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.Comment: 58 page