Abstract
We present new automata-theoretical characterizations for the regularity of the iterated shuffle on commutative regular languages. Using these characterizations we show that, for a fixed alphabet, it is tractable to decide whether the iterated shuffle of a regular commutative language is itself regular when the input language is given by a deterministic automaton. Additionally, we introduce two new subclasses of commutative regular languages, called Type I and Type II languages, on which the iterated shuffle is regularity-preserving and show that the iterated shuffle of a commutative language is a Type I language if it is regular. Additionally, we establish various closure properties and show that we can decide if a language given by deterministic automaton is in one of these classes in polynomial time.
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Notes
- 1.
As noted by a reviewer, this is a more compact way of expressing the fact that there exist \(n \ge 0\) and \(p > 0\) such that \(q = \delta (q_0 ua^n)\), \(q = \delta (q, a^p)\) and n is divisible by p.
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I thank the anonymous reviewers for careful reading and helping me identifying some unclear formulations and typos throughout the text.
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Hoffmann, S. (2022). Automata-Theoretical Regularity Characterizations for the Iterated Shuffle on Commutative Regular Languages. In: Diekert, V., Volkov, M. (eds) Developments in Language Theory. DLT 2022. Lecture Notes in Computer Science, vol 13257. Springer, Cham. https://doi.org/10.1007/978-3-031-05578-2_13
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