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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 ..."
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Cited by 18 (6 self)
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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 ..."
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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 ..."
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Cited by 11 (5 self)
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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.
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 ..."
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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
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, ..."
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Cited by 4 (3 self)
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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.
Scholars Journal of Physics, Mathematics and Statistics On the bounds for the main proof measures in some propositional proof systems
"... Abstract: Various proof complexity characteristics are investigated in three propositional proof systems, based on determinative disjunctive normal forms. The comparative analysis for size, time, space, width of proofs is given. For some formula family we obtain in our systems simultaneously bounds ..."
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Abstract: Various proof complexity characteristics are investigated in three propositional proof systems, based on determinative disjunctive normal forms. The comparative analysis for size, time, space, width of proofs is given. For some formula family we obtain in our systems simultaneously bounds for different proof complexity measures (asymptotically the same upper and lower bounds for each measures). These results can be generalized for the other formulas and for the other systems also.. Key Words: Determinative conjunct, determinative disjunctive normal form, elimination rule, size, time, space, width of proofs. INTRODUCTION One of the most fundamental problems of the proof complexity theory is to find an efficient proof system for propositional calculus. During the last decade an active line of research in classical propositional proof complexity has been to study space complexity and sizetimespacewidth tradeoffs for proofs. The space of proving a formula corresponds to the minimal size of a blackboard needed to verify all steps in the proof. Besides being an interesting natural complexity measure, space has connection to the memory consumption of SATISFIABILITY (SAT) problem solving, and so research has mostly focused on weak systems that are used by SAT solvers. Using the notion of determinative disjunctive normal form (dDNF), introduced by first coauthor in [1] and two proof systems introduced in [2] on the base of dDNF, we describe a new propositional proof systems also, and investigate the comparative analysis for mentioned proof complexity characteristics in them. First two systems are polynomially equivalent to wellknown resolution system R and cutfree sequent system LK (see in It is known that some of complexity measures (for example space and time) sometimes display a tradeoff: there are formulas having proofs in both short length and small space, but for which there can not exist proofs in short length and small space simultaneously The upper bounds for size, time, space and width are obtained on the base of some normal forms of proofs in mentioned systems. The "good" lower bounds are obtained using the properties of dDNF of our tautologies. Using the notion of strong equality of tautologies and comparative analysis for their proof complexities, given in [6], we can generalize our results for the other formulas and for some other systems. The results can be used for SAT problem solving.
Narrow proofs may be . . .
"... 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 nΩ(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size nO(w) is essentially tight. Moreover, our lo ..."
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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 nΩ(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size nO(w) is essentially tight. Moreover, our lower bounds can be generalized to polynomial calculus resolution (PCR) and SheraliAdams, 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—the formulas we study have Lasserre proofs of constant rank and size polynomial in both n and w.
Research Statement
, 2012
"... torics. Of course, these two subjects are no strangers: combinatorics is often used as a tool in theoretical computer science. The questions in theoretical computer science that I find most attractive are those which have a strong combinatorial flavor. Moreover, for me, combinatorics has an innate a ..."
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torics. Of course, these two subjects are no strangers: combinatorics is often used as a tool in theoretical computer science. The questions in theoretical computer science that I find most attractive are those which have a strong combinatorial flavor. Moreover, for me, combinatorics has an innate appeal, and I pursue it in and of itself. Combinatorics is a vast subject. My research has concentrated on using spectral methods to answer two types of questions: those of extremal combinatorics, of the ErdősKoRado type, and those of the analysis of Boolean functions, following the seminal work of FriedgutKalaiNaor. Together with my coauthors, we have proved a decadesold conjecture in extremal combinatorics concerning the maximal size of triangleintersecting families of graphs. In more recent work, we have generalized FriedgutKalaiNaor to Boolean functions on Sn. My contributions in theoretical computer science (with various coauthors) span several areas. In the sequel, I will focus on three areas encompassing my main contributions. First, I have designed a combinatorial algorithm for monotone submodular maximization over a matroid. Second, I have generalized a simulation result in circuit complexity to a corresponding result in proof complexity. Finally, I have studied the complexity class of comparator circuits, constructing a universal comparator circuit, and proving oracle separation results. In the future, I hope to combine the two threads of my research. Analysis of Boolean functions has led in the past to deep results in hardness of approximation. I believe that my expertise puts me in an excellent position to pursue similar questions. 1