Behavioural differential equations: a coinductive calculus of streams, automata, and power series

We present a theory of streams (infinite sequences), automata and languages, and formal power series, in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This coalgebraic perspective leads to a unified theory, in which the observation that each of the aforementioned sets carries a so-called final automaton structure, plays a central role. Finality forms the basis for both definitions and proofs by coinduction, the coalgebraic counterpart of induction. Coinductive definitions take the shape of what we have called behavioural differential equations, after Brzozowski's notion of input derivative. A calculus is developed for coinductive reasoning about all of the afore mentioned structures, closely resembling calculus from classical analysis.