Representations of Stream Processors Using Nested Fixed Points

TL;DR

This paper introduces a novel framework for representing continuous functions on infinite streams using nested fixed points, combining well-founded and non-wellfounded trees to model discrete and stream-valued functions.

Contribution

It presents a unified approach to representing stream processors with nested fixed points, extending existing theories with coinductive and inductive structures.

Findings

Representations use well-founded trees for discrete functions

Non-wellfounded trees model stream-valued functions coinductively

An operation for composing continuous functions is defined

Abstract

We define representations of continuous functions on infinite streams of discrete values, both in the case of discrete-valued functions, and in the case of stream-valued functions. We define also an operation on the representations of two continuous functions between streams that yields a representation of their composite. In the case of discrete-valued functions, the representatives are well-founded (finite-path) trees of a certain kind. The underlying idea can be traced back to Brouwer's justification of bar-induction, or to Kreisel and Troelstra's elimination of choice-sequences. In the case of stream-valued functions, the representatives are non-wellfounded trees pieced together in a coinductive fashion from well-founded trees. The definition requires an alternating fixpoint construction of some ubiquity.