Boolean Circuit Complexity of Regular Languages

In this paper we define a new descriptional complexity measure for Deterministic Finite Automata, BC-complexity, as an alternative to the state complexity. We prove that for two DFAs with the same number of states BC-complexity can differ exponentially. In some cases minimization of DFA can lead to an exponential increase in BC-complexity, on the other hand BC-complexity of DFAs with a large state space which are obtained by some standard constructions (determinization of NFA, language operations), is reasonably small. But our main result is the analogue of the "Shannon effect" for finite automata: almost all DFAs with a fixed number of states have BC-complexity that is close to the maximum.