Lecture 1 Polynomial Time Hierarchy April 1 , 2008 Lecturer : Paul Beame

Lecture 1 : Course Overview and Turing machine complexity

1. Basic properties of Turing Machines (TMs), Circuits & Complexity 2. P, NP, NP-completeness, Cook-Levin Theorem. 3. Hierarchy theorems, Circuit lower bounds. 4. Space complexity: PSPACE, PSPACE-completeness, L, NL, Closure properties 5. Polynomial-time hierarchy 6. #P and counting problems 7. Randomized complexity 8. Circuit lower bounds 9. Interactive proofs & PCP Theorem Hardness of approxi...

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 ...

Lecture 15 : Unique - SAT , Toda ’ s Theorem , Circuit Lower

Definition 1.1. Let USAT ⊆ SAT be the set of formulas φ such that φ has a unique satisfying assignment. Let UP be the set of languages A such that there is a polytime TM M and a polynomial p such that x ∈ A⇔ ∃!y ∈ {0, 1}p(|x|) (M(x, y) = 1). Lemma 1.2 (Valiant-Vazirani). There is a randomized polynomial-computable reduction f from SAT to USAT with the following properties: [φ] ∈ SAT → [f(φ)] ∈ ...

Lecture 11 : Randomization and Complexity

Definition 1.1. A probabilistic Turing machine (PTM) M is a TM with two transition functions in place of the usual one: δ0 : Q × Γ → Q × (Γ × L, S,R) and δ1 : Q × Γ → Q× (Γ× L, S,R). At each time step, independently, the algorithm chooses b from {0, 1} with equal probability 1/2 and makes its move using transition function δb. The running time of M is the maximum number of steps before M halts ...

Proof Complexity Lecturer : Paul Beame Notes : Ashish Sabharwal