LP Distance and Equivalence of Probabilistic Automata

This paper presents an exhaustive analysis of the problem of computing the Lp distance of two probabilistic automata. It gives efficient exact and approximate algorithms for computing these distances for p even and proves the problem to be NP-hard for all odd values of p, thereby completing previously known hardness results. It further proves the hardness of approximating the Lp distance of two probabilistic automata for odd values of p. Similar techniques to those used for computing the Lp distance also yield efficient algorithms for computing the Hellinger distance of two unambiguous probabilistic automata both exactly and approximately. A problem closely related to the computation of a distance between probabilistic automata is that of testing their equivalence. This paper also describes an efficient algorithm for testing the equivalence of two arbitrary probabilistic automata A1 and A2 in time O(|Σ| (|A1|+ |A2|)), a significant improvement over the previously best reported algorithm for this problem.