Multicomputation with Numbers: The Case of Simple Multiway Systems

TL;DR

This paper explores simple integer iteration rules as models of multicomputation, revealing complex behaviors like emergent geometry and confluence issues, with extensions to non-integer rules and connections to physics and number theory.

Contribution

It introduces minimal multicomputation models using integer rules, analyzes their complex multiway graphs, and discusses generalizations and interdisciplinary connections.

Findings

Multiway graphs exhibit emergent geometric structures.

Confluence can be complex or fail in simple integer rules.

Extensions to non-integer functions broaden the scope of multicomputation.

Abstract

Integer iteration rules such as n |-> {a n + b, c n +d} are studied as minimal examples of the general process of multicomputation. Despite the simplicity of such rules, their multiway graphs can be complex, exhibiting, for example, emergent geometry and difficult questions of confluence. Generalizations to rules involving non-integers and other functions are also considered. Connections with physics and with various number-theoretic and other questions are made.