The Skolem machine is a Turing-complete machine model where the instructions are first-order form... more The Skolem machine is a Turing-complete machine model where the instructions are first-order formulas of a specific form. We introduce Skolem machines and prove their logical correctness and completeness. Skolem machines compute queries for the Geolog language, a rich fragment of first-order logic. The concepts of Geolog trees and complete Geolog trees are defined, and these tree concepts are used to show logical correctness and completeness of Skolem machine computations. The universality of Skolem machine computations is demonstrated. Lastly, the paper outlines implementation design issues using an abstract machine model approach.
We provide here a computational interpretation of first-order logic based on a constructive inter... more We provide here a computational interpretation of first-order logic based on a constructive interpretation of satisfiability w.r.t. a fixed but arbitrary interpretation. In this approach the formulas themselves are programs. This contrasts with the so-called formulas as types approach in which the proofs of the formulas are typed terms that can be taken as programs. This view of computing is inspired by logic programming and constraint logic programming but differs from them in a number of crucial aspects. Formulas as programs is argued to yield a realistic approach to programming that has been realized in the implemented programming language Alma-0 Apt, Brunekreef, Partington & Schaerf (1998) that combines the advantages of imperative and logic programming. The work here reported can also be used to reason about the correctness of non-recursive Alma-0 programs that do not include destructive assignment. 1991 Mathematics Subject Classification: 68N05, 68N17, 68Q60 1991 Comput...
Moore's seminal paper (9) can be taken as the starting point of Algorithmic Learning Theory. ... more Moore's seminal paper (9) can be taken as the starting point of Algorithmic Learning Theory. Moore studied the problem of unraveling the inner structure of a (minimum state) deterministic finite automaton (DFA) from its input-output behaviour. In this note we pursue Moore's line of research, studying conditions under which it is possible to compute a grammar and/or an automaton for a given language L from a language class C. It doesn't come as a surprise that such conditions must be quite strong. We improve on the algorithms in some cases where computing the grammar/automaton is possible. We correct some mistakes in the literature and come up with some new results, positive and negative, for (subclasses of) context-free languages.
ABSTRACT The Skolem machine is a Turing-complete machine model where the instructions are first-o... more ABSTRACT The Skolem machine is a Turing-complete machine model where the instructions are first-order formulas of a specific form. We introduce Skolem machines and prove their logical completeness. Skolem machines compute queries for the Geolog language, a rich fragment of first-order logic. The concept of complete Geolog trees is defined, and this tree concept is used to show logical completeness for Skolem machines: If the query for a Geolog theory is a logical consequence of the axioms then the corresponding Skolem machine halts succesfully in a configuration that supports the query.
ABSTRACT Inspired by the wonderful design and implementation of the Prolog language afforded by t... more ABSTRACT Inspired by the wonderful design and implementation of the Prolog language afforded by the Warren Abstract Machine (WAM), this paper describes an extended logical language which can compute larger realms of first-order logic, based upon theories for finitary geometric logic. The paper describes a Geolog language for expressing first-order geometric logic in tidy closed form, a mathematical Skolem Machine that computes the language, and an implementation prototype that intimately mimics the abstract machine, and which also reformulates expensive bottom-up inference into efficient top-down inference. There are promising mathematical theorem proving applications for geometric logic systems, collected on the website [5]. The emphasis of this paper is theory, abstract machine design and direct implementation of the abstract machine.
We show that the even linear languages are characterised by a certain extension of the signature ... more We show that the even linear languages are characterised by a certain extension of the signature of the monadic second-order logic used by Büchi (1960) and Elgot (1961) to characterise the regular languages.
... BibTeX. @INPROCEEDINGS{Nakata11aproof, author = {Keiko Nakata and Tarmo Uustalu and Marc Beze... more ... BibTeX. @INPROCEEDINGS{Nakata11aproof, author = {Keiko Nakata and Tarmo Uustalu and Marc Bezem}, title = {A proof pearl with the fan theorem and bar induction ... 1. Citations. 998, Temporal and modal logic - Emerson - 1990. 321, Constructive Analysis - Bishop - 1967. ...
... the length of the initial segment of the zero function such that M is constant on the corresp... more ... the length of the initial segment of the zero function such that M is constant on the corresponding ... One elegant approach is to use deriva-tions in infinitary propositional logic [18, 23 ... is a general cut-elimination result for this logic which gives a computational interpretation of proofs ...
We present a possible computational content of the negative translation of classical analysis wit... more We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Interestingly, this interpretation uses a refinement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis, and how to extract algorithms from proofs of statements.
