Decomposition of Automata recognizing Ideals
Mathias Berry, Ismaël Jecker, Pierre-Cyrille Héam
Minimizing the size of finite automata is a fundamental problem in theoretical computer science. Beyond standard minimization, further reductions can be achieved by decomposing an automaton into smaller components whose languages combine via union or intersection to recover the original language. However, in general, no polynomial-time algorithm is known for computing such decompositions. In this paper, we focus on automata that recognize ideals, that is, languages at level 1/2 in the Straubing–Thérien hierarchy. Equivalently, these are the languages expressible as a finite union of languages of the form $A^a_1A^\cdots A^a_nA^$ where $A$ is an alphabet and $a_i$ are letters of A. We show that the two problems of deciding whether such a language can be decomposed into an intersection or a union of smaller automata are decidable in NL. Moreover, we provide a polynomial-time algorithm that computes a decomposition into an intersection, if one exists, while ensuring that the resulting components also recognize ideal languages.