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 |
Lecture 23: | Toda's 2nd Theorem & lower bounds for uniform ACC | Reading: Arora--Barak Chapters 17.4.4, 14.4.2; and, B.2 of the Web Addendum (with correction) |
Lecture 24: | Hardness vs. Randomness I | Reading: Arora--Barak Chapters 20.0, 20.1 |
Lecture 25: | Hardness vs. Randomness II | Reading: Arora--Barak Chapters 20.2 |
Lecture 26: | Hardness amplification | Reading: Arora--Barak Chapters 19.0, 19.1 |
Lecture 27: | Ironic Complexity | Reading: Arora--Barak Web Addendum |
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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