Relating Proof Complexity Measures and Practical Hardness of SAT

  • Matti Järvisalo
  • Arie Matsliah
  • Jakob Nordström
  • Stanislav Živný
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7514)


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.


Space Complexity Resolution Space General Resolution Substitution Function Resolution Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Matti Järvisalo
    • 1
  • Arie Matsliah
    • 2
  • Jakob Nordström
    • 3
  • Stanislav Živný
    • 4
  1. 1.Department of Computer Science & HIITUniversity of HelsinkiFinland
  2. 2.IBM Research and TechnionHaifaIsrael
  3. 3.KTH Royal Institute of TechnologyStockholmSweden
  4. 4.University of OxfordUnited Kingdom

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