Proof search in constructive logics

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 similar calculi for other logics. We also point out to some related work on permutations in intuitionistic Gentzen sequent calculi that clarifies the relationship between such calculi and natural deduction.Centro de Matemática da Universidade do Minho (CMAT).União Europeia (UE) - Programa ESPRIT - BRA 7232 Gentzen

Cut-elimination and a permutation-free sequent calculus for intuitionistic logic

We describe a sequent calculus, based on work of Herbelin's, of which the cut-free derivations are in 1-1 correspondence with normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cut-elimination theorem for the calculus, using the recursive path oredering theorem of Dershowitz.Junta Nacional de Investigação Científica e Tecnológica (JNICT).União Europeia (UE) - Programa ESPRIT - grant BRA 7232 GENTZEN

Truth-value semantics and functional extensions for classical logic of partial terms based on equality

We develop a bottom-up approach to truth-value semantics for classical logic of partial terms based on equality and apply it to prove the conservativity of the addition of partial description and partial selection functions, independently of any strictness assumption.Comment: 15 pages, to appear in the Notre Dame Journal of Formal Logi

Generic Modal Cut Elimination Applied to Conditional Logics

We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity were explicitly stated as open in the literature