A Semantical Method To Reduce Branching In Tableau Proofs Alessandro Avellone, Guido Fiorino, and Ugo Moscato Dipartimento di Metodi Quantitativi per l’Economia, Universit`a Milano-Bicocca,Piazza dell’Ateneo Nuovo, 1, 20126 Milano, Italy, {alessandro.avellone,guido.fiorino,ugo.moscato}@unimib.it Abstract. We describe a technique, we call it Semantical Constraint Propagation (SCP), to reduce branching in tableau proofs. SCP is appli- cable to classical logic and the same ideas can adapted to other logics. At the end of this note we give the results obtained with our theorem prover PITP for propositional intuitionistic logic with SCP turned on and off. 1 Introduction The main concern in proofs for classical, modal and description logics is to bound branching, which is the source of inefficiency. Both for clausal and non-clausal theorem proving, a great deal of research has been done in order to bound branching and many techniques employed in clausal theorem proving have been generalized or adapted to non-clausal theorem proving ([3, 2, 5, 4, 1]). Here we are interested in tableau systems. The contribution of this note is a technique, we call it Semantical Branching Propagation (SCP for short), to bound branching in tableau proofs. Although in this note we consider the Smullyan tableau calculus for propositional classical logic, the same ideas can be applied to other logics. SCP is a strategy to select branching formulas in tableau proofs and is justified by semantical considerations. It can be inserted in a the- orem prover together with the well known optimization techniques. We present some experimental results of a C++ implementation (PITP) for propositional intuitionistic logic. We have tested our implementation on the ILTP Library ([6]) and on formulas generated at random. 2 Preliminary and Notation In order to describe SCP, we take as main reference the Smullyan calculus for propositional classical logic (Cl) provided in [7]. The rules of the calculus are given in Figure 1. The meaning of the signs T and F is explained in terms of realizability as follows. Let σ be a classical model, then σ ✄ TA (σ realizes TA) iff σ | = A (where | = is the usual binary relation between models and formulas) and σ ✄ FA iff σ | = A. A signed formula (swff for short) is a formula prefixed with T or F. Given a set S, σ ✄ S iff σ realizes every swff in S. A proof table