The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct reference to polynomials, time, or even computation. Complexity classes characterized in this way include polynomial time, the functional polytime hierarchy, the logspace decidable problems, and NC. After developing these "resource free" definitions, we apply them to redeveloping the feasible logical system of Cook and Urquhart, and show how this first-order system relates to the second-order system of Leivant. The connection is an interesting one since the systems were defined independently and have what appear to be very different rules for the principle of induction. Furthermore it is interesting to see, albeit in a very specific context, how to retract a second order statement, ("inducti...

It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of combinatorial tautologies that we consider seem to give at most a quasipolynomial speed-up of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl.

The combinatorial parity principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a fixed bi-partition of the vertices, there is no perfect matching between them. Therefore, it follows from recent lower bounds for the pigeonhole principle that the parity principle requires exponential-size bounded-depth Frege proofs. Ajtai [Ajt90] previously showed that the parity principle does not have polynomial-size bounded-depth Frege proofs even with the pigeonhole principle as an axiom schema. His proof utilizes nonstandard model theory and is nonconstructive. We improve Ajtai's lower bound from barely superpolynomial to exponential and eliminate the nonstandard model theory. Our lower bound is also related to the inherent complexity of particular search classes (see [Pap91]). In particular, oracle separations between the complexity classes PPA and PPAD, and between PPA and PPP also follow from ou...

Propositional Proof Complexity is an active area of research whose main focus is the study of the length of proofs in propositional logic. There are several motivations for such a study, the main of which is probably its connection to the P vs NP problem in Computational Complexity. The experience

It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatoriM tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of combinatorial tautologies that we consider seem to give at most a quasipolynomial speed-up of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl.