On comparing deterministic finite automata and the shuffle of words

Abstract

We continue the study of the shuffle of individual words, and the problem of decomposing a finite automaton into the shuffle on words. There is a known polynomial time algorithm to decide whether the shuffle of two words is a subset of the language accepted by a deterministic finite automaton [5]. In this paper, we consider the converse problem of determining whether or not the language accepted by a deterministic finite automaton is a subset of the shuffle of two words. We provide a polynomial time algorithm to decide whether the language accepted by a deterministic finite automaton is a subset of the shuffle of two words, for the special case when the skeletons of the two words are of fixed length. Therefore, for this special case, we can decide equality in polynomial time as well. However, we then show that this problem is coNP-Complete in general, as conjectured in [2]. © 2014 Springer International Publishing Switzerland.