We introduce a three-tiered framework for modelling and reasoning about agents who (i) can use po... more We introduce a three-tiered framework for modelling and reasoning about agents who (i) can use possibly complete reasoning systems without any restrictions but who nevertheless are (ii) bounded in the sense that they never reach infinitely many results and, finally, who (iii) perform their reasoning in time. This last aspect does not concern so much the time it takes for
The Skolem machine is a Turing-complete machine model where the instructions are first-order form... more The Skolem machine is a Turing-complete machine model where the instructions are first-order formulas of a specific form. We introduce Skolem machines and prove their logical correctness and completeness. Skolem machines compute queries for the Geolog language, a rich fragment of first-order logic. The concepts of Geolog trees and complete Geolog trees are defined, and these tree concepts are used to show logical correctness and completeness of Skolem machine computations. The universality of Skolem machine computations is demonstrated. Lastly, the paper outlines implementation design issues using an abstract machine model approach.
We provide here a computational interpretation of first-order logic based on a constructive inter... more We provide here a computational interpretation of first-order logic based on a constructive interpretation of satisfiability w.r.t. a fixed but arbitrary interpretation. In this approach the formulas themselves are programs. This contrasts with the so-called formulas as types approach in which the proofs of the formulas are typed terms that can be taken as programs. This view of computing is inspired by logic programming and constraint logic programming but differs from them in a number of crucial aspects. Formulas as programs is argued to yield a realistic approach to programming that has been realized in the implemented programming language Alma-0 Apt, Brunekreef, Partington & Schaerf (1998) that combines the advantages of imperative and logic programming. The work here reported can also be used to reason about the correctness of non-recursive Alma-0 programs that do not include destructive assignment. 1991 Mathematics Subject Classification: 68N05, 68N17, 68Q60 1991 Comput...
Moore's seminal paper (9) can be taken as the starting point of Algorithmic Learning Theory. ... more Moore's seminal paper (9) can be taken as the starting point of Algorithmic Learning Theory. Moore studied the problem of unraveling the inner structure of a (minimum state) deterministic finite automaton (DFA) from its input-output behaviour. In this note we pursue Moore's line of research, studying conditions under which it is possible to compute a grammar and/or an automaton for a given language L from a language class C. It doesn't come as a surprise that such conditions must be quite strong. We improve on the algorithms in some cases where computing the grammar/automaton is possible. We correct some mistakes in the literature and come up with some new results, positive and negative, for (subclasses of) context-free languages.
ABSTRACT The Skolem machine is a Turing-complete machine model where the instructions are first-o... more ABSTRACT The Skolem machine is a Turing-complete machine model where the instructions are first-order formulas of a specific form. We introduce Skolem machines and prove their logical completeness. Skolem machines compute queries for the Geolog language, a rich fragment of first-order logic. The concept of complete Geolog trees is defined, and this tree concept is used to show logical completeness for Skolem machines: If the query for a Geolog theory is a logical consequence of the axioms then the corresponding Skolem machine halts succesfully in a configuration that supports the query.
ABSTRACT Inspired by the wonderful design and implementation of the Prolog language afforded by t... more ABSTRACT Inspired by the wonderful design and implementation of the Prolog language afforded by the Warren Abstract Machine (WAM), this paper describes an extended logical language which can compute larger realms of first-order logic, based upon theories for finitary geometric logic. The paper describes a Geolog language for expressing first-order geometric logic in tidy closed form, a mathematical Skolem Machine that computes the language, and an implementation prototype that intimately mimics the abstract machine, and which also reformulates expensive bottom-up inference into efficient top-down inference. There are promising mathematical theorem proving applications for geometric logic systems, collected on the website [5]. The emphasis of this paper is theory, abstract machine design and direct implementation of the abstract machine.
We show that the even linear languages are characterised by a certain extension of the signature ... more We show that the even linear languages are characterised by a certain extension of the signature of the monadic second-order logic used by Büchi (1960) and Elgot (1961) to characterise the regular languages.
... BibTeX. @INPROCEEDINGS{Nakata11aproof, author = {Keiko Nakata and Tarmo Uustalu and Marc Beze... more ... BibTeX. @INPROCEEDINGS{Nakata11aproof, author = {Keiko Nakata and Tarmo Uustalu and Marc Bezem}, title = {A proof pearl with the fan theorem and bar induction ... 1. Citations. 998, Temporal and modal logic - Emerson - 1990. 321, Constructive Analysis - Bishop - 1967. ...
... the length of the initial segment of the zero function such that M is constant on the corresp... more ... the length of the initial segment of the zero function such that M is constant on the corresponding ... One elegant approach is to use deriva-tions in infinitary propositional logic [18, 23 ... is a general cut-elimination result for this logic which gives a computational interpretation of proofs ...
We present a possible computational content of the negative translation of classical analysis wit... more We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Interestingly, this interpretation uses a refinement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis, and how to extract algorithms from proofs of statements.
We introduce a three-tiered framework for modelling and reasoning about agents who (i) can use po... more We introduce a three-tiered framework for modelling and reasoning about agents who (i) can use possibly complete reasoning systems without any restrictions but who nevertheless are (ii) bounded in the sense that they never reach infinitely many results and, finally, who (iii) perform their reasoning in time. This last aspect does not concern so much the time it takes for
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