Approximate polymorphisms

For a function g∶0,1m→0,1, a function f∶ 0,1n→0,1 is called a g-polymorphism if their actions commute: f(g(row1(Z)),…,g(rown(Z))) = g(f(col1(Z)),…,f(colm(Z))) for all Z∈0,1n× m. The function f is called an approximate g-polymorphism if this equality holds with probability close to 1, when Z is sampled uniformly. A pair of functions f0,f1∶ 0,1n → 0,1 are called a skew g-polymorphism if f0(g(row1(Z)),…,g(rown(Z))) = g(f1(col1(Z)),…,f1(colm(Z))) for all Z∈0,1n× m.