Abstract
What are the limits of computation? Can physical systems compute?
Cris Moore associated universal Turing machines with transformations of the square Cantor set. Using Poincaré sections, we can extend this to dimension 3. These constructions gain significance when connected with a contact geometrical structure, as they "mirror" specific solutions of the Euler equation. Consequently, a universal Turing machine capable of simulating fluid motions emerges from this framework, providing a physical system capable of performing computations. The undecidability of the halting problem, demonstrated by Alan Turing, implies the existence of undecidable fluid paths, introducing a new notion of "chaos" in the logical sense ([1] and [2]).
This procedure of "contact-ing" 2D Turing complete systems associates fluid computers with transformations of the Cantor set which we call a "flubit." In joint work in progress with Ángel González and Daniel Peralta we refine this construction to build a hybrid machine where the basic units of computation are flubits. The way to assemble these ·pieces" is inspired by Feynman and can be formalized as a Topological Field Theory. We call it TKFT (Topological Kleene Field Theory) in honor of Stephen Kleene.
Will this hybrid machine challenge quantum supremacy?
[1] Robert Cardona, Eva Miranda, Daniel Peralta-Salas and Fran Presas, Constructing Turing complete Euler flows in dimension 3, Proc. Natl. Acad. Sci. USA 118 (2021), no. 19, Paper No. 2026818118, 9 pp
[2] Renzo Bruera, Robert Cardona, Eva Miranda, Daniel Peralta-Salas, Topological entropy of Turing complete dynamics, arXiv:2404.07288
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