de mi durante la tesis. Ellas tuvieron la habilidad de hacer mi trabajo más fácil.
Primero, quiero agradecer a mi antigua compañera de despacho Alicia Villanueva
su ayuda y paciencia. Segundo, quiero dar las gracias a mi colega José Guadalupe
Ramos que me ha dado buenas ideas durante nuestras conversaciones, y que siempre
me ha escuchado con increible paciencia, incluso cuando estaba seguro de que él tenı́a
razón y yo me equivocaba. Estoy seguro de que tendrás éxito con tus proyectos. Te
echaré mucho de menos. Por último, le estoy enormemente agradecido a mi amigo
Vicent Estruch que es un modelo de persona y al cual siempre admiraré. Me siento
muy afortunado de ser su amigo.
Todos ellos constituyen el grupo de investigación ELP al cual estoy orgulloso de
pertenecer.
También quiero agradecer a mis coautores y colegas por sus valiosas contribuciones
a esta tesis; especialmente a Demis Ballis y Claudio Ochoa por ser como son. Durante
el primer año de tesis me dieron energias suficientes para el resto de la tesis. También
a Michael Hanus, Frank Huch, Bernd Braßel y Sebastian Fischer por darme el primer
empujón en Kiel. Finalmente, a Olaf Chitil, Thomas Davie y Yong Luo por invitarme
amablemente a su universidad y hacer que mi periodo de investigación allı́ fuera un
placer.
Cuando algo termina, siempre piensas en cómo empezó. Siempre agradeceré a
Isidro Ramos y Pepe Carsı́ el haberme introducido en el mundo de la investigación.
Finalmente, ha habido una mano sujetándome más fuerte que todas las demás.
Es el infinito y tierno amor que continuamente recibo de Vanessa. Ella me ha hecho
ignorar los problemas porque siempre he sabido que no importaba si al final iba a
tener éxito o no en mi trabajo. Después de todo, ella iba a estar ahı́. Orgullosa de
mı́...
Valencia, 2007
Josep Silva
212
To German Vidal,
my advisor, my instructor, my mentor, my director;
my tutor, my monitor, my teacher, my professor;
my leader, my guide, my trend and my friend.
The man who stepped on my work:
fetter my theorems, counterexampled my proofs,
criticized my presentations,
and never valued my implementations.
Who made me read when I wanted to write.
Who shut me up when I wanted to talk.
Who always told me that I was wrong,
who made me leave my country for long
and live where I never wanted to go.
Who introduced me to people I admired or so.
Who made me stop when I wanted to run.
Who was serious when I wanted fun.
Who forced me to study things I thought were useless the most.
The man who found me when I was lost.
Without any apparent reason
converted my life into a hell, into a prison,
and made me feel like a slave.
This man made me enter into a black cave,
a maze without any exit: research.
He made me mind, and search,
where there was nothing to find...
And, when I couldn’t resist,
when I was about to desist,
he took my hand and told me that
I was a doctor.
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