ar X iv : 1 40 1 . 82 09 v 2 [ cs . L O ] 2 M ay 2 01 4 Propositional Logics Complexity and the Sub - Formula Property Edward

In 1979 Richard Statman proved, using proof-theory, that the purely implicational fragment of Intuitionistic Logic (M→) is PSPACEcomplete. He showed a polynomially bounded translation from full Intuitionistic Propositional Logic into its implicational fragment. By the PSPACE-completeness of S4, proved by Ladner, and the Gödel translation from S4 into Intuitionistic Logic, the PSPACE-completeness of M→ is drawn. The sub-formula principle for a deductive system for a logic L states that whenever {γ1, . . . , γk} ⊢L α there is a proof in which each formula occurrence is either a sub-formula of α or of some of γi. In this work we extend Statman’s result and show that any propositional (possibly modal) structural logic satisfying a particular statement of the sub-formula principle is PSPACE-complete. As a consequence, EXPTIME-complete propositional logics, such as PDL and the commonknowledge epistemic logic with at least 2 agents satisfy this particular sub-formula principle, if and only if, PSPACE=EXPTIME.