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On the Complexity of Timed Pattern Matching

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12860)

Abstract

Timed pattern matching consists in finding occurrences of a timed regular expression in a timed word. This problem has been addressed using several techniques, its solutions are implemented in tools (quite efficient in practice), and used, for example in log analysis and runtime verification. In this article, we explore computational complexity of timed pattern matching, and prove P, NP and PSPACE bounds, depending on connectives used in expressions and other details. We conclude with a couple of open questions.

Keywords

  • Timed regular expressions
  • Pattern matching
  • Complexity

E. Asarin—Supported by ANR-JST grant CyPhAI.

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Fig. 1.
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Fig. 3.

Notes

  1. 1.

    To avoid confusion, we recall that high complexity bounds, such as the famous non-elementary bound of [18], deal with quite a different problem of emptiness checking.

  2. 2.

    We avoid empty matches, and for this reason use \(^+\) rather than \(^*\) for Kleene iteration.

  3. 3.

    Note that the present bound, unlike the ones in [21], does not rely on expressions having integer interval bounds.

  4. 4.

    This would not work for Kleene \(^+\).

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Acknowledgement

We would like to thank Thao Dang, Nicolas Basset, and Akshay Mambakam for motivating and technically relevant discussions.

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Asarin, E., Ferrère, T., Ničković, D., Ulus, D. (2021). On the Complexity of Timed Pattern Matching. In: Dima, C., Shirmohammadi, M. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2021. Lecture Notes in Computer Science(), vol 12860. Springer, Cham. https://doi.org/10.1007/978-3-030-85037-1_2

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  • DOI: https://doi.org/10.1007/978-3-030-85037-1_2

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