[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content

Advertisement

Springer Nature Link
Log in

Quantified maximum satisfiability

  • Published:
Constraints Aims and scope Submit manuscript

Abstract

In recent years, there have been significant improvements in algorithms both for Quantified Boolean Formulas (QBF) and for Maximum Satisfiability (MaxSAT). This paper studies an optimization extension of QBF and considers the problem in a quantified MaxSAT setting. More precisely, the general QMaxSAT problem is defined for QBFs with a set of soft clausal constraints and consists in finding the largest subset of the soft constraints such that the remaining QBF is true. Two approaches are investigated. One is based on relaxing the soft clauses and performing an iterative search on the cost function. The other approach, which is the main contribution of the paper, is inspired by recent work on MaxSAT, and exploits the iterative identification of unsatisfiable cores. The paper investigates the application of these approaches to the two concrete problems of computing smallest minimal unsatisfiable subformulas (SMUS) and smallest minimal equivalent subformulas (SMES), decision versions of which are well-known problems in the second level of the polynomial hierarchy. Experimental results, obtained on representative problem instances, indicate that the core-guided approach for the SMUS and SMES problems outperforms the use of iterative search over the values of the cost function. More significantly, the core-guided approach to SMUS also outperforms the state-of-the-art SMUS extractor Digger.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Andraus, Z.S., Liffiton, M.H., & Sakallah, K.A. (2006). Refinement strategies for verification methods based on datapath abstraction. In ASP-DAC (pp. 19–24). doi:10.1145/1118299.1118306.

  2. Andraus, Z.S., Liffiton, M.H., & Sakallah, K.A. (2008). Reveal: a formal verification tool for verilog designs. In LPAR (pp. 343–352). doi:10.1007/978-3-540-89439-1_25.

  3. Ansótegui, C., Bonet, M.L., & Levy, J. (2009). Solving (weighted) partial MaxSAT through satisfiability testing. In SAT (pp. 427–440).

  4. Ansótegui, C., Bonet, M.L., & Levy, J. (2010). A new algorithm for weighted partial MaxSAT. In AAAI.

  5. Ansótegui, C., Bonet, M.L., & Levy, J. (2013). SAT-based MaxSAT algorithms. Artificial Intelligence, 196, 77–105.

    Article  MathSciNet  MATH  Google Scholar 

  6. Ansotegui, C., Gomes, C.P., & Selman, B. (2005). The Achilles’ heel of QBF. In AAAI (pp. 275–281).

  7. Bailey, J., & Stuckey, P.J. (2005). Discovery of minimal unsatisfiable subsets of constraints using hitting set dualization. In PADL (pp. 174–186).

  8. Balabanov, V., & Jiang, J.H.R. (2011). Resolution proofs and Skolem functions in QBF evaluation and applications. In G. Gopalakrishnan, S. Qadeer (Eds.), CAV (pp. 149–164).

  9. Balabanov, V., & Jiang, J.H.R. (2012). Unified QBF certification and its applications. Formal Methods in System Design, 41(1), 45–65.

    Article  MATH  Google Scholar 

  10. Balabanov, V., Widl, M., & Jiang, J.H.R. (2014). QBF resolution systems and their proof complexities. In C. Sinz, U. Egly (Eds.), SAT (pp. 154–169). Springer.

  11. Belov, A., Janota, M., Lynce, I., & Marques-Silva, J. (2012). On computing minimal equivalent subformulas. In M. Milano (Ed.), CP (Vol. 7514, pp. 158–174). Springer.

  12. Belov, A., Janota, M., Lynce, I., & Marques-Silva, J. (2014). Algorithms for computing minimal equivalent subformulas. Artificial Intelligence, 216, 309–326. doi:10.1016/j.artint.2014.07.011.

    Article  MathSciNet  MATH  Google Scholar 

  13. Belov, A., Lynce, I., & Marques-Silva, J. (2012). Towards efficient MUS extraction. AI Communication, 25(2), 97–116.

    MathSciNet  MATH  Google Scholar 

  14. Belov, A., Morgado, A., & Marques-Silva, J. (2013). SAT-based preprocessing for MaxSAT. In LPAR (pp. 96–111). doi:10.1007/978-3-642-45221-5_7.

