Narrow Proofs May Be Maximally Long

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 O(w) 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. Our results do not extend all the way to Lasserre, however, where the formulas we study have proofs of constant rank and size polynomial in both n and w. IntroductionProof complexity studies how hard it is to prove that propositional logic formulas are tautologies. While the original motivation for this line of research, as discussed in [CR79], was to prove superpolynomial lower bounds on proof size for increasingly stronger proof systems as a way towards establishing NP = co-NP (and hence P = NP), it is probably fair to say that most current research in proof complexity is driven by other concerns.One such concern is the connection to SAT solving. By a standard transformation any propositional logic formula can be converted to another formula in conjunctive normal form (CNF) that has the same size up to constant factors and is unsatisfiable if and only if the original formula is a tautology. Any algorithm for solving SAT defines a proof system in the sense that the execution trace of the algorithm constitutes a polynomial-time verifiable witness of unsatisfiability. 1 In fact, most modern-day SAT solvers can be seen to search for proofs in systems at fairly low levels in the proof complexity hierarchy, and upper and lower bounds for these proof systems hence give information about the potential and limitations of the corresponding SAT solvers. In this work, we focus on such proof systems. BackgroundThe dominant strategy in applied SAT solving today is so-called conflict-driven clause learning (CDCL) [BS97, MS99, MMZ + 01], which is ultimately based on the resolution proof system [Bla37]. The most studied complexity measure for resolution is size (also referred to as * This is the full-length version of the paper [ALN14], which appeared in Proceedings of the 29th Annual IEEE Conference on Computational Complexity (CCC '14).1 Such a witness is often referred to as a refutation rather than a proof , and these two terms are sometimes used interchangeably.

Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

“…This is consistent with how size is defined for the "dynamic version" of Nullstellensatz known as polynomial calculus [26,1], and also with the general size definitions for so-called algebraic and semialgebraic proof systems in [4,14,5].…”

“…By way of background, it is not hard to show that for all three proof systems upper bounds on degree/width imply upper bounds on size, in the sense that if a CNF formula over n variables can be refuted in degree/width d, then such a refutation can be carried out in size n O(d) . Furthermore, this upper bound has been proven to be tight up to constant factors in the exponent for resolution and polynomial calculus [4], and it follows from [44] that this also holds for Nullstellensatz. In the other direction, it has been shown for resolution and polynomial calculus that strong enough lower bounds on degree/width imply lower bounds on size [36,11].…”

Automating Resolution is NP-Hard

“…This follows from the size-width trade-off of Ben-Sasson and Wigderson [10]. Indeed, it is trivial to find a refutation of width at most w in time n O(w) if there is one (and, in general, time n Ω(w) is necessary [7]). When s is subexponential the runtime of this algorithm is the non-trivial 2 n 1/2+o (1) .…”

A Cardinal Improvement to Pseudo-Boolean Solving

“…Another possible direction is to perform the cardinality detection in an adaptive fashion, where the solver would spend more or less time on detection depending on how much this work has paid off so far during search (provided that a meaningful way of measuring this could be found). We already know of crafted CNF formulas for which our approach does not work, although the formulas contain cardinality constraints (e.g., the so-called relativized pigeonhole principle (RPHP) formulas in (Atserias, Lauria, and Nordström 2016)), and as a first step would want to be able to solve also such formulas.…”