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Theoretical Aspects of Symbolic Automata

  • Hellis Tamm
  • Margus Veanes
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10706)

Abstract

Symbolic finite automata extend classical automata by allowing infinite alphabets given by Boolean algebras and having transitions labeled by predicates over such algebras. Symbolic automata have been intensively studied recently and they have proven useful in several applications. We study some theoretical aspects of symbolic automata. Especially, we study minterms of symbolic automata, that is, the set of maximal satisfiable Boolean combinations of predicates of automata. We define canonical minterms of a language accepted by a symbolic automaton and show that these minterms can be used to define symbolic versions of some known classical automata. Also we show that canonical minterms have an important role in finding minimal nondeterministic symbolic automata. We show that Brzozowski’s double-reversal method for minimizing classical deterministic automata as well as its generalization is applicable for symbolic automata.

References

  1. 1.
    Brzozowski, J.A., Tamm, H.: Theory of átomata. Theor. Comput. Sci. 539, 13–27 (2014)CrossRefzbMATHGoogle Scholar
  2. 2.
    Brzozowski, J.A.: Canonical regular expressions and minimal state graphs for definite events. In: Proceedings of the Symposium on Mathematical Theory of Automata, MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Press, Polytechnic Institute of Brooklyn, NY (1963)Google Scholar
  3. 3.
    Brzozowski, J.A.: Derivatives of regular expressions. J. ACM 11(4), 481–494 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D’Antoni, L., Veanes, M.: The power of symbolic automata and transducers. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10426, pp. 47–67. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63387-9_3 CrossRefGoogle Scholar
  5. 5.
    D’Antoni, L., Veanes, M.: Minimization of symbolic automata. In: The 41st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2014, San Diego, CA, USA, 20–21 January 2014, pp. 541–554 (2014)Google Scholar
  6. 6.
    D’Antoni, L., Veanes, M.: Forward bisimulations for nondeterministic symbolic finite automata. In: Tools and Algorithms for the Construction and Analysis of Systems - 23rd International Conference, TACAS 2017, ETAPS 2017, Proceedings, Part I, Uppsala, Sweden, 22–29 April 2017, pp. 518–534 (2017)Google Scholar
  7. 7.
    Denis, F., Lemay, A., Terlutte, A.: Residual finite state automata. Fund. Informaticae 51, 339–368 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Denis, F., Lemay, A., Terlutte, A.: Learning regular languages using RFSAs. Theor. Comput. Sci. 313(2), 267–294 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Drews, S., D’Antoni, L.: Learning symbolic automata. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 173–189. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-662-54577-5_10 CrossRefGoogle Scholar
  10. 10.
    Iván, S.: Complexity of atoms, combinatorially. Inf. Process. Lett. 116, 356–360 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Keil, M., Thiemann, P.: Symbolic solving of extended regular expression inequalities. In: 34th International Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2014, 15–17 December 2014, New Delhi, India, pp. 175–186 (2014)Google Scholar
  12. 12.
    Owens, S., Reppy, J., Turon, A.: Regular-expression derivatives re-examined. J. Funct. Programm. 19(2), 173–190 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Tamm, H.: Generalization of the double-reversal method of finding a canonical residual finite state automaton. In: Shallit, J., Okhotin, A. (eds.) DCFS 2015. LNCS, vol. 9118, pp. 268–279. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-19225-3_23 CrossRefGoogle Scholar
  14. 14.
    Tamm, H.: New interpretation and generalization of the Kameda-Weiner method. In: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), vol. 55, Dagstuhl, Germany, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, pp. 116:1–116:12 (2016)Google Scholar
  15. 15.
    Thompson, K.: Regular expression search algorithm. Commun. ACM 11(6), 419–422 (1968)CrossRefzbMATHGoogle Scholar
  16. 16.
    Veanes, M., de Halleux, P., Tillmann, N.: Rex: symbolic regular expression explorer. In: Third International Conference on Software Testing, Verification and Validation, ICST 2010, pp. 498–507. IEEE Computer Society (2010)Google Scholar
  17. 17.
    Watson, B.W.: A taxonomy of finite automata construction algorithms. Computing science report 93/43. Eindhoven University of Technology (1995)Google Scholar
  18. 18.
    Watson, B.W.: Implementing and using finite automata toolkits. In: Extended finite state models of language, pp. 19–36. Cambridge University Press (1999)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Tallinn University of TechnologyTallinnEstonia
  2. 2.Microsoft ResearchRedmondUSA

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