It is well-known that G3i, the sequent calculus for intuitionistic propositional logic where weakening and contraction are absorbed into the rules, is not terminating. Indeed, due to the contraction in the rule for left implication, the naı̈ve goal-oriented proof-search strategy, consisting in applying the rules of the calculus bottom-up until possible, can generate branches of infinite length. ...

The calculus of structures is a proof theoretical formalism which generalizes sequent calculus with the feature of deep inference: in contrast to sequent calculus, the calculus of structures does not rely on the notion of main connective and, like in term rewriting, it permits the application of inference rules at any depth inside a formula. Tom is a pattern matching processor that integrates t...

Abstract In Schwichtenberg (Studies in logic and the foundations of mathematics, vol 90, Elsevier, pp 867–895, 1977), fine-tuned Tait’s technique (Tait The syntax semantics infinitary languages, Springer, 204–236, 1968) so as to provide a simplified version Gentzen’s original cut-elimination procedure for first-order classical (Gallier Logic computer science: automatic theorem proving, Courier ...

Sequent calculi usually provide a general deductive setting that uniformly embeds other proof-theoretical approaches, such as tableaux methods, resolution techniques, goal-directed proofs, etc. Unfortunately, in temporal logic, existing sequent calculi make use of a kind of inference rules that prevent the effective mechanization of temporal deduction in the general setting. In particular, temp...

This note is a summary of the main result contained in the paper [9], which will be published elsewhere. In that paper we study abstract logics corresponding to the classical connectives of conjunction and disjunction, in order to clarify the relations between these logics and distributive lattices, and give some lattice-theoretical applications. Given any algebra A = 〈A,∧,∨〉 of type (2, 2) we ...

Bi-intuitionistic logic is the extension of intuitionistic logic with exclusion, a connective dual to implication. Cut-elimination in biintuitionistic logic is complicated due to the interaction between these two connectives, and various extended sequent calculi, including a display calculus, have been proposed to address this problem. In this paper, we present a new extended sequent calculus D...

When defined with general elimination/application rules, natural deduction and λ-calculus become closer to sequent calculus. In order to get real isomorphism, normalisation has to be defined in a “multiary” variant, in which reduction rules are necessarily non-local (reason: nomalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of s...

We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasi-completion of...

Proof theory reveals a wide mosaic of possibilities for sub-classical logics. These are logics that are strictly contained (as consequence relations) in classical logic. Thus, by choosing a subset of axioms and derivation rules that are derivable in (some proof system for) classical logic, one easily obtains a (proof system for a) sub-classical logic. Various important and useful non-classical ...

In M. McRobbie and J. Slaney, editors, 13 International Conference on Automated Deduction, LNAI 1104, pp. 418–432, c ©Springer Verlag, 1996. Abstract. We present a uniform algorithm for transforming matrix proofs in classical, constructive, and modal logics into sequent style proofs. Making use of a similarity between matrix methods and Fitting’s prefixed tableaus we first develop a procedure f...