We prove a folklore theorem, that two derivations in a cut-free sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are inter-permutable (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 inter-permutable, 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 ...

We exploit a system of realizers for classical logic, and a translation from resolution into the sequent calculus, to assess the intuitionistic force of classical resolution for a fragment of intuitionistic logic. This approach is in contrast to formulating locally intuitionistically sound resolution rules. The techniques use the ffl-calculus, a development of Parigot's -calculus.

. Feferman has proposed FS 0 , a theory of finitary inductive systems, as a framework theory that allows a user to reason both in and about an encoded theory. I look here at how practical FS 0 really is. To this end I formalise a sequent calculus presentation of classical propositional logic, and show this can be used for work in both the theory and the metatheory. the latter is illustrated with a discussion of a proof of Gentzen's Hauptsatz. Contents x 1 Introduction 2 x 1.1 Background : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 x 1.2 Outline of paper : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 x 2 The theory FS 0 and notational conventions 4 x 2.1 What is FS 0 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 x 3 An informal description of Gentzen's calculus 5 x 3.1 The language : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 x 3.2 The calculus for classical propositional logic : : : : : : : : : : : : 6 x 4 Formalising the ...

We present an overview of some sequent calculi organised not for "theorem-proving" 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 Tarski-style semantics (answering questions "What judgments are provable?") rather than a Heyting-style 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...

. In this paper we present a mechanism to define names for proof-witnesses of formulae and thus to use Gentzen's cut-rule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proof-search by indicating when an instance of the cut-rule should be attempted. By using the cut-rule there are proofs that can be made dramatically shorter. We explain how this idea of using the cut-rule can be applied to the logic of hereditary Harrop formulae. 1 Introduction The computation mechanisms both for logic and for functional programming are searches for cut-free proofs. First, in pure logic programming the achievement of a goal G w.r.t. a program P can be seen 1 as the search for a proof in Gentzen's intuitionistic sequent calculus LJ [Gen69], of the sequent P ) G, that by Gentzen's cut-elimination theorem can be cut-free [Bee89], [Mil90]; a -term found as a witness to a proof contains among...

this paper is three-fold:

In this work we present a type system called HH def that extends the theory of simply typed hereditary Harrop formulae [Mil90] with definitions and strong \Sigma-types. The use of definitions permits the construction of clearer programs and of shorter proofs by using a rule (the def rule) similar to Gentzen's cut rule. Proof-search for HH def is performed in a goal-directed manner with occurrences of defined constants in a goal triggering instances of the def rule. Such a search procedure is shown to be complete for HH def . 1 Introduction The motivation for development of the calculus HH def is to provide a logical foundation on which to develop a programming language that integrates logic and functional programming. This work develops the ideas outlined in [Pin94], which are intended to be a first step towards a proof-theoretic characterisation of such programming. The central idea is that the execution mechanisms both for logic and for functional programming can be seen as ...

. This paper presents two sequent calculi, requiring no clausal form for types, whose typable terms are in 1-1 correspondence with the normal terms of the \Pi- and \Pi\Sigma-calculi. Such sequent calculi allow no permutations in the order in which inference rules occur on derivations of typable terms and are thus appropriate for proof search. In these calculi proof search can be performed in a root-first fashion and type conversions are solely required in axiom formation. 1 The \Pi-calculus The \Pi-calculus [6, 9] is a theory extending the simply typed -calculus [1] with first-order \Pi-types, i.e. with dependent types. It is the type system of the Edinburgh Logical Framework (LF) [6], a logic for presenting logics, and it is the basis of Elf [7, 8], a language giving to types an operational interpretation similar to that given to formulae in logic programming. The \Pi-calculus essentially corresponds to the P system of the -cube [1] and it is also closely related to the \Pi-fr...