Dyson-Schwinger equations in the theory of computation

TL;DR

This paper explores Dyson-Schwinger equations within algebraic structures like Hopf algebras, operads, and properads to model self-similarity in computation and analyze fundamental problems such as the halting problem.

Contribution

It introduces a novel algebraic framework for understanding self-similarity in computation using Dyson-Schwinger equations on advanced algebraic structures.

Findings

Dyson-Schwinger equations encode self-similarity in computational structures.

Application to flow charts models recursive and halting behaviors.

Provides new algebraic insights into the halting problem.

Abstract

Following Manin's approach to renormalization in the theory of computation, we investigate Dyson-Schwinger equations on Hopf algebras, operads and properads of flow charts, as a way of encoding self-similarity structures in the theory of algorithms computing primitive and partial recursive functions and in the halting problem.