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44
Strong Normalisation of CutElimination in Classical Logic
, 2000
"... In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and D ..."
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Cited by 44 (4 self)
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In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and Danos et al., our procedure requires no special annotations on formulae and allows cutrules to pass over other cutrules. In order to adapt the notion of symmetric reducibility candidates for proving the strong normalisation property, we introduce a novel term assignment for sequent proofs of classical logic and formalise cutreductions as term rewriting rules.
A Linear Spine Calculus
 Journal of Logic and Computation
, 2003
"... We present the spine calculus S ##&# as an efficient representation for the linear #calculus # ##&# which includes unrestricted functions (#), linear functions (#), additive pairing (&), and additive unit (#). S ##&# enhances the representation of Church's simply typed # ..."
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Cited by 41 (9 self)
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We present the spine calculus S ##&# as an efficient representation for the linear #calculus # ##&# which includes unrestricted functions (#), linear functions (#), additive pairing (&), and additive unit (#). S ##&# enhances the representation of Church's simply typed #calculus by enforcing extensionality and by incorporating linear constructs. This approach permits procedures such as unification to retain the efficient head access that characterizes firstorder term languages without the overhead of performing #conversions at run time. Applications lie in proof search, logic programming, and logical frameworks based on linear type theories. It is also related to foundational work on term assignment calculi for presentations of the sequent calculus. We define the spine calculus, give translations of # ##&# into S ##&# and viceversa, prove their soundness and completeness with respect to typing and reductions, and show that the typable fragment of the spine calculus is strongly normalizing and admits unique canonical, i.e. ##normal, forms.
Permutability of Proofs in Intuitionistic Sequent Calculi
, 1996
"... We prove a folklore theorem, that two derivations in a cutfree sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are interpermutable (using a set of basic "permutation reduction rules" derived from Kleene's work in 1952) iff they determine the sa ..."
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Cited by 30 (4 self)
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We prove a folklore theorem, that two derivations in a cutfree sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are interpermutable (using a set of basic "permutation reduction rules" derived from Kleene's work in 1952) iff they determine the same natural deduction. The basic rules form a confluent and weakly normalising rewriting system. We refer to Schwichtenberg's proof elsewhere that a modification of this system is strongly normalising. Key words: intuitionistic logic, proof theory, natural deduction, sequent calculus. 1 Introduction There is a folklore theorem that two intuitionistic sequent calculus derivations are "really the same" iff they are interpermutable, using permutations as described by Kleene in [13]. Our purpose here is to make precise and prove such a "permutability theorem". Prawitz [18] showed how intuitionistic sequent calculus derivations determine natural deductions, via a mapping ' from LJ to NJ (here we consider only ...
LJQ: a strongly focused calculus for intuitionistic logic
 COMPUTABILITY IN EUROPE 2006, VOLUME 3988 OF LNCS
, 2006
"... LJQ is a focused sequent calculus for intuitionistic logic, with a simple restriction on the first premisss of the usual left introduction rule for implication. We discuss its history (going back to about 1950, or beyond), present the underlying theory and its applications both to terminating proof ..."
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Cited by 21 (1 self)
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LJQ is a focused sequent calculus for intuitionistic logic, with a simple restriction on the first premisss of the usual left introduction rule for implication. We discuss its history (going back to about 1950, or beyond), present the underlying theory and its applications both to terminating proofsearch calculi and to callbyvalue reduction in lambda calculus.
Revisiting the correspondence between cutelimination and normalisation
 In Proceedings of ICALP’2000
, 2000
"... Abstract. Cutfree proofs in Herbelin’s sequent calculus are in 11 correspondence with normal natural deduction proofs. For this reason Herbelin’s sequent calculus has been considered a privileged middlepoint between Lsystems and natural deduction. However, this bijection does not extend to pro ..."
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Cited by 14 (3 self)
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Abstract. Cutfree proofs in Herbelin’s sequent calculus are in 11 correspondence with normal natural deduction proofs. For this reason Herbelin’s sequent calculus has been considered a privileged middlepoint between Lsystems and natural deduction. However, this bijection does not extend to proofs containing cuts and Herbelin observed that his cutelimination procedure is not isomorphic to βreduction. In this paper we equip Herbelin’s system with rewrite rules which, at the same time: (1) complete in a sense the cut elimination procedure firstly proposed by Herbelin; and (2) perform the intuitionistic “fragment ” of the tqprotocol a cutelimination procedure for classical logic defined by Danos, Joinet and Schellinx. Moreover we identify the subcalculus of our system which is isomorphic to natural deduction, the isomorphism being with respect not only to proofs but also to normalisation. Our results show, for the implicational fragment of intuitionistic logic, how to embed natural deduction in the much wider world of sequent calculus and what a particular cutelimination procedure normalisation is. 1
Completing Herbelin’s programme
"... In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a CurryHoward interpretation of sequent calculus into a variant of the λcalculus, specifically a variant which manipulates formally “applicative contexts” and inverts t ..."
