Approximate polymorphisms of predicates
Yaroslav Alekseev, Yuval Filmus
TL;DR
The paper generalizes the BLR linearity test to approximate generalized polymorphisms of arbitrary predicates P ⊆ {0,1}^m under full-support distributions μ, proving that such near-polymorphisms are close to actual generalized polymorphisms when measured on μ’s marginals. The approach unifies and extends multiple results—linearity testing, Arrow-type theorems, intersecting families, and promise CSPs—via a combination of Jones regularity lemmas, It Ain’t Over Till It’s Over, and a two-step rounding scheme. It treats monotone predicates first (via a triangle-removal–style framework) and then handles general predicates by differentiating coordinates as flexible or inflexible, using two-step sampling to construct rounding that yields a generalized polymorphism. The work further develops regularity tools for general alphabets and provides open questions about extending the framework to all predicates and optimizing the ε–δ relationship.
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 $μ$ on $P$), then $f_1,\dots,f_m$ are close to a generalized polymorphism of $P$ (with respect to the marginals of $μ$). Our main result generalizes several results in the literature: linearity testing, quantitative Arrow theorems, approximate intersecting families, AND testing, and more generally $f$-testing.