|Lecture 01:||Overview of the course||Review: Arora--Barak Chapters 1 (except 1.7), 2, and 4|
|Lecture 02:||Hierarchy theorems: time, space, and nondeterministic versions||Reading: Arora--Barak Chapters 3.1, 3.2; also 1.7 if you're interested in the O(T log T) simulation|
|Lecture 03:||Hopcroft--Paul--Valiant Theorem||Reading: The original paper|
|Lecture 04:||Circuits||Reading: Arora--Barak Chapters 6.1--6.7|
|Lecture 05:||Probabilistic complexity classes||Reading: Arora--Barak Chapters 7.1--7.5 (except not 7.5.2)|
|Lecture 06:||Quasilinear Cook--Levin Theorem||Reading: Section 2.3.1 in this survey by van Melkebeek, these slides by Viola|
|Lecture 07:||The Polynomial Time Hierarchy and alternation||Reading: Arora--Barak Chapters 5.1--5.3|
|Lecture 08:||Oracles, and the Polynomial Time Hierarchy vs. circuits||Reading: Arora--Barak Chapters 5.5, 6.4. Bonus: improving Kannan's Theorem.|
|Lecture 09:||Time/space tradeoffs for SAT||Reading: Arora--Barak Chapter 5.4|
|Lecture 10:||Intro to Merlin-Arthur protocols: MA and MA||Reading: Arora--Barak Chapter 8.2.0|
|Lecture 11:||More on constant-round interactive proof systems||Reading: Arora--Barak Chapter 8.2.4, Chapter 8 exercises|
|Lecture 12:||Approximate counting||Reading: Arora--Barak Chapter 8.2.1, 8.2.2|
|Lecture 13:||Valiant--Vazirani Theorem and exact counting (#P)||Reading: Arora--Barak Chapters 17.0, 17.1, 17.2.1, 17.3.2, 17.4.1|
|Lecture 14:||Toda's 1st Theorem, and the Permanent||Reading: Arora--Barak Chapters 17.4, 8.6.2, 17.3.1|
|Lecture 20 (sic):||Permanent is #P-complete||Reading: PowerPoint slides|
|Lecture 15:||Algebraic circuit complexity||Reading: Arora--Barak Chapter 16.1. Bonus: "algebraic NP vs. P" vs. "Boolean NP vs. P".|
|Lecture 16:||Instance checking and the Permanent||Reading: Arora--Barak Chapter 8.6|
|Lecture 17:||IP = PSPACE||Reading: Arora--Barak Chapters 8.3, 8.4|
|Lecture 18:||Random restrictions and AC0 lower bounds||Reading: Arora--Barak Chapter 14.1|
|Lecture 19:||The Switching Lemma||Reading: My old notes on Razborov's proof|
|Lecture 21:||Monotone circuit lower bounds||Reading: Arora--Barak Chapter 14.3|
|Lecture 22:||Razborov-Smolensky lower bounds for AC0[p]||Reading: Arora--Barak Chapter 14.2|
Prerequisite: An undergraduate course in computational complexity theory, covering most of "Part III" of Sipser and/or most of Carnegie Mellon's 15-455.
Potential topics: Models and Time Hierarchy Theorem. Nondeterminism, padding, Hopcroft-Paul-Valiant Theorem. Circuits and advice. Randomized classes. Cook-Levin Theorem and SAT. Nondeterministic Time Hierarchy Theorem, and nondeterministic models. Oracles, alternation, and the Polynomial Time Hierarchy. Kannan's Theorem, Karp-Lipton, and PH vs. constant-depth circuits. Time-Space tradeoffs for SAT. Randomized classes vs. PH. Interactive proofs and the AM hierarchy. NP in BPP implies PH in BPP, and Boppana-Hastad-Zachos. BCGKT Theorem and Cai's Theorem. Counting classes and the permanent. Valiant's Theorem. Algebraic Complexity. IP = PSPACE and interactive proofs. Instance checkers and Santhanam's Theorem. Random restrictions and AC0 lower bounds for parity. Monotone circuit lower bounds. Razborov-Smolensky lower bounds for AC0[p]. Valiant-Vazirani and Toda Theorems. Beigel-Tarui Theorem. Hardness vs. Randomness and Nisan-Wigderson. Hardness amplification and derandomization. Williams's Theorem. Natural proofs and barriers.
There will be 11 homeworks, and two take-home "tests".
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