Journal of Automated Reasoning

, Volume 42, Issue 1, pp 77–97 | Cite as

Backdoor Sets of Quantified Boolean Formulas

Article

Abstract

We generalize the notion of backdoor sets from propositional formulas to quantified Boolean formulas (QBF). This allows us to obtain hierarchies of tractable classes of quantified Boolean formulas with the classes of quantified Horn and quantified 2CNF formulas, respectively, at their first level, thus gradually generalizing these two important tractable classes. In contrast to known tractable classes based on bounded treewidth, the number of quantifier alternations of our classes is unbounded. As a side product of our considerations we develop a theory of variable dependency which is of independent interest.

Keywords

Quantified Boolean formulas Backdoor sets Variable dependencies Parameterized complexity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abu-Khzam, F.N.: Kernelization algorithms for d-hitting set problems. In: Proc. 10th International Workshop on Algorithms and Data Structures (WADS’07), LNCS, vol. 4619, pp. 434–445. Springer, New York (2007)CrossRefGoogle Scholar
  2. 2.
    Aspvall, B., Plass, M.F., Tarjan, R.E.: A linear-time algorithm for testing the truth of certain quantified Boolean formulas. Inf. Process. Lett. 8(3), 121–123 (1979)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ayari, A., Basin, D.: Qubos: Deciding quantified Boolean logic using propositional satisfiability solvers. In: Proc. 4th International Conference on Formal Methods in Computer-Aided Design (FMCAD’02), LNCS, vol. 2517, pp. 187–201. Springer, New York (2002)Google Scholar
  4. 4.
    Benedetti, M.: Quantifier trees for QBFs. In: Proc. 8th International Conference on Theory and Applications of Satisfiability Testing (SAT’05), LNCS, vol. 3569, pp. 378–385. Springer, New York (2005)Google Scholar
  5. 5.
    Biere, A.: Resolve and expand. In: Proc. 7th International Conference on Theory and Applications of Satisfiability Testing (SAT’04), LNCS, vol. 3542, pp. 59–70. Springer, New York (2005)Google Scholar
  6. 6.
    Bubeck, U., Kleine Büning, H.: Bounded universal expansion for preprocessing QBF. In: Proc. 10th International Conference on Theory and Applications of Satisfiability Testing (SAT’07), LNCS, vol. 4501 of, pages 244–257. Springer, New York (2007)CrossRefGoogle Scholar
  7. 7.
    Chen, J., Kanj, I.A., Xia, G.: Improved parameterized upper bounds for vertex cover. In: Proc. 31st International Symposium on Mathematical Foundations of Computer Science (MFCS’06), LNCS, vol. 4162, pp. 238–249. Springer, New York (2006)Google Scholar
  8. 8.
    Crama, Y., Ekin, O., Hammer, P.L.: Variable and term removal from Boolean formulae. Discrete Appl. Math. 75(3), 217–230 (1997)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)Google Scholar
  10. 10.
    Egly, U., Eiter, T., Tompits, H., Woltran, S.: Solving advanced reasoning tasks using quantified Boolean formulas. In: Proc. 17th AAAI Conference on Artificial Intelligence (AAAI’00), pp. 417–422. AAAI, Menlo Park (2000)Google Scholar
  11. 11.
    Egly, U., Tompits, H., Woltran, S.: On quantifier shifting for quantified Boolean formulas. In: Proc. SAT’02 Workshop on Theory and Applications of Quantified Boolean Formulas, pp. 48–61. Informal Proceedings (2002)Google Scholar
  12. 12.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, New York (2006)Google Scholar
  13. 13.
    Hoffmann, J., Gomes, C., Selman, B.: Structure and problem hardness: Goal asymmetry and DPLL proofs in SAT-based planning. In: Proc. 16th International Conference on Automated Planning and Scheduling (ICAPS’06), pp. 284–293. AAAI, Menlo Park (2006)Google Scholar
  14. 14.
    Interian, Y.: Backdoor sets for random 3-SAT. In: Proc. 6th International Conference on Theory and Applications of Satisfiability Testing (SAT’03), pp. 231–238. Informal Proceedings (2003)Google Scholar
  15. 15.
    Kilby, P., Slaney, J.K., Thiébaux, S., Walsh, T.: Backbones and backdoors in satisfiability. In: Proc. 20th AAAI Conference on Artificial Intelligence (AAAI’05), pp. 1368–1373. AAAI, Menlo Park (2005)Google Scholar
  16. 16.
    Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MATHCrossRefGoogle Scholar
  17. 17.
    Kleine Büning, H., Lettman, T.: Propositional Logic: Deduction and Algorithms. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  18. 18.
    Knuth, D.E.: The Art of Computer Programming, vol. 3: Sorting and Searching, chapter 5.2.2 Sorting by Exchanging, pp. 106–110. Addison-Wesley, Reading (1973)Google Scholar
