Succinct Representation of Regular Languages by Boolean Automata II

Boolean automata are a generalization of finite automata in the sense that the next state (the result of the transition function, given a state and a letter) is not just a single state (deterministic automata) or a set of states (nondeterministic automata) but a boolean function of the states. Boolean automata accept precisely the regular languages; also, they correspond in a natural way to certain language equation involving complementation as well as to sequential networks. In a previous note we showed that for every n 2 1, there exists a boolean automaton B,, with n states such that the smallest deterministic automaton for the same language has 22” states. In the present note we will show a precisely attainable lower bound on the succinctness of representing regular languages by boolean automata; namely, we will show that, for every n 3 1, there exists a reduced automaton D, with n states such that the smallest boolean automaton accepting the same language has also n states.