Inequality of finite behaviors of rational weight finite automata is in R

Inequivalence of finite automata accepting finite languages over a non-unary alphabet is NP-complete. However, the inequality of their behaviors does not appear to have been carefully investigated. In the simplest case, the behavior of a finite automaton is the formal series f such that the coefficient f(w) of a word w is the number of distinct accepting computations on w. This notion will be generalized in the paper to finite automata with rational weights. The main result is that inequality of rational weight finite automata with finite behaviors is in R, random polynomial time.