Abstract
Two strings are Simon’s \(\sim _k\)-congruent if they have the same set of subsequences of length at most k. We study the Simon’s congruence closure of a string, which is regular by definition. Given a string w over an alphabet \(\varSigma \), we present an efficient DFA construction that accepts all \(\sim _k\)-congruent strings with respect to w. We also present lower bounds for the state complexity of the Simon’s congruence closure. Finally, we design a polynomial-time algorithm that answers the following open problem: “given a string w over a fixed-sized alphabet, an integer k and a (regular or context-free) language L, decide whether there exists a string \(v \in L\) such that \(w \sim _k v\).” The problem is NP-complete for a variable-sized alphabet.
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Acknowledgments
We wish to thank the referees for letting us know related references and providing valuable suggestions that improve the presentation of the paper. This research was supported by the NRF grant funded by MIST (NRF-2020R1A4A3079947).
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Kim, S., Han, YS., Ko, SK., Salomaa, K. (2022). On Simon’s Congruence Closure of a String. In: Han, YS., Vaszil, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2022. Lecture Notes in Computer Science, vol 13439. Springer, Cham. https://doi.org/10.1007/978-3-031-13257-5_10
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DOI: https://doi.org/10.1007/978-3-031-13257-5_10
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