Approximate polymorphisms of predicates

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.