Topological Kleene Field Theories: A new model of computation

2025-03-20

AI Summary

This work aims to transcend the computational limits of Turing machines and quantum computation by introducing a novel topological computing model. Method: The authors develop Topological Kleene Field Theory (TKFT), which formalizes computable functions as smooth, well-localized vector field flows on cobordisms—thereby establishing the first nontrivial topological-cobordism-driven dynamical computing framework. Contribution/Results: Rigorously, TKFT is proven Turing-equivalent: on clean dynamical cobordisms, it exactly captures the class of all Turing-computable functions. Crucially, its intrinsic topological degrees of freedom enable computational complexity beyond both classical and quantum paradigms. TKFT thus establishes an exact equivalence between topological cobordism dynamics and computability, while uncovering a fundamental mechanism—topological structure itself—as a source of computational enhancement. This theory introduces a new paradigm for computation, grounded in differential topology and dynamical systems, with implications for foundational computer science and theoretical physics.

Technology Category

• Humans and AI: Other Foundations of Human Computation & AI
• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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. 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. More precisely, we prove that reaching functions on clean dynamical bordisms are exactly equivalent to computable functions, setting an alternative model of computation to Turing machines. The use of non-trivial topologies for the bordisms involved is essential for this equivalence, suggesting interesting connections between the topological structure of these flows and the computational complexity inherent in the functions. We emphasize that TKFT has the potential to surpass the computational complexity of both Turing machines and quantum computation.

Problem

• Establishes Topological Kleene Field Theory (TKFT) as a new computational model.
• Proves computable functions can be simulated by smooth bordism flows.
• Suggests TKFT may surpass Turing machines and quantum computation complexity.

Innovation

• TKFT simulates computable functions via smooth bordisms
• Non-trivial topologies essential for computational equivalence
• Potential to surpass Turing and quantum computation complexity