CS 880 : Advanced Complexity Theory 2 / 1 / 2008 Lecture 5 : Dictatorship Testing

Today we will finish developing a tester for dictator functions and then explore connections of this test (and the linearity test discussed last lecture) with error correcting codes and probabilisti-cally checkable proofs. In the last lecture, we introduced and analyzed a local tester for linearity and began to reverse engineer how it could be modified to yield a test for dictatorships. Recall that the linearity test operated by picking two points uniformly at random and then verifying that the function indeed satisfies linearity at these points. – Accept if and only if f (x)f (y) = f (z). T easily satisfies the local tester property of making a constant number of queries— it makes three, to be exact. We must also show that it always accepts linear functions, and that the probability that it rejects any function f is at least the order of the distance of f from a linear function. More formally, we need to show that (∀f ∈ LINEAR) Pr[T f accepts] = 1, and (∀f) Pr[T f rejects] ≥ Ω(D(f, LINEAR)). Our analysis of T showed that if it accepts with high probability, then the given function f must have a large Fourier coefficient. In particular, we showed that Pr[T f accepts] = 1 2 + 1 2 S (ˆ f (S)) 3 (1) ≤ 1 2 + 1 2 max S ˆ f (S). We have already seen that the existence of a large Fourier coefficient implies that f must be close to some linear function (namely, the character corresponding to the large Fourier coefficient), so 1