Results 1  10
of
21
Linear Gaps Between Degrees for the Polynomial Calculus Modulo Distinct Primes
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1999
"... This paper gives nearly optimal lower bounds on the minimum degree of polynomial calculus refutations of Tseitin's graph tautologies and the mod p counting principles, p 2. The lower bounds apply to the polynomial calculus over fields or rings. These are the first linear lower bounds for ..."
Abstract

Cited by 36 (9 self)
 Add to MetaCart
This paper gives nearly optimal lower bounds on the minimum degree of polynomial calculus refutations of Tseitin's graph tautologies and the mod p counting principles, p 2. The lower bounds apply to the polynomial calculus over fields or rings. These are the first linear lower bounds for the polynomial calculus for kCNF formulas. As a
PEBBLE GAMES, PROOF COMPLEXITY AND TIMESPACE TRADEOFFS
, 2010
"... 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 resolutionbased proof systems when compari ..."
Abstract

Cited by 18 (6 self)
 Add to MetaCart
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 resolutionbased proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing sizespace tradeoffs. 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 tradeoffs between proof size and proof space.
Some Tradeoff Results for Polynomial Calculus [Extended Abstract]
"... We present sizespace tradeoffs for the polynomial calculus (PC) and polynomial calculus resolution (PCR) proof systems. These are the first true sizespace tradeoffs in any algebraic proof system, showing that size and space cannot be simultaneously optimized in these models. We achieve this by e ..."
Abstract

Cited by 11 (6 self)
 Add to MetaCart
(Show Context)
We present sizespace tradeoffs for the polynomial calculus (PC) and polynomial calculus resolution (PCR) proof systems. These are the first true sizespace tradeoffs 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 sizespace tradeoffs for resolution to PC and PCR. As such, our results cover space complexity from constant all the way up to exponential and yield mostly superpolynomial or even exponential size blowups. Since the upper bounds in our tradeoffs hold for resolution, our work shows that there are formulas for which adding algebraic reasoning on top of resolution does not improve the tradeoff properties in any significant way. As byproducts of our analysis, we also obtain tradeoffs between space and degree in PC and PCR exactly matching analogous results for space versus width in resolution, and strengthen the resolution tradeoffs in [Beame, Beck, and Impagliazzo ’12] to apply also to kCNF formulas.
On the Virtue of Succinct Proofs: Amplifying Communication Complexity Hardness to TimeSpace Tradeoffs in Proof Complexity (Extended Abstract)
 STOC’12, MAY 19–22
, 2012
"... An active line of research in proof complexity over the last decade has been the study of proof space and tradeoffs 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 intr ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
An active line of research in proof complexity over the last decade has been the study of proof space and tradeoffs 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 [BenSasson and Nordström 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 socalled 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 timespace tradeoffs for PC and CP. Namely, we exhibit a family of modified pebbling formulas {Fn} ∞ n=1 such that: • The formulas Fn have size Θ(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 twoplayer communication complexity lower bound for composed search problems in terms of block sensitivity, a contribution that we believe to be of independent interest.
Space complexity in polynomial calculus
 In Proceedings of the 27th Conference on Computational Complexity, CCC 2012
"... AbstractDuring the last decade, an active line of research in proof complexity has been to study space complexity and timespace tradeoffs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving. For the polynomial calculus proo ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
(Show Context)
AbstractDuring the last decade, an active line of research in proof complexity has been to study space complexity and timespace tradeoffs 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 et al. '02], where the lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could refute any kCNF formula in constant space. We prove several new results on space in polynomial calculus (PC) and in the extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich et al. '02]. 1) For PCR, we prove an Ω(n) space lower bound for a bitwise encoding of the functional pigeonhole principle with m pigeons and n holes. These formulas have width O(log n), and hence this is an exponential improvement over [Alekhnovich et al. '02] measured in the width of the formulas. 2) We then present another encoding of the pigeonhole principle that has constant width, and prove an Ω(n) space lower bound in PCR for these formulas as well. 3) We prove an Ω(n) space lower bound in PC for the canonical 3CNF version of the pigeonhole principle formulas PHP m n with m pigeons and n holes, and show that this is tight. 4) We prove that any kCNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not known to be the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3CNF in the canonical way.
Towards an Understanding of Polynomial Calculus: New Separations and Lower Bounds
"... 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 polynomi ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
(Show Context)
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 sizespace 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 4regular graphs almost surely require space at least Ω ` √ n ´. Our proofs use techniques recently introduced in [BonacinaGalesi ’13]. Our final contribution, however, is to show that these techniques provably cannot yield nonconstant 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. 1
Optimality of SizeDegree Tradeoffs for Polynomial Calculus
, 2010
"... There are methods to turn short refutations in Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR) into refutations of low degree. Bonet and Galesi [8, 10] asked if such sizedegree tradeoffs for PC [13, 17] and PCR [2] are optimal. We answer this question by showing a polynomia ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
There are methods to turn short refutations in Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR) into refutations of low degree. Bonet and Galesi [8, 10] asked if such sizedegree tradeoffs for PC [13, 17] and PCR [2] are optimal. We answer this question by showing a polynomial encoding of Graph Ordering Principle on m variables which requires PC and PCR refutations of degree Ω( m). Tradeoffs optimality follows from our result and from the short refutations of Graph Ordering Principle in [8, 9]. We then introduce the algebraic proof system PCRk which combines together Polynomial Calculus (PC) and kDNF Resolution (RESk). We show a size hierarchy theorem for PCRk: PCRk is exponentially separated from PCRk+1. This follows from the previous degree lower bound and from techniques developed for RESk. Finally we show that random formulas in conjunctive normal form (3CNF) are hard to refute in PCRk. 1
On the Automatizability of Polynomial Calculus
 THEORY COMPUT SYST
, 2009
"... We prove that Polynomial Calculus and Polynomial Calculus with Resolution are not automatizable, unless W [P]hard problems are fixed parameter tractable by oneside error randomized algorithms. This extends to Polynomial Calculus the analogous result obtained for Resolution by Alekhnovich and Razbo ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
We prove that Polynomial Calculus and Polynomial Calculus with Resolution are not automatizable, unless W [P]hard problems are fixed parameter tractable by oneside error randomized algorithms. This extends to Polynomial Calculus the analogous result obtained for Resolution by Alekhnovich and Razborov (SIAM J. Comput. 38(4):1347–1363, 2008).
Narrow proofs may be maximally long.
 In Proceedings of the 29th Annual IEEE Conference on Computational Complexity (CCC ’14),
, 2014
"... Abstract We prove that there are 3CNF 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 is essentially tight. Moreover, ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
Abstract We prove that there are 3CNF 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 is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution (PCR) and SheraliAdams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.
Algebraic proofs over noncommutative formulas
 Information and Computation
"... Abstract We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege, yielding a semantic way to define a CookReckhow (i.e., polynomially verifiable) algebraic ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
Abstract We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege, yielding a semantic way to define a CookReckhow (i.e., polynomially verifiable) algebraic analog of Frege proofs, different from that given in The motivation behind this work is developing techniques incorporating rank arguments (similar to those used in arithmetic circuit complexity) for establishing lower bounds on propositional proofs.