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Topological Kleene Field Theories as a model of computation

Ángel González-Prieto, Eva Miranda, Daniel Peralta-Salas

TL;DR

This work proposes Topological Kleene Field Theory (TKFT), a framework in which computable functions are realized as the flow of volume-preserving vector fields on smooth bordisms, effectively providing a single-pass dynamical model of computation for partial recursive functions. The authors develop a constructive approach that encodes Turing machine computations inside 3D bordisms via Cantor-based tape encodings, basic read/write/shift bordisms, and a thickening process yielding a smooth, computable dynamics whose reaching function $Z_0(W,X)$ equals the target function $f$. They formalize this inside a 2-categorical setting, introducing 2-categories of partial/partial-recursive functions and of dynamical bordisms, and define TKFT as a monoidal subcategory whose functor to $\textup{PRF}$ is full and surjective on objects, thereby linking computation with topological field-theoretic structures. The results imply that the class of clean dynamical bordisms is Turing complete and that computation via dynamical systems inherits undecidability from the halting problem, while also opening avenues for speedups and beyond-Turing models through topological and categorical perspectives.

Abstract

In this article, we establish the foundations of a computational field theory, which we term Topological Kleene Field Theory (TKFT), inspired by Stephen Kleene's seminal work on partial recursive functions and drawing parallels with Topological Field Theory. Our central result shows that any computable function can be simulated by the flow on a smooth bordism of a vector field with good local properties, setting an alternative model of computation to Turing machines. We thus establish that a computable function can be fully realized within a single go of a dynamical system, differing from previous works where computation is encoded as an iterative process. The output of the computable function emerges directly, laying the groundwork for potential applications that accelerate the physical realization of computation.

Topological Kleene Field Theories as a model of computation

TL;DR

This work proposes Topological Kleene Field Theory (TKFT), a framework in which computable functions are realized as the flow of volume-preserving vector fields on smooth bordisms, effectively providing a single-pass dynamical model of computation for partial recursive functions. The authors develop a constructive approach that encodes Turing machine computations inside 3D bordisms via Cantor-based tape encodings, basic read/write/shift bordisms, and a thickening process yielding a smooth, computable dynamics whose reaching function equals the target function . They formalize this inside a 2-categorical setting, introducing 2-categories of partial/partial-recursive functions and of dynamical bordisms, and define TKFT as a monoidal subcategory whose functor to is full and surjective on objects, thereby linking computation with topological field-theoretic structures. The results imply that the class of clean dynamical bordisms is Turing complete and that computation via dynamical systems inherits undecidability from the halting problem, while also opening avenues for speedups and beyond-Turing models through topological and categorical perspectives.

Abstract

In this article, we establish the foundations of a computational field theory, which we term Topological Kleene Field Theory (TKFT), inspired by Stephen Kleene's seminal work on partial recursive functions and drawing parallels with Topological Field Theory. Our central result shows that any computable function can be simulated by the flow on a smooth bordism of a vector field with good local properties, setting an alternative model of computation to Turing machines. We thus establish that a computable function can be fully realized within a single go of a dynamical system, differing from previous works where computation is encoded as an iterative process. The output of the computable function emerges directly, laying the groundwork for potential applications that accelerate the physical realization of computation.

Paper Structure

This paper contains 15 sections, 13 theorems, 24 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The Church-Turing Thesis
  3. Turing Machines
  4. Partial Recursive Functions
  5. Turing Machines as finite state machines
  6. Dynamical computation
  7. Dynamical bordisms
  8. Cantor-based encodings
  9. Partial functions from dynamical bordisms
  10. Proof of the main theorem
  11. Reversible Turing machines
  12. The basic bordism
  13. Computational thickening
  14. Categorical interpretation: Topological Kleene Field Theories
  15. Conclusions

Key Result

Theorem A

For any partial recursive function $f: \mathbb{N} \dashrightarrow \mathbb{N}$, there exists a clean dynamical bordism $(W,X)$ between disks, such that $X$ is a volume-preserving vector field and $Z_0(W, X) = f$. Conversely, the reaching function of any clean dynamical bordism is a partial recursive

Figures (11)

  • Figure 1: Dynamical bordism with generated flow.
  • Figure 2: Thickening process to represent a partial recursive function as a dynamical bordism.
  • Figure 3: Simulation of a partial recursive function through a dynamical bordism, reproduced from Ramos_2025.
  • Figure 4: Finite state machine for the parity detector Turing Machine.
  • Figure 5: Different possibilities occurring in the definition of $\psi_{(W, X)}: C^n \dashrightarrow C^n$.
  • ...and 6 more figures

Theorems & Definitions (34)

  • Theorem A
  • Remark 2.1
  • Example 2.2
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Definition 3.4
  • Definition 3.5
  • Remark 3.6
  • Remark 3.7
  • ...and 24 more