Trade-offs Between Time and Memory in a Tighter Model of CDCL SAT Solvers

  • Jan Elffers
  • Jan Johannsen
  • Massimo Lauria
  • Thomas Magnard
  • Jakob NordströmEmail author
  • Marc Vinyals
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9710)


A long line of research has studied the power of conflict-driven clause learning (CDCL) and how it compares to the resolution proof system in which it searches for proofs. It has been shown that CDCL can polynomially simulate resolution even with an adversarially chosen learning scheme as long as it is asserting. However, the simulation only works under the assumption that no learned clauses are ever forgotten, and the polynomial blow-up is significant. Moreover, the simulation requires very frequent restarts, whereas the power of CDCL with less frequent or entirely without restarts remains poorly understood. With a view towards obtaining results with tighter relations between CDCL and resolution, we introduce a more fine-grained model of CDCL that captures not only time but also memory usage and number of restarts. We show how previously established strong size-space trade-offs for resolution can be transformed into equally strong trade-offs between time and memory usage for CDCL, where the upper bounds hold for CDCL without any restarts using the standard 1UIP clause learning scheme, and the (in some cases tightly matching) lower bounds hold for arbitrarily frequent restarts and arbitrary clause learning schemes.


Memory Usage Proof System Unit Clause General Resolution Conflict Analysis 
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.



We are grateful to the anonymous SAT conference reviewers for detailed comments that helped improve the exposition in this paper.

The third author performed this work while at KTH Royal Institute of Technology, and most of the work of the second and fourth author was done while visiting KTH. The first, third, fifth, and sixth author were funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC grant agreement no. 279611 as well as by Swedish Research Council grant 621-2012-5645. The third author was also supported by the European Research Council under the European Union’s Horizon 2020 Research and Innovation Programme/ERC grant agreement no. 648276.


