Jakob Nordstrom’s research while affiliated with KTH Royal Institute of Technology and other places

Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolution-based proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing size-space trade-offs. This is a survey of research in proof complexity drawing on results and tools from pebbling, with a focus on proof space lower bounds and trade-offs between proof size and proof space.

We present size-space trade-offs for the polynomial calculus (PC) and polynomial calculus resolution (PCR) proof systems. These are the first true size-space trade-offs in any algebraic proof system, showing that size and space cannot be simultaneously optimized in these models. We achieve this by extending essentially all known size-space trade-offs for resolution to PC and PCR. As such, our results cover space complexity from constant all the way up to exponential and yield mostly superpolynomial or even exponential size blow-ups. Since the upper bounds in our trade-offs hold for resolution, our work shows that there are formulas for which adding algebraic reasoning on top of resolution does not improve the trade-off properties in any significant way. As byproducts of our analysis, we also obtain trade-offs between space and degree in PC and PCR exactly matching analogous results for space versus width in resolution, and strengthen the resolution trade-offs in [Beame, Beck, and Impagliazzo '12] to apply also to k-CNF formulas.

During the last decade, an active line of research in proof complexity has been to study space complexity and time space 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 PHPmn 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.

An active line of research in proof complexity over the last decade has been the study of proof space and trade-offs between size and space. Such questions were originally motivated by practical SAT solving, but have also led to the development of new theoretical concepts in proof complexity of intrinsic interest and to results establishing nontrivial relations between space and other proof complexity measures. By now, the resolution proof system is fairly well understood in this regard, as witnessed by a sequence of papers leading up to [Ben-Sasson and Nordstrom 2008, 2011] and [Beame, Beck, and Impagliazzo 2012]. However, for other relevant proof systems in the context of SAT solving, such as polynomial calculus (PC) and cutting planes (CP), very little has been known. Inspired by [BN08, BN11], we consider CNF encodings of so-called pebble games played on graphs and the approach of making such pebbling formulas harder by simple syntactic modifications. We use this paradigm of hardness amplification to make progress on the relatively longstanding open question of proving time-space trade-offs for PC and CP. Namely, we exhibit a family of modified pebbling formulas {F_n} such that: - The formulas F_n have size O(n) and width O(1). - They have proofs in length O(n) in resolution, which generalize to both PC and CP. - Any refutation in CP or PCR (a generalization of PC) in length L and space s must satisfy s log L >≈ √[4]{n}. A crucial technical ingredient in these results is a new two-player communication complexity lower bound for composed search problems in terms of block sensitivity, a contribution that we believe to be of independent interest.