Journal of Physics: Complexity

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Journal of Physics: Complexity

What does it mean for a system to compute?

Abstract

Many real-world dynamic systems, both natural and artificial, are understood to be performing computations. For artificial dynamic systems, explicitly designed to perform computation-such as digital computers-by construction, we can identify which aspects of the dynamic system match the input and output of the computation that it performs, as well as the aspects of the dynamic system that match the intermediate logical variables of that computation. In contrast, in many naturally occurring dynamic systems that we understand to be computers, even though we neither designed nor constructed them-such as the human brain-it is not a priori clear how to identify the computation we presume to be encoded in the dynamic system. Regardless of their origin, dynamic systems capable of computation can, in principle, be mapped onto corresponding abstract computational machines that perform the same operations. In this paper, we begin by surveying a wide range of dynamic systems whose computational properties have been studied. We then introduce a very broadly applicable framework for identifying what computation(s) are emulated by a given dynamic system. After an introduction, we summarize key examples of dynamic systems whose computational properties have been studied. We then introduce a very broadly applicable framework that defines the computation performed by a given dynamic system in terms of maps between that system's evolution and the evolution of an abstract computational machine. We illustrate this framework with several examples from the literature, in particular discussing why some of those examples do not fully fall within the remit of our framework. We also briefly discuss several related issues, such as uncomputability in dynamic systems, and how to quantify the 'value of computation' in naturally occurring computers. We conclude with a discussion of some of the promising directions for future research.

One. Introduction

One point two. Constructed versus non-constructed computers

Two. Focus of this paper

Three. Examples of dynamic systems that implement computation

Four. Universality of emulating computational machines

Four point one. There are many dynamic systems that can be viewed as performing an infinite set of computations

Four point two. Formal definition of emulating a computational machine with a dynamic system

Four point four. Constraints on the computational power of the encoding and decoding maps

Five. Illustrations of the formal definition of dynamic systems emulating computational machines

Five point one. Billiard ball model of computer

Five point two. Computation with chemical reaction networks

Five point three. Fluid computers

Six. Uncomputability in dynamic systems

Seven. Value of computation

Eight. Future work

Data availability statement

Author contributions

Overview

The study examines what it means for dynamic systems, both natural and artificial, to perform computations. It aims to formalize how we can decode states of these systems into logical variables and discusses the implications for understanding non-constructed computers.

Key Points

1Dynamic systems, both natural and artificial, can be seen as performing computations
2A framework is proposed for identifying the computations of dynamic systems
3The distinction between constructed and non-constructed computers is explored
4Non-constructed computers can behave as different computations depending on decoding choices
5The paper discusses future research directions regarding computation in dynamic systems.

Details

Authors: David H Wolpert, Jan Korbel
Category: Physical Sciences

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