k-Universality of Regular Languages.
CoRR(2023)
摘要
A subsequence of a word $w$ is a word $u$ such that $u = w[i_1] w[i_2] \dots
w[i_{k}]$, for some set of indices $1 \leq i_1 < i_2 < \dots < i_k \leq \lvert
w\rvert$. A word $w$ is $k$-subsequence universal over an alphabet $\Sigma$ if
every word in $\Sigma^k$ appears in $w$ as a subsequence. In this paper, we
study the intersection between the set of $k$-subsequence universal words over
some alphabet $\Sigma$ and regular languages over $\Sigma$. We call a regular
language $L$ \emph{$k$-$\exists$-subsequence universal} if there exists a
$k$-subsequence universal word in $L$, and \emph{$k$-$\forall$-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
\emph{$k$-$\exists$-subsequence universal} and, respectively, if it is
\emph{$k$-$\forall$-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
\emph{$k$-$\exists$-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.
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