On Complexity Measures in Polynomial Calculus

Proof complexity is the study of non-deterministic computational models, called proof systems, for proving that a given formula of propositional logic is unsatisfiable. As one of the subfields of computational complexity theory, the main questions of study revolve around the amount of resources needed to prove the unsatisfiability of various formulas in different proof systems. This line of inquiry has ties to some of the fundamental questions in theoretical computer science, as showing superpolynomial lower bounds on proof size for an arbitrary proof system would separate P from NP. However, while this was the original motivation for the field, that goal of separating P and NP still remains far out of our reach. In this thesis, we study two simple proof systems: resolution and polynomial calculus. In resolution we reason using clauses, while in polynomial calculus we can use polynomials over some fixed field. We have two main measures of complexity of proofs: size and space. Formally, size is the number of clauses or monomials that appear in a resolution or polynomial calculus proof, respectively. Space is the maximum number of clauses/monomials we need to keep at each time step if we view the proof as being presented as a sequence of configurations of limited memory. A third measure, which turns out to be very important in understanding the others, is width/degree. Width is the size of the largest clause in a resolution proof, while degree is an analogous measure for polynomial calculus that measures the size of a largest monomial in a proof. One reason that width is important in resolution is that width is a lower bound for space. The original proof of this claim focused on proving a characterization of resolution width in finite model theory and using this characterization to prove the relation with space. In this thesis we give a direct proof of the space-width relation, thereby improving our understanding of it. In the case of polynomial calculus we can pose the question whether the same relation holds between space and degree. We make some progress on this front by showing that if a formula F requires resolution width w then the XORified version of F requires space Ω(w). On the other hand we show that space lower bounds do not imply degree lower bounds in polynomial calculus, which was already known in resolution. The second reason why width/degree is an important measure is that strong lower bounds for width/degree imply strong lower bounds for size in both resolution and polynomial calculus. By now, proving width lower bounds in resolution follows a standard process with a developed machinery behind it. However, the situation in polynomial calculus was quite different and degree was much more poorly understood. We improve this situation by providing a unified framework for almost all previous degree lower bounds. Using this framework we also prove a few new degree and size lower bounds. In addition, we explore the relation between theory and practice by running experiments on some current state-of-the-art SAT solvers that are based on resolution.