On the Descriptional Complexity of the Direct Product of Finite Automata

In [4] the descriptional complexity of certain automata products of two finite state devices, for reset, permutation, permutation-reset, and finite automata was investigated. Although an almost complete picture emerged for the magic number problem, there were several open problems related to the direct product, also called cross product, of finite automata, in particular for permutation and permutation-reset devices. We solve these left open problems and show (i) that for two permutation-reset automata of n- and m-states the whole range [1, nm] of state complexities is obtainable for the direct product, if the automata have at least a quaternary input alphabet, while (ii) for binary input alphabet this is not the case, and (iii) for the direct product of a permutation and a permutation-reset automaton the number alpha = 2 is always magic if n and m fulfill some property, i.e., cannot be obtained by the direct product of any automata of this kind. Moreover, our results can be seen as a generalization of previous results in [7] for the intersection operation on automata.