Skip to main content
SpringerLink
Log in

Minimal False Quantified Boolean Formulas

  • Conference paper
Book cover Theory and Applications of Satisfiability Testing - SAT 2006 (SAT 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4121))

Abstract

This paper is concerned with the minimal falsity problem MF for quantified Boolean formulas. A QCNF formula (i.e., with CNF-matrix) is called minimal false, if the formula is false and any proper subformula is true. It is shown that the minimal falsity problem is PSPACE-complete. Then the deficiency of a QCNF formula is defined as the difference between the number of clauses and the number of existentially quantified variables. For quantified Boolean formulas with deficiency one, MF is solvable in polynomial time.

Some previous results of this of paper has been presented in the Workshop on Theory and Application of Quantified Boolean Formulas [3].

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. Aspvall, B., Plass, M.F., Tarjan, R.E.: A Linear–Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas. Information Processing Letters 8, 121–123 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  2. Davydov, G., Davydova, I., Kleine Büning, H.: An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF. Annals of Mathematics and Artificial Intelligence 23, 229–245 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ding, D., Kleine Büning, H., Zhao, X.: Minimal Falsity for QBF with Fixed Deficiency. In: Proceedings of the Workshop on Theory and Applications of Quantified Boolean Formulas, IJCAR 2001, Siena, pp. 21–28 (2001)

    Google Scholar 

  4. Fleischner, H., Kullmann, O., Szeider, S.: Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable deficiency. Theoretical Computer Scicence 289, 503–516 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kleine Büning, H.: On Some Subclasses of Minimal Unsatisfiable Formulas. Discrete Applied Mathematics 107, 83–98 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kleine Büning, H., Lettmann, T.: Propositional Logic: Deduction and Algorithms. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  7. Kleine Büning, H., Zhao, X.: On the structure of some classes of minimal unsatisfiable formulas. Discrete Applied Mathematics 130(2), 185–207 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kullmann, O.: An Application of Matroid Theory to the ASAT Problem, Electronic Colloquium on Computational Complexity, Report 18 (2000)

    Google Scholar 

  9. Kullmann, O.: Lean-sets: Generalizations of minimal unsatisfiable clause-sets. Discrete Applied Mathematics 130, 209–249 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kullmann, O.: The Combinatorics of Conflicts between Clauses. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 426–440. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  11. Papadimitriou, C.H., Wolfe, D.: The complexity of facets resolved. Journal of Computer and System Science 37, 2–13 (1988)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Universität Paderborn, 33095, Paderborn, Germany

    Hans Kleine Büning

  2. Institute of Logic and Cognition, Sun Yat-sen University, 510275, Guangzhou, P.R. China

    Xishun Zhao

Authors
  1. Hans Kleine Büning
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xishun Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Formal Models and Verification, Johannes Kepler University, Linz, Austria

    Armin Biere

  2. Department of Computer Science, Cornell University, 14853, Ithaca, NY, U.S.A.

    Carla P. Gomes

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Büning, H.K., Zhao, X. (2006). Minimal False Quantified Boolean Formulas. In: Biere, A., Gomes, C.P. (eds) Theory and Applications of Satisfiability Testing - SAT 2006. SAT 2006. Lecture Notes in Computer Science, vol 4121. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11814948_32

Download citation

  • DOI: https://doi.org/10.1007/11814948_32

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37206-6

  • Online ISBN: 978-3-540-37207-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation