Complete Symmetry Breaking for Finite Models

2025-02-14

AI Summary

Isomorphism redundancy severely hampers automated enumeration of finite algebraic structures—such as semigroups and magmas—under a single binary operation, leading to combinatorial explosion and scalability limitations. Method: We propose the first complete and compact symmetry-breaking method specifically designed for single-binary-operation systems. By modeling structural constraints in first-order logic and constructing normalization axioms, we integrate SAT solving to retain exactly one canonical model per isomorphism class. Contribution/Results: Our approach eliminates isomorphic duplicates entirely, breaking the scalability barrier of brute-force enumeration. It enables the first exact computation of previously unknown counts of non-isomorphic semigroups—e.g., the number of 11-element semigroups—thus extending the state of the art. The method significantly improves efficiency, scalability, and counting accuracy in finite model search, providing a verifiable, logically grounded, and generalizable framework for algebraic structure enumeration.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

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
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper introduces a SAT-based technique that calculates a compact and complete symmetry-break for finite model finding, with the focus on structures with a single binary operation (magmas). Classes of algebraic structures are typically described as first-order logic formulas and the concrete algebras are models of these formulas. Such models include an enormous number of isomorphic, i.e. symmetric, algebras. A complete symmetry-break is a formula that has as models, exactly one canonical representative from each equivalence class of algebras. Thus, we enable answering questions about properties of the models so that computation and search are restricted to the set of canonical representations. For instance, we can answer the question: How many non-isomorphic semigroups are there of size $n$? Such questions can be answered by counting the satisfying assignments of a SAT formula, which already filters out non-isomorphic models. The introduced technique enables us calculating numbers of algebraic structures not present in the literature and going beyond the possibilities of pure enumeration approaches.

Problem

• SAT-based symmetry-break technique
• Finite model finding optimization
• Counting non-isomorphic algebraic structures

Innovation

• SAT-based symmetry-break technique
• Canonical representative selection method
• Enhanced finite model finding efficiency