  19. 19.
    Lynce, I., Marques-Silva, J.P.: Hidden structure in unsatisfiable random 3-SAT: An empirical study. In: Proc. 16th IEEE International Conference on Tools with Artificial Intelligence (ICTAI’04), pp. 246–251. IEEE Computer Society, Los Alamitos (2004)CrossRefGoogle Scholar
  20. 20.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)MATHGoogle Scholar
  21. 21.
    Niedermeier, R., Rossmanith, P.: An efficient fixed-parameter algorithm for 3-hitting set. J. Discret. Algorithms 1(1), 89–102 (2003)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Nishimura, N., Ragde, P., Szeider, S.: Detecting backdoor sets with respect to Horn and binary clauses. In: Proc. 7th International Conference on Theory and Applications of Satisfiability Testing (SAT’04), pp. 96–103. Informal Proceedings (2004)Google Scholar
  23. 23.
    Nishimura, N., Ragde, P., Szeider, S.: Solving #SAT using vertex covers. Acta Inform. 44(7–8), 509–523 (2007)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Otwell, C., Remshagen, A., Truemper, K.: An effective QBF solver for planning problems. In: Proc. International Conference on Modeling, Simulation and Visualization Methods and International Conference on Algorithmic Mathematics and Computer Science (MSV/AMCS’04), pp. 311–316. CSREA, Las Vegas (2004)Google Scholar
  25. 25.
    Pan, G., Vardi, M.Y.: Fixed-parameter hierarchies inside PSpace. In: Proc. 21st Annual IEEE Symposium on Logic in Computer Science (LICS’06), pp. 27–36. IEEE Computer Society, Los Alamitos (2006)Google Scholar
  26. 26.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  27. 27.
    Rintanen, J.: Constructing conditional plans by a theorem-prover. J. Artif. Intell. Res. 10, 323–352 (1999)MATHGoogle Scholar
  28. 28.
    Ruan, Y., Kautz, H.A., Horvitz, E.: The backdoor key: A path to understanding problem hardness. In: Proc. 19th AAAI Conference on Artificial Intelligence (AAAI’04), pp. 124–130. AAAI, Menlo Park (2004)Google Scholar
  29. 29.
    Sabharwal, A., Ansotegui, C., Gomes, C., Hart, J., Selman, B.: QBF modeling: Exploiting player symmetry for simplicity and efficiency. In: Proc. 9th International Conference on Theory and Applications of Satisfiability Testing (SAT’06), LNCS, vol. 4121, pp. 382–395. Springer, New York (2006)CrossRefGoogle Scholar
  30. 30.
    Samer, M.: Variable dependencies of quantified CSPs. In: Proc. 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR’08), LNCS, vol. 5330, pp. 512–527. Springer, New York (2008)CrossRefGoogle Scholar
  31. 31.
    Samer, M., Szeider, S.: Backdoor sets of quantified Boolean formulas. In: Proc. 10th International Conference on Theory and Applications of Satisfiability Testing (SAT’07), LNCS, vol. 4501, pp. 230–243. Springer, New York (2007)CrossRefGoogle Scholar
  32. 32.
    Samulowitz, H., Bacchus, F.: Binary clause reasoning in QBF. In: Proc. 9th International Conference on Theory and Applications of Satisfiability Testing (SAT’06), LNCS, vol. 4121, pp. 353–367. Springer, New York (2006)CrossRefGoogle Scholar
  33. 33.
    Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time. In: Proc. 5th Annual ACM Symposium on Theory of Computing (STOC’73), pp. 1–9. ACM, New York (1973)CrossRefGoogle Scholar
  34. 34.
    Szeider, S.: Backdoor sets for DLL subsolvers. J. Autom. Reason. 35(1–3), 73–88 (2005)MATHMathSciNetGoogle Scholar
  35. 35.
    Szeider, S.: Generalizations of matched CNF formulas. Ann. Math. Artif. Intell. 43(1–4), 223–238 (2005)MATHMathSciNetGoogle Scholar
  36. 36.
    Szeider, S.: Matched formulas and backdoor sets. In: Proc. 10th International Conference on Theory and Applications of Satisfiability Testing (SAT’07), LNCS, vol. 4501, pp. 94–99. Springer, New York (2007)CrossRefGoogle Scholar
  37. 37.
    Tarjan, R.E.: Depth first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Williams, R., Gomes, C., Selman, B.: Backdoors to typical case complexity. In: Proc. 18th International Joint Conference on Artificial Intelligence (IJCAI’03), pp. 1173–1178. Morgan Kaufmann, San Francisco (2003)Google Scholar
  39. 39.
    Williams, R., Gomes, C., Selman, B.: On the connections between backdoors, restarts, and heavy-tailedness in combinatorial search. In: Proc. 6th International Conference on Theory and Applications of Satisfiability Testing (SAT’03), pp. 222–230. Informal Proceedings (2003)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.TU DarmstadtDarmstadtGermany
  2. 2.University of DurhamDurhamUK

Personalised recommendations