Efficient Algorithm for Checking Multiplicity Equivalence for the Finite Z - Σ^*-Automata

Abstract

We represent a new fast algorithm for checking multiplicity equivalence for finite Z - Σ^* - automata. The classical algorithm of Eilenberg [1]is exponential one. On the other hand, for the finite deterministic automata an analogous Aho - Hopcroft - Ullman [2] algorithm has “almost” linear time complexity. Such a big gap leads to the idea of the existence of an algorithm which is more faster than an exponential one. Hunt and Stearns [3] announced the existence of polynomial algorithm for the Z-Σ^* - automata of finite degree of ambiguity only. Tzeng [4] created O(n ⁴) algorithm for the general case. Diekert [5] informed us Tzeng's algorithm can be implemented in O(n ³) using traingular matrices. Any way, we propose a new O(n ³) algorithm (for the general case). Our algorithm utilizes recent results of Siberian mathematical school (Gerasimov [6], Valitckas [7]) about the structure of rings and does not share common ideas with Eilenberg [1] and Tzeng [4].