  15. Benedetti, M., Lallouet, A., & Vautard, J. (2008). Quantified constraint optimization. In CP (pp. 463–477).

  16. Berre, D.L., & Parrain, A. (2010). The Sat4j library, release 2.2. JSAT, 7(2-3), 59–6.

    Google Scholar 

  17. Birnbaum, E., & Lozinskii, E.L. (2003). Consistent subsets of inconsistent systems: structure and behaviour. Journal of Experimental & Theoretical Artificial Intelligence, 15(1), 25–46.

    Article  MATH  Google Scholar 

  18. Chen, H., Janota, M., & Marques-Silva, J. (2012). QBF-based Boolean function bi-decomposition. In DATE (pp. 816–819).

  19. Chen, H., & Pál, M. (2004). Optimization, games, and quantified constraint satisfaction. In Mathematical foundations of computer science (pp. 239–250).

  20. Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., & Veith, H. (2003). Counterexample-guided abstraction refinement for symbolic model checking. Journal of ACM, 50(5), 752–794.

    Article  MathSciNet  Google Scholar 

  21. Condon, A., Feigenbaum, J., Lund, C., & Shor, P.W. (1995). Probabilistically checkable debate systems and nonapproximability of PSPACE-hard functions. Chicago Journal of Theoretical Computer Science, 1995.

  22. Eén, N., & Sörensson, N. (2003). An extensible SAT-solver. In SAT (pp. 502–518).

  23. Egly, U., Lonsing, F., & Widl, M. (2013). Long-distance resolution: Proof generation and strategy extraction in search-based QBF solving. In K. L. McMillan, A. Middeldorp, A. Voronkov (Eds.), LPAR (Vol. 8312, pp. 291–308). Springer.

  24. Fu, Z., & Malik, S. (2006). On solving the partial MAX-SAT problem. In A. Biere, C. P. Gomes (Eds.), SAT, lecture notes in computer science (Vol. 4121, pp. 252–265). Springer.

  25. Goldberg, E.I., & Novikov, Y. (2003). Verification of proofs of unsatisfiability for CNF formulas. In DATE (pp. 10,886–10,891). IEEE Computer Society.

  26. Goultiaeva, A., & Bacchus, F. (2010). Exploiting QBF duality on a circuit representation. AAAI.

  27. Goultiaeva, A., & Bacchus, F. (2013). Recovering and utilizing partial duality in QBF. In SAT (pp. 83–99). doi:10.1007/978-3-642-39071-5_8.

  28. Goultiaeva, A., Seidl, M., & Biere, A. (2013). Bridging the gap between dual propagation and CNF-based QBF solving. In DATE (pp. 811–814).

  29. Gupta, A. (2006). Learning abstractions for model checking, Ph.D. thesis, Carnegie Mellon University.

  30. Han, B., & Lee, S.J. (1999). Deriving minimal conflict sets by CS-trees with mark set in diagnosis from first principles. IEEE Transactions on Systems, Man, and Cybernetics: Part B, 29(2), 281–286.

    Article  Google Scholar 

  31. Heras, F., Morgado, A., & Marques-Silva, J. (2011). Core-guided binary search algorithms for maximum satisfiability. In W. Burgard, D. Roth (Eds.), AAAI. AAAI Press.

  32. Heule, M., Seidl, M., & Biere, A. (2014). A unified proof system for QBF preprocessing. In S. Demri, D. Kapur, C. Weidenbach (Eds.), IJCAR (Vol. 8562, pp. 91–106).

  33. Ignatiev, A., Janota, M., & Marques-Silva, J. (2013). Quantified maximum satisfiability: A core-guided approach. In M. Järvisalo, A. Van Gelder (Eds.), SAT (Vol. 7962, pp. 250–266). Springer.

  34. Ignatiev, A., Morgado, A., Manquinho, V.M., Lynce, I., & Marques-Silva, J. (2014). Progression in maximum satisfiability. In ECAI (pp. 453–458). doi:10.3233/978-1-61499-419-0-453.

  35. Jain, H., Kroening, D., Sharygina, N., & Clarke, E.M. (2005). Word level predicate abstraction and refinement for verifying RTL verilog. In DAC (pp. 445–450). doi:10.1145/1065579.1065697.

  36. Janota, M., Grigore, R., & Marques-Silva, J. (2013). On QBF proofs and preprocessing. In K.L. McMillan, A. Middeldorp, A. Voronkov (Eds.), LPAR (Vol. 8312, pp. 473–489). Springer.

