Sequent

About: Sequent is a research topic. Over the lifetime, 1427 publications have been published within this topic receiving 19045 citations.

Structural proof theory

Abstract: Introduction 1. From natural deduction to sequent calculus 2. Sequent calculus for institutionistic logic 3. Sequent calculus for classical logic 4. The quantifiers 5. Variants of sequent calculi 6. Structural proof analysis of axiomatic theories 7. Intermediate logical systems 8. Back to natural deduction Conclusion: diversity and unity in structural proof theory Appendix A. Simple type theory and categorical grammar Appendix B. Proof theory and constructive type theory Appendix C. A proof editor for sequent calculus.

Contraction-Free Sequent Calculi for Intuitionistic Logic

Abstract: ?0. Prologue. Gentzen's sequent calculus LJ, and its variants such as G3 [21], are (as is well known) convenient as a basis for automating proof search for IPC (intuitionistic propositional calculus). But a problem arises: that of detecting loops, arising from the use (in reverse) of the rule : => for implication introduction on the left. We describe below an equivalent calculus, yet another variant on these systems, where the problem no longer arises: this gives a simple but effective decision procedure for IPC. The underlying method can be traced back forty years to Vorob'ev [33], [34]. It has been rediscovered recently by several authors (the present author in August 1990, Hudelmaier [18], [19], Paulson [27], and Lincoln et al. [23]). Since the main idea is not plainly apparent in Vorob'ev's work, and there are mathematical applications [28], it is desirable to have a simple proof. We present such a proof, exploiting the Dershowtiz-Manna theorem [4] on multiset orderings.

A system of interaction and structure

Abstract: This article introduces a logical system, called BV, which extends multiplicative linear logic by a noncommutative self-dual logical operator. This extension is particularly challenging for the sequent calculus, and so far, it is not achieved therein. It becomes very natural in a new formalism, called the calculus of structures, which is the main contribution of this work. Structures are formulas subject to certain equational laws typical of sequents. The calculus of structures is obtained by generalizing the sequent calculus in such a way that a new top-down symmetry of derivations is observed, and it employs inference rules that rewrite inside structures at any depth. These properties, in addition to allowing the design of BV, yield a modular proof of cut elimination.

A Term Calculus for Intuitionistic Linear Logic

Abstract: In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky [1]) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is well-typed). We also consider term reduction arising from cut-elimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these and consider their computational content.

A Lambda-Calculus Structure Isomorphic to Gentzen-Style Sequent Calculus Structure

Abstract: We consider a λ-calculus for which applicative terms have no longer the form (...((u u1) u2)... un) but the form (u [u1;...;un]), for which [u1;...;un] is a list of terms. While the structure of the usual λ-calculus is isomorphic to the structure of natural deduction, this new structure is isomorphic to the structure of Gentzen-style sequent calculus. To express the basis of the isomorphism, we consider intuitionistic logic with the implication as sole connective. However we do not consider Gentzen's calculus LJ, but a calculus LJT which leads to restrict the notion of cut-free proofs in LJ. We need also to explicitly consider, in a simply typed version of this λ-calculus, a substitution operator and a list concatenation operator. By this way, each elementary step of cutelimination exactly matches with a β-reduction, a substitution propagation step or a concatenation computation step.