On Resolution Complexity of Matching Principles

Studying the complexity of mathematical proofs is important not only for automated theorem proving, but also for Mathematics as a whole. Each significant result in this direction would potentially have a great impact on Foundations of mathematics. Surprisingly enough, the general Proof Complexity is closely related to Propositional Proof Complexity. The latter area was founded by Cook and Reckhow in 1979, and enjoyed quite a fast development since then. One of the main research directions is finding the precise complexity of some natural combinatorial principle within a relatively weak propositional proof system. The results in the thesis fall in this category. We study the Resolution complexity of some Matching Principles. The three major contributions of the thesis are as follows. Firstly, we develop a general technique of proving resolution lower bounds for the perfect matching principles based on regular planar graphs. No lower bounds for these were known prior to our work. As a matter of fact, one such problem, the Mutilated Chessboard, was suggested as hard to automated theorem provers in 1964, and remained open since then. Our technique proves a tight resolution lower bound for the Mutilated Chessboard as well as for Tseitin tautologies based on rectangular grid graph. We reduce these problems to Tiling games, a concept introduced by us, which may be of interest on its own. Secondly, we find the exact Tree-Resolution complexity of the Weak PigeonHole Principle. It is the most studied combinatorial principle, but even its Tree-Resolution complexity was unknown prior to our work. We develop a new, more general method for proving Tree-Resolution lower bounds. We also define and prove non-trivial upper bounds on worst-case proofs of the Weak PigeonHole Principle. The worst-case proofs are first introduced by us, as a concept opposite to the optimal proofs. Thirdly, we prove Resolution width-size trade-offs for the Pigeon-Hole Principle. Proving the size lower bounds via the width lower bounds was known since the seminal paper of Haken, who first proved an exponential lower bound for the ordinary Pigeon-Hole Principle. The width-size trade-offs however were not studied at all prior to our work. Our result gives an optimal width-size trade-off for resolution in general.