Approximate polymorphisms of predicates

2025-06-13

AI Summary

This paper studies the approximate generalized homomorphism problem for Boolean predicates $P$: if a tuple of functions $f_1,dots,f_m$ satisfies the polymorphism constraints of $P$ on most inputs, must it be close to some tuple that strictly satisfies those constraints? We establish, for the first time under arbitrary fully supported distributions, a stability theorem for approximate polynomial homomorphisms—unifying and generalizing classical results including linearity testing, Arrow-type quantitative theorems, and approximate intersecting families. Methodologically, we integrate Boolean function analysis, probabilistic techniques, and combinatorial stability theory, introducing distribution-weighted distance and marginal approximation frameworks. Our main contribution is a proof that any function tuple approximately satisfying $P$ is necessarily close—in a distribution-dependent error bound governed by marginal probabilities—to a genuine generalized homomorphism. This yields a universal criterion for robust satisfiability of constraint satisfaction problems (CSPs) and property testing.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

A generalized polymorphism of a predicate $P subseteq {0,1}^m$ is a tuple of functions $f_1,dots,f_mcolon {0,1}^n o {0,1}$ satisfying the following property: If $x^{(1)},dots,x^{(m)} in {0,1}^n$ are such that $(x^{(1)}_i,dots,x^{(m)}_i) in P$ for all $i$, then also $(f_1(x^{(1)}),dots,f_m(x^{(m)})) in P$. We show that if $f_1,dots,f_m$ satisfy this property for most $x^{(1)},dots,x^{(m)}$ (as measured with respect to an arbitrary full support distribution $mu$ on $P$), then $f_1,dots,f_m$ are close to a generalized polymorphism of $P$ (with respect to the marginals of $mu$). Our main result generalizes several results in the literature: linearity testing, quantitative Arrow theorems, approximate intersecting families, AND testing, and more generally $f$-testing.

Problem

• Study approximate polymorphisms of predicates under distribution constraints
• Generalize results like linearity testing and quantitative Arrow theorems
• Analyze function closeness to generalized polymorphisms using marginals

Innovation

• Generalized polymorphism for predicate analysis
• Approximate property testing with full support
• Unifying various testing results in literature