Algebraic series and context-free languages

Abstract

In this chapter we will discuss both the general theory of algebraic formal power series and its applications to context-free languages. Since many notions concerning series are very closely related to familiar notions concerning languages, we prefer to present the general theory in connection with applications rather than trying to separate strictly the two things. Perhaps the most significant single result established in this chapter is Shamir’s Theorem which gives a characterization similar to the one given in the Representation Theorem of Schutzenberger. Shamir’s Theorem can also be viewed as a formalized version of the intuitive ideas connected with the notion of a pushdown automaton. The reader will also notice that in many cases, for instance as regards sequences, the algebraic counterpart of the theory developed for rational series is missing.