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Title: Testing Simon’s congruence
Authors: Fleischer, Lukas
Kufleitner, Manfred
Keywords: Regular language
Scattered subword
Piecewise testability
String algorithm
Issue Date: 2018
Publisher: © Lukas Fleischer and Manfred Kufleitner
Citation: FLEISCHER, L. and KUFLEITNER, M., 2018.. Testing Simon’s congruence. IN: Potapov, I., Spirakis, P. and Worrell, J. (Eds.) 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018) Volume 117, Article No. 62; pp. 62:1–62:13, August 27-31, 2018, Liverpool, UK.
Series/Report no.: Leibniz International Proceedings in Informatics, LIPIcs;117
Abstract: Piecewise testable languages are a subclass of the regular languages. There are many equivalent ways of defining them; Simon’s congruence ∼kis one of the most classical approaches. Two words are ∼k-equivalent if they have the same set of (scattered) subwords of length at most k. A language L is piecewise testable if there exists some k such that L is a union of ∼k-classes. For each equivalence class of ∼k, one can define a canonical representative in shortlex normal form, that is, the minimal word with respect to the lexicographic order among the shortest words in ∼k. We present an algorithm for computing the canonical representative of the ∼k-class of a given word w ∈ A∗of length n. The running time of our algorithm is in O(|A|n) even if k ≤ n is part of the input. This is surprising since the number of possible subwords grows exponentially in k. The case k > n is not interesting since then, the equivalence class of w is a singleton. If the alphabet is fixed, the running time of our algorithm is linear in the size of the input word. Moreover, for fixed alphabet, we show that the computation of shortlex normal forms for ∼kis possible in deterministic logarithmic space. One of the consequences of our algorithm is that one can check with the same complexity whether two words are ∼k-equivalent (with k being part of the input).
Description: Licensed under Creative Commons License CC-BY
Version: Published
DOI: 10.4230/LIPIcs.MFCS.2018.62
URI: https://dspace.lboro.ac.uk/2134/36029
Publisher Link: https://doi.org/10.4230/LIPIcs.MFCS.2018.62
ISBN: 9783959770866
ISSN: 1868-8969
Appears in Collections:Conference Papers and Presentations (Computer Science)

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