Elementary Complexity and von Neumann Algebras

TL;DR

This paper explores how von Neumann algebra concepts, particularly the hyperfinite II_1 factor, can model classical computation and complexity, linking operator algebras with computational complexity theory.

Contribution

It introduces a novel approach to understanding classical computation through the lens of von Neumann algebras, especially the hyperfinite II_1 factor, connecting operator algebras with complexity.

Findings

A construction of an implicit complexity model using von Neumann algebra concepts.

A new perspective on the relation between computation and combinatorial aspects in operator algebras.

Insights into how discrete invariants in operator algebras relate to computational complexity phenomena.

Abstract

In this paper, we show how a construction of an implicit complexity model can be implemented using concepts coming from the core of von Neumann algebras. Namely, our aim is to gain an understanding of classical computation in terms of the hyperfinite $\mathrm{II}_1$ factor, starting from the class of Kalmar recursive functions. More methodologically, we address the problem of finding the right perspective from which to view the new relation between computation and combinatorial aspects in operator algebras. The rich structure of discrete invariants may provide a mathematical setting able to shed light on some basic combinatorial phenomena that are at the basis of our understanding of complexity.