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M-equivalence of Parikh Matrix over a Ternary Alphabet

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Implementation and Application of Automata (CIAA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14151))

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Abstract

The Parikh matrix, an extension of the Parikh vector for words, is a fundamental concept in combinatorics on words. We investigate M-unambiguity that identifies words with unique Parikh matrices. While the problem of identifying M-unambiguous words for a binary alphabet is solved using a palindromicly amicable relation, it is open for larger alphabets. We propose substitution rules that establish M-equivalence and solve the problem of M-unambiguity for a ternary alphabet. Our rules build on the principles of the palindromicly amicable relation and enable tracking of the differences of length-3 ordered scattered-factors. We characterize the set of M-unambiguous words and obtain a regular expression for the set.

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Notes

  1. 1.

    \(x\equiv _{pa}10211201 \equiv _{pa}11200211\equiv _{pa}02111120\equiv _{pa}y\).

  2. 2.

    The regular expression is incorrect since it misses some M-unambiguous words illustrated in Fig. 4 in Sect. 4.

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Acknowledgments

We thank all the reviewers for their valuable comments. This research was supported by the NRF grant funded by MIST (NRF-RS-2023-00208094). The first two authors contributed equally.

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Authors and Affiliations

  1. Department of Computer Science, Yonsei University, Seoul, 03722, Republic of Korea

    Joonghyuk Hahn, Hyunjoon Cheon & Yo-Sub Han

Authors
  1. Joonghyuk Hahn
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  2. Hyunjoon Cheon
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  3. Yo-Sub Han
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Correspondence to Yo-Sub Han .

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Editors and Affiliations

  1. Eastern Mediterranean University, Famagusta, Cyprus

    Benedek Nagy

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Hahn, J., Cheon, H., Han, YS. (2023). M-equivalence of Parikh Matrix over a Ternary Alphabet. In: Nagy, B. (eds) Implementation and Application of Automata. CIAA 2023. Lecture Notes in Computer Science, vol 14151. Springer, Cham. https://doi.org/10.1007/978-3-031-40247-0_10

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  • DOI: https://doi.org/10.1007/978-3-031-40247-0_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-40246-3

  • Online ISBN: 978-3-031-40247-0

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