Proving at Scale for Universal Algebra

2026-10-01

AI Summary

This study addresses the long-standing open problem of deciding the finite basis property for identities of finite semigroups, a challenge further compounded by the substantial human effort required for large-scale formal verification. To overcome these bottlenecks, this work introduces SemiBase, a novel automated proof framework that integrates large language model (LLM) guidance with formal verification. Specifically, it employs a multi-agent collaborative architecture to drive LLM-based proof search and reconstruct proof paths, which are subsequently subjected to rigorous auditing by the Lean theorem prover kernel. As a result, this research successfully achieves the formal certification of identity bases for all semigroups of order six and below, establishing the first machine-verified catalog of algebraic classifications. Ultimately, this work provides a new paradigm for trustworthy computation over large-scale algebraic structures.

Abstract

We introduce SemiBase, a project that computes and formally certifies finite identity bases for small semigroups. Deciding finite basability is undecidable for finite algebras and remains open for finite semigroups. The task requires a proof that a candidate basis is complete, or a proof that none exists, rather than a single first-order validity query. LLM-guided agents search for these proofs; a referee agent rebuilds them from source, and the Lean kernel checks the resulting corpus in a final audit. Humans choose targets and approve final outcomes. We certify every semigroup of order at most 6: all 1309 semigroups of order at most 5 and all 15973 of order 6, including proofs that the four known nonfinitely based semigroups have no finite basis. The bases for order 6 define 505 distinct varieties, whose inclusion order Vampire determines except for four pairs. The resulting catalogue is a machine-checked account of results scattered across the literature and a tested foundation for order 7.

Problem

• finite identity bases
• finite basability
• semigroups
• universal algebra

Innovation

• Formal verification
• LLM-guided agents
• Finite identity bases
• Lean kernel
• Automated theorem proving