Jakob Nordström

Short Proofs May Be Spacious: An Optimal Separation of Space and Length in Resolution

A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space.In this paper we resolve the question by answering it negatively in the strongest possible way. We show that there are families of 6-CNF formulas of size n, for arbitrarily large n, that have resolution proofs of length O(n) but for which any proof requires space Omega(n/log n). This is the strongest asymptotic separation possible since any proof of length O(n) can always be transformed into a proof in space O(n / log n).Our result follows by reducing the space complexity of so called pebbling formulas over a directed acyclic graph to the black-white pebbling price of the graph.The proof is somewhat simpler than previous results (in particular, those reported in [Nordstrom 2006, Nordstrom and Hastad 2008]) as it uses a slightly different flavor of pebbling formulas which allows for a rather straightforward reduction of proof space to standard black-white pebbling price.

Narrow proofs may be spacious: separating space and width in resolution

The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously been studied and related to the resolution refutation size of unsatisfiable CNF formulas. Also, the refutation space of a formula has been proven to be at least as large as the refutation width, but it has been open whether space can be separated from width or the two measures coincide asymptotically. We prove that there is a family of k-CNF formulas for which the refutation width in resolution is constant but the refutation space is non-constant, thus solving a problem mentioned in several previous papers.

Pebble Games, Proof Complexity, and Time-Space Trade-offs

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.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.

On the virtue of succinct proofs: amplifying communication complexity hardness to time-space trade-offs in proof complexity

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.

Towards an optimal separation of space and length in resolution

Most state-of-the-art satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these measures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Theta(√(n)) on the space needed for so-called pebbling contradictions over pyramid graphs of size n. Also, continuing the line of research initiated by (Ben-Sasson 2002) into trade-offs between different proof complexity measures, we present a simplified proof of the recent length-space trade-off result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential trade-offs in resolution.

Relating proof complexity measures and practical hardness of SAT

Boolean satisfiability (SAT) solvers have improved enormously in performance over the last 10---15 years and are today an indispensable tool for solving a wide range of computational problems. However, our understanding of what makes SAT instances hard or easy in practice is still quite limited. A recent line of research in proof complexity has studied theoretical complexity measures such as length, width, and space in resolution, which is a proof system closely related to state-of-the-art conflict-driven clause learning (CDCL) SAT solvers. Although it seems like a natural question whether these complexity measures could be relevant for understanding the practical hardness of SAT instances, to date there has been very limited research on such possible connections. This paper sets out on a systematic study of the interconnections between theoretical complexity and practical SAT solver performance. Our main focus is on space complexity in resolution, and we report results from extensive experiments aimed at understanding to what extent this measure is correlated with hardness in practice. Our conclusion from the empirical data is that the resolution space complexity of a formula would seem to be a more fine-grained indicator of whether the formula is hard or easy than the length or width needed in a resolution proof. On the theory side, we prove a separation of general and tree-like resolution space, where the latter has been proposed before as a measure of practical hardness, and also show connections between resolution space and backdoor sets.

On the interplay between proof complexity and SAT solving

This paper is intended as an informal and accessible survey of proof complexity for non-experts, focusing on some comparatively weak proof systems of particular interest in connection with SAT solving. We review resolution, polynomial calculus, and cutting planes (related to conflict-driven clause learning, Gröbner basis computations, and pseudo-Boolean solving, respectively) and some complexity measures that have been studied for these proof systems. We also discuss briefly to what extent proof complexity could provide insights into SAT solver performance, and how concerns related to applied SAT solving can give rise to interesting complexity-theoretic questions. Along the way, we highlight a number of current research challenges.

From Small Space to Small Width in Resolution

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of a Conjunctive Normal Form (CNF) formula is always an upper bound on the width needed to refute the formula. Their proof is beautiful but uses a nonconstructive argument based on Ehrenfeucht-Fraïssé games. We give an alternative, more explicit, 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.

Long Proofs of (Seemingly) Simple Formulas

In 2010, Spence and Van Gelder presented a family of CNF formulas based on combinatorial block designs. They showed empirically that this construction yielded small instances that were orders of magnitude harder for state-of-the-art SAT solvers than other benchmarks of comparable size, but left open the problem of proving theoretical lower bounds. We establish that these formulas are exponentially hard for resolution and even for polynomial calculus, which extends resolution with algebraic reasoning. We also present updated experimental data showing that these formulas are indeed still hard for current CDCL solvers, provided that these solvers do not also reason in terms of cardinality constraints (in which case the formulas can become very easy). Somewhat intriguingly, however, the very hardest instances in practice seem to arise from so-called fixed bandwidth matrices, which are provably easy for resolution and are also simple in practice if the solver is given a hint about the right branching order to use. This would seem to suggest that CDCL with current heuristics does not always search efficiently for short resolution proofs, despite the theoretical results of [Pipatsrisawat and Darwiche 2011] and [Atserias, Fichte, and Thurley 2011].

Near-Optimal Lower Bounds on Quantifier Depth and Weisfeiler--Leman Refinement Steps

We prove near-optimal trade-offs for quantifier depth versus number of variables in first-order logic by exhibiting pairs of n-element structures that can be distinguished by a k-variable first-order sentence but where every such sentence requires quantifier depth at least nΩ(k/ log k). Our trade-offs also apply to first-order counting logic, and by the known connection to the k-dimensional Weisfeiler–Leman algorithm imply near-optimal lower bounds on the number of refinement iterations. A key component in our proof is the hardness condensation technique recently introduced by [Razborov ’16] in the context of proof complexity. We apply this method to reduce the domain size of relational structures while maintaining the quantifier depth required to distinguish them.Categories and Subject Descriptors F.4.1 [Mathematical Logic]: Computational Logic, Model theory; F.2.3 [Tradeoffs between Complexity Measures]