Notes for Lecture 4

The polynomial hierarchy starts with familiar classes on level one: Σ1 = NP and Π1 = coNP. For all i ≥ 1, it includes two classes, Σi and Πi, which are defined as follows: Definition 2 Σk is the class of all problems such that there is a polynomial time computable F (·, ..., ·) and k polynomials p1(), ..., pk() such that x is a YES-instance⇔ ∃y1 ∈ {0, 1}1.∀y2 ∈ {0, 1}p2(|x|). . . . . . . ∀/∃ k is odd/even yk ∈ {0, 1}pk(|x|). F (x, y1, . . . , yk)