k-Universality of Regular Languages
Year: 2025
Type: article
Abstract: A subsequence of a word w is a word u such that u = w [ i 1 ] w [ i 2 ] … w [ i k ] , for some set of indices 1 ≤ i 1 < i 2 < … < 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, non-deterministic) finite automaton, and ranking an input word (path) within the set of k -subsequence universal words accepted by a given finite automaton.
Source: Information and Computation
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,840
OpenAlex ID: W4414264943