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Katalin Bimbó - One of the best experts on this subject based on the ideXlab platform.
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Extracting BB′IW Inhabitants of Simple Types From Proofs in the Sequent Calculus $${LT_\to^{t}}$$ L
Logica Universalis, 2014Co-Authors: Katalin Bimbó, J. Michael DunnAbstract:The decidability of the logic of pure ticket entailment means that the problem of inhabitation of simple types by Combinators over the base { B , B ′, I , W } is decidable too. Type-assignment systems are often formulated as natural deduction systems. However, our decision procedure for this logic, which we presented in earlier papers, relies on two sequent calculi and it does not yield directly a Combinator for a theorem of $${T_\to}$$ T → . Here we describe an algorithm to extract an inhabitant from a sequent calculus proof —without translating the proof into another proof system.
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extracting bb iw inhabitants of simple types from proofs in the sequent calculus lt_ to t for implicational ticket entailment
Logica Universalis, 2014Co-Authors: Katalin Bimbó, Michael J DunnAbstract:The decidability of the logic of pure ticket entailment means that the problem of inhabitation of simple types by Combinators over the base { B, B′, I, W } is decidable too. Type-assignment systems are often formulated as natural deduction systems. However, our decision procedure for this logic, which we presented in earlier papers, relies on two sequent calculi and it does not yield directly a Combinator for a theorem of \({T_\to}\) . Here we describe an algorithm to extract an inhabitant from a sequent calculus proof—without translating the proof into another proof system.
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Admissibility of Cut in LC with Fixed Point Combinator
Studia Logica, 2005Co-Authors: Katalin BimbóAbstract:The fixed point Combinator (Y) is an important non-proper Combinator, which is defhable from a Combinatorially complete base. This Combinator guarantees that recursive equations have a solution. Structurally free logics (LC) turn Combinators into formulas and replace structural rules by Combinatory ones. This paper introduces the fixed point and the dual fixed point Combinator into structurally free logics. The admissibility of (multiple) cut in the resulting calculus is not provable by a simple adaptation of the similar proof for LC with proper Combinators. The novelty of our proof—beyond proving the cut for a newly extended calculus–is that we add a fourth induction to the by-and-large Gentzen-style proof.
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Investigation into Combinatory Systems with Dual Combinators
Studia Logica, 2000Co-Authors: Katalin BimbóAbstract:Combinatory logic is known to be related to substructural logics. Algebraic considerations of the latter, in particular, algebraic considerations of two distinct implications (→, ←), led to the introduction of dual Combinators in Dunn & Meyer 1997. Dual Combinators are "mirror images" of the usual Combinators and as such do not constitute an interesting subject of investigation by themselves. However, when combined with the usual Combinators (e.g., in order to recover associativity in a sequent calculus), the whole system exhibits new features. A dual Combinatory system with weak equality typically lacks the Church-Rosser property, and in general it is inconsistent. In many subsystems terms "unexpectedly" turn out to be weakly equivalent. The paper is a preliminary attempt to investigate some of these issues, as well as, briefly compare function application in symmetric λ-calculus (cf. Barbanera & Berardi 1996) and dual Combinatory logic.
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The Paideia Archive: Twentieth World Congress of Philosophy - Dual Identity Combinators
The Paideia Archive: Twentieth World Congress of Philosophy, 1998Co-Authors: Katalin BimbóAbstract:This paper offers an analysis of the effect of the identity Combinators in dual systems. The result is based on an easy technical trick, namely, that the identity Combinators collapse all the Combinators which are dual with respect to them. (Dual Combinators were introduced in Dunn & Meyer 1997, a related system, the symmetric l-calculus was introduced by Barbanera & Berardi 1996.) After reviewing dual Combinators I consider the possible Combinatory systems and l-calculi in which the functions and/or the application operation are bidirectional. The last section of the paper shows the devastating effect the identity Combinators have for a dual system: they half trivialize simple Combinatory bases, although they are not sufficient to cause real triviality for what cancellative Combinators are needed.
Mireille Ducassé - One of the best experts on this subject based on the ideXlab platform.
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an abstract interpretation based Combinator for modelling while loops in constraint programming
Principles and Practice of Constraint Programming, 2007Co-Authors: Tristan Denmat, Arnaud Gotlieb, Mireille DucasséAbstract:We present the w constraint Combinator that models while loops in Constraint Programming. Embedded in a finite domain constraint solver, it allows programmers to develop non-trivial arithmetical relations using loops, exactly as in an imperative language style. The deduction capabilities of this Combinator come from abstract interpretation over the polyhedra abstract domain. This Combinator has already demonstrated its utility in constraint-based verification and we argue that it also facilitates the rapid prototyping of arithmetic constraints (e.g. power, gcd or sum).
