On Davis-Putnam Reductions for Minimally Unsatisfiable Clause-Sets

  • Oliver Kullmann
  • Xishun Zhao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7317)

Abstract

DP-reduction \(F \rightsquigarrow DP_{v}(F)\), applied to a clause-set F and a variable v, replaces all clauses containing v by their resolvents (on v). A basic case, where the number of clauses is decreased (i.e., c(DP v (F)) < c(F)), is singular DP-reduction (sDP-reduction), where v must occur in one polarity only once. For minimally unsatisfiable \(F \in \mathcal{M\hspace{0.8pt}U}\), sDP-reduction produces another \(F' := DP_{v}(F) \in \mathcal{M\hspace{0.8pt}U}\) with the same deficiency, that is, δ(F′) = δ(F); recall δ(F) = c(F) − n(F), using n(F) for the number of variables. Let sDP(F) for \(F \in \mathcal{M\hspace{0.8pt}U}\) be the set of results of complete sDP-reduction for F; so F′ ∈ sDP(F) fulfil \(F' \in \mathcal{M\hspace{0.8pt}U}\), are nonsingular (every literal occurs at least twice), and we have δ(F′) = δ(F). We show that for \(F \in \mathcal{M\hspace{0.8pt}U}\) all complete reductions by sDP must have the same length, establishing the singularity index of F. In other words, for F′, F′′ ∈ sDP(F) we have n(F′) = n(F′′). In general the elements of sDP(F) are not even (pairwise) isomorphic. Using the fundamental characterisation by Kleine Büning, we obtain as application of the singularity index, that we have confluence modulo isomorphism (all elements of sDP(F) are pairwise isomorphic) in case δ(F) = 2. In general we prove that we have confluence (i.e., |sDP(F) = 1) for saturated F (i.e., \(F \in \mathcal{S}\mathcal{M\hspace{0.8pt}U} \)). More generally, we show confluence modulo isomorphism for eventually saturated F, that is, where we have \(sDP(F) \subseteq \mathcal{S}\mathcal{M\hspace{0.8pt}U} \), yielding another proof for confluence modulo isomorphism in case of δ(F) = 2.

Keywords

Constraint Satisfaction Problem Conjunctive Normal Form Singularity Index Main Clause Isomorphism Type 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aharoni, R., Linial, N.: Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. Journal of Combinatorial Theory A 43, 196–204 (1986)MathSciNetGoogle Scholar
  2. 2.
    Davydov, G., Davydova, I., Büning, H.K.: An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF. Annals of Mathematics and Artificial Intelligence 23, 229–245 (1998)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Fleischner, H., Kullmann, O., Szeider, S.: Polynomial–time recognition of minimal unsatisfiable formulas with fixed clause–variable difference. Theoretical Computer Science 289(1), 503–516 (2002)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Fliti, T., Reynaud, G.: Sizes of minimally unsatisfiable conjunctive normal forms. Faculté des Sciences de Luminy, Dpt. Mathematique-Informatique, 13288 Marseille, France (November 1994)Google Scholar
  5. 5.
    Büning, H.K.: On subclasses of minimal unsatisfiable formulas. Discrete Applied Mathematics 107, 83–98 (2000)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Büning, H.K., Kullmann, O.: Minimal unsatisfiability and autarkies. In: Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, ch. 11, pp. 339–401. IOS Press (February 2009)Google Scholar
  7. 7.
    Kullmann, O.: Obere und untere Schranken für die Komplexität von aussagenlogischen Resolutionsbeweisen und Klassen von SAT-Algorithmen. Master’s thesis, Johann Wolfgang Goethe-Universität Frankfurt am Main (Upper and lower bounds for the complexity of propositional resolution proofs and classes of SAT algorithm; Diplomarbeit am Fachbereich Mathematik) (April 1992) (in German)Google Scholar
  8. 8.
    Kullmann, O.: An application of matroid theory to the SAT problem. In: Fifteenth Annual IEEE Conference on Computational Complexity, pp. 116–124. IEEE Computer Society (July 2000)Google Scholar
  9. 9.
    Kullmann, O.: Lean clause-sets: Generalizations of minimally unsatisfiable clause-sets. Discrete Applied Mathematics 130, 209–249 (2003)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kullmann, O.: Constraint satisfaction problems in clausal form I: Autarkies and deficiency. Fundamenta Informaticae 109(1), 27–81 (2011)MathSciNetGoogle Scholar
  11. 11.
    Kullmann, O.: Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structure. Fundamenta Informaticae 109(1), 83–119 (2011)MathSciNetGoogle Scholar
  12. 12.
    Kullmann, O., Luckhardt, H.: Deciding propositional tautologies: Algorithms and their complexity. Preprint, 82 pages; the ps-file can be obtained (January 1997), http://cs.swan.ac.uk/~csoliver/Artikel/tg.ps
  13. 13.
    Kullmann, O., Luckhardt, H.: Algorithms for SAT/TAUT decision based on various measures. Preprint, 71 pages; the ps-file can be obtained (February 1999), http://cs.swan.ac.uk/~csoliver/Artikel/TAUT.ps
  14. 14.
    Kullmann, O., Zhao, X.: On Variables with Few Occurrences in Conjunctive Normal Forms. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 33–46. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Kullmann, O., Zhao, X.: On Davis-Putnam reductions for minimally unsatisfiable clause-sets. Technical Report arXiv:1202.2600v3 [cs.DM], arXiv (May 2012)Google Scholar
  16. 16.
    Marques-Silva, J.: Computing minimally unsatisfiable subformulas: State of the art and future directions. Journal of Multiple-Valued Logic and Soft Computing (to appear, 2012)Google Scholar
  17. 17.
    Szeider, S.: Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. Journal of Computer and System Sciences 69(4), 656–674 (2004)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Zhao, X., Decheng, D.: Two tractable subclasses of minimal unsatisfiable formulas. Science in China (Series A) 42(7), 720–731 (1999)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Oliver Kullmann
    • 1
  • Xishun Zhao
    • 2
  1. 1.Computer Science DepartmentSwansea UniversityUK
  2. 2.Institute of Logic and CognitionSun Yat-sen UniversityGuangzhouP.R.C.

Personalised recommendations