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We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov'03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further
We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size nΩ(d) for values of d = d(n) from constant all the way up to nδ for some universal constant δ. This shows that the nO(d) running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs
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
During the last 10 to 15 years, 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 concern in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers. There has been a relatively long sequence of paper
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
This talk is intended as a selective survey of proof complexity, focusing on some comparatively weak proof systems that are of particular 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, respectively) and some proof complexity measures that
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 syntact
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(ω) is essentially tight. Moreover, our lower bounds can be generalized to polynomial calculus resolution (PCR) and Sherali-Adams, implying tha
During the last decade, an active line of research in proof complexity has been into the space complexity of proofs and how space is related to other measures. By now these aspects of resolution are fairly well understood, but many open problems remain for the related but stronger polynomial calculus (PC/PCR) proof system. For instance, the space complexity of many standard "benchmark formulas" is
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 resu
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 tr
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,
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
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 comple
The last decade has seen a revival of interest in pebble games in the context 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. The typical approach has been to encode the pebble game played on a graph as a CNF formula an
A well-known theorem by Tarsi states that a minimally unsatisfiable CNF formula with m clauses can have at most m - 1 variables, and this bound is exact. In the context of proving lower bounds on proof space in k-DNF resolution, [Ben-Sasson and Nordström 2009] extended the concept of minimal unsatisfiability to sets of k-DNF formulas and proved that a minimally unsatisfiable k-DNF set with m formu
The last decade has seen a revival of interest in pebble games in the context 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. The typical approach has been to encode the pebble game played on a graph as a CNF formula an
We present a greatly simplified proof of the length-space trade-off result for resolution in [P. Hertel, T. Pitassi, Exponential time/space speedups for resolution and the PSPACE-completeness of black-white pebbling, in: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS '07), Oct. 2007, pp. 137-149], and also prove a couple of other theorems in the same vein. W
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 conjunctive normal form
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, fo
