Results on the Average State and Transition Complexity of Finite Automata Accepting Finite Languages

We investigate the average-case state and transition complexity of deterministic and nondeterministic finite automata, when choosing a finite language of given maximum word length n uniformly at random. The case where all words are of equal length is also taken into account. It is shown that almost all deterministic finite automata accepting finite languages over a binary input alphabet have state complexity Θ( n n ). Moreover, we develop a framework that allows us to investigate the average-case complexity of operations like union, intersection, complementation, and reversal on finite languages. Nondeterministic finite automata are shown to perform better than deterministic ones, namely their state complexity is in Θ( √ 2) on the average. Interestingly, in both cases the aforementioned bounds are asymptotically like in the worst case. However, the nondeterministic transition complexity is shown to be again Θ( n n ). The case of unary finite languages is also considered.