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2016 | OriginalPaper | Chapter

Online Timed Pattern Matching Using Derivatives

Authors : Dogan Ulus, Thomas Ferrère, Eugene Asarin, Oded Maler

Published in: Tools and Algorithms for the Construction and Analysis of Systems

Publisher: Springer Berlin Heidelberg

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Abstract

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.

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Footnotes
1
To keep the survey within a reasonable size and avoid tedious repetitions, the description here is not fully rigorous, using the same notation for the semantic notion of a left quotient, which is unique for every language and word, and the syntactic notion of a derivative of a regular expression. The derivation of the minimal automaton from a regular expression, for example, requires additional rewrite rules to detect equivalence between different regular expressions.
 
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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)
Metadata
Title
Online Timed Pattern Matching Using Derivatives
Authors
Dogan Ulus
Thomas Ferrère
Eugene Asarin
Oded Maler
Copyright Year
2016
Publisher
Springer Berlin Heidelberg
Book
Tools and Algorithms for the Construction and Analysis of Systems

Print ISBN: 978-3-662-49673-2

Electronic ISBN: 978-3-662-49674-9

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-662-49674-9_47

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