Algebra and coalgebra of stream products

TL;DR

This paper explores the algebraic and coalgebraic structures of stream products, establishing a framework to reason about streams via polynomial representations and applying it to decision algorithms and differential equations.

Contribution

It introduces (F,G)-products on streams, linking polynomial and stream coalgebra, enabling algebraic reasoning and decision algorithms for stream equivalence.

Findings

Established a canonical transition function for (F,G)-products

Developed an algebraic decision algorithm for stream equivalence

Derived closed forms for generating functions and solutions of differential equations

Abstract

We study connections among polynomials, differential equations and streams over a field K, in terms of algebra and coalgebra. We first introduce the class of (F,G)-products on streams, those where the stream derivative of a product can be expressed as a polynomial of the streams themselves and their derivatives. Our first result is that, for every (F,G)-product, there is a canonical way to construct a transition function on polynomials such that the induced unique final coalgebra morphism from polynomials into streams is the (unique) K-algebra homomorphism -- and vice versa. This implies one can reason algebraically on streams, via their polynomial representation. We apply this result to obtain an algebraic-geometric decision algorithm for polynomial stream equivalence, for an underlying generic (F,G)-product. As an example of reasoning on streams, we focus on specific products (convolution, shuffle, Hadamard) and show how to obtain closed forms of algebraic generating functions of combinatorial sequences, as well as solutions of nonlinear ordinary differential equations.