  In 1979 Richard Statman proved, using proof-theory, that the purely
implicational fragment of Intuitionistic Logic (M-imply) is PSPACE-complete. 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 Goedel translation from S4 into Intuitionistic
Logic, the PSPACE- completeness of M-imply is drawn. The sub-formula principle
for a deductive system for a logic L states that whenever F1,...,Fk proves A,
there is a proof in which each formula occurrence is either a sub-formula of A
or of some of Fi. In this work we extend Statman result and show that any
propositional (possibly modal) structural logic satisfying a particular
formulation of the sub-formula principle is in PSPACE. If the logic includes
the minimal purely implicational logic then it is PSPACE-complete. As a
consequence, EXPTIME-complete propositional logics, such as PDL and the
common-knowledge epistemic logic with at least 2 agents satisfy this particular
sub-formula principle, if and only if, PSPACE=EXPTIME. We also show how our
technique can be used to prove that any finitely many-valued logic has the set
of its tautologies in PSPACE.
