A coinductive calculus of streams

Abstract

We develop a coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers). The main ingredient is the notion of stream derivative, which can be used to formulate both coinductive proofs and definitions. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics.

Well-Definedness of Streams by Termination

“…By Theorem 2 we can conclude it is the only one, since for R d = ∅ and R s consisting of the above four rules, the resulting TRS Obs(R s ) is terminating as can be proved by AProVE [4] or TTT2 [7]. Both [3] and [11] fail to prove that the identity is the only stream function satisfying the equation for f . By essentially choosing Obs(R s ) as the input and adding information about special contexts, the tool Circ [8] is able to prove that f is the identity.…”

“…Proving well-definedness in stream specification is closely related to proving equality of streams. A standard approach for this is co-induction [11]: two streams or stream functions are equal if a bisimulation can be found between them. Finding such an arbitrary bisimulation is a hard problem in the general setting, but restricting to circular co-induction [6] finding this automatically is tractable.…”

Left-Handed Completeness

“…Silva [16] calls the following result the fundamental theorem in analogy to a similar result proved for infinite streams by Rutten [14], closely related to the fundamental theorem of calculus. It is fundamental in the sense that it connects the differential structure, given by δ a and ε, with the axioms of LKA.…”

Presenting Distributive Laws

“…We illustrate this with an example. [30,31] to define streams and stream operations. Here, the behaviour functor is F X = R×X whose final coalgebra R ω , ζ consists of streams over the real numbers together with the map ζ(σ) = σ(0), σ ′ which maps a stream σ to its initial value σ(0) and derivative σ ′ .…”