Lecture 9 : Polynomial - Time Hierarchy , Time - Space Tradeoffs

Last time we defined problems EXACT -INDSET = {[G, k] | the largest independent set of G has size = k}, MINDNF = {[φ, k] | φ is a DNF that has an equivalent DNF of size ≤ k}, and the complexity classes Σ2 and its dual Π P 2 . Σ P 2 and Π P 2 were defined in analogy with NP and coNP except that there are two levels of quantifiers with alternation ∃∀ and ∀∃, respectively. We observed that MINDNF ∈ Σ2 and EXACT -INDSET ∈ Σ2 ∩ Π2 . More generally we have the definition: Definition 1.1. Σk is the set of A ⊆ {0, 1}∗ such that there exist polynomials p1, . . . , pk and polynomial-time verifier V such that x ∈ A ⇔ ∃y1 ∈ {0, 1}1∀y2 ∈ {0, 1}p2(|x|) . . . Qkyk ∈ {0, 1}pk(|x| (V (x, y1, . . . , yk) = 1) where Qk = ∃ if k is odd and Qk = ∀ if k is even. Πk = {L | L ∈ Σk }; alternatively, it is the set of B ⊆ {0, 1}∗ such that there exist p1, . . . , pk and V such that x ∈ B ⇔ ∀y1 ∈ {0, 1}1∃y2 ∈ {0, 1}p2(|x|) . . . Qkyk ∈ {0, 1}pk(|x| (V (x, y1, . . . , yk) = 1) where Qk = ∀ if k is odd and Qk = ∃ if k is even.