Physically-Relativized Church-Turing Hypotheses

TL;DR

This paper formalizes the Church-Turing Hypothesis relative to specific physical theories, clarifying misconceptions and providing a framework that combines physical structuralism with complexity theory, with applications in computational physics.

Contribution

It introduces a formal framework to study the Church-Turing Hypothesis relative to physical theories, improving clarity and rigor in the field.

Findings

Examples of computability in physics

Complexity results in physical models

Clarification of misconceptions about CTH

Abstract

We turn `the' Church-Turing Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and well-defined scientific problem(s): Examining recent controversies, and causes for misunderstanding, concerning the state of the Church-Turing Hypothesis (CTH), suggests to study the CTH relative to an arbitrary but specific physical theory--rather than vaguely referring to ``nature'' in general. To this end we combine (and compare) physical structuralism with (models of computation in) complexity theory. The benefit of this formal framework is illustrated by reporting on some previous, and giving one new, example result(s) of computability and complexity in computational physics.