Regular Inference as Vertex Coloring

  • Christophe Costa Florêncio
  • Sicco Verwer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7568)

Abstract

This paper is concerned with the problem of supervised learning of deterministic finite state automata, in the technical sense of identification in the limit from complete data, by finding a minimal DFA consistent with the data (regular inference).

We solve this problem by translating it in its entirety to a vertex coloring problem. Essentially, such a problem consists of two types of constraints that restrict the hypothesis space: inequality and equality constraints.

Inequality constraints translate to the vertex coloring problem in a very natural way. Equality constraints however greatly complicate the translation to vertex coloring. In previous coloring-based translations, these were therefore encoded either dynamically by modifying the vertex coloring instance on-the-fly, or by encoding them as satisfiability problems. We provide the first translation that encodes both types of constraints together in a pure vertex coloring instance. This offers many opportunities for applying insights from combinatorial optimization and graph theory to regular inference. We immediately obtain new complexity bounds, as well as a family of new learning algorithms which can be used to obtain both exact hypotheses, as well as fast approximations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abela, J., Coste, F., Spina, S.: Mutually Compatible and Incompatible Merges for the Search of the Smallest Consistent DFA. In: Paliouras, G., Sakakibara, Y. (eds.) ICGI 2004. LNCS (LNAI), vol. 3264, pp. 28–39. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Akram, H.I., Batard, A., de la Higuera, C., Eckert, C.: PSMA: A parallel algorithm for learning regular languages. In: NIPS Workshop on Learning on Cores, Clusters and Clouds (2010)Google Scholar
  3. 3.
    Angluin, D.: On the complexity of minimum inference of regular sets. Information and Control 39(3), 337–350 (1978)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Beigel, R., Eppstein, D.: 3-coloring in time O(1.3289n). Journal of Algorithms 54(2), 168–204 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press (February 2009)Google Scholar
  6. 6.
    Biermann, A.W., Feldman, J.A.: On the synthesis of finite-state machines from samples of their behavior. IEEE Trans. Comput. 21(6), 592–597 (1972)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39(2), 546–563 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bugalho, M., Oliveira, A.L.: Inference of regular languages using state merging algorithms with search. Pattern Recognition 38, 1457–1467 (2005)MATHCrossRefGoogle Scholar
  9. 9.
    Byskov, J.M.: Enumerating maximal independent sets with applications to graph colouring. Operations Research Letters 32(6), 547–556 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Coste, F., Nicolas, J.: Regular inference as a graph coloring problem. In: Workshop on Grammatical Inf., Automata Ind., and Language Acq., ICML 1997 (1997)Google Scholar
  11. 11.
    de la Higuera, C.: Grammatical Inference: Learning Automata and Grammars. Cambridge University Press (2010)Google Scholar
  12. 12.
    García, P., Vázquez de Parga, M., López, D., Ruiz, J.: Learning Automata Teams. In: Sempere, J.M., García, P. (eds.) ICGI 2010. LNCS (LNAI), vol. 6339, pp. 52–65. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co. (1979)Google Scholar
  14. 14.
    Gold, E.M.: Complexity of automaton identification from given data. Information and Control 37(3), 302–320 (1978)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Grinchtein, O., Leucker, M., Piterman, N.: Inferring Network Invariants Automatically. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 483–497. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Heule, M.J.H., Verwer, S.: Exact DFA Identification Using SAT Solvers. In: Sempere, J.M., García, P. (eds.) ICGI 2010. LNCS (LNAI), vol. 6339, pp. 66–79. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Karger, D., Motwani, R., Sudan, M.: Approximate graph coloring by semidefinite programming. J. ACM 45, 246–265 (1998)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kearns, M.J., Valiant, L.: Cryptographic limitations on learning Boolean formulae and finite automata. J. ACM 41, 67–95 (1994)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Lang, K.J.: Faster algorithms for finding minimal consistent DFAs. Technical report, NEC Research Institute (1999)Google Scholar
  20. 20.
    Lang, K.J., Pearlmutter, B.A., Price, R.A.: Results of the Abbadingo One DFA Learning Competition and a New Evidence-Driven State Merging Algorithm. In: Honavar, V.G., Slutzki, G. (eds.) ICGI 1998. LNCS (LNAI), vol. 1433, pp. 1–12. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  21. 21.
    Malaguti, E., Toth, P.: A survey on vertex coloring problems. International Transactions in Operational Research 17(1), 1–34 (2010)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Oliveira, A.L., Marques-Silva, J.P.: Efficient search techniques for the inference of minimum sized finite state machines. In: String Processing and Information Retrieval, pp. 81–89 (1998)Google Scholar
  23. 23.
    Oncina, J., Garcia, P.: Inferring regular languages in polynomial update time. In: Pattern Recognition and Image Analysis. Series in Machine Perception and Artificial Intelligence, vol. 1, pp. 49–61. World Scientific (1992)Google Scholar
  24. 24.
    Pitt, L., Warmuth, M.K.: The minimum consistent DFA problem cannot be approximated within any polynomial. In: STOC, pp. 421–432 (1989)Google Scholar
  25. 25.
    Sudkamp, T.A.: Languages and Machines: an introduction to the theory of computer science, 3rd edn. Addison-Wesley (2006)Google Scholar
  26. 26.
    Walkinshaw, N., Lambeau, B., Damas, C., Bogdanov, K., Dupont, P.: STAMINA: a competition to encourage the development and assessment of software model inference techniques. Empirical Software Engineering, 1–34 (to appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Christophe Costa Florêncio
    • 1
  • Sicco Verwer
    • 2
  1. 1.Department of Computer ScienceUniversity of AmsterdamThe Netherlands
  2. 2.Institute for Computing and Information SciencesRadboud University NijmegenThe Netherlands

Personalised recommendations