Coinductive Streams in Monoidal Categories

TL;DR

This paper introduces monoidal streams, a generalization of causal stream functions in symmetric monoidal categories, providing a new semantic framework for dataflow programming with process theories and feedback structures.

Contribution

It extends the concept of streams to arbitrary symmetric monoidal categories, enabling semantics for dataflow languages with complex process interactions.

Findings

Monoidal streams form a feedback monoidal category.

Coinductive string diagrams facilitate reasoning about feedback monoidal categories.

Semantics for a stochastic dataflow language are developed.

Abstract

We introduce monoidal streams. Monoidal streams are a generalization of causal stream functions, which can be defined in cartesian monoidal categories, to arbitrary symmetric monoidal categories. In the same way that streams provide semantics to dataflow programming with pure functions, monoidal streams provide semantics to dataflow programming with theories of processes represented by a symmetric monoidal category. Monoidal streams also form a feedback monoidal category. In the same way that we can use a coinductive stream calculus to reason about signal flow graphs, we can use coinductive string diagrams to reason about feedback monoidal categories. As an example, we study syntax for a stochastic dataflow language, with semantics in stochastic monoidal streams.