Rational streams coalgebraically

TL;DR

This paper provides a coalgebraic framework to understand rational streams over a field, demonstrating their equivalence across four different finite representations.

Contribution

It introduces a uniform coalgebraic proof showing the equivalence of four notions of rational stream representability, unifying various models.

Findings

Equivalence of four notions of rational stream representation

Elementary proof leveraging the finality of the set of streams

Unified coalgebraic perspective on rational streams

Abstract

We study rational streams (over a field) from a coalgebraic perspective. Exploiting the finality of the set of streams, we present an elementary and uniform proof of the equivalence of four notions of representability of rational streams: by finite dimensional linear systems; by finite stream circuits; by finite weighted stream automata; and by finite dimensional subsystems of the set of streams.