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.

翻译：一个谓词 $P \\subseteq \\{0,1\\}^m$ 的广义多态性是指满足以下性质的函数元组 $f_1,\\dots,f_m\\colon \\{0,1\\}^n \\to \\{0,1\\}$：若 $x^{(1)},\\dots,x^{(m)} \\in \\{0,1\\}^n$ 满足对所有 $i$ 均有 $(x^{(1)}_i,\\dots,x^{(m)}_i) \\in P$，则 $(f_1(x^{(1)}),\\dots,f_m(x^{(m)})) \\in P$ 亦成立。本文证明，若 $f_1,\\dots,f_m$ 对大多数 $x^{(1)},\\dots,x^{(m)}$（基于 $P$ 上任意全支撑分布 $μ$ 的测度）满足该性质，则 $f_1,\\dots,f_m$ 接近 $P$ 的某个广义多态性（相对于 $μ$ 的边缘分布）。我们的主要结果推广了文献中的若干结论：线性测试、定量阿罗定理、近似相交族、AND 测试，以及更一般的 $f$ 测试。