On generalized language equations

AbstractA system of generalized language equations over an alphabet A is a set of n equations in n variables: Xi = Gi(X1,..., Xn), i = 1,...,n, where the Gi are functions from [P(A*)]n into P(A*), i=1,..., n, P(A*) denoting the set of all languages over A. Furthermore the Gi are expressible in terms of set-operations, concatenations, and stars which involve the variable Xi as well as certain mixed languages. In this note we investigate existence and uniqueness of solutions of a certain subclass of generalized language equations. Furthermore we show that a solution is regular if all fixed languages are regular

Succinct representation of regular languages by boolean automata II

AbstractBoolean 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 ⩾ 1, there exists a boolean automaton Bn with n states such that the smallest deterministic automaton for the same language has 22n 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 ⩾ 1, there exists a reduced automaton Dn with n states such that the smallest boolean automaton accepting the same language has also n states