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CutElimination and a PermutationFree Sequent Calculus for Intuitionistic Logic
, 1998
"... We describe a sequent calculus, based on work of Herbelin, of which the cutfree derivations are in 11 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cutelimination theorem for the calculus, using the recursive path ..."
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Cited by 40 (6 self)
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We describe a sequent calculus, based on work of Herbelin, of which the cutfree derivations are in 11 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cutelimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.
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 same natural deductio ..."
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Cited by 23 (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 ...
Theorem Proving for Conditional Logics: CondLean and GoalDuck
 Journal of Applied NonClassical Logics (JANCL
"... ABSTRACT. In this paper we focus on theorem proving for conditional logics. First, we give a detailed description of CondLean, a theorem prover for some standard conditional logics. CondLean is a SICStus Prolog implementation of some labeled sequent calculi for conditional logics recently introduced ..."
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Cited by 5 (4 self)
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ABSTRACT. In this paper we focus on theorem proving for conditional logics. First, we give a detailed description of CondLean, a theorem prover for some standard conditional logics. CondLean is a SICStus Prolog implementation of some labeled sequent calculi for conditional logics recently introduced. It is inspired to the so called “lean ” methodology, even if it does not fit this style in a rigorous manner. CondLean also comprises a graphical interface written in Java. Furthermore, we introduce a goaldirected proof search mechanism, derived from the above mentioned sequent calculi based on the notion of uniform proofs. Finally, we describe GOALDUCK, a simple SICStus Prolog implementation of the goaldirected calculus mentioned here above. Both the programs CondLean and GOALDUCK, together with their source code, are available for free download at.
Polymorphic Lemmas and Definitions in λProlog and Twelf
 THEORY AND PRACTICE OF LOGIC PROGRAMMING
, 2004
"... λProlog is known to be wellsuited for expressing and implementing logics and inference systems. We show that lemmas and definitions in such logics can be implemented with a great economy of expression. We encode a higherorder logic using an encoding that maps both terms and types of the object log ..."
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Cited by 2 (0 self)
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λProlog is known to be wellsuited for expressing and implementing logics and inference systems. We show that lemmas and definitions in such logics can be implemented with a great economy of expression. We encode a higherorder logic using an encoding that maps both terms and types of the object logic (higherorder logic) to terms of the metalanguage (AProlog). We discuss both the Terzo and Teyjus implementations of AProlog. We also encode the same logic in Twelf and compare the features of these two metalanguages for our purposes.
Strategies for Logic Programming Languages
"... . Logic programs consist of formulas of mathematical logic and various prooftheoretic techniques can be used to design and analyse execution models for such programs. We briefly review the main problems, which are questions that are still elusive in the design of logic programming languages, from a ..."
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. Logic programs consist of formulas of mathematical logic and various prooftheoretic techniques can be used to design and analyse execution models for such programs. We briefly review the main problems, which are questions that are still elusive in the design of logic programming languages, from a prooftheoretic point of view. Existing strategies which lead to the various languages are all rather sophisticated and involve complex manipulations of proofs. All are designed for analysis on paper by a human and many of them are ripe for automation. We aim to perform the automation of some aspects of strategies for logic programming language, in order to assist in the design of these languages. In this paper we describe the first steps towards the design of such an automatic analysis tool. We investigate the usage of particular proof manipulations for the analysis of logic programming strategies. We propose a more precise specification of sequent calculi inference rules that we use as a ...
Theorem Proving for Untyped Constructive λCalculus: Implementation and Application
"... This paper presents a theorem prover for a highly intensional logic, namely a constructive version of property theory [25] (this language essentially provides a combination of constructive firstorder logic and the #calculus). The paper presents the basic theorem prover, which is a higherorder ext ..."
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This paper presents a theorem prover for a highly intensional logic, namely a constructive version of property theory [25] (this language essentially provides a combination of constructive firstorder logic and the #calculus). The paper presents the basic theorem prover, which is a higherorder extension of Manthey and Bry's model generation theorem prover for firstorder logic [14]; considers issues relating to the compiletime optimisations that are often used with firstorder theorem provers; and shows how the resulting system can be used in a natural language understanding system.
Search Calculi for Classical and Intuitionistic Logic
"... Abstract. It is wellknown that inference rules in the sequent calculus can be interpreted as both proof construction rules (i.e. constructing proofs from the leaves towards the root of the tree) and proof search rules (i.e. finding proofs by starting at the (supposed) root and working towards the l ..."
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Abstract. It is wellknown that inference rules in the sequent calculus can be interpreted as both proof construction rules (i.e. constructing proofs from the leaves towards the root of the tree) and proof search rules (i.e. finding proofs by starting at the (supposed) root and working towards the leaves). However, the information used in each case is different: the former constructs larger proofs from smaller ones, whereas the latter constructs search trees, from which, if the search is successful, a proof can be recovered. Thus during search the intermediate stages are at best partial proofs. In this paper we explore a variation of the sequent calculus in which search information, in the form of Boolean constraints, is added to each sequent. In particular, we show how this can be done for the sequent calculus LK for classical logic and the multipleconclusioned sequent LM for intuitionistic logic. In addition, we show how a judicious use of hypersequents can improve the search properties of LM. 1