  37. Janota, M., Klieber, W., Marques-Silva, J., & Clarke, E.M. (2012). Solving QBF with counterexample guided refinement. In A. Cimatti, R. Sebastiani (Eds.), SAT (Vol. 7317, pp. 114–128). Springer.

  38. Janota, M., & Marques-Silva, J. (2011). Abstraction-based algorithm for 2QBF. In SAT (pp. 230–244).

  39. Kleine Büning, H., & Bubeck, U. (2009). Theory of quantified Boolean formulas. In A. Biere, M. Heule, H. van Maaren, T. Walsh (Eds.), Handbook of satisfiability, frontiers in artificial intelligence and applications (Vol. 185, pp. 735–760). IOS Press.

  40. Kleine Büning, H., Karpinski, M., & Flögel, A. (1995). Resolution for quantified Boolean formulas. Information and Computation, 117(1), 12–18.

    Article  MathSciNet  MATH  Google Scholar 

  41. Kleine Büning, H., Subramani, K., & Zhao, X. (2007). Boolean functions as models for quantified Boolean formulas. Journal of Automated Reasoning, 39(1), 49–75.

    Article  MathSciNet  MATH  Google Scholar 

  42. Klieber, W., Sapra, S., Gao, S., & Clarke, E.M. (2010). A non-prenex, non-clausal QBF solver with game-state learning. In SAT (pp. 128–142). doi:10.1007/978-3-642-14186-7_12.

  43. Kullmann, O. (2011). Constraint satisfaction problems in clausal form II: minimal unsatisfiability and conflict structure. Fundamental and Information, 109(1), 83–119.

    MathSciNet  MATH  Google Scholar 

  44. Liberatore, P. (2005). Redundancy in logic I: CNF propositional formulae. Artificial Intelligence, 163(2), 203–232.

    Article  MathSciNet  MATH  Google Scholar 

  45. Liffiton, M.H., Mneimneh, M.N., Lynce, I., Andraus, Z.S., Marques-Silva, J., & Sakallah, K.A. (2009). A branch and bound algorithm for extracting smallest minimal unsatisfiable subformulas. Constraints, 14(4), 415–442.

    Article  MathSciNet  MATH  Google Scholar 

  46. Liffiton, M.H., & Sakallah, K.A. (2005). On finding all minimally unsatisfiable subformulas. In SAT (pp. 173–186).

  47. Liffiton, M.H., & Sakallah, K.A. (2008). Algorithms for Computing Minimal Unsatisfiable Subsets of Constraints. Journal of Automated Reasoning, 40(1), 1–33.

    Article  MathSciNet  MATH  Google Scholar 

  48. Lonsing, F., & Egly, U. (2014). Incremental QBF solving. In CP (pp. 514–530).

  49. Lynce, I., & Marques-Silva, J.P. (2004). On computing minimum unsatisfiable cores. In SAT.

  50. Manquinho, V.M., Marques-Silva, J., & Planes, J. (2009). Algorithms for weighted Boolean optimization. In SAT (pp. 495–508).

  51. Marques-Silva, J. (2010). Minimal unsatisfiability: models, algorithms and applications (invited paper). In ISMVL (pp. 9–14). IEEE Computer Society.

  52. Marques-Silva, J., Heras, F., Janota, M., Previti, A., & Belov, A. (2013). On computing minimal correction subsets. In IJCAI.

  53. Mneimneh, M.N., Lynce, I., Andraus, Z.S., Marques-Silva, J.P., & Sakallah, K.A. (2005). A branch-and-bound algorithm for extracting smallest minimal unsatisfiable formulas. In SAT (pp. 467–474).

  54. Morgado, A., Dodaro, C., & Marques-Silva, J. (2014). Core-guided MaxSAT with soft cardinality constraints. In CP (pp. 564–573). doi:10.1007/978-3-319-10428-7_41.

  55. Morgado, A., Heras, F., Liffiton, M.H., Planes, J., & Marques-Silva, J. (2013). Iterative and core-guided MaxSAT solving: a survey and assessment. Constraints, 18 (4), 478–534.

