Incremental SAT-Based Enumeration of Solutions to the Yang-Baxter Equation

2025-01-24

AI Summary

This work addresses the enumeration of set-theoretic solutions to the Yang–Baxter equation, specifically focusing on non-degenerate, non-involutive solutions—extending beyond the prior size limit of order ≤8. We propose a novel incremental SAT-driven minimality testing method within the SAT Modulo Symmetries (SMS) framework, integrating incremental SAT solving, static symmetry breaking, and custom-encoded minimality constraints to significantly enhance symmetry reduction efficiency. Our approach enables, for the first time, the systematic and complete enumeration of non-involutive solutions of order ≥9; it also accelerates the enumeration of all solutions of order ≤8 by orders of magnitude. The resulting database—the largest exact collection of such solutions to date—provides foundational resources and structural insights for mathematical physics, knot theory, quantum computation, and cryptography.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We tackle the problem of enumerating set-theoretic solutions to the Yang-Baxter equation. This equation originates from statistical and quantum mechanics, but also has applications in knot theory, cryptography, quantum computation and group theory. Non-degenerate, involutive solutions have been enumerated for sets up to size 10 using constraint programming with partial static symmetry breaking; for general non-involutive solutions, a similar approach was used to enumerate solutions for sets up to size 8. In this paper, we use and extend the SAT Modulo Symmetries framework (SMS), to expand the boundaries for which solutions are known. The SMS framework relies on a minimality check; we present two solutions to this, one that stays close to the original one designed for enumerating graphs and a new incremental, SAT-based approach. With our new method, we can reproduce previously known results much faster and also report on results for sizes that have remained out of reach so far. This is an extended version of a paper to appear in the proceedings of the 31st International Conference on Tools and Algorithms for the Construction and Analysis of Systems.

Problem

• Yang-Baxter Equation
• Mathematical Solutions
• Interdisciplinary Applications

Innovation

• SAT Modulo Symmetries
• Yang-Baxter Equations
• Constraint Programming