Physically-Relativized Church-Turing Hypotheses

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.

💡 Research Summary

The paper “Physically‑Relativized Church‑Turing Hypotheses” re‑examines the Church‑Turing Hypothesis (CTH) by anchoring it to specific physical theories rather than to an ill‑defined notion of “nature”. The author argues that the traditional formulation of CTH—“every function that can be computed in the physical world is computable by a Turing machine, and efficiently computable functions coincide with polynomial‑time Turing‑computable functions”—is ambiguous because it does not specify which physical laws are being invoked. To resolve this, the paper proposes a family of relativized hypotheses, denoted CTH Φ, each tied to a concrete physical theory Φ (e.g., Newtonian mechanics, quantum mechanics, general relativity).

The core of the proposal is a formal framework that treats a physical theory as a computational model. A theory Φ is described by three components: (1) its basic entities (particles, fields, spacetime points), (2) its governing laws (differential equations, operator algebras, etc.), and (3) its measurement and input/output protocols. By making these elements explicit, one can define precisely what it means for a physical system described by Φ to compute a function, and then ask whether that function can be simulated by a Turing machine (or by a more powerful abstract machine).

The paper surveys several “hyper‑computation” proposals—relativistic spacetimes that allow infinite proper time in finite external time, quantum superposition exploiting infinite Hilbert‑space dimensions, non‑collision singularities in Newtonian many‑body systems, and idealized optical setups with perfect mirrors. In each case the author shows that the claimed computational advantage relies on idealized assumptions that are not part of any empirically validated version of the underlying theory. Consequently, the alleged violations of CTH are better understood as challenges to the realism of the physical assumptions rather than as genuine counter‑examples to the hypothesis.

A philosophical contribution is the adoption of a constructivist interpretation of existence within a physical theory: asserting that a system “exists” must be accompanied by an explicit construction procedure, a preparation protocol for inputs, and a concrete read‑out mechanism for outputs. This requirement turns the question “Does nature admit a system more powerful than a Turing machine?” into a well‑posed scientific problem that can be answered by examining the feasibility of the required constructions within Φ.

The paper then outlines a research program based on CTH Φ. It illustrates how, for each theory, one can map its computational capabilities onto standard complexity classes. In celestial mechanics, Newtonian gravitation permits polynomial‑time prediction of planetary orbits but may require exponential resources for general N‑body simulations; in geometric optics, idealized ray tracing corresponds to a deterministic finite automaton, while wave optics introduces continuous‑state dynamics akin to analog computation; in quantum mechanics, bounded‑qubit quantum circuits are polynomially simulable by classical Turing machines, whereas unrestricted infinite‑dimensional superpositions would place the theory beyond the classical hierarchy. By systematically comparing these classes, researchers can identify precisely where a given physical theory extends, matches, or falls short of Turing‑computability.

Overall, the work reframes the Church‑Turing Hypothesis from a vague philosophical claim into a family of concrete, theory‑relative statements. This relativization clarifies longstanding debates, isolates the role of physical assumptions in alleged hyper‑computation, and provides a structured pathway for interdisciplinary research between physics and theoretical computer science. It suggests that the true scientific content of CTH lies not in a universal axiom but in the systematic analysis of each physical theory’s computational limits.