In this paper we present LSJ, a contraction-free sequent calculus for Intuitionistic propositional logic whose proofs are linearly bounded in the length of the formula to be proved and satisfy the subformula property. We also introduce a sequent calculus RJ for intuitionistic unprovability with the same properties of LSJ. We show that from a refutation of RJ of a sequent σ we can extract a Kripke countermodel for σ . Finally, we provide a procedure that given a sequent σ returns either a proof of σ in LSJ or a refutation in RJ such that the extracted counter-model is of minimal depth.

Abstract. We present fCube, a theorem prover for Intuitionistic propositional logic based on a tableau calculus. The main novelty of fCube is that it implements several optimization techniques that allow to prune the search space acting on different aspects of proof-search. We tested the efficiency of our techniques by comparing fCube with other theorem provers. We found that our prover outperforms the other provers on several interesting families of formulas.

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In this paper we present BCDL, a description logic based on information terms semantics, which allows a constructive interpretation of ALC formulas. In the paper we describe the information terms semantics, we define a natural deduction calculus for BCDL and we show it is sound and complete. As a first application of proof-theoretical properties of the calculus, we show how it fulfills the proofs-asprograms paradigm. Finally, we discuss the role of generators, the main element distinguishing our formalisation from the usual ones.

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In this article we study the complexity of disjunction property for intuitionistic logic, the modal logics S4, S4.1, Grzegorczyk logic, Gödel-Löb logic, and the intuitionistic counterpart of the modal logic K. For S4 we even prove the feasible interpolation theorem and we provide a lower bound for the length of proofs. The techniques we use do not require proving structural properties of the calculi in hand, such as the cut-elimination theorem or the normalization theorem. This is a key point of our approach, since it allows us to treat logics for which only Hilbert-style characterizations are known.

The implementation of a logic requires, besides the definition of a calculus and a decision procedure, the development of techniques to reduce the search space. In this paper we introduce some simplification rules for Intuitionistic propositional logic that try to replace a formula with an equi-satisfiable ``simpler'' one with the aim to reduce the search space. Our results are proved via semantical techniques based on Kripke models. We also provide an empirical evaluation of their impact on implementations

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