Michal Walicki
University of Bergen, Department of Informatics, Faculty Member
Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models... more
Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models for inconsistent theories, specializing to the classical semantics for the consistent ones. Direct (instead of refutational) reasoning with classical resolution is sound and complete for this semantics, when augmented with a specific weakening which, in particular, excludes Ex Falso. Dropping all forms of weakening yields reasoning which also avoids typical fallacies of relevance.
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Kernels of digraphs provide an alternative presentation of the classical propositional semantics, [3, 2, 9]. Semikernels generalize kernels and, with some additional restrictions, provide semantics also for inconsistent theories, reducing... more
Kernels of digraphs provide an alternative presentation of the classical propositional semantics, [3, 2, 9]. Semikernels generalize kernels and, with some additional restrictions, provide semantics also for inconsistent theories, reducing to the classical kernels whenever the theory is consistent. The used semikernels are kernels of a maximal consistent subtheory (defined in the paper), so that semantics of any (also inconsistent) theory is the classical semantics of such a subtheory. We show that a classical resolution system, introduced in [9], is sound and complete (modulo weakening) for this new semantics, when used for direct, instead of refutational, reasoning. Natural interpretation of the Graph Normal Form, underlying the presentation, gives immediate applications to the analysis of paradoxes, which are used in the examples. §
Graph normal form, GNF, [1], was used in [2, 3] for analyzing paradoxes in propositional discourses, with the semantics—equivalent to the classical one—defined by kernels of digraphs. The paper presents infinitary, resolution-based... more
Graph normal form, GNF, [1], was used in [2, 3] for analyzing paradoxes in propositional discourses, with the semantics—equivalent to the classical one—defined by kernels of digraphs. The paper presents infinitary, resolution-based reasoning with GNF theories, which is refutationally complete for the classical semantics. Used for direct (not refutational) deduction it is not explosive and allows to identify in an inconsistent discourse, a maximal consistent subdiscourse with its classical consequences. Semikernels, generalizing kernels, provide the semantic interpretation.
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xxiv, 489 p. ; 24 cm
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History of Logic Basic Set Theory and Induction Turing Machines and Computability Propositional Logic: Hilbert's and Gentzen's Proof Systems Boolean and Set-Valued Semantics DNF and CNF Soundness and Completeness First-Order... more
History of Logic Basic Set Theory and Induction Turing Machines and Computability Propositional Logic: Hilbert's and Gentzen's Proof Systems Boolean and Set-Valued Semantics DNF and CNF Soundness and Completeness First-Order Logic: Hilbert's and Gentzen's Proof Systems Tarski's Semantics Prenex NF Term Models and Prolog Soundness and Completeness Exercises to Each Topic.
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Research Interests: Engineering, Cognitive Science, Computer Science, Software Engineering, Library Science, and 13 moreSoftware Development, Software Evolution, Transformational Leadership, Formal methods, Theoretical Computer Science, Software, Computer Software, Rewriting, Code Refactoring, Informatik, Correctness, Programming language, and Zeitschrift
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Multialgebras generalise algebraic semantics to handle nondeterminism. They model relational structures, representing relations as multivalued functions by selecting one argument as the “result”. This leads to strong algebraic properties... more
Multialgebras generalise algebraic semantics to handle nondeterminism. They model relational structures, representing relations as multivalued functions by selecting one argument as the “result”. This leads to strong algebraic properties missing in the case of relational structures. However, such strong properties can be obtained only by first choosing appropriate notion of homomorphism. We summarize earlier results on the possible notions of compositional homomorphisms of multialgebras and investigate in detail one of them, the outer-tight homomorphisms which yield rich structural properties not offered by other alternatives. The outer-tight homomorphisms are different from those obtained when relations are modeled as coalgebras and the associated congruence is the converse bisimulation equivalence. The category is cocomplete but initial objects are of little interest (essentially empty). On the other hand, the category does not, in general, possess final objects for the usual cardinality reasons. The main objective of the paper is to show that Aczel's construction of final coalgebras for set-based functors can be modified and applied to multialgebras. We therefore extend the category admitting also structures over proper classes and show the existence of final objects in this category.
