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Computer Science > Logic in Computer Science

arXiv:2505.16963 (cs)
[Submitted on 22 May 2025]

Title:A Formal Proof of Complexity Bounds on Diophantine Equations

Authors:Jonas Bayer, Marco David
View a PDF of the paper titled A Formal Proof of Complexity Bounds on
Diophantine Equations, by Jonas Bayer and 1 other authors
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    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.

Comments: 16 pages, 1 figure
Subjects: Logic in Computer Science (cs.LO); Number Theory (math.NT)
Cite as:  arXiv:2505.16963 [cs.LO]
          (or arXiv:2505.16963v1 [cs.LO] for this version)
          https://doi.org/10.48550/arXiv.2505.16963
          Focus to learn more
          arXiv-issued DOI via DataCite (pending registration)

Submission history

From: Marco David [view email]
[v1] Thu, 22 May 2025 17:45:33 UTC (229 KB)
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