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Cited by 13 (5 self)
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In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a CurryHoward interpretation of sequent calculus into a variant of the λcalculus, specifically a variant which manipulates formally “applicative contexts” and inverts the associativity of “applicative terms”. Herbelin worked with a fragment of sequent calculus with constraints on left introduction. In this paper we complete Herbelin’s programme for full sequent calculus, that is, sequent calculus without the mentioned constraints, but where permutative conversions necessarily show up. This requires the introduction of a lambdalike calculus for full sequent calculus and an extension of natural deduction that gives meaning to “applicative contexts” and “applicative terms”. Such extension is a calculus with modus ponens and primitive substitution that refines von Plato’s natural deduction; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. The prooftheoretical outcome is noteworthy: the puzzling relationship between cut and substitution is settled; and cutelimination in sequent calculus is proven isomorphic to normalisation in the proposed natural deduction system. The isomorphism is the mapping that inverts the associativity of applicative terms.
Proof Search in Constructive Logics
 In Sets and proofs
, 1998
"... We present an overview of some sequent calculi organised not for "theoremproving" but for proof search, where the proofs themselves (and the avoidance of known proofs on backtracking) are objects of interest. The main calculus discussed is that of Herbelin [1994] for intuitionistic lo ..."
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Cited by 10 (2 self)
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We present an overview of some sequent calculi organised not for "theoremproving" but for proof search, where the proofs themselves (and the avoidance of known proofs on backtracking) are objects of interest. The main calculus discussed is that of Herbelin [1994] for intuitionistic logic, which extends methods used in hereditary Harrop logic programming; we give a brief discussion of some similar calculi for other logics. We also point to some related work on permutations in intuitionistic Gentzen sequent calculi that clarifies the relationship between such calculi and natural deduction. 1 Introduction It is widely held that ordinary logic programming is based on classical logic, with a Tarskistyle semantics (answering questions "What judgments are provable?") rather than a Heytingstyle semantics (answering questions like "What are the proofs, if any, of each judgment?"). If one adopts the latter style (equivalently, the BHK interpretation: see [35] for details) by regardi...
GraphBased Proof Counting and Enumeration with Applications for Program Fragment Synthesis
 in &quot;International Symposium on Logicbased Program Synthesis and Transformation 2004 (LOPSTR 2004)&quot;, S. ETALLE (editor)., Lecture Notes in Computer Science
, 2004
"... Abstract. For use in earlier approaches to automated module interface adaptation, we seek a restricted form of program synthesis. Given some typing assumptions and a desired result type, we wish to automatically build a number of program fragments of this chosen typing, using functions and values av ..."
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Cited by 9 (0 self)
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Abstract. For use in earlier approaches to automated module interface adaptation, we seek a restricted form of program synthesis. Given some typing assumptions and a desired result type, we wish to automatically build a number of program fragments of this chosen typing, using functions and values available in the given typing environment. We call this problem term enumeration. To solve the problem, we use the CurryHoward correspondence (propositionsastypes, proofsasprograms) to transform it into a proof enumeration problem for an intuitionistic logic calculus. We formally study proof enumeration and counting in this calculus. We prove that proof counting is solvable and give an algorithm to solve it. This in turn yields a proof enumeration algorithm. 1
An isomorphism between a fragment of sequent calculus and an extension of natural deduction
"... ..."
Strong normalisation for a gentzenlike cutelimination procedure
 In TLCA
, 2001
"... Abstract. In this paper we introduce a cutelimination procedure for classical logic, which is both strongly normalising and consisting of local proof transformations. Traditional cutelimination procedures, including the one by Gentzen, are formulated so that they only rewrite neighbouring inferenc ..."
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Cited by 8 (0 self)
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Abstract. In this paper we introduce a cutelimination procedure for classical logic, which is both strongly normalising and consisting of local proof transformations. Traditional cutelimination procedures, including the one by Gentzen, are formulated so that they only rewrite neighbouring inference rules; that is they use local proof transformations. Unfortunately, such local proof transformation, if defined naïvely, break the strong normalisation property. Inspired by work of Bloo and Geuvers concerning the λxcalculus, we shall show that a simple trick allows us to preserve this property in our cutelimination procedure. We shall establish this property using the recursive path ordering by Dershowitz.