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1. Filmus, Y.

et al.

Lauria, Massimo

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

Mikša, Mladen

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

Nordström, Jakob

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

Vinyals, M.

From small space to small width in resolution2015In: ACM Transactions on Computational Logic, ISSN 1529-3785, E-ISSN 1557-945X, Vol. 16, no 4, article id 28Article in journal (Refereed)

Abstract [en]

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 by simple syntactic manipulations of resolution refutations. As a by-product, we develop a "black-box" technique for proving space lower bounds via a "static" complexitymeasure that works against any resolution refutation-previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similarmethods.

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

Mikša, Mladen

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

Nordström, Jakob

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

Vinyals, Marc

KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.

From small space to small width in resolution2014In: 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing , 2014, Vol. 25, p. 300-311Conference paper (Refereed)

Abstract [en]

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 syntactic manipulations of resolution refutations. As a by-product, we develop a "black-box" technique for proving space lower bounds via a "static" complexity measure that works against any resolution refutation-previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similar methods.

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 [Bonacina-Galesi '13]. 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.

Aggregation of location estimates from multiple services for provisioning of location information enhances the accuracy and robustness of the final location information. Recent Internet of Things (IoT)-based localization service architectures therefore envision a "manager" for selecting and invoking provisioning services and, in the later step, aggregating the received information and providing it to location-based applications. The selection of provisioning services should take into account the accuracy and latency requirements from the applications and accuracy, latency, and power consumption characteristics of provisioning services. However, it is yet unclear how such selection should be made. We propose two algorithms for the selection of provisioning services aiming at meeting latency and subsequently accuracy requirements from the applications, one subject to minimizing per-request power consumption, while the other subject to a per-time bucket power minimization. In the considered examples, we show that the per-time bucket optimization achieves around 25% better performance in terms of power consumption, while trading-off accuracy satisfaction.

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 hard for resolution and even for polynomial calculus, which extends resolution with algebraic reasoning. We also present updated experimental data showing that these formulas are indeed still hard for current CDCL solvers, provided that these solvers do not also reason in terms of cardinality constraints (in which case the formulas can become very easy). Somewhat intriguingly, however, the very hardest instances in practice seem to arise from so-called fixed bandwidth matrices, which are provably easy for resolution and are also simple in practice if the solver is given a hint about the right branching order to use. This would seem to suggest that CDCL with current heuristics does not always search efficiently for short resolution proofs, despite the theoretical results of [Pipatsrisawat and Darwiche 2011] and [Atserias, Fichte, and Thurley 2011].

Proof complexity is the study of different resources that a proof needs in different proof systems for propositional logic. This line of inquiry relates to the fundamental questions in theoretical computer science, as lower bounds on proof size for an arbitrary proof system would separate P from NP.

We study two simple proof systems: resolution and polynomial calculus. In resolution we reason using clauses, while in polynomial calculus we use polynomials. We study three measures of complexity of proofs: size, space, and width/degree. Size is the number of clauses or monomials that appear in a resolution or polynomial calculus proof, respectively. Space is the maximum number of clauses/monomials we need to keep at each time step of the proof. Width/degree is the size of the largest clause/monomial in a proof.

Width is a lower bound for space in resolution. The original proof of this claim used finite model theory. In this thesis we give a different, more direct proof of the space-width relation. We can ask whether a similar relation holds between space and degree in polynomial calculus. We make some progress on this front by showing that when a formula F requires resolution width w then the XORified version of F requires polynomial calculus space Ω(w). We also show that space lower bounds do not imply degree lower bounds in polynomial calculus.

Width/degree and size are also related, as strong lower bounds for width/degree imply strong lower bounds for size. Currently, proving width lower bounds has a well-developed machinery behind it. However, the degree measure is much less well-understood. We provide a unified framework for almost all previous degree lower bounds. We also prove some new degree and size lower bounds. In addition, we explore the relation between theory and practice by running experiments on some current state-of-the-art SAT solvers.

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 develop the techniques in [AR03] to show that if one can "cluster" clauses and variables in a way that "respects the structure" of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. As a corollary of this, we prove that the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree when restricted to constant-degree expander graphs. This answers an open question in [Razborov’02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as shown in [Riis’93], both FPHP and Onto-PHP formulas are hard even when restricted to bounded-degree expanders.