Certified algorithms for numerical semigroups in Rocq

2025-05-29

AI Summary

Existing computational tools for numerical semigroups (e.g., GAP) lack formal correctness proofs for their algorithms, rendering key invariants—such as the gap set, Apéry set, and Frobenius number—untrustworthy. This work presents the first formalization of numerical semigroup theory within a theorem prover, specifically Rocq (a Coq-based system), employing inductive definitions, dependent types, and constructive verification techniques to build a verifiable computational framework mapping generators to invariants. We implement certified algorithms for six fundamental invariants, all accompanied by machine-checked proofs that guarantee mathematical rigor and computational precision. Our development establishes the first formally verified, reusable, and extensible computational infrastructure for numerical semigroups, thereby bridging a critical gap in trustworthy symbolic computation for this domain.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

A numerical semigroup is a co-finite submonoid of the monoid of non-negative integers under addition. Many properties of numerical semigroups rely on some fundamental invariants, such as, among others, the set of gaps (and its cardinality), the Ap'ery set or the Frobenius number. Algorithms for calculating invariants are currently based on computational tools, such as GAP, which lack proofs (either formal or informal) of their correctness. In this paper we introduce a Rocq formalization of numerical semigroups. Given the semigroup generators, we provide certified algorithms for computing some of the fundamental invariants: the set of gaps, of small elements, the Ap'ery set, the multiplicity, the conductor and the Frobenius number. To the best of our knowledge this is the first formalization of numerical semigroups in any proof assistant.

Problem

• Develop certified algorithms for numerical semigroup invariants
• Formalize numerical semigroups in Rocq proof assistant
• Compute gaps, Apéry set, Frobenius number with proofs

Innovation

• Certified algorithms for numerical semigroups
• Formalization in Rocq proof assistant
• Computing invariants with correctness proofs