Sufficient conditions for first order based sequent calculi to admit cut elimination by a Schütte-Tait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related with the calculus. The conditions are general enough to be satisfied by a wide class of sequent calculi encompassing, among others, some sequent calculi presentations for the first order and the propositional versions of classical and intuitionistic logic, classical and intuitionistic modal logic S4, and classical and intuitionistic linear logic and some of its fragments. Moreover the conditions are such that there is an algorithm for checking if they are satisfied by a sequent calculus.

Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural two-sorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result

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This paper considers the computational complexity of the disjunction and existential properties of intuitionistic logic. We prove that the disjunction property holds feasibly for intuitionistic propositional logic; i.e., from a proof of A B , a proof either of A or of B can be found in polynomial time. For intuitionistic predicate logic, we prove superexponential lower bounds for the disjunction property, namely, there is a superexponential lower bound on the time required, given a proof of A B , to produce one of A and B which is true. In addition, there is superexponential lower bound on the size of terms which fulfill the existential property of intuitionistic predicate logic. There are superexponential upper bounds for these problems, so the lower bounds are essentially optimal. 1 Introduction It is a well-known fact [1] that intuitionistic logic satisfies the following disjunction and existential properties: (throughout this paper, ` represents intuitionistic provab...

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