Topological Kleene Field Theories as a model of computation

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.