Simplification Rules for Intuitionistic Propositional Tableaux

Abstract

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

fCube: An Efficient Prover for Intuitionistic Propositional Logic

“…The simplification rules of Fig. 2 also have a deep impact on the proof strategy as we showed in [4].…”

“…The first kind of simplification implemented in fCube exploits the well-known boolean simplification rules [4,7]. These rules simplify formulas containing the constants and ⊥ using Intuitionistic equivalences; e.g., (A ∨ ) ∧ B simplifies to B, by the equivalences A ∨ ≡ and B ∧ ≡ B.…”

“…As far as we know, the only works that address these issues in the context of tableau calculi are [5,7] that essentially refer to classical and modal logics, and [4] which addresses the case of Intuitionistic tableau calculi. The optimization rules implemented in fCube are those presented in [4]. We remark that fCube is a prototype Prolog implementation of the above techniques and we did very little work to "optimize the implementation"; e.g., fCube is based on a very rough implementation of the relevant data structures.…”

Intuitionistic Decision Procedures Since Gentzen

“…Several other easy optimisations are to be found in Franzén's [29], Ferrari et al's [25] and Weich's [72].…”

An Evaluation-Driven Decision Procedure for G3i

“…The evaluation function is defined by means of the simplifications displayed in Figures 1 and 2. The function B : L → L defined in Figure 1 is the usual Boolean simplification of formulas [Massacci 1998;Ferrari et al 2012]. The functions R L , R R , and R defined in Figure 2 take as arguments a formula in L and a sequent in Seq L and yield a formula in L .…”