Skip to main content
SpringerLink
Log in

On Approximate Horn Formula Minimization

  • Conference paper
Book cover Automata, Languages and Programming (ICALP 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6198))

Included in the following conference series:

Abstract

The minimization problem for Horn formulas is to find a Horn formula equivalent to a given Horn formula, using a minimum number of clauses. A \(2^{\log^{1 - \epsilon}(n)}\)-inapproximability result is proven, which is the first inapproximability result for this problem. We also consider several other versions of Horn minimization. The more general version which allows for the introduction of new variables is known to be too difficult as its equivalence problem is co-NP-complete. Therefore, we propose a variant called Steiner-minimization, which allows for the introduction of new variables in a restricted manner. Steiner-minimization of Horn formulas is shown to be MAX-SNP-hard. In the positive direction, a o(n), namely, O(nloglogn/(logn)1/4)-approximation algorithm is given for the Steiner-minimization of definite Horn formulas. The algorithm is based on a new result in algorithmic extremal graph theory, on partitioning bipartite graphs into complete bipartite graphs, which may be of independent interest. Inapproximability results and approximation algorithms are also given for restricted versions of Horn minimization, where only clauses present in the original formula may be used.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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. Albert, R., DasGupta, B., Dondi, R., Sontag, E.: Inferring (biological) signal transduction networks via transitive reductions of directed graphs. Algorithmica 51(2), 129–159 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alon, N., Duke, R.A., Lefmann, H., Rödl, V., Yuster, R.: The algorithmic aspects of the regularity lemma. J. of Algorithms 16, 80–109 (1994)

    Article  MATH  Google Scholar 

  3. Angluin, D., Frazier, M., Pitt, L.: Learning conjunctions of Horn clauses. Machine Learning 9, 147–164 (1992)

    MATH  Google Scholar 

  4. Ausiello, G., D’atri, A., Saccà, D.: Minimal representations of directed hypergraphs. SIAM J. Comput. 15(2), 418–431 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berman, P., DasGupta, B., Karpinski, M.: Approximating transitive reduction problems for directed networks. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tòth, C.D. (eds.) 11th Algorithms and Data Structures Symposium. LNCS, vol. 5664, pp. 74–85. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  6. Berman, P., Karpinski, M., Scott, A.D.: Approximation hardness of short symmetric instances of MAX-3SAT. Electronic Colloquium on Computational Complexity, Report TR03-049 (2003)

    Google Scholar 

  7. Boros, E.: Horn functions. In: Crama, Y., Hammer, P.L. (eds.) Boolean Functions: Theory, Algorithms and Applications, Cambridge Univ. Press, Cambridge (forthcoming, 2010)

    Google Scholar 

  8. Boros, E., Čepek, O.: On the complexity of Horn minimization, Rutcor Research Report 1-94 (1994)

    Google Scholar 

  9. Boros, E., Crama, Y., Hammer, P.L.: Polynomial-time inference of all valid implications for Horn and related formulae. Ann. Math. Artif. Intell. 1, 21–32 (1990)

    Article  MATH  Google Scholar 

  10. Bublitz, S.: Decomposition of graphs and monotone formula size of homogeneous fuctions. Acta Informatica 23, 689–696 (1996)

    Article  MathSciNet  Google Scholar 

  11. Caspard, N., Monjardet, B.: The lattice of closure systems, closure operators and implications on a finite set: a survey. Discrete Appl. Math. 127, 241–269 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chandra, A.K., Markowsky, G.: On the number of prime implicants. Discrete Math. 24, 7–11 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chung, F.R.K., Erdős, P., Spencer, J.: On the decomposition of graphs into complete bipartite graphs. Studies in Pure Mathematics, To the Memory of Paul Turán, pp. 95-101. Akadémiai Kiadó (1983)

    Google Scholar 

  14. Clote, P., Kranakis, E.: Boolean Functions and Models of Computation. Springer, Heidelberg (2003)

    Google Scholar 

  15. Cook, S.A.: Feasibly constructive proofs and the propositional calculus. In: STOC, pp. 83–97 (1975)

    Google Scholar 

  16. Feige, U., Kogan, S.: Hardness of approximation of the balanced complete bipartite subgraph problem, Tech. Rep. MCS04-04, Dept. of Comp. Sci. and Appl. Math., The Weizmann Inst. of Science (2004)

    Google Scholar 

  17. Flögel, A., Kleine Büning, H., Lettmann, T.: On the restricted equivalence subclasses of propositional logic. ITA 27, 327–340 (1993)

    MATH  Google Scholar 

  18. Frazier, M.D.: Matters Horn and other features in the computational learning theory landscape: the notion of membership, Ph.D. thesis, Univ. of Illinois at Urbana-Champaign (1994)

    Google Scholar 

  19. Guigues, J.L., Duquenne, V.: Familles minimales d’implications informatives résultant d’un tableau de données binaires. Math. Sci. Humaines 95, 5–18 (1986)

    MathSciNet  Google Scholar 

  20. Hammer, P.L., Kogan, A.: Horn functions and their DNFs. Information Processing Letters 44, 23–29 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hammer, P.L., Kogan, A.: Optimal compression of propositional Horn knowledge bases: complexity and approximation. Artificial Intelligence 64, 131–145 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hammer, P.L., Kogan, A.: Quasi-acyclic Horn knowledge bases: optimal compression. IEEE Trans. on Knowledge and Data Engineering 7, 751–762 (1995)

