A Formal Proof of Complexity Bounds on Diophantine Equations

2025-05-22

AI Summary

This work addresses the decidability of integer solvability for Diophantine equations under bounded complexity—specifically, with constraints on the number of variables ν and the degree δ—and investigates whether a universal parameter pair (ν, δ) exists such that all Diophantine sets can be encoded within these bounds, thereby implying undecidability for the corresponding equation class. Method: We formalize and mechanize multivariate polynomial algebra, classical number-theoretic results (including Matiyasevich’s theorem and related reductions), and supporting metaprogramming infrastructure in Isabelle/HOL. Contribution/Results: We establish, for the first time, a formally verified nontrivial minimal universal parameter pair (ν, δ) = (9, 1.6 × 10⁴⁷), proving the undecidability of integer Diophantine equations under bounded complexity. Our development constitutes a reusable, modular formal framework for Diophantine equations; it significantly extends the Archive of Formal Proofs (AFP) polynomial library and advances a new paradigm of collaborative co-evolution between mathematicians and interactive theorem provers.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $ u$ and the degree $delta$. If every Diophantine set can be represented within the bounds $( u, delta)$, 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.

Problem

• Formal proof of complexity bounds on Diophantine equations
• Verification of universal pairs for undecidable Diophantine classes
• Extension of Isabelle/HOL for Diophantine equation theory

Innovation

• Formal verification of universal Diophantine equations
• Extends Isabelle AFP with multivariate polynomials
• Metaprogramming for efficient polynomial handling