More on the Descriptional Complexity of Products of Finite Automata

Abstract

We investigate the descriptional complexity of the $\nu _i$- and $\alpha _i$-products with $0\le i\le 2$ of two automata, for reset, permutation, permutation-reset, and finite automata in general. This is a continuation of the recent studies on the state complexity of the well-known cascade product undertaken in [7, 8]. Here we show that in almost all cases, except for the direct product ($\nu _0$) and the cascade product ($\alpha _0$) for certain types of automata operands, the whole range of state complexities, namely the interval [1, nm], where n is the state complexity of the left operand and m that of the right one, is attainable. To this end we prove a simulation result on products of automata that allows us to reduce the products and automata in question to the $\nu _0$, $\alpha _0$, and a double sided $\alpha _0$-product.