AMR-RMA

Can nature compute, or are there intrinsic limits to what physical systems can predict and realize?

In 1936, Alan Turing established the undecidability of the halting problem, revealing a fundamental boundary of algorithmic reasoning. A natural question is whether such limits belong only to logic, or whether they are already encoded in the laws of the physical world.

In the 1960s, Stephen Smale gave chaos a precise geometric form through the horseshoe map, a structure in which trajectories encode symbolic sequences and dynamics acquire an informational content. In this sense, the horseshoe provides the shape of chaos. This naturally leads to a deeper question: what is the shape of the undecidable?

 Later, Christopher Moore introduced generalized shifts, extending symbolic dynamics to systems capable of universal computation and, thereby, undecidable behavior.

We present recent developments showing that such phenomena arise naturally in fluid dynamics and other physical systems. Using symplectic and cosymplectic geometry, we construct vector fields that are stationary solutions of the Euler and Navier–Stokes equations and that realize generalized shift dynamics, and thus universal computation. Classical billiard systems offer another setting in which simple mechanical configurations exhibit the same computational power.

A unifying conceptual framework is provided by Topological Kleene Field Theories, which interpret vector field flows as computational processes. This perspective explains how logical complexity can emerge from continuous dynamics and identifies the structural ingredients required to encode computation in physical systems once symbolic structures are present.

In celestial mechanics, Jürgen Moser showed that the Sitnikov restricted three-body problem exhibits shift dynamics. This raises a fundamental question: can more general n-body systems realize generalized shifts? If so, certain trajectories would exhibit provably undecidable long-term behavior. This leads to a striking open question: does the three-body problem, long understood as chaotic, in fact reach the threshold of computational universality?

This work is based on several collaborations, including ongoing joint work with Urs Frauenfelder, Ángel González-Prieto, and Daniel Peralta-Salas.