Abstract-During the last decade, an active line of research in proof complexity has been to study space complexity and timespace trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving. For the polynomial calculus proof system, the only previously known space lower bound is for CNF formulas of unbounded width in [Alekhnovich et al. '02], where the lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could refute any k-CNF formula in constant space. We prove several new results on space in polynomial calculus (PC) and in the extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich et al. '02]. 1) For PCR, we prove an Ω(n) space lower bound for a bitwise encoding of the functional pigeonhole principle with m pigeons and n holes. These formulas have width O(log n), and hence this is an exponential improvement over [Alekhnovich et al. '02] measured in the width of the formulas. 2) We then present another encoding of the pigeonhole principle that has constant width, and prove an Ω(n) space lower bound in PCR for these formulas as well. 3) We prove an Ω(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas PHP m n with m pigeons and n holes, and show that this is tight. 4) We prove that any k-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not known to be the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way.

Abstract We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n Ω(w) . This shows that the simple counting argument that any formula refutable in width w must have a proof in size n is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.

This talk is intended as a selective survey of proof complexity, focusing on some comparatively weak proof systems that are of partic-ular interest in connection with SAT solving. We will review resolution, polynomial calculus, and cutting planes (related to conflict-driven clause learning, Gröbner basis computations, and pseudo-Boolean solvers, re-spectively) and some proof complexity measures that have been studied for these proof systems. We will also briefly discuss if and how these proof complexity measures could provide insights into SAT solver performance.

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools from finite model theory. We give an alternative, completely elementary, proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a “black-box ” technique for proving space lower bounds via a “static ” complexity measure that works against any resolution refutation—previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similar methods.

We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size nΩ(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size nO(w) is essentially tight. Moreover, our lower bounds can be generalized to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. Our results do not extend all the way to Lasserre, however—the formulas we study have Lasserre proofs of constant rank and size polynomial in both n and w.

torics. Of course, these two subjects are no strangers: combinatorics is often used as a tool in theoretical computer science. The questions in theoretical computer science that I find most attractive are those which have a strong combinatorial flavor. Moreover, for me, combinatorics has an innate appeal, and I pursue it in and of itself. Combinatorics is a vast subject. My research has concentrated on using spectral methods to answer two types of questions: those of extremal combinatorics, of the Erdős-Ko-Rado type, and those of the analysis of Boolean functions, following the seminal work of Friedgut-Kalai-Naor. Together with my coauthors, we have proved a decades-old conjecture in ex-tremal combinatorics concerning the maximal size of triangle-intersecting families of graphs. In more recent work, we have generalized Friedgut-Kalai-Naor to Boolean functions on Sn. My contributions in theoretical computer science (with various coauthors) span several areas. In the sequel, I will focus on three areas encompassing my main contributions. First, I have designed a combinatorial algorithm for monotone submodular maximization over a matroid. Second, I have generalized a simulation result in circuit complexity to a correspond-ing result in proof complexity. Finally, I have studied the complexity class of comparator circuits, constructing a universal comparator circuit, and proving oracle separation results. In the future, I hope to combine the two threads of my research. Analysis of Boolean functions has led in the past to deep results in hardness of approximation. I believe that my expertise puts me in an excellent position to pursue similar questions. 1