  1. 1.
    Alekhnovich, M., Ben-Sasson, E., Razborov, A.A., Wigderson, A.: Space complexity in propositional calculus. SIAM J. Comput. 31(4), 1184–1211 (2002). Preliminary version in STOC 2000MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alekhnovich, M., Razborov, A.A.: Resolution is not automatizable unless W[P] is tractable. SIAM J. Comput. 38(4), 1347–1363 (2008). Preliminary version in FOCS 2001MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Atserias, A., Fichte, J.K., Thurley, M.: Clause-learning algorithms with many restarts and bounded-width resolution. J. Artif. Intell. Res. 40, 353–373 (2011). Preliminary version in SAT 2009MathSciNetzbMATHGoogle Scholar
  4. 4.
    Audemard, G., Simon, L.: Predicting learnt clauses quality in modern SAT solvers. In: Proceedings of the 21st International Joint Conference on Artificial Intelligence (IJCAI 2009), pp. 399–404, July 2009Google Scholar
  5. 5.
    Bayardo Jr., R.J., Schrag, R.: Using CSP look-back techniques to solve real-world SAT instances. In: Proceedings of the 14th National Conference on Artificial Intelligence (AAAI 1997), pp. 203–208, July 1997Google Scholar
  6. 6.
    Beame, P., Beck, C., Impagliazzo, R.: Time-space tradeoffs in resolution: superpolynomial lower bounds for superlinear space. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC 2012), pp. 213–232, May 2012Google Scholar
  7. 7.
    Beame, P., Kautz, H., Sabharwal, A.: Towards understanding and harnessing the potential of clause learning. J. Artif. Intell. Res. 22, 319–351 (2004). Preliminary version in IJCAI 2003MathSciNetzbMATHGoogle Scholar
  8. 8.
    Beame, P., Sabharwal, A.: Non-restarting SAT solvers with simple preprocessing can efficiently simulate resolution. In: Proceedings of the 28th National Conference on Artificial Intelligence (AAAI 2014), pp. 2608–2615. AAAI Press, July 2014Google Scholar
  9. 9.
    Beck, C., Nordström, J., Tang, B.: Some trade-off results for polynomial calculus. In: Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC 2013), pp. 813–822, May 2013Google Scholar
  10. 10.
    Ben-Sasson, E., Galesi, N.: Space complexity of random formulae in resolution. Random Struct. Algorithms 23(1), 92–109 (2003). Preliminary version in CCC 2001MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ben-Sasson, E., Nordström, J.: Understanding space in proof complexity: separations and trade-offs via substitutions. In: Proceedings of the 2nd Symposium on Innovations in Computer Science (ICS 2011). pp. 401–416, January 2011Google Scholar
  12. 12.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow–resolution madesimple. J. ACM 48(2), 149–169 (2001). Preliminary version in STOC 1999MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bennett, P., Bonacina, I., Galesi, N., Huynh, T., Molloy, M., Wollan, P.: Space proof complexity for random 3-CNFs. Technical report, April 2015
  14. 14.
    Blake, A.: Canonical Expressions in Boolean Algebra. Ph.D. thesis, University of Chicago (1937)Google Scholar
  15. 15.
    Bonacina, I.: Total space in resolution is at least width squared. In: Proceedings of the 43rd International Colloquium on Automata, Languages and Programming (ICALP 2016), (to appear, July 2016)Google Scholar
  16. 16.
    Bonacina, I., Galesi, N., Thapen, N.: Total space in resolution. In: Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2014), pp. 641–650, October 2014Google Scholar
  17. 17.
    Bonet, M.L., Buss, S., Johannsen, J.: Improved separations of regular resolution from clause learning proof systems. J. Artif. Intell. Res. 49, 669–703 (2014)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Buss, S.R., Hoffmann, J., Johannsen, J.: Resolution trees with lemmas: resolution refinements that characterize DLL-algorithms with clause learning. Logical Meth. Comput. Sci. 4(4:13), 1–28 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Buss, S.R., Kołodziejczyk, L.: Small stone in pool. Logical Meth. Comput. Sci. 10, 1–22 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Chvátal, V., Szemerédi, E.: Many hard examples for resolution. J. ACM 35(4), 759–768 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cook, S.A., Reckhow, R.: The relative efficiency of propositional proof systems. J. Symbolic Logic 44(1), 36–50 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Commun. ACM 5(7), 394–397 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. J. ACM 7(3), 201–215 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Esteban, J.L., Torán, J.: Space bounds for resolution. Inf. Comput. 171(1), 84–97 (2001). Preliminary versions of these results appeared in STACS 1999 and CSL 1999MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Haken, A.: The intractability of resolution. Theor. Comput. Sci. 39(2–3), 297–308 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hertel, P., Bacchus, F., Pitassi, T., Van Gelder, A.: Clause learning can effectively P-simulate general propositional resolution. In: Proceedings of the 23rd National Conference on Artificial Intelligence (AAAI 2008), pp. 283–290, July 2008Google Scholar
  27. 27.
    Marques-Silva, J.P., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999). Preliminary version in ICCAD 1996MathSciNetCrossRefGoogle Scholar
  28. 28.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Proceedings of the 38th Design Automation Conference (DAC 2001). pp. 530–535, June 2001Google Scholar
  29. 29.
    Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT modulo theories: from an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T). J. ACM 53(6), 937–977 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Pipatsrisawat, K., Darwiche, A.: On the power of clause-learning SAT solvers as resolution engines. Artif. Intell. 175, 512–525 (2011). Preliminary version in CP 2009MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Urquhart, A.: Hard examples for resolution. J. ACM 34(1), 209–219 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Van Gelder, A.: Pool resolution and its relation to regular resolution and DPLL with clause learning. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 580–594. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  33. 33.
    Zhang, L., Madigan, C.F., Moskewicz, M.W., Malik, S.: Efficient conflict driven learning in boolean satisfiability solver. In: Proceedings of the IEEE/ACM International Conference on Computer-Aided Design (ICCAD 2001), pp. 279–285, November 2001Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Jan Elffers
    • 1
  • Jan Johannsen
    • 2
  • Massimo Lauria
    • 3
  • Thomas Magnard
    • 4
  • Jakob Nordström
    • 1
    Email author
  • Marc Vinyals
    • 1
  1. 1.KTH Royal Institute of TechnologyStockholmSweden
  2. 2.Ludwig-Maximilians-Universität MünchenMunichGermany
  3. 3.Universitat Politècnica de CatalunyaBarcelonaSpain
  4. 4.École Normale SupérieureParisFrance

Personalised recommendations