Measures of Nondeterminism in Finite Automata

While deterministic nite automata seem to be well understood, surprisingly many important problems concerning nondeterministic nite automata (nfa's) remain open. One such problem area is the study of di erent measures of nondeterminism in nite automata and the estimation of the sizes of minimal nondeterministic nite automata. In this paper the concept of communication complexity is applied in order to achieve progress in this problem area. The main results are as follows: 1. Deterministic communication complexity provides lower bounds on the size of unambiguous nfa's. Applying this fact, the proofs of several results about nfa's with limited ambiguity can be simpli ed. 2. For an nfa A we consider the complexity measures adviceA(n) as the number of advice bits, ambigA(n) as the number of accepting computations, and leafA(n) as the number of computations for worst case inputs of size n. These measures are correlated as follows (assuming that the nfa A is minimal): adviceA(n); ambigA(n) leafA(n) O(adviceA(n) ambigA(n)). 3. leafA(n) is always either a constant, between linear and polynomial in n, or exponential in n. 4. There is a family of languagesKONk2 with an exponential size gap between nfa's with polynomial leaf number/ambiguity and nfa's with ambiguity k. This partially provides an answer to the open problem posed by Ravikumar and Ibarra [SIAM J. Comput. 18 (1989), 1263{1282], and Hing Leung [SIAM J. Comput. 27 (1998), 1073{1082].