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This paper investigates connections between these measures and real solver performance, focusing on space complexity and presenting extensive experiments.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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and Stanislav ˇZivn´y4  \n1 Department of Computer Science & HIIT, University of Helsinki, Finland  \n2 IBM Research and Technion, Haifa, Israel  \n3 KTH Royal Institute of Technology, Stockholm, Sweden  \n4 University of Oxford, United Kingdom  \nAbstract. Boolean satisfiability (SAT) solvers have improved enormously in performance over the last 10–15 years and are today an indispensable tool for solving a wide range of computational problems. However, our understanding of what makes SAT instances hard or easy in practice is still quite limited. A recent line of research in proof complexity has studied theoretical complexity measures such as length, width, and space in resolution, which is a proof system closely related to state-of-the-art conflict-driven clause learning (CDCL) SAT solvers. Although it seems like a natural question whether these complexity measures could be relevant for understanding the practical hardness of SAT instances, to date there has been very limited research on such possible connections. This paper sets out on a systematic study of the interconnections between theoretical complexity and practical SAT solver performance. Our main focus is on space complexity in resolution, and we report results from extensive experiments aimed at understanding to what extent this measure is correlated with hardness in practice. Our conclusion from the empirical data is that the resolution space complexity of a formula would seem to be a more fine-grained indicator of whether the formula is hard or easy than the length or width needed in a resolution proof. On the theory side, we prove a separation of general and tree-like resolution space, where the latter has been proposed before as a measure of practical hardness, and also show connections between resolution space and backdoor sets.  \n1 Introduction  \nIn the last 10–15 years, SAT solvers have become a standard tool for solving a wide variety of real-world computational problems [1] . Although all known SAT solvers have exponential running time in the worst case, dramatic improvements in performance have led to modern SAT solvers that can handle formulas with millions of variables. At the same time, very small formulas with just a few hundred variables are known which are completely beyond the reach of even the very best solvers. Understanding what makes a SAT instance hard or easy for state-of-the-art SAT solvers is therefore a fundamental problem. In particular, a natural, but not at all well-understood, question is whether one can find a good measure on the practical hardness of SAT instances.  \nThe current work addresses this question from the viewpoint of conflict-driven clause learning (CDCL) solvers [2, 3, 4], which—applying efficient data structures,  \nclause learning, forgetting, restarting, phase saving, and other important schemes—form the most prominent SAT solver paradigm today. Our goal is to explore possible connections between practical hardness—as witnessed by running times of CDCL solvers on SAT instances—and proof complexity measures employed in the formal study of the resolution proof systems that can be seen to underlie CDCL SAT solvers.  \nThe main bottleneck for CDCL solvers—apart from the obvious exponential worst case behavior—is the amount of memory used. In practice, it is completely infeasible to store all clauses learned during a CDCL run, and one therefore needs to design a highly selective and efficient clause caching scheme that learns and keeps the clauses needed for the CDCL solver to finish fast. Thus, understanding time and memory requirements for clause learning algorithms, and how these resources are related to each other, is a question of great practical importance.  \nProof complexity provides a possible approach for analyzing the potential and limitations of SAT solvers by studying the formal systems of reasoning which th","cbCaisc4eKo0bcyj","https://ap.wps.com/l/cbCaisc4eKo0bcyj","pdf",208406,17,"English","# Introduction\n## Background: SAT solvers and the challenge of hardness\n## Proof complexity and resolution measures\n## Focus of this paper: resolution space complexity","[{\"question\":\"What does the paper investigate about SAT instance hardness?\",\"answer\":\"It studies how theoretical proof complexity measures relate to the practical hardness of SAT instances for CDCL solvers, with a primary focus on resolution space complexity.\"},{\"question\":\"Which resolution proof measures are discussed?\",\"answer\":\"The paper discusses length (size), width, and space in resolution, explaining each measure and its informal meaning for proof requirements.\"},{\"question\":\"What conclusion does the paper reach from its experiments?\",\"answer\":\"Empirical results indicate that resolution space complexity is a finer-grained indicator of whether a formula is hard or easy than length or width.\"}]","Relating Proof Complexity Measures and Practical Hardness of SAT | PDF",43]