A Formal Proof of Complexity Bounds on Diophantine Equations

TL;DR

This paper formalizes a construction of Diophantine equations with bounded complexity in Isabelle/HOL, advancing the understanding of undecidability in number theory and demonstrating the synergy between formal verification and mathematical research.

Contribution

It provides a formal proof of the main construction for universal pairs of Diophantine equations with bounded complexity, extending Isabelle's capabilities in number theory formalization.

Findings

Formal verification of the construction of universal pairs.

Extension of Isabelle AFP with multivariate polynomial formalization.

Development of infrastructure for handling complex polynomial definitions.

Abstract

We present a universal construction of Diophantine equations with bounded complexity in Isabelle/HOL. This is a formalization of our own work in number theory. Hilbert's Tenth Problem was answered negatively by Yuri Matiyasevich, who showed that there is no general algorithm to decide whether an arbitrary Diophantine equation has a solution. However, the problem remains open when generalized to the field of rational numbers, or contrarily, when restricted to Diophantine equations with bounded complexity, characterized by the number of variables $\nu$ and the degree $\delta$. If every Diophantine set can be represented within the bounds $(\nu, \delta)$, we say that this pair is universal, and it follows that the corresponding class of equations is undecidable. In a separate mathematics article, we have determined the first non-trivial universal pair for the case of integer unknowns. In this paper, we contribute a formal verification of the main construction required to establish said universal pair. In doing so, we markedly extend the Isabelle AFP entry on multivariate polynomials, formalize parts of a number theory textbook, and develop classical theory on Diophantine equations in Isabelle. Additionally, our work includes metaprogramming infrastructure designed to efficiently handle complex definitions of multivariate polynomials. Our mathematical draft has been formalized while the mathematical research was ongoing, and benefitted largely from the help of the theorem prover. We reflect how the close collaboration between mathematician and computer is an uncommon but promising modus operandi.