The study of the connections between mathematical automata and formal logic is as old as theoretical computer science itself. In the founding paper of the subject, published in 1936, Turing showed how to describe the behavior of a universal computing machine with a formula of first-order predicate logic, and thereby concluded that there is no algorithm for deciding the validity of sentences in this logic. Research on the logical aspects of the theory of finite-state automata, which is the subject of this book, began in the early 1960's with the work of J. Richard Biichi on monadic second-order logic.