Towards an Understanding of Polynomial Calculus: New Separations and Lower Bounds

Abstract

Abstract. 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 still open, as well as the relation of space to size and degree in PC/PCR. We prove that if a formula requires large resolution width, then making XOR substitution yields a formula requiring large PCR space, providing some circumstantial evidence that degree might be a lower bound for space. More importantly, this immediately yields formulas that are very hard for space but very easy for size, exhibiting a size-space separation similar to what is known for resolution. Using related ideas, we show that if a graph has good expansion and in addition its edge set can be partitioned into short cycles, then the Tseitin formula over this graph requires large PCR space. In particular, Tseitin formulas over random 4-regular graphs almost surely require space at least Ω`√n´. Our proofs use techniques recently introduced in . Our final contribution, however, is to show that these techniques provably cannot yield non-constant space lower bounds for the functional pigeonhole principle, delineating the limitations of this framework and suggesting that we are still far from characterizing PC/PCR space.

From Small Space to Small Width in Resolution

“…In Filmus et al [2013] we obtained a result analogous to Ben-Sasson and Nordström [2008] that there are formulas of worst-case space complexity that require only constant degree. The question of whether degree lower bounds imply space lower bounds remains open, however, and other results in Filmus et al [2013] can be interpreted as implying that the techniques in Bonacina and Galesi [2013] probably are not sufficient to resolve this question. Unfortunately, as discussed toward the end of this article, we also show that it appears unlikely that this problem can be addressed by methods similar to our proof of the corresponding inequality for resolution.…”

“…Or could perhaps even the stronger claim hold that polynomial calculus space is an upper bound on resolution width? These questions remain wide open, but in the recent paper by Filmus et al [2013] we made some limited progress by showing that if a formula requires large resolution width, then the "XORified version" of the formula requires large polynomial calculus space. We refer to the introductory section of Filmus et al [2013] for a more detailed discussion of these issues.…”

“…A possible approach for attacking this question was proposed in Bonacina and Galesi [2013]. In Filmus et al [2013] we obtained a result analogous to Ben-Sasson and Nordström [2008] that there are formulas of worst-case space complexity that require only constant degree. The question of whether degree lower bounds imply space lower bounds remains open, however, and other results in Filmus et al [2013] can be interpreted as implying that the techniques in Bonacina and Galesi [2013] probably are not sufficient to resolve this question.…”

A (Biased) Proof Complexity Survey for SAT Practitioners

“…Regarding the relation between space and degree, it is open whether degree is a lower bound for space (which would be the analogue of what holds in resolution), but some limited results in this direction were proven in [21]. The same paper also established that the two measures can be separated in the sense that there are formulas of minimal (i.e., constant) degree complexity requiring maximal (i.e., linear) space.…”

“…As to size versus space in PC, it is open whether small space complexity implies small size complexity, but [21] showed that small size does not imply small space, just as for resolution. Strong size-space trade-offs have been shown in [7], essentially extending the results for resolution in [6,10] but with slightly weaker parameters.…”