We consider the problem about the length of proofs of the sentences $\operatorname{Con}_S(\underline{n})$ saying that there is no proof of contradiction in S whose length is ≤ n. We show the relation of this problem to some problems about propositional proof systems.

We show that Smullyan's analytic tableaux cannot p-simulate the truth-tables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance between cut-free proofs and the Principle of Bivalence. Finally we discuss some ways in which this principle can be built into a tableau-like method without affecting its analytic nature.

LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives ¬ and $\bigwedge, \bigvee$. Then for every d ≥ 0 and n ≥ 2, there is a set Td n of depth d sequents of total size O which are refutable in LK by depth d + 1 proof of size exp) but such that every depth d refutation must have the size at least exp). The sets Td n express a weaker form of the pigeonhole (...) principle. (shrink)

Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic.

A notion of feasible function of finite type based on the typed lambda calculus is introduced which generalizes the familiar type 1 polynomial-time functions. An intuitionistic theory IPVω is presented for reasoning about these functions. Interpretations for IPVω are developed both in the style of Kreisel's modified realizability and Gödel's Dialectica interpretation. Applications include alternative proofs for Buss's results concerning the classical first-order system S12 and its intuitionistic counterpart IS12 as well as proofs of some of Buss's conjectures concerning IS12, (...) and a proof that IS12 cannot prove that extended Frege systems are not polynomially bounded. (shrink)

The method of analytic tableaux is discussed with respect to the notion of computational complexity. The class of formulas that was first identified to be intractable for the tableau method I. examined, and an explicit proof of this intractability is presented. A minor addition that leads to fast proofs of these formulas is described. Several results demonstrating that tableau deductions can be substantially speeded up with applications of path dissolution, which is an inference medianism that generalises the method of analytic (...) tableaux, are summarised. (shrink)