The idea of feasible interpolation for propositional proof systems is to derive lower bounds for propositional proofs using circuit lower bounds for Craig’s interpolant. However, as far as we know, proof systems such as constant-depth Frege do not admit feasible interpolation. We extend the notion of feasible interpolation so that it is admitted by a number of treelike propositional proof syste...

One particular topic in the literature on Frege’s conception of sense relates to two apparently contradictory theses held by Frege: the isomorphism of thought and language on one hand and the expressibility of a thought by different sentences on the other. I will divide the paper into five sections. In (1) I introduce the problem of the tension in Frege’s thought. In (2) I discuss the main atte...

We present a simple propositional proof system which consists of a single axiom schema and a single rule, and use this system to construct a sequence of combinatorial tautologies that, when added to any Frege system, p-simulates extended-Frege systems.

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 bounde...

We extend results of Bonet, Buss and Pitassi on Bondy’s Theorem and of Nozaki, Arai and Arai on Bollobás’ Theorem by proving that Frankl’s Theorem on the trace of sets has quasipolynomial size Frege proofs. For constant values of the parameter t, we prove that Frankl’s Theorem has polynomial size AC-Frege proofs from instances of the pigeonhole principle. §