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Horn Upper Bounds and Renaming

  • Marina Langlois
  • Robert H. Sloan
  • György Turán
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4501)

Abstract

We consider the problem of computing tractable approximations to CNF formulas, extending the approach of Selman and Kautz to compute the Horn-LUB to involve renaming of variables. Negative results are given for the quality of approximation in this extended version. On the other hand, experiments for random 3-CNF show that the new algorithms improve both running time and approximation quality. The output sizes and approximation errors exhibit a ‘Horn bump’ phenomenon: unimodal patterns are observed with maxima in some intermediate range of densities. We also present the results of experiments generating pseudo-random satisfying assignments for Horn formulas.

Keywords

Approximation Quality Truth Assignment Horn Clause Output Size Prime Implicants 
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.

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References

  1. 1.
    Aspvall, B.: Recognizing disguised NR(1) instances of the satisfiability problem. Journal of Algorithms 1, 97–103 (1980)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Bollobás, B., Kohayakawa, Y., Łuczak, T.: On the evolution of random Boolean functions. In: Frankl, P., et al. (eds.) Extremal Problems for Finite Sets (Visegrád). Bolyai Society Mathematical Studies, vol. 3, pp. 137–156. János Bolyai Mathematical Society, Budapest (1994)Google Scholar
  3. 3.
    Boros, E.: Maximum renamable Horn sub-CNFs. Discrete Appl. Math. 96-97, 29–40 (1999)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Boufkhad, Y.: Algorithms for propositional KB approximation. In: Proceedings of the 15th National Conference on Artificial Intelligence (AAAI-98) and of the 10th Conference on Innovative Applications of Artificial Intelligence (IAAI-98), pp. 280–285 (1998)Google Scholar
  5. 5.
    Cadoli, M., Donini, F.M.: A survey on knowledge compilation. AI Communications 10(3–4), 137–150 (1997)Google Scholar
  6. 6.
    Crama, Y., Ekin, O., Hammer, P.L.: Variable and term removal from Boolean formulae. Discrete Applied Mathematics 75(3), 217–230 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Darwiche, A., Marquis, P.: A knowledge compilation map. Journal of Artificial Intelligence Research 17, 229–264 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    del Val, A.: First order LUB approximations: characterization and algorithms. Artificial Intelligence 162(1-2), 7–48 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Erdős, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci 5, 17–61 (1960)Google Scholar
  10. 10.
    Horn, A.: On sentences which are true on direct unions of algebras. J. Symbolic Logic 16, 14–21 (1951)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Jerrum, M.R., Valiant, L.G., Vazirani, V.V.: Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci. 43, 169–188 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Kautz, H., Selman, B.: An empirical evaluation of knowledge compilation by theory approximation. In: Proceedings of the 12th National Conference on Artificial Intelligence, pp. 155–161 (1994)Google Scholar
  13. 13.
    Kearns, M.J., Vazirani, U.V.: An Introduction to Computational Learning Theory. MIT Press, Cambridge (1994)Google Scholar
  14. 14.
    Langlois, M., Sloan, R.H., Turán, G.: Horn upper bounds of random 3-CNF: A computational study. In: Ninth Int. Symp. Artificial Intelligence and Mathematics (2006), Available on-line from URL http://anytime.cs.umass.edu/aimath06/
  15. 15.
    Levin, A.A.: Comparative complexity of disjunctive normal forms (in Russian). Metody Discret. Analiz. 36, 23–38 (1981)zbMATHGoogle Scholar
  16. 16.
    Lewis, H.R.: Renaming a set of clauses as a Horn set. J. ACM 25, 134–135 (1978)zbMATHGoogle Scholar
  17. 17.
    McKinsey, J.C.C.: The decision problem for some classes without quantifiers. J. Symbolic Logic 8, 61–76 (1943)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Mézard, M., Zecchina, R.: The random K-satisfiability problem: from an analytic solution to an efficient algorithm. Physical Review E 66, 56–126 (2002)CrossRefGoogle Scholar
  19. 19.
    Mora, T., Mézard, M., Zecchina, R.: Pairs of SAT assignments and clustering in random Boolean formulae. Submitted to Theoretical Computer Science (2005), available from URL http://www.citebase.org/cgi-bin/citations?id=oai:arXiv.org:cond-mat/0506053
  20. 20.
    Roth, D.: On the hardness of approximate reasoning. Artificial Intelligence 82, 273–302 (1996)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Vardi, M.Y., Aguirre, A.S.M.: Random 3-SAT and BDDs: The Plot Thickens Further. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 121–136. Springer, Heidelberg (2001)Google Scholar
  22. 22.
    Selman, B., Kautz, H.: Knowledge compilation and theory approximation. J. ACM 43, 193–224 (1996)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Sloan, R.H., Szörényi, B., Turán, G.: On k-term DNF with the maximal number of prime implicants. Submitted for publication. Preliminary version available as Electronic Colloquium on Computational Complexity (ECCC) Technical Report TR05-023, available on-line at http://www.eccc.uni-trier.de/eccc/
  24. 24.
    Truemper, K.: Effective Logic Computation. Wiley-Interscience, Chichester (1998)zbMATHGoogle Scholar
  25. 25.
    van Norden, L., van Maaren, H.: Hidden Threshold Phenomena for Fixed-Density SAT-formulae. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 135–149. Springer, Heidelberg (2004)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Marina Langlois
    • 1
  • Robert H. Sloan
    • 1
  • György Turán
    • 1
  1. 1.University of Illinois at Chicago, Chicago, IL 60607USA

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