Marc Vinyals

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.

Cumulative Space in Black-White Pebbling and Resolution

We study space complexity and time-space trade-offs with a focus not on peak memory usage but on overall memory consumption throughout the computation. Such a cumulative space measure was introduced for the computational model of parallel black pebbling by [Alwen and Serbinenko 2015] as a tool for obtaining results in cryptography. We consider instead the nondeterministic black-white pebble game and prove optimal cumulative space lower bounds and trade-offs, where in order to minimize pebbling time the space has to remain large during a significant fraction of the pebbling. We also initiate the study of cumulative space in proof complexity, an area where other space complexity measures have been extensively studied during the last 10-15 years. Using and extending the connection between proof complexity and pebble games in [Ben-Sasson and Nordstrom 2008, 2011], we obtain several strong cumulative space results for (even parallel versions of) the resolution proof system, and outline some possible future directions of study of this, in our opinion, natural and interesting space measure.

Using Combinatorial Benchmarks to Probe the Reasoning Power of Pseudo-Boolean Solvers

We study cdcl-cuttingplanes, Open-WBO, and Sat4j, three successful solvers from the Pseudo-Boolean Competition 2016, and evaluate them by performing experiments on crafted benchmarks designed to be trivial for the cutting planes (CP) proof system underlying pseudo-Boolean (PB) proof search but yet potentially tricky for PB solvers. Our experiments demonstrate severe shortcomings in state-of-the-art PB solving techniques. Although our benchmarks have linear-size tree-like CP proofs, and are thus extremely easy in theory, the solvers often perform quite badly even for very small instances. We believe this shows that solvers need to employ stronger rules of cutting planes reasoning. Even some instances that lack not only Boolean but also real-valued solutions are very hard in practice, which indicates that PB solvers need to get better not only at Boolean reasoning but also at linear programming. Taken together, our results point to several crucial challenges to be overcome in the quest for more efficient pseudo-Boolean solvers, and we expect that a further study of our benchmarks could shed more light on the potential and limitations of current state-of-the-art PB solving.

Trade-offs Between Time and Memory in a Tighter Model of CDCL SAT Solvers

A long line of research has studied the power of conflict-driven clause learning (CDCL) and how it compares to the resolution proof system in which it searches for proofs. It has been shown that CDCL can polynomially simulate resolution even with an adversarially chosen learning scheme as long as it is asserting. However, the simulation only works under the assumption that no learned clauses are ever forgotten, and the polynomial blow-up is significant. Moreover, the simulation requires very frequent restarts, whereas the power of CDCL with less frequent or entirely without restarts remains poorly understood. With a view towards obtaining results with tighter relations between CDCL and resolution, we introduce a more fine-grained model of CDCL that captures not only time but also memory usage and number of restarts. We show how previously established strong size-space trade-offs for resolution can be transformed into equally strong trade-offs between time and memory usage for CDCL, where the upper bounds hold for CDCL without any restarts using the standard 1UIP clause learning scheme, and the (in some cases tightly matching) lower bounds hold for arbitrarily frequent restarts and arbitrary clause learning schemes.

In Between Resolution and Cutting Planes: A Study of Proof Systems for Pseudo-Boolean SAT Solving

We initiate a proof complexity theoretic study of subsystems of cutting planes (CP) modelling proof search in conflict-driven pseudo-Boolean (PB) solvers. These algorithms combine restrictions such as that addition of constraints should always cancel a variable and/or that so-called saturation is used instead of division. It is known that on CNF inputs cutting planes with cancelling addition and saturation is essentially just resolution. We show that even if general addition is allowed, this proof system is still polynomially simulated by resolution with respect to proof size as long as coefficients are polynomially bounded.

CNFgen: A Generator of Crafted Benchmarks

We present CNFgen, a generator of combinatorial benchmarks in DIMACS and OPB format. The proof complexity literature is a rich source not only of hard instances but also of instances that are theoretically easy but “extremal” in different ways, and therefore of potential interest in the context of SAT solving. Since most of these formulas appear not to be very well known in the SAT community, however, we propose CNFgen as a resource to make them readily available for solver development and evaluation. Many formulas studied in proof complexity are based on graphs, and CNFgen is also able to generate, parse and do basic manipulation of such objects. Furthermore, it includes a library cnfformula giving access to the functionality of CNFgen to Python programs.