    Article  Google Scholar 

  23. Hwang, F.K., Richards, D.S., Winter, P.: The Steiner Tree Problem. North-Holland, Amsterdam (1992)

    MATH  Google Scholar 

  24. Kirchner, S.: Lower bounds for Steiner tree algorithms and the construction of bicliques in dense graphs, Ph.D. Dissertation, Humboldt-Universität zu Berlin (2008) (in German)

    Google Scholar 

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

    MATH  Google Scholar 

  26. Kortsarz, G., Krauthgamer, R., Lee, J.R.: Hardness of approximating vertex-connectivity network design problems. SIAM J. Comput. 33(3), 704–720 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kővári, T., Sós, V.T., Turán, P.: On a problem of K. Zarankiewicz. Colloq. Math. 3, 50–57 (1954)

    Google Scholar 

  28. Langlois, M., Sloan, R.H., Turán, G.: Horn upper bounds and renaming. J. Satisfiability, Boolean Modelling and Computation 7, 1–15 (2009)

    Google Scholar 

  29. Langlois, M., Sloan, R.H., Szörényi, B., Turán, G.: Horn complements: towards Horn-to-Horn belief revision, pp. 466–471. AAAI, Menlo Park (2008)

    Google Scholar 

  30. Langlois, M., Mubayi, D., Sloan, R.H., Turán, G.: Combinatorial Problems for Horn Clauses. In: Lipshteyn, M., Levit, V.E., McConnell, R.M. (eds.) Graph Theory, Computational Intelligence and Thought. LNCS, vol. 5420, pp. 54–65. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  31. Lovász, L.: Coverings and colorings of hypergraphs. In: Proc. 4th Southeastern Conf. on Combinatorics, Graph Theory and Computing, Utilitas Math., pp. 3–12 (1973)

    Google Scholar 

  32. Maier, D.: The Theory of Relational Databases. Comp. Sci. Press, Rockville (1983)

    MATH  Google Scholar 

  33. Mubayi, D., Turán, G.: Finding bipartite subgraphs efficiently. Inf. Proc. Lett. 110, 174–177 (2010)

    Article  Google Scholar 

  34. Orlin, J.: Contentment in graph theory: covering graphs with cliques. Indag. Math. 80, 406–424 (1977)

    MATH  MathSciNet  Google Scholar 

  35. Russell, S., Norvig, P.: Artificial Intelligence: A Modern Approach, 2nd edn. Prentice Hall, Englewood Cliffs (2002)

    Google Scholar 

  36. Singh, P., Lin, T., Mueller, E.T., Lim, G., Perkins, T., Zhu, W.L.: Open Mind Common Sense: knowledge acquisition from the general public., CoopIS/DOA/ODBASE, 1223–1237 (2002)

    Google Scholar 

  37. Tarján, T.: Complexity of lattice-configurations. Studia Sci. Math. Hung. 10, 203–211 (1975)

    Google Scholar 

  38. Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Seminars in Mathematics, V. A. Steklov Mathematical Institut, Leningrad, vol. 8 (1968)

    Google Scholar 

  39. Tuza, Z.: Covering of graphs by complete bipartite subgraphs; complexity of 0-1 matrices. Combinatorica 4, 111–116 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  40. Umans, C.: Hardness of approximating \(\Sigma_2^p\) minimization problems. In: FOCS, pp. 465-474 (1999)

    Google Scholar 

  41. Umans, C.: The Minimum Equivalent DNF problem and shortest implicants. J. Comput. Syst. Sci. 63, 597–611 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  42. Vetta, A.: Approximating the minimum strongly connected subgraph via a matching lower bound. In: 12th ACM-SIAM Symposium on Discrete Algorithms, pp. 417–426 (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Tata Institute of Fundamental Research, Colaba, Mumbai, 400 005, India

    Amitava Bhattacharya

  2. Department of Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7053

    Bhaskar DasGupta

  3. Department of Mathematics, Statistics & Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045

    Dhruv Mubayi & György Turán

  4. Research Group on Artificial Intelligence, University of Szeged, Hungary

    György Turán

Authors
  1. Amitava Bhattacharya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bhaskar DasGupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dhruv Mubayi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. György Turán
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Oxford University Computing Laboratory, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Samson Abramsky

  2. Université de Bordeaux (LaBRI) & INRIA, 351, cours de la Libération, 33405, Talence Cedex, France

    Cyril Gavoille

  3. INRIA, Centre de Recherche Bordeaux – Sud-Ouest, 351 cours de la Libération, 33405, Talence Cedex, France

    Claude Kirchner

  4. Heinz Nixdorf Institute, University of Paderborn, Fürstenallee 11, 33102, Paderborn, Germany

    Friedhelm Meyer auf der Heide

  5. University of Patras and RACTI, 26500, Patras, Greece

    Paul G. Spirakis

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bhattacharya, A., DasGupta, B., Mubayi, D., Turán, G. (2010). On Approximate Horn Formula Minimization. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds) Automata, Languages and Programming. ICALP 2010. Lecture Notes in Computer Science, vol 6198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14165-2_38

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-14165-2_38

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14164-5

  • Online ISBN: 978-3-642-14165-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation