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Structural Cut Elimination  I. Intuitionistic and Classical Logic
 Information and Computation
, 2000
"... this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91]. Multisets are avoided altogether in these proofs, and termination measures are replaced b ..."
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this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91]. Multisets are avoided altogether in these proofs, and termination measures are replaced by three nested structural inductions. Parameters are treated as variables bound in derivations, thus naturally capturing occurrence conditions. A starting point for the proofs is Kleene's sequent system G 3 [Kle52], which we derive systematically from the point of view that a sequent calculus should be a calculus of proof search for natural deductions. It can easily be related to Gentzen's original and other sequent calculi. We augment G 3 with proof terms that are stable under weakening. These proof terms enable the structural induction and furthermore form the basis of the representation of the proof in LF. The most closely related work on cut elimination is MartinLo# f 's proof of admissibility [ML68]. In MartinLo# f 's system the cut rule incorporates aspects of both weakening and contraction which enables a structural induction argument closely related to ours. However, without the introduction of proof terms, the implicit weakening in the cut rule makes it difficult to implement this proof directly. Herbelin [Her95] restates this proof and proceeds by assigning proof terms only to restricted sequent calculi LJT and LKT which correspond more immediately to
Ordered Linear Logic and Applications
, 2001
"... This work is dedicated to my parents. Acknowledgments Firstly, and foremost, I would like to thank my principal advisor, Frank Pfenning, for his patience with me, and for teaching me most of what I know about logic and type theory. I would also like to acknowledge some useful discussions with Kevin ..."
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Cited by 39 (0 self)
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This work is dedicated to my parents. Acknowledgments Firstly, and foremost, I would like to thank my principal advisor, Frank Pfenning, for his patience with me, and for teaching me most of what I know about logic and type theory. I would also like to acknowledge some useful discussions with Kevin Watkins which led me to simplify some of this work. Finally, I would like to thank my other advisor, John Reynolds, for all his kindness and support over the last five years. Abstract This thesis introduces a new logical system, ordered linear logic, which combines reasoning with unrestricted, linear, and ordered hypotheses. The logic conservatively extends (intuitionistic) linear logic, which contains both unrestricted and linear hypotheses, with a notion of ordered hypotheses. Ordered hypotheses must be used exactly once, subject to the order in which they were assumed (i.e., their order cannot be changed during the course of a derivation). This ordering constraint allows for logical representations of simple data structures such as stacks and queues. We construct ordered linear logic in the style of MartinL&quot;of from the basic notion of a hypothetical judgement. We then show normalization for the system by constructing a sequent calculus presentation and proving cutelimination of the sequent system.
Relating Natural Deduction and Sequent Calculus for Intuitionistic NonCommutative Linear Logic
, 1999
"... We present a sequent calculus for intuitionistic noncommutative linear logic (INCLL) , show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cutfree derivations, and arbitrary natura ..."
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We present a sequent calculus for intuitionistic noncommutative linear logic (INCLL) , show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cutfree derivations, and arbitrary natural deductions to sequent derivations with cut. This gives us a syntactic proof of normalization for a rich system of noncommutative natural deduction and its associated calculus. INCLL conservatively extends linear logic with means to express sequencing, which has applications in functional programming, logical frameworks, logic programming, and natural language parsing. 1 Introduction Linear logic [11] has been described as a logic of state because it views linear hypotheses as resources which may be consumed in the course of a deduction. It thereby significantly extends the expressive power of both classical and intuitionistic logics, yet it does not offer means to express sequencing. Th...
Cut Rules and Explicit Substitutions
, 2000
"... this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the l ..."
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Cited by 16 (0 self)
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this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the logical connectives which operate on the right, i.e., they manipulate the succedent formula. Examples are Gentzen's NJ and NK (Gentzen 1935). Logical deduction systems are given via leftintroduction and rightintroduction rules for the logical connectives. Although others have called these systems "sequent calculi", we call them Lsystems to avoid confusion with other systems given in sequent style. Examples are Gentzen's LK and LJ (Gentzen 1935). In this paper we are primarily interested in Lsystems. The advantage of Nsystems is that they seem closer to actual reasoning, while Lsystems on the other hand seem to have an easier proof theory. Lsystems are often extended with a "cut" rule as part of showing that for a given Lsystem and Nsystem, the derivations of each system can be encoded in the other. For example, NK proves the same as LK + cut (Gentzen 1935). Proof Normalization. A system is consistent when it is impossible to prove false, i.e., derive absurdity from zero assumptions. A system is analytic (has the analycity property) when there is an e#ective method to decompose any conclusion sequent into simpler premise sequents from which the conclusion can be obtained by some rule in the system such that the conclusion is derivable i# the premises are derivable (Maenpaa 1993). To achieve the goals of consistency and analycity, it has been customary to consider
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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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
A sequent calculus for type theory
 CSL 2006. LNCS
, 2006
"... Based on natural deduction, Pure Type Systems (PTS) can express a wide range of type theories. In order to express proofsearch in such theories, we introduce the Pure Type Sequent Calculi (PTSC) by enriching a sequent calculus due to Herbelin, adapted to proofsearch and strongly related to natural ..."
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Cited by 5 (0 self)
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Based on natural deduction, Pure Type Systems (PTS) can express a wide range of type theories. In order to express proofsearch in such theories, we introduce the Pure Type Sequent Calculi (PTSC) by enriching a sequent calculus due to Herbelin, adapted to proofsearch and strongly related to natural deduction. PTSC are equipped with a normalisation procedure, adapted from Herbelin’s and defined by local rewrite rules as in Cutelimination, using explicit substitutions. It satisfies Subject Reduction and it is confluent. A PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising if and only if the latter is. We show how the conversion rules can be incorporated inside logical rules (as in syntaxdirected rules for type checking), so that basic proofsearch tactics in type theory are merely the rootfirst application of our inference rules.
Total Functionals and Wellfounded Strategies (Extended Abstract)
, 1999
"... In existing game models, total functionals have no simple characterization neither in term of game strategies, nor in term of the total settheoretical functionals they define. We show that the situation changes if we extend the usual notion of game by allowing infinite plays. Total functionals a ..."
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In existing game models, total functionals have no simple characterization neither in term of game strategies, nor in term of the total settheoretical functionals they define. We show that the situation changes if we extend the usual notion of game by allowing infinite plays. Total functionals are
An Explicit Natural Deduction
, 1998
"... The typed calculus and the natural deduction are isomorph. There is a mapping (the CurryHoward's isomorphism) between each others. This paper shows that the typed calculi with explicit substitutions are isomorph to a logical deduction system which is equivalent to the natural deduction. This ..."
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The typed calculus and the natural deduction are isomorph. There is a mapping (the CurryHoward's isomorphism) between each others. This paper shows that the typed calculi with explicit substitutions are isomorph to a logical deduction system which is equivalent to the natural deduction. This logical formalism inherits the explicit substitutions properties: confluence on open terms, firstorder calculus and no PSN. 1 Introduction What is called in mathematics a proof or a demonstration is in fact a more or less informal speech which is founded on some axioms and which follows the greek logic tradition. The aim of the demonstration theory is to supply a formal framework in order to describe a raisoning; the proof becomes a rigorous object. It can be handled like any other mathematical object. In a proof, we have to distinguish what is proved from the demonstration itself. In the following, we will use formalisms where the proved objects are propositional formulas 1 . More precisely,...