Understanding, Improving and Parallelizing MUS Finding Using Model Rotation

  • Siert Wieringa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7514)


Recently a new technique for improving algorithms for extracting Minimal Unsatisfiable Subsets (MUSes) from unsatisfiable CNF formulas called “model rotation” was introduced [Marques-Silva et. al. SAT2011]. The technique aims to reduce the number of times a MUS finding algorithm needs to call a SAT solver. Although no guarantees for this reduction are provided the technique has been shown to be very effective in many cases. In fact, such model rotation algorithms are now arguably the state-of-the-art in MUS finding.

This work analyses the model rotation technique in detail and provides theoretical insights that help to understand its performance. These new insights on the operation of model rotation lead to several modifications and extensions that are empirically evaluated. Moreover, it is demonstrated how such MUS extracting algorithms can be effectively parallelized using existing techniques for parallel incremental SAT solving.


Model Rotation Satisfying Assignment Bound Model Check Input Formula 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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  1. 1.
    Belov, A., Lynce, I., Marques-Silva, J.P.: Towards efficient MUS extraction (2012) (to appear in AI Communications)Google Scholar
  2. 2.
    Belov, A., Marques-Silva, J.P.: Accelerating MUS extraction with recursive model rotation. In: FMCAD, pp. 37–40 (2011)Google Scholar
  3. 3.
    Belov, A., Marques-Silva, J.P.: MUSer2: An efficient MUS extractor (2012), to appear in proceedings of Pragmatics of SAT (POS)Google Scholar
  4. 4.
    Biere, A., Cimatti, A., Clarke, E., Zhu, Y.: Symbolic Model Checking without BDDs. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, pp. 193–207. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Cook, S.A.: The complexity of theorem-proving procedures. In: Harrison, M.A., Banerji, R.B., Ullman, J.D. (eds.) STOC, pp. 151–158. ACM (1971)Google Scholar
  6. 6.
    Eén, N., Sörensson, N.: An Extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Goldberg, E.I., Novikov, Y.: Verification of proofs of unsatisfiability for CNF formulas. In: DATE, pp. 10886–10891. IEEE Computer Society (2003)Google Scholar
  8. 8.
    Grégoire, É., Mazure, B., Piette, C.: On approaches to explaining infeasibility of sets of boolean clauses. In: ICTAI (1), pp. 74–83. IEEE Computer Society (2008)Google Scholar
  9. 9.
    Junker, U.: QUICKXPLAIN: Preferred explanations and relaxations for over-constrained problems. In: McGuinness, D.L., Ferguson, G. (eds.) AAAI, pp. 167–172. AAAI Press/The MIT Press (2004)Google Scholar
  10. 10.
    van Maaren, H., Wieringa, S.: Finding Guaranteed MUSes Fast. In: Kleine Büning, H., Zhao, X. (eds.) SAT 2008. LNCS, vol. 4996, pp. 291–304. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Marques-Silva, J.P., Planes, J.: Algorithms for maximum satisfiability using unsatisfiable cores. In: DATE, pp. 408–413. IEEE (2008)Google Scholar
  12. 12.
    Marques-Silva, J.P.: Minimal unsatisfiability: Models, algorithms and applications (invited paper). In: ISMVL, pp. 9–14. IEEE Computer Society (2010)Google Scholar
  13. 13.
    Marques-Silva, J., Lynce, I.: On Improving MUS Extraction Algorithms. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 159–173. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Marques-Silva, J.P., Sakallah, K.A.: GRASP: A search algorithm for propositional satisfiability. IEEE Trans. Computers 48(5), 506–521 (1999)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Oh, Y., Mneimneh, M.N., Andraus, Z.S., Sakallah, K.A., Markov, I.L.: AMUSE: a minimally-unsatisfiable subformula extractor. In: Malik, S., Fix, L., Kahng, A.B. (eds.) DAC, pp. 518–523. ACM (2004)Google Scholar
  16. 16.
    Papadimitriou, C.H., Wolfe, D.: The complexity of facets resolved. J. Comput. Syst. Sci. 37(1), 2–13 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Sinz, C.: Visualizing SAT instances and runs of the DPLL algorithm. J. Autom. Reasoning 39(2), 219–243 (2007)CrossRefzbMATHGoogle Scholar
  18. 18.
    Sinz, C., Kaiser, A., Küchlin, W.: Formal methods for the validation of automotive product configuration data. AI EDAM 17(1), 75–97 (2003)Google Scholar
  19. 19.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Wieringa, S., Niemenmaa, M., Heljanko, K.: Tarmo: A framework for parallelized bounded model checking. In: Brim, L., van de Pol, J. (eds.) PDMC. EPTCS, vol. 14, pp. 62–76 (2009)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Siert Wieringa
    • 1
  1. 1.School of Science, Department of Information and Computer ScienceAalto UniversityAaltoFinland

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