Skip to main content
SpringerLink
Log in

On Unification of QBF Resolution-Based Calculi

  • Conference paper
Mathematical Foundations of Computer Science 2014 (MFCS 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8635))

Abstract

Several calculi for quantified Boolean formulas (QBFs) exist, but relations between them are not yet fully understood. This paper defines a novel calculus, which is resolution-based and enables unification of the principal existing resolution-based QBF calculi, namely Q-resolution, long-distance Q-resolution and the expansion-based calculus ∀Exp+Res. All these calculi play an important role in QBF solving. This paper shows simulation results for the new calculus and some of its variants. Further, we demonstrate how to obtain winning strategies for the universal player from proofs in the calculus. We believe that this new proof system provides an underpinning necessary for formal analysis of modern QBF solvers.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    Article  MATH  Google Scholar 

  2. Beame, P., Kautz, H.A., Sabharwal, A.: Towards understanding and harnessing the potential of clause learning. J. Artif. Intell. Res. (JAIR) 22, 319–351 (2004)

    MATH  MathSciNet  Google Scholar 

  3. Benedetti, M.: Evaluating QBFs via symbolic Skolemization. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 285–300. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  4. Benedetti, M., Mangassarian, H.: QBF-based formal verification: Experience and perspectives. JSAT 5(1-4), 133–191 (2008)

    MathSciNet  Google Scholar 

  5. Biere, A.: Resolve and expand. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 59–70. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  6. Buss, S.R.: Towards NP-P via proof complexity and search. Ann. Pure Appl. Logic 163(7), 906–917 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Buss, S.R., Hoffmann, J., Johannsen, J.: Resolution trees with lemmas: Resolution refinements that characterize DLL algorithms with clause learning. Logical Methods in Computer Science 4(4) (2008)

    Google Scholar 

  8. Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Log. 44(1), 36–50 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  9. Egly, U.: On sequent systems and resolution for QBFs. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 100–113. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  10. Egly, U., Lonsing, F., Widl, M.: Long-distance resolution: Proof generation and strategy extraction in search-based QBF solving. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR-19 2013. LNCS, vol. 8312, pp. 291–308. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  11. Giunchiglia, E., Narizzano, M., Tacchella, A.: Clause/term resolution and learning in the evaluation of quantified Boolean formulas. JAIR 26(1), 371–416 (2006)

    MATH  MathSciNet  Google Scholar 

  12. Giunchiglia, E., Marin, P., Narizzano, M.: Reasoning with quantified boolean formulas. In: Handbook of Satisfiability, pp. 761–780. IOS Press (2009)

    Google Scholar 

  13. Goultiaeva, A., Van Gelder, A., Bacchus, F.: A uniform approach for generating proofs and strategies for both true and false QBF formulas. In: IJCAI (2011)

    Google Scholar 

  14. Hertel, P., Bacchus, F., Pitassi, T., Van Gelder, A.: Clause learning can effectively p-simulate general propositional resolution. In: AAAI (2008)

    Google Scholar 

  15. Janota, M., Grigore, R., Marques-Silva, J.: On QBF proofs and preprocessing. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR-19 2013. LNCS, vol. 8312, pp. 473–489. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  16. Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.: Solving QBF with counterexample guided refinement. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 114–128. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  17. Janota, M., Marques-Silva, J.: ∀Exp+Res does not P-Simulate Q-resolution. In: International Workshop on Quantified Boolean Formulas (2013)

    Google Scholar 

  18. Janota, M., Marques-Silva, J.: On propositional QBF expansions and Q-resolution. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 67–82. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  19. Kleine Büning, H., Bubeck, U.: Theory of quantified boolean formulas. In: Handbook of Satisfiability, pp. 735–760. IOS Press (2009)

    Google Scholar 

  20. Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)

    Article  MATH  Google Scholar 

  21. Kleine Büning, H., Subramani, K., Zhao, X.: Boolean functions as models for quantified boolean formulas. J. Autom. Reasoning 39(1), 49–75 (2007)

    Article  MATH  Google Scholar 

  22. Klieber, W., Janota, M., Marques-Silva, J., Clarke, E.: Solving QBF with free variables. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 415–431. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  23. Klieber, W., Sapra, S., Gao, S., Clarke, E.: A non-prenex, non-clausal QBF solver with game-state learning. In: Strichman, O., Szeider, S. (eds.) SAT 2010. LNCS, vol. 6175, pp. 128–142. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  24. Krajíček, J., Pudlák, P.: Quantified propositional calculi and fragments of bounded arithmetic. Mathematical Logic Quarterly 36(1), 29–46 (1990)

    Article  MATH  Google Scholar 

  25. Marques Silva, J.P., Lynce, I., Malik, S.: Conflict-driven clause learning SAT solvers. In: Handbook of Satisfiability, IOS Press (2009)

    Google Scholar 

  26. Pipatsrisawat, K., Darwiche, A.: On the power of clause-learning SAT solvers as resolution engines. Artif. Intell. 175(2), 512–525 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rintanen, J.: Asymptotically optimal encodings of conformant planning in QBF. In: AAAI, pp. 1045–1050. AAAI Press (2007)

    Google Scholar 

  28. Robinson, J.A.: A machine-oriented logic based on the resolution principle. J. ACM 12(1), 23–41 (1965)

    Article  MATH  Google Scholar 

  29. Samer, M., Szeider, S.: Backdoor sets of quantified Boolean formulas. J. Autom. Reasoning 42(1), 77–97 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  30. Slivovsky, F., Szeider, S.: Variable dependencies and Q-Resolution. In: International Workshop on Quantified Boolean Formulas (2013)

    Google Scholar 

  31. Van Gelder, A.: Variable independence and resolution paths for quantified Boolean formulas. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 789–803. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  32. Van Gelder, A.: Contributions to the theory of practical quantified Boolean formula solving. In: Milano, M. (ed.) CP 2012. LNCS, vol. 7514, pp. 647–663. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computing, University of Leeds, United Kingdom

    Olaf Beyersdorff & Leroy Chew

  2. INESC-ID, Lisbon, Portugal

    Mikoláš Janota

Authors
  1. Olaf Beyersdorff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leroy Chew
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mikoláš Janota
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Informatics, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary

    Erzsébet Csuhaj-Varjú

  2. Fakultät für Informatik und Automatisierung, Technische Universität Ilmenau, Helmholtzplatz, 598693, Ilmenau, Germany

    Martin Dietzfelbinger

  3. Institute of Informatics, Szeged University, Tisza Lajos krt. 103, 6720, Szeged, Hungary

    Zoltán Ésik

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Beyersdorff, O., Chew, L., Janota, M. (2014). On Unification of QBF Resolution-Based Calculi. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds) Mathematical Foundations of Computer Science 2014. MFCS 2014. Lecture Notes in Computer Science, vol 8635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44465-8_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-44465-8_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44464-1

  • Online ISBN: 978-3-662-44465-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation