Branching Measures and Nearly Acyclic NFAs

To get a more comprehensive understanding of the amount of branching in computations of a nondeterministic finite automaton (NFA), we introduce and study the string path width and depth path width measures. For a given NFA, the string path width on a string w counts the number of all complete computations on w, and the depth path width on an integer l counts the number of complete computations on all strings of length l. We give an algorithm to decide the finiteness of the depth path width of an NFA. Deciding finiteness of string path width can be reduced to the corresponding question on ambiguity. An NFA is nearly acyclic if any computation can pass through at most one cycle. The class of nearly acyclic NFAs consists of exactly all NFAs with finite depth path width. Using this characterization we show that the finite depth path width of an m-state NFA over a k-letter alphabet is at most (k + 1)(m-1) and that this bound is tight. The nearly acyclic NFAs recognize exactly the class of constant density regular languages.