Efficient SAT-based Proof Search in Intuitionistic Propositional Logic

Abstract

We present an efficient proof search procedure for Intuitionistic Propositional Logic which involves the use of an incremental SAT-solver. Basically, it is obtained by adding a restart operation to the system by Claessen and Rosén, thus we call our implementation . We gain some remarkable advantages: derivations have a simple structure; countermodels are in general small; using a standard benchmarks suite, we outperform and other state-of-the-art provers.

SAT-Based Proof Search in Intermediate Propositional Logics

“…1 exhibits how to extract a set of instances Ψ α of the L-axioms such that Ψ α i α. If D does not contain applications of rule Claus 1 , Ψ α is empty, and this ascertains that α is IPL-valid; actually, D can be immediately embedded into the calculus for IPL introduced in [8]. As an immediate consequence of Prop.…”

“…A bottom-up application of cpl 1 requires the choice of an implication clause λ = (a → b) → c from X, we call the main formula, and the selection of a set of atoms A ⊆ V R,X,g such that R, A c b, where b is the middle variable in λ. As discussed in [8,9], cpl 1 is a sort of generalization of the rule L →→ of the sequent calculus LJT/G4ip for IPL [5,18]. Rules Claus 0 and Claus 1 exploit the clausification procedure.…”

“…For L = IPL, Step (S4) is trivial, since the set Ax(IPL, V ) is empty. Actually, intuitRIL applied to IPL has the same behaviour as the procedure intuitR introduced in [8].…”

“…Actually, the decision procedure mimics the standard root-first proof search strategy for a sequent calculus strongly connected with Dyckhoff's calculus LJT [5] (alias G4ip). To improve performances, we have re-designed the prover by adding a restart operation, thus obtaining intuitR [8] (intuit with Restart). Differently from intuit, the intuitR procedure has a simple structure, consisting of two nested loops.…”

“…Differently from intuit, the intuitR procedure has a simple structure, consisting of two nested loops. Given a formula α, if α is provable in IPL the call intuitR(α) yields a derivation of α in the sequent calculus introduced in [8], a plain calculus where derivations have a single branch. If α is not provable in IPL, the outcome of intuitR(α) is a (typically small) countermodel for α, namely a Kripke model falsifying α.…”