Propositional Logics Complexity and the Sub-Formula Property

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