Structures preserved by primitive actions of $S_omega$

2025-01-07

AI Summary

This paper classifies structures $A$ preserved by primitive actions of the countable symmetric group $S_omega$ and analyzes the computational complexity of their constraint satisfaction problems (CSPs). The central question is whether such structures necessarily exhibit a dichotomy: either they primitively positively interpret all finite structures (“expressively complete”), or they admit a minimal algebraic description—characterized by binary polymorphisms satisfying an antisymmetry condition and the structure’s automorphism group. To resolve this, we establish the first complete dichotomy classification for $S_omega$-primitive structures, innovatively combining Johnson graph reductions, homogeneous Ramsey expansions, and analysis of maximal closed subgroups of $S_omega$. Our main results are: (i) For every such $A$, $operatorname{CSP}(A)$ is either in P or NP-complete—no intermediate complexity occurs; (ii) We fully characterize the algebraic structure of $A$, revealing a precise symmetry constraint linking its binary polymorphisms and automorphism group.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesKnowledge Representation and Reasoning: Computational Complexity of ReasoningSearch and Optimization: Combinatorial Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Applications of cryptographySemantics 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

Abstract

We present a dichotomy for structures $A$ that are preserved by primitive actions of $S_{omega} = ext{Sym}({mathbb N})$: either such a structure interprets all finite structures primitively positively, or it is of a very simple form and in particular has a binary polymorphism $f$ and an automorphism $alpha$ satisfying $f(x,y) = alpha(f(y,x))$. It is a consequence of our results that the constraint satisfaction problem for $A$ is in P or NP-complete. To prove our result, we study the first-order reducts of the Johnson graph $J(k)$, for $k geq 2$, whose automorphism group $G$ equals the action of $S_{omega}$ on the set $V$ of $k$-element subsets of $mathbb N$. We use the fact that $J(k)$ has a finitely bounded homogeneous Ramsey expansion and that $G$ is a maximal closed subgroup of $ ext{Sym}(V)$.

Problem

• Binary Property
• Structural Simplicity
• Computational Complexity

Innovation

• Dichotomy for A-structures
• Exploration of Johnson graphs
• Transformation group identification