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Descriptional Complexity of Power and Positive Closure on Convex Languages

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Implementation and Application of Automata (CIAA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11601))

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Abstract

We study the descriptional complexity of the k-th power and positive closure operations on the classes of prefix-, suffix-, factor-, and subword-free, -closed, and -convex regular languages, and on the classes of right, left, two-sided, and all-sided ideal languages. We show that the upper bound kn on the nondeterministic complexity of the k-th power in the class of regular languages is tight for closed and convex classes, while in the remaining classes, the tight upper bound is \(k(n-1)+1\). Next we show that the upper bound n on the nondeterministic complexity of the positive closure operation in the class of regular languages is tight in all considered classes except for classes of factor-closed and subword-closed languages, where the complexity is one. All our worst-case examples are described over a unary or binary alphabet, except for witnesses for the k-th power on subword-closed and subword-convex languages which are described over a ternary alphabet. Moreover, whenever a binary alphabet is used for describing a worst-case example, it is optimal in the sense that the corresponding upper bounds cannot be met by a language over a unary alphabet. The most interesting result is the description of a binary factor-closed language meeting the upper bound kn for the k-th power. To get this result, we use a method which enables us to avoid tedious descriptions of fooling sets. We also provide some results concerning the deterministic state complexity of these two operations on the classes of free, ideal, and closed languages.

Research supported by VEGA grant 2/0132/19 and grant APVV-15-0091.

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  1. Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01, Košice, Slovakia

    Michal Hospodár

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  1. Michal Hospodár
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  1. Slovak Academy of Sciences, Košice, Slovakia

    Michal Hospodár

  2. Slovak Academy of Sciences, Košice, Slovakia

    Galina Jirásková

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Hospodár, M. (2019). Descriptional Complexity of Power and Positive Closure on Convex Languages. In: Hospodár, M., Jirásková, G. (eds) Implementation and Application of Automata. CIAA 2019. Lecture Notes in Computer Science(), vol 11601. Springer, Cham. https://doi.org/10.1007/978-3-030-23679-3_13

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  • DOI: https://doi.org/10.1007/978-3-030-23679-3_13

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