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ABSTRACT
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... Dr. Cavaco Silva, 2744-016 Porto Salvo, Portugal (phone: +351 21 423 3552; email:diogo.ferreira@ist.utl.pt) *This work was performed while the first author was visiting the Technical University of Lisbon. 978-1-4244-8126-2/101$26.00... more
... Dr. Cavaco Silva, 2744-016 Porto Salvo, Portugal (phone: +351 21 423 3552; email:diogo.ferreira@ist.utl.pt) *This work was performed while the first author was visiting the Technical University of Lisbon. 978-1-4244-8126-2/101$26.00 ©201 0 IEEE ...
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We discuss briefly the duality (or rather, complementarity) of system descriptions based on actions and transitions, on the one hand, and states and their changes, on the other. We settle for the latter and present a simple language, for... more
We discuss briefly the duality (or rather, complementarity) of system descriptions based on actions and transitions, on the one hand, and states and their changes, on the other. We settle for the latter and present a simple language, for describing state changes, which is parameterized by an arbitrary language for describing properties of the states. The language can be viewed as a simple fragment of step logic, admitting however various extensions by appropriate choices of the underlying logic. Alternatively, it can be seen as a very specific fragment of temporal logic (with a variant of 'until' or 'chop' operator), and is interpreted over dense (possibly continuous) linear time. The reasoning system presented here is sound, as well as strongly complete and decidable (provided that so is the parameter logic for reasoning about a single state). We give the main idea of the completeness proof and suggest a wide range of possible applications (action based descript...
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We introduce a variant of pointer structures with denotational semantics and show its equivalence to systems of boolean equations: both have the same solutions. Taking paradoxes to be statements represented by systems of equations (or... more
We introduce a variant of pointer structures with denotational semantics and show its equivalence to systems of boolean equations: both have the same solutions. Taking paradoxes to be statements represented by systems of equations (or pointer structures) having no solutions, we thus obtain two alternative means of deciding paradoxical character of statements, one of which is the standard theory of solving boolean equations. To analyze more adequately statements involving semantic predicates, we extend propositional logic with the assertion operator and give its complete axiomatization. This logic is a sub-logic of statements in which the semantic predicates become internalized (for instance, counterparts of Tarski’s definitions and T-schemata become tautologies). Examples of analysis of self-referential paradoxes are given and the approach is compared to the alternative ones.
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We introduce a framework that generalizes algebraic specifications by equipping algebras with descriptions of evaluation strategies. The resulting abstract mathematical description allows one to model the implementation of algebras on... more
We introduce a framework that generalizes algebraic specifications by equipping algebras with descriptions of evaluation strategies. The resulting abstract mathematical description allows one to model the implementation of algebras on various platforms in a way that is independent of the function-oriented specifications.We study algebras with associated data dependencies. The latter provide separate means for modelling computational aspects apart from the functional specifications captured by an algebra. The formalization of evaluation strategies (1) introduces increased portability among different hardware platforms, and (2) allows a potential increase in execution efficiency, since a chosen evaluation strategy may be tailored to a particular platform. We present the development process where algebraic specifications are equipped with data dependencies, the latter are refined, and, finally, mapped to actual hardware architectures.
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The Problem Epistemic logics are logics for reasoning about knowledge in systems of agents. Traditional modal epistemic logics are information-theoretical, in the sense that they consider knowledge about propositions and that agents know... more
The Problem Epistemic logics are logics for reasoning about knowledge in systems of agents. Traditional modal epistemic logics are information-theoretical, in the sense that they consider knowledge about propositions and that agents know everything that follows from the information they possess. Although proved useful in many applications, this view is unrealistic when modeling explicit knowledge – i.e. knowledge that agents have computed and can act upon. Hintikka (1975) called this the logical omniscience problem, and when epistemic logic is used to explain concepts in game theory, the idea of logical non-omniscience is closely related to the idea of bounded rationality. Finding more realistic epistemic logics modeling how explicit knowledge is obtained has been a popular research topic. Our goal in this paper is different. First, instead of describing closure conditions on explicit knowledge, we assume that explicit knowledge has been obtained, and we construct a logic for reasoning about static explicit knowledge in a group of agents. A framework for reasoning about static knowledge is useful for analyzing the knowledge in a group of agents at an instant in time (or a time span when no epistemic changes are made), for example between computing deductions. Second, we consider ascribing knowledge of propositions to agents in multiagent systems to be unrealistic. Instead, we assume that the agents posses and process syntactical objects. (Of course, neither of these ideas are new; we briefly discuss related work in the last section). We make the following semantic assumptions:
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Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models... more
Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models for inconsistent theories, specializing to the classical semantics for the consistent ones. Direct (instead of refutational) reasoning with classical resolution is sound and complete for this semantics, when augmented with a specific weakening which, in particular, excludes Ex Falso. Dropping all forms of weakening yields reasoning which also avoids typical fallacies of relevance.
