k-Universality of Regular Languages

Authors Duncan Adamson , Pamela Fleischmann , Annika Huch, Tore Koß , Florin Manea , Dirk Nowotka

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Duncan Adamson
  • Leverhulme Centre for Functional Material Design, University of Liverpool, UK
Pamela Fleischmann
  • Department of Computer Science, Kiel University, Germany
Annika Huch
  • Department of Computer Science, Kiel University, Germany
Tore Koß
  • Department of Computer Science, University of Göttingen, Germany
Florin Manea
  • Department of Computer Science, University of Göttingen, Germany
Dirk Nowotka
  • Department of Computer Science, Kiel University, Germany

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Duncan Adamson, Pamela Fleischmann, Annika Huch, Tore Koß, Florin Manea, and Dirk Nowotka. k-Universality of Regular Languages. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 4:1-4:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


A subsequence of a word w is a word u such that u = w[i₁] w[i₂] … w[i_k], for some set of indices 1 ≤ i₁ < i₂ < … < i_k ≤ |w|. A word w is k-subsequence universal over an alphabet Σ if every word in Σ^k appears in w as a subsequence. In this paper, we study the intersection between the set of k-subsequence universal words over some alphabet Σ and regular languages over Σ. We call a regular language L k-∃-subsequence universal if there exists a k-subsequence universal word in L, and k-∀-subsequence universal if every word of L is k-subsequence universal. We give algorithms solving the problems of deciding if a given regular language, represented by a finite automaton recognising it, is k-∃-subsequence universal and, respectively, if it is k-∀-subsequence universal, for a given k. The algorithms are FPT w.r.t. the size of the input alphabet, and their run-time does not depend on k; they run in polynomial time in the number n of states of the input automaton when the size of the input alphabet is O(log n). Moreover, we show that the problem of deciding if a given regular language is k-∃-subsequence universal is NP-complete, when the language is over a large alphabet. Further, we provide algorithms for counting the number of k-subsequence universal words (paths) accepted by a given deterministic (respectively, nondeterministic) finite automaton, and ranking an input word (path) within the set of k-subsequence universal words accepted by a given finite automaton.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • String Algorithms
  • Regular Languages
  • Finite Automata
  • Subsequences


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