Abstract
Given an integer k, Simon’s congruence relation says that two strings u and v are \(\sim _k\)-congruent if they have the same set of subsequences of length at most k. We extend Simon’s congruence to languages. First, we define the Simon’s congruence neighborhood of a language L to be a set of strings that have a \(\sim _k\)-congruent string in L. Next, we define two languages \(L_1\) and \(L_2\) to be \(\equiv _k\)-congruent if both have the same Simon’s congruence neighborhood. We prove that it is PSPACE-complete to check \(\equiv _k\)-congruence of two regular languages and decidable up to recursive languages. Moreover, we tackle the problem of computing the maximum k that makes two given languages \(\equiv _k\)-congruent. This problem is PSPACE-complete for two regular languages, and undecidable for context-free languages.
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Acknowledgements
We thank the reviewers that pointed us to the book “Varieties of Formal Languages”. Kim, Han and Ko were supported by the NRF grant (RS-2023-00208094) and Salomaa was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Kim, S., Han, YS., Ko, SK., Salomaa, K. (2023). On the Simon’s Congruence Neighborhood of Languages. In: Drewes, F., Volkov, M. (eds) Developments in Language Theory. DLT 2023. Lecture Notes in Computer Science, vol 13911. Springer, Cham. https://doi.org/10.1007/978-3-031-33264-7_14
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