Coinductive properties of Lipschitz functions on streams

TL;DR

This paper explores the hierarchical structure of Lipschitz functions on streams, demonstrating their closure under coiterative constructions and constructing new stream coalgebras, including applications to the Collatz function.

Contribution

It introduces a hierarchical framework for Lipschitz functions on streams and proves their closure properties under coiterative processes, enabling new coalgebra constructions.

Findings

Sets of non-expanding and contractive functions are closed under coiteration.

Constructed new final stream coalgebras over finite alphabets.

Applied the framework to the 2-adic extension of the Collatz function.

Abstract

A simple hierarchical structure is imposed on the set of Lipschitz functions on streams (i.e. sequences over a fixed alphabet set) under the standard metric. We prove that sets of non-expanding and contractive functions are closed under a certain coiterative construction. The closure property is used to construct new final stream coalgebras over finite alphabets. For an example, we show that the 2-adic extension of the Collatz function and certain variants yield final bitstream coalgebras.