On the Complexity of Disjunction and Explicit Definability Properties in Some Intermediate Logics

Abstract

Abstract. In this paper we provide a uniform framework, based on extraction calculi, where to study the complexity of the problem to decide the disjunction and the explicit definability properties for Intuitionistic Logic and some Superintuitionistic Logics. Unlike the previous approaches, our framework is independent of structural properties of the proof systems and it can be applied to Natural Deduction systems, Hilbert style systems and Gentzen sequent systems.

On the complexity of the disjunction property in intuitionistic and modal logics

“…So far, the main issues have been concerned with the length of classical propositional proofs, a problem related to the NP = coNP question [Buss 1999;Krajíček 1995;Pudlák 1999]. Recently, some studies [Buss and Mints 1999;Buss and Pudlák 2001;Ferrari et al 2002] have also been devoted to the complexity of some properties of intuitionistic logic, in particular the Authors' current addresses: M. Ferrari, DICOM, Università degli Studi dell'Insubria, Via Mazzini 5, 21100 Varese, Italy; email: mauro.ferrari@uninsubria.it; C.Fiorentini, Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Via Comelico 39/41, 20135 Milan, Italy; email: fiorentini@dsi.unimi,it; G. Fiorino, Dipartimento di Metodi Quantitativi, Università degli Studi di Milano-Bicocca, Via P. dell'Ateneo Nuovo 1, 20126 Milan, Italy; email: guido.fiorino@unimib.it Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored.…”

“…This makes our proof of the feasible disjunction property essentially independent of structural properties of C. This has some advantages: first of all, it allows us to get the result without proving cut-elimination or normalization; second, it allows us to extend the proof to calculi with "weak structural properties" such as Hilbert-style calculi, and hence to treat also logics for which only Hilbert-style characterizations are known. This framework has already been applied in Ferrari et al [2002] to study the complexity of the disjunction property and the explicit definability property (the predicate counterpart of disjunction property) of some predicate intermediate logics.…”

A polynomial time complete disjunction property in intuitionistic propositional logic

“…Buss and Mints [2] gave a polynomial time algorithm, which extracts an i from a given derivation of α 0 ∨ α 1 in natural deduction such that α i is intuitionistically valid. Such a feasible algorithm based on sequent calculi is given in Buss and Pudlák [3], and Ferrari, Fiorentini and Fiorino [4] provides an algorithm for derivable sequents Γ ⇒ α 0 ∨ α 1 with sets Γ of Harrop formulas.…”

“…The idea in these algorithms, which comes from [2], is to prove that one of formulas α 0 and α 1 is in a small set of sequents (immediately derivable sequents) relative to a given intuitionistic derivation of the disjunction α 0 ∨ α 1 (Boundedness), for which there is a polynomial time algorithm testing the membership of sequents in the set, and any sequent in the set is readily seen to be intuitionistically valid. In [4] the authors introduce extraction calculi to generate the set.…”

“…In [2] the proof of the Boundedness is done through a partial normalization in natural deduction, and the proof in [3] through cut-elmination. On the other side, one in [4] is based on an evaluation relation, a variant of Aczel's slash [1], cf. [6].…”