Online Timed Pattern Matching Using Derivatives

  • Dogan Ulus
  • Thomas Ferrère
  • Eugene Asarin
  • Oded Maler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9636)


Timed pattern matching consists in finding all segments of a dense-time Boolean signal that match a pattern defined by a timed regular expression. This problem has been formulated and solved in [17] via an offline algorithm that takes the signal and expression as inputs and produces the set of all matches, represented as a finite union of two-dimensional zones. In this work we develop an online version of this approach where the input signal is presented incrementally and the matching is computed incrementally as well.

Naturally, the concept of derivatives of regular expressions due to Brzozowski [6] can play a role in defining what remains to match after having read a prefix of the signal. However the adaptation of this concept is not a straightforward for two reasons: the dense infinite-state nature of timed behaviors and the fact that we are interested in matching, not only in prefix acceptance. To resolve these issues we develop an alternative theory of signals and expressions based on absolute time and show how derivatives are defined and computed in this setting. We then implement an online timed pattern matching algorithm based on these results.


Temporal Logic Regular Expression Constant Signal Propositional Variable Empty Word 
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.



This work was partially supported by the French ANR projects EQINOCS and CADMIDIA and benefitted from useful comments made by anonymous referees.


  1. 1.
    Antimirov, V.M.: Partial derivatives of regular expressions and finite automaton constructions. Theor. Comput. Sci. 155(2), 291–319 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antimirov, V.M., Mosses, P.D.: Rewriting extended regular expressions. Theor. Comput. Sci. 143(1), 51–72 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Asarin, E., Caspi, P., Maler, O.: A Kleene theorem for timed automata. In: Logic in Computer Science (LICS), pp. 160–171 (1997)Google Scholar
  4. 4.
    Asarin, E., Caspi, P., Maler, O.: Timed regular expressions. J. ACM 49(2), 172–206 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berry, G., Sethi, R.: From regular expressions to deterministic automata. Theor. Comput. Sci. 48(3), 117–126 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brzozowski, J.A.: Derivatives of regular expressions. J. ACM 11(4), 481–494 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ferrère, T., Maler, O., Ničković, D., Ulus, D.: Measuring with timed patterns. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9207, pp. 322–337. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  8. 8.
    Giavitto, J.-L., Echeveste, J.: Real-time matching of antescofo temporal patterns. In: Principles and Practice of Declarative Programming (PPDP), pp. 93–104 (2014)Google Scholar
  9. 9.
    Havlicek, J., Little, S.: Realtime regular expressions for analog and mixed-signal assertions. In: Formal Methods in Computer-Aided Design (FMCAD), pp. 155–162 (2011)Google Scholar
  10. 10.
    Koymans, R.: Specifying real-time properties with metric temporal logic. Real-Time Syst. 2(4), 255–299 (1990)CrossRefGoogle Scholar
  11. 11.
    Maler, O., Nickovic, D., Pnueli, A.: Checking temporal properties of discrete, timed and continuous behaviors. In: Avron, A., Dershowitz, N., Rabinovich, A. (eds.) Pillars of Computer Science. LNCS, vol. 4800, pp. 475–505. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Morin-Allory, K., Borrione, D.: On-line monitoring of properties built on regular expressions. In: Forum on specification and Design Languages, (FDL), pp. 249–255 (2006)Google Scholar
  13. 13.
    Owens, S., Reppy, J.H., Turon, A.: Regular-expression derivatives re-examined. J. Funct. Program. 19(2), 173–190 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rosu, G., Viswanathan, M.: Testing extended regular language membership incrementally by rewriting. In: Rewriting Techniques and Applications (RTA), pp. 499–514 (2003)Google Scholar
  15. 15.
    Sen, K., Rosu, G.: Generating optimal monitors for extended regular expressions. Electron. Notes Theor. Comput. Sci. 89(2), 226–245 (2003)CrossRefGoogle Scholar
  16. 16.
    Sulzmann, M., van Steenhoven, P.: A flexible and efficient ML lexer tool based on extended regular expression submatching. In: Cohen, A. (ed.) CC 2014 (ETAPS). LNCS, vol. 8409, pp. 174–191. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  17. 17.
    Ulus, D., Ferrère, T., Asarin, E., Maler, O.: Timed pattern matching. In: Legay, A., Bozga, M. (eds.) FORMATS 2014. LNCS, vol. 8711, pp. 222–236. Springer, Heidelberg (2014)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Dogan Ulus
    • 1
  • Thomas Ferrère
    • 1
  • Eugene Asarin
    • 2
  • Oded Maler
    • 1
  1. 1.VERIMAG Université Grenoble-Alpes/CNRSGrenobleFrance
  2. 2.IRIF, Université Paris Diderot/CNRSParisFrance

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