L _p distance and equivalence of probabilistic automata

This paper presents an exhaustive analysis of the problem of computing the L _p 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 L _p distance of two probabilistic automata for odd values of p. Similar techniques to those used for computing the L _p 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 A ₁ and A ₂ in time O(|Σ| (|A ₁| + |A ₂|) ³), a significant improvement over the previously best reported algorithm for this problem.