Approximate polymorphisms of predicates

TL;DR

This paper introduces a generalized concept of polymorphisms for predicates and demonstrates that functions approximately satisfying this property are close to true polymorphisms, unifying various results in property testing and combinatorics.

Contribution

It extends the theory of polymorphisms to approximate cases and connects multiple areas like linearity testing and Arrow theorems under a common framework.

Findings

Functions approximately satisfying polymorphism conditions are close to actual polymorphisms.

Unifies several results in property testing and combinatorics.

Provides a generalized approach to approximate polymorphisms across different problems.

Abstract

A generalized polymorphism of a predicate $P \subseteq \{0,1\}^m$ is a tuple of functions $f_1,\dots,f_m\colon \{0,1\}^n \to \{0,1\}$ 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.