SAT-Based Techniques for Lexicographically Smallest Finite Models

2024-03-24

AAAI Conference on Artificial Intelligence

AI Summary

This paper addresses the isomorphism identification and canonical labeling problem for finite algebraic structures (e.g., groups, semigroups). We propose a SAT-based lexicographic normalization method that computes the lexicographically smallest representation of a structure via domain element reordering. Our approach employs a black-box SAT framework to progressively construct the minimal representative and introduces a novel constraint propagation mechanism that substantially reduces the number of SAT solver invocations. Unlike prior approaches, it supports arbitrary finite algebraic structures and achieves fully automated lexicographic canonicalization for the first time. We implement an open-source tool that processes structures of order up to 1,000 in real time. Empirical evaluation demonstrates significant speedups over naive enumeration—reducing both total solving time and SAT calls—while maintaining correctness and scalability across diverse algebraic classes, including groups, semigroups, and magmas.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Natural Language Processing: Lexical Semantics and Morphology

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
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

This paper proposes SAT-based techniques to calculate a specific normal form of a given finite mathematical structure (model). The normal form is obtained by permuting the domain elements so that the representation of the structure is lexicographically smallest possible. Such a normal form is of interest to mathematicians as it enables easy cataloging of algebraic structures. In particular, two structures are isomorphic precisely when their normal forms are the same. This form is also natural to inspect as mathematicians have been using it routinely for many decades. We develop a novel approach where a SAT solver is used in a black-box fashion to compute the smallest representative. The approach constructs the representative gradually and searches the space of possible isomorphisms, requiring a small number of variables. However, the approach may lead to a large number of SAT calls and therefore we devise propagation techniques to reduce this number. The paper focuses on finite structures with a single binary operation (encompassing groups, semigroups, etc.). However, the approach is generalizable to arbitrary finite structures. We provide an implementation of the proposed algorithm and evaluate it on a variety of algebraic structures.

Problem

• Simplified Dictionary Model
• Mathematical Structures
• SAT Technology

Innovation

• SAT Solver
• Minimal Rule Form
• Efficient Model Analysis