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An Abstract Interpretation-based Combinator for Modelling While Loops in Constraint Programming
2007Co-Authors: Tristan Denmat, Arnaud Gotlieb, Mireille DucasséAbstract:We present the w constraint Combinator that models while loops in Constraint Programming. Embedded in a finite domain constraint solver, it allows programmers to develop non-trivial arithmetical relations using loops, exactly as in an imperative language style. The deduction capabilities of this Combinator comes from abstract interpretation over the polyhedra abstract domain. This Combinator has already demonstrated its utility in constraint-based verification and we argue that it also facilitates the rapid prototyping of arithmetic constraints (power, gcd, sum, ...).
Tristan Denmat - One of the best experts on this subject based on the ideXlab platform.
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an abstract interpretation based Combinator for modelling while loops in constraint programming
Principles and Practice of Constraint Programming, 2007Co-Authors: Tristan Denmat, Arnaud Gotlieb, Mireille DucasséAbstract:We present the w constraint Combinator that models while loops in Constraint Programming. Embedded in a finite domain constraint solver, it allows programmers to develop non-trivial arithmetical relations using loops, exactly as in an imperative language style. The deduction capabilities of this Combinator come from abstract interpretation over the polyhedra abstract domain. This Combinator has already demonstrated its utility in constraint-based verification and we argue that it also facilitates the rapid prototyping of arithmetic constraints (e.g. power, gcd or sum).
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An Abstract Interpretation-based Combinator for Modelling While Loops in Constraint Programming
2007Co-Authors: Tristan Denmat, Arnaud Gotlieb, Mireille DucasséAbstract:We present the w constraint Combinator that models while loops in Constraint Programming. Embedded in a finite domain constraint solver, it allows programmers to develop non-trivial arithmetical relations using loops, exactly as in an imperative language style. The deduction capabilities of this Combinator comes from abstract interpretation over the polyhedra abstract domain. This Combinator has already demonstrated its utility in constraint-based verification and we argue that it also facilitates the rapid prototyping of arithmetic constraints (power, gcd, sum, ...).
Henk Barendregt - One of the best experts on this subject based on the ideXlab platform.
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Completeness of the propositions-as-types interpretation of intuitionistic logic into illative Combinatory logic
Journal of Symbolic Logic, 1998Co-Authors: Wil Dekkers, Martin W. Bunder, Henk BarendregtAbstract:Illative Combinatory logic consists of the theory of Combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative Combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become Combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions. ?
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Completeness of two systems of illative Combinatory logic for first-order propositional and predicate calculus
Archive for Mathematical Logic, 1998Co-Authors: Wil Dekkers, Martin W. Bunder, Henk BarendregtAbstract:Illative Combinatory logic consists of the theory of Combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative Combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become Combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of \(\lambda\)-terms or Combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent).
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Systems of illative Combinatory logic complete for first-order propositional and predicate calculus
Journal of Symbolic Logic, 1993Co-Authors: Henk Barendregt, Martin W. Bunder, Wil DekkersAbstract:Illative Combinatory logic consists of the theory of Combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative Combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become Combinators or, in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of A-terms or Combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). ?
Wil Dekkers - One of the best experts on this subject based on the ideXlab platform.
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Completeness of the propositions-as-types interpretation of intuitionistic logic into illative Combinatory logic
Journal of Symbolic Logic, 1998Co-Authors: Wil Dekkers, Martin W. Bunder, Henk BarendregtAbstract:Illative Combinatory logic consists of the theory of Combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative Combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become Combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions. ?
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Completeness of two systems of illative Combinatory logic for first-order propositional and predicate calculus
Archive for Mathematical Logic, 1998Co-Authors: Wil Dekkers, Martin W. Bunder, Henk BarendregtAbstract:Illative Combinatory logic consists of the theory of Combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative Combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become Combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of \(\lambda\)-terms or Combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent).
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Systems of illative Combinatory logic complete for first-order propositional and predicate calculus
Journal of Symbolic Logic, 1993Co-Authors: Henk Barendregt, Martin W. Bunder, Wil DekkersAbstract:Illative Combinatory logic consists of the theory of Combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative Combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become Combinators or, in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of A-terms or Combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). ?