    Article  MathSciNet  Google Scholar 

  56. Narodytska, N., & Bacchus, F. (2014). Maximum satisfiability using core-guided MaxSAT resolution. In AAAI (pp. 2717–2723).

  57. Niemetz, A., Preiner, M., Lonsing, F., Seidl, M., & Biere, A. (2012). Resolution-based certificate extraction for QBF - (tool presentation). In A. Cimatti, R. Sebastiani (Eds.), SAT (Vol. 7317, pp. 430–435). Springer.

  58. Nöhrer, A., Biere, A., & Egyed, A. (2012). Managing SAT inconsistencies with HUMUS. In VAMOS (pp. 83–91).

  59. Plaisted, D.A., & Greenbaum, S. (1986). A structure-preserving clause form translation. Journal of Symbolic Computation, 2(3), 293–304. doi:10.1016/S0747-7171(86)80028-1.

    Article  MathSciNet  MATH  Google Scholar 

  60. Reimer, S., Sauer, M., Marin, P., & Becker, B. (2014). QBF with soft variables. ECEASST 70.

  61. Reiter, R. (1987). A theory of diagnosis from first principles. Artificial Intelligence, 32(1), 57–95.

    Article  MathSciNet  MATH  Google Scholar 

  62. Roussel, O., & Manquinho, V. (2009). Pseudo-Boolean and cardinality constraints. In A. Biere, M. Heule, H. van Maaren, T. Walsh (Eds.), Handbook of satisfiability, frontiers in artificial intelligence and applications (Vol. 185, pp. 695–733). IOS Press.

  63. Ryvchin, V., & Strichman, O. (2011). Faster extraction of high-level minimal unsatisfiable cores. In SAT (pp. 174–187).

  64. Schaefer, M., & Umans, C. (2002). Completeness in the polynomial-time hierarchy: a compendium. SIGACT News, 33(3), 32–49.

    Article  Google Scholar 

  65. Sinz, C., Kaiser, A., & Küchlin, W. (2003). Formal methods for the validation of automotive product configuration data. AI EDAM, 17(1), 75–97.

    Google Scholar 

  66. Stuckey, P.J. (2013). There are no CNF problems. In SAT (pp. 19–21). doi:10.1007/978-3-642-39071-5_3.

  67. Stuckey, P.J., Sulzmann, M., & Wazny, J. (2003). Interactive type debugging in Haskell. In ACM SIGPLAN workshop on Haskell (pp. 72–83). ACM.

  68. Tseitin, G.S. (1968). On the complexity of derivations in the propositional calculus. Studies in Mathematics and Mathematical Logic Part II, 115–125.

  69. Van Gelder, A. (2012). Contributions to the theory of practical quantified Boolean formula solving. In M. Milano (Ed.), CP (Vol. 7514, pp. 647–663). Springer.

  70. Yu, Y., & Malik, S. (2005). Validating the result of a quantified Boolean formula (QBF) solver: theory and practice. In ASP-DAC (pp. 1047–1051).

  71. Zhang, J., Li, S., & Shen, S. (2006). Extracting minimum unsatisfiable cores with a greedy genetic algorithm. In AUS-AI (pp. 847–856).

  72. Zhang, L. (2006). Solving QBF by combining conjunctive and disjunctive normal forms. In AAAI (pp. 143–150).

  73. Zhang, L., & Malik, S. (2002). Conflict driven learning in a quantified Boolean satisfiability solver. In ICCAD (pp. 442–449).

  74. Zhang, L., & Malik, S. (2003). Validating SAT solvers using an independent resolution-based checker: practical implementations and other applications. In DATE (pp. 10,880–10,885). IEEE Computer Society.

Download references

Author information

Authors and Affiliations

  1. INESC-ID, IST, University of Lisbon, Lisbon, Portugal

    Alexey Ignatiev, Mikoláš Janota & Joao Marques-Silva

  2. ISDCT SB RAS, Irkutsk, Russia

    Alexey Ignatiev

  3. University College Dublin, Dublin, Ireland

    Joao Marques-Silva

Authors
  1. Alexey Ignatiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikoláš Janota
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joao Marques-Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexey Ignatiev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ignatiev, A., Janota, M. & Marques-Silva, J. Quantified maximum satisfiability. Constraints 21, 277–302 (2016). https://doi.org/10.1007/s10601-015-9195-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10601-015-9195-9

Keywords