9 research outputs found
On lengths of proofs in non-classical logics
AbstractWe give proofs of the effective monotone interpolation property for the system of modal logic K, and others, and the system IL of intuitionistic propositional logic. Hence we obtain exponential lower bounds on the number of proof-lines in those systems. The main results have been given in [P. Hrubeš, Lower bounds for modal logics, Journal of Symbolic Logic 72 (3) (2007) 941–958; P. Hrubeš, A lower bound for intuitionistic logic, Annals of Pure and Applied Logic 146 (2007) 72–90]; here, we give considerably simplified proofs, as well as some generalisations
A simplified lower bound for implicational logic
We present a streamlined and simplified exponential lower bound on the length
of proofs in intuitionistic implicational logic, adapted to Gordeev and
Haeusler's dag-like natural deduction.Comment: 31 page
On the proof complexity of logics of bounded branching
We investigate the proof complexity of extended Frege (EF) systems for basic
transitive modal logics (K4, S4, GL, ...) augmented with the bounded branching
axioms . 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 or under some
assumptions weaker than . 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
, and we show conditional separations between EF and SF for all
intermediate logics contained in .Comment: 58 page
Proof complexity of intuitionistic implicational formulas
Abstract We study implicational formulas in the context of proof complexity of intuitionistic propositional logic (IPC). On the one hand, we give an efficient transformation of tautologies to implicational tautologies that preserves the lengths of intuitionistic extended Frege (EF ) or substitution Frege (SF ) proofs up to a polynomial. On the other hand, EF proofs in the implicational fragment of IPC polynomially simulate full intuitionistic logic for implicational tautologies. The results also apply to other fragments of other superintuitionistic logics under certain conditions. In particular, the exponential lower bounds on the length of intuitionistic EF proofs by Hrubeš [9], generalized to exponential separation between EF and SF systems in superintuitionistic logics of unbounded branching by Jeřábe