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This paper presents efficient algorithms for testing the finite, polynomial, and exponential ambiguity of finite automata with 茂戮驴-transitions. It gives an algorithm for testing the exponential ambiguity of an automaton Ain time $O(|A|_E^2)$, and finite or polynomial ambiguity in time $O(|A|_E^3)$, where |A|Edenotes the number of transitions of A. These complexities significantly improve over the previous best complexities given for the same problem. Furthermore, the algorithms presented are simple and based on a general algorithm for the composition or intersection of automata. We also give an algorithm to determine in time $O(|A|_E^3)$ the degree of polynomial ambiguity of a polynomially ambiguous automaton A. Finally, we present an application of our algorithms to an approximate computation of the entropy of a probabilistic automaton.