On the Computational Complexity of Problems Related to Distinguishability Sets

Abstract

We study the computational complexity of problems related to distinguishability sets for regular languages. Roughly speaking, the distinguishability set $\mathsf {D}(L)$ for a (not necessarily regular) language $L$ consists of all words $w$ that are a common suffix of a word $xw$ in $L$ and of a word $yw$ that does not belong to $L$. In particular, we investigate the complexity of the representation problem, i.e., deciding for given automata $A$ and $B$, whether $B$ accepts the distinguishability set of $L(A)$. It is shown that this problem and some of its variants are highly intractable, namely PSPACE-complete. In fact, determining the size of an automaton for $\mathsf {D}(L(A))$ is already PSPACE-complete. On the other hand, questions related to the hierarchy induced by iterated application of the $\mathsf {D}$-operator turn out to be much easier. For instance, the question whether for a given automaton $A$, the accepted language is equal to its own distinguishability set, i.e., whether $L(A)=\mathsf {D}(L(A))$ holds, is shown to be NL-complete. As a byproduct of our investigations, we found a compelling characterization of synchronizing automata, namely that a (minimal) automaton $A$ is synchronizing if and only if $\mathsf {D}(L(A))=\mathsf {D}^2(L(A))$.