SAT-Based Techniques for Lexicographically Smallest Finite Models

TL;DR

This paper introduces SAT-based methods to compute lexicographically smallest normal forms of finite structures, aiding in structure cataloging and isomorphism detection, with an implementation tested on algebraic structures.

Contribution

It presents a novel SAT-based approach for computing minimal normal forms of finite structures, including propagation techniques to reduce SAT calls.

Findings

Effective in computing normal forms for various algebraic structures

Reduces the number of SAT calls through propagation techniques

Implementation demonstrates practical applicability

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.