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A major motivating force behind research into abstract data types and algebraic specifications is the realization that software in general and types in particular should be described ("specified") in an abstract manner. The... more
A major motivating force behind research into abstract data types and algebraic specifications is the realization that software in general and types in particular should be described ("specified") in an abstract manner. The objective is to give specifications at some level of abstraction: on the one hand leaving open decisions regarding further refinement and on the other allowing for substitutivity of modules as long as they satisfy a particular specification. The use of nondeterministic operators is an appropriate and useful abstraction tool, and more: nondeter- minism is a natural abstraction concept whenever there is a hidden state or other components of a system de- scription which are, methodologically, conceptually or technically, inaccessible at a particular level of abstrac- tion. In this report we explore the various approaches to dealing with nondeterminism within the framework of algebraic specifications. The basic concepts involved in the study of nondetermini...
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Relational syntax is presented for nondeterministic algebraic specifications called the language of relational terms. It is shown how nondeterministic specifications can be translated to the relational terms. A complete inference system... more
Relational syntax is presented for nondeterministic algebraic specifications called the language of relational terms. It is shown how nondeterministic specifications can be translated to the relational terms. A complete inference system is presented for derivation of valid clauses of relational atoms. Finally, a -abstraction construct is introduced extending the syntax of relational terms. It is weaker than the functional -abstraction, but allows translation of first-order predicate language into the language of relational terms. A slight extension of the inference system remains sound and complete for this extended syntax. Introduction The paper presents work in progress on establishing connections between the theory of nondeterministic specifications and other relevant theories, in particular, relational algebra [ABHVW], -calculus and first-order predicate calculus. In our earlier work [KW95, WM95a, WM95b] nondeterministic specifications are sets of clauses, where literals...
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Research Interests: Computer Science and NJC
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ABSTRACT properties of the intended structure may allow different implementations, and one should avoid overspecifying the problems as this may exclude some, otherwise acceptable, implementations. Limiting a language to Horn formulae may... more
ABSTRACT properties of the intended structure may allow different implementations, and one should avoid overspecifying the problems as this may exclude some, otherwise acceptable, implementations. Limiting a language to Horn formulae may restrict the possibility of assuming such a "don't care" attitude. The paradigmatic example of this makes use of disjunction: assume that we want an operation c:SxS-S which, for arguments a,b, should equal either a or b. The only way to capture this intention without using disjunction is to leave c partly specified or unspecified. For instance: SP is F: { a,b,h: - S c: SxS - S P: S - Bool } Ih { 1. P(a) = P(b) = T 2. P(h) = F 3. x=c(a,b) P(x)=T end 4. c(x,x) = x 5. c(x,y) = c(y,x) 6. c(x,c(y,z) ) = c( c(x,y),z ) where a, b, and h are the three intended elements of sort S. Axiom 3. excludes h from being the result of c(a,b) (provided that TF). 4.-6. are included primarily to simplify the model below. They assert general properties of c as join. Underspecification means that the specification possesses several nonisomorphic models, and all of them are taken as the semantics of the specification. SP will allow models where c(a,b) = a, and others where c(a,b) = b. Unfortunately, loose semantics will typically admit structures which violate the demands of "no junk" and "no confusion". Not only will SP have models where a=b, but also such where the result of c(a,b) is neither a nor b.