Growth Functions, Rates and Classes of String-Based Multiway Systems

TL;DR

This paper analyzes the growth behavior of string-based multiway systems within the Wolfram Physics framework, revealing their diverse computational capabilities and proposing a classification based on growth rates.

Contribution

It introduces the concepts of growth functions, rates, and classes for multiway systems, establishing their bounds and diversity, and develops a classification scheme based on growth properties.

Findings

Multiway systems can grow slower than all computable functions but not faster than exponential functions.

Multiway growth functions are computationally diverse, capable of approximating various mathematical functions.

Methods are provided to construct multiway systems with specific growth functions.

Abstract

In context of the Wolfram Physics Project, a certain class of abstract rewrite systems known as "multiway systems" have played an important role in discrete models of spacetime and quantum mechanics. However, as abstract mathematical entities, these rewrite systems are interesting in their own right. This paper undertakes the effort to establish computational properties of multiway systems. Specifically, we investigate growth rates and growth classes of string-based multiway systems. After introducing the concepts of "growth functions", "growth rates" and "growth classes" to quantify a system's state-space growth over "time" (successive steps of evolution) on different levels of precision, we use them to show that multiway systems can, in a specific sense, grow slower than all computable functions while never exceeding the growth rate of exponential functions. In addition, we start developing a classification scheme for multiway systems based on their growth class. Furthermore, we find that multiway growth functions are not trivially regular but instead "computationally diverse", meaning that they are capable of computing or approximating various commonly encountered mathematical functions. We discuss several implications of these properties as well as their physical relevance. Apart from that, we present and exemplify methods for explicitly constructing multiway systems to yield desired growth functions.