Succinct representation of regular languages by boolean automata

Boolean automata are a generalization of finite automata in the sense that the ‘next state’, i.e. the result of the transition function given a state and a letter, is not just a single state (deterministic automata) or a union of states (nondeterministic automata) but a boolean function of states. Boolean automata accept precisely regular languages; furthermore they correspond in a natural way to certain language equations as well as to sequential networks. We investigate the succinctness of representing regular languages by boolean automata. In particular, we show that for every deterministic automaton A with m states there exists a boolean automaton with [log₂m] states which accepts the reverse of the language accepted by A (m≥1). We also show that for every n≥1 there exists a boolean automation with n states such that the smallest deterministic automaton accepting the same language has 2⁽²ⁿ⁾ states; moreover this holds for an alphabet with only two letters.