Abstract
We investigate the proof complexity of extended Frege (EF) systems for basic transitive modal logics (K4, S4, GL, …) augmented with the bounded branching axioms BBk. First, we study feasibility of the disjunction property and more general extension rules in EF systems for these logics: we show that the corresponding decision problems reduce to total coNP search problems (or equivalently, disjoint NP pairs, in the binary case); more precisely, the decision problem for extension rules is equivalent to a certain special case of interpolation for the classical EF system. Next, we use this characterization to prove superpolynomial (or even exponential, with stronger hypotheses) separations between EF and substitution Frege (SF) systems for all transitive logics contained in S4.2GrzBB2 or GL.2BB2 under some assumptions weaker than PSPACE≠NP. We also prove analogous results for superintuitionistic logics: we characterize the decision complexity of multi-conclusion Visser's rules in EF systems for Gabbay–de Jongh logics Tk, and we show conditional separations between EF and SF for all intermediate logics contained in T2+KC.
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