An implementation of a O ( n logn ) - SPACEDecision Procedure for PropositionalIntuitionistic

Extended Abstract In this paper we present an implementation of a tableau calculus for propositional Intuitionistic Logic in which the depth of the deductions is linearly bounded in the length of the formula to be proved and giving rise to a O(n log n)-SPACE decision procedure. This implementation is based on the tableau calculus given in 1], which is an improvement of that given in 3]. In T, diierently from 2], the eecient calculus is obtained by modifying some rules of the original one. With these new rules the depth of deductions of T is linear in the length of the formula to be proved, a peculiarity that the calculus of 3] lacks and that is essential to develop a O(n log n)-SPACE decision procedure. In T the connectives _, ^ and ! have less and simpler rules than the calculus given in 2] and two non-invertible rules; on the contrary, the sequent calculus quoted above has three non-invertible rules. Another diierence is in the treatment of the negation, in T considered a primitive symbol. In the implementation considered here, we have carefully developed the strategy of our decision procedure so to lower its degree of nondeterminism (that implies to shrink its search space). Such a nondeterminism is related to the non-invertibility of the rules; for the case of propositional Intuitionistic Logic this nondeterminism is unavoidable and justiies the PSPACE-completeness (proved in 4]) of the corresponding decision problem. The development of such strategies requires a deep interaction between the development of the decision procedure and the construction of the counter model. Indeed, a strategy for the decision procedure can be extracted from the proof of the Completeness Theorem and, on the other hand, any change in the strategy of the decision procedure must be proved complete justifying it in the proof of the Completeness Theorem. In this section we present our calculus T for Int working on sets of sww's. The calculus uses three signs: T; F and F c : For the meaning of these signs, the deenitions of Kripke model, connguration, contradictory set of sww's, proof table of T, closed proof table of and consistent set of sww's we refer to 3] and 1].