Approximate polymorphisms

For a function \(g\colon\{0,1\}^m\to\{0,1\}\), a function \(f\colon \{0,1\}^n\to\{0,1\}\) is called a \(g\)-polymorphism if their actions commute: \(f(g(\mathsf{row}_1(Z)),\ldots,g(\mathsf{row}_n(Z))) = g(f(\mathsf{col}_1(Z)),\ldots,f(\mathsf{col}_m(Z)))\) for all \(Z\in\{0,1\}^{n\times m}\). The function \(f\) is called an approximate polymorphism if this equality holds with probability close to \(1\), when \(Z\) is sampled uniformly. We study the structure of exact polymorphisms as well as approximate polymorphisms. Our results include: - We prove that an approximate polymorphism \(f\) must be close to an exact polymorphism; - We give a characterization of exact polymorphisms, showing that besides trivial cases, only the functions \(g = \mathsf{AND}, \mathsf{XOR}, \mathsf{OR}, \mathsf{NXOR}\) admit non-trivial exact polymorphisms. We also study the approximate polymorphism problem in the list-decoding regime (i.e., when the probability equality holds is not close to \(1\), but is bounded away from some value). We show that if \(f(x \land y) = f(x) \land f(y)\) with probability larger than \(s_\land \approx 0.815\) then \(f\) correlates with some low-degree character, and \(s_\land\) is the optimal threshold for this property. Our result generalize the classical linearity testing result of Blum, Luby and Rubinfeld, that in this language showed that the approximate polymorphisms of \(g = \mathsf{XOR}\) are close to XOR's, as well as a recent result of Filmus, Lifshitz, Minzer and Mossel, showing that the approximate polymorphisms of AND can only be close to AND functions.