On the proof complexity of logics of bounded branching

04/23/2020 ∙ by Emil Jeřábek, et al. ∙ 0

We investigate the proof complexity of extended Frege (EF) systems for basic transitive modal logics (K4, S4, GL, ...) augmented with the bounded branching axioms 𝐁𝐁_k. 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 𝐒4.2𝐆𝐫𝐳𝐁𝐁_2 or 𝐆𝐋.2𝐁𝐁_2 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 𝐓_k, and we show conditional separations between EF and SF for all intermediate logics contained in 𝐓_2 + 𝐊𝐂.

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