Bidirectional Decision Procedures for the Intuitionistic Propositional Modal Logic IS4

We present a multi-context focused sequent calculus whose derivations are in bijective correspondence with normal natural deductions in the propositional fragment of the intuitionistic modal logic IS4 . This calculus, suitable for the enumeration of normal proofs, is the starting point for the development of a sequent calculus-based bidirectional decision procedure for propositional IS4 . In this system, relevant derived inference rules are constructed in a forward direction prior to proof search, while derivations constructed using these derived rules are searched over in a backward direction. We also present a variant which searches directly over normal natural deductions. Experimental results show that on most problems, the bidirectional prover is competitive with both conventional backward provers using loop-detection and inverse method provers, significantly outperforming them in a number of cases.