Proof Complexity of Intuitionistic Propositional Logic

We explore the proof complexity of intuitionistic propositional logic (IPL). The problem of determining whether or not an intuitionistic formula is valid is PSPACE-Complete via a reduction from QBF . In view of this reduction (due to Statman), it is natural to compare the proof-theoretic strength of a standard axiomatic system for IPL with a similar proof system for classical quantified Boolean logic. In fact, the intuitionistic system seems to be weak in comparison with the latter, in the following sense – unless a variant of Gentzen’s proof system LK is super (and therefore NP = coNP), Statman’s reduction from QBF to IPL cannot even translate trivial classical instances of the law of excluded middle into intuitionistic formulas having proofs with polynomial size upper bounds. An immediate implication of this result is that unless the variant of LK for classical logic is a super proof system, there is a superpolynomial separation between it and a similar variant of Gentzen’s system LJ for Intuitionistic Logic.