Lecture Notes on Testing Dictatorships, Juntas, and Monomials

1. Dictatorship: The case where the value of f depends on a single Boolean variable (i.e., f(x) = xi ⊕ σ for some i ∈ [l] and σ ∈ {0, 1}). 2. Junta (of size k): The case where the value of f depends on at most k Boolean variables (i.e., f(x) = f (xI) for some k-subset I ⊂ [l] and f ′ : {0, 1}k → {0, 1}). 3. Monomial (of size k): The case where the value of f is the conjunction of exactly k Boolean variables (i.e., f(x) = ∧i∈I(xi ⊕ σi) for some k-subset I ⊆ [l] and σ1, ..., σl ∈ {0, 1}).