Automata, Power Series, and Coinduction: Taking Input Derivatives Seriously (Extended Abstract)

Abstract

Formal power series are functions σ: A^* → k from the set of words over some alphabet A to some semiring k. Examples include formal languages (k = {0; 1}) and power series in classical analysis (k=ℝ, viewing the elements of A as variables). Because of their relevance to many different scientific areas, both in mathematics and computer science, a large body of literature on power series exists. Most approaches to the subject are essentially algebraic. The aim of this paper is to show that it is worthwhile to view power series from a dual per- spective, called coalgebra (see [Rut96] for a general account). In summary, this amounts to supplying the set of all power series with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduction, which is the coalgebraic counterpart of induction.