Context-free recognition with weighted automata

We introduce the definition of language recognition with weighted automata, a generalization of the classical definition of recognition with unweighted acceptors. We show that, with our definition of recognition, weighted automata can be used to recognize a class of languages that strictly includes regular languages. The class of languages accepted depends on the weight set which has the algebraic structure of a semiring. We give a generic linear time algorithm for recognition with weighted automata and describe examples with various weight sets illustrating the recognition of several classes of context-free languages. We prove, in particular, that the class of languages equivalent to the language of palindromes can be recognized by weighted automata over the (+, ·)-semiring, and that the class of languages equivalent to the Dyck language of first order D′₁* can be recognized by weighted automata over the real tropical semiring. We also prove that weighted automata over the real tropical semiring can be used to recognize regular expressions.