Comp 531 Advanced Theory of Computation

Instructor   Hamed Hatami
TA's   Teaching assistant: Ben Cheung (
Lecture   Monday, Wednesday 11:35-12:55 at BURN 708
Course Outline Download
Office hours:   Wednesday 14:40-15:40. If you want to meet outside office hours, the best thing is to send me an email, but you can also just drop by my office, and if I'm not busy I will answer your questions.
Lecture notes:   Here is a draft of the Lecture Notes. It will be updated as we go.


Part I
Lecture Topic Reading
1 (9/7) Computational model, Time and Space classes Chapter 1, Chapter 2 of notes until the beginning of time Hierarchy Theorems.
2 (9/12) Time Hierarchy Theorems, Polynomial Hierarchy. Chapter 2, Chapter 3
3 (9/14) Space complexity. Chapter 4
4 (9/19) Savitches theorem. Chapter 5
5 (9/21) NL equals coNL. Chapter 6
6 (9/26) Oracles and relativization Chapter 8
7 (9/28) Probabilistic complexity classes: RP and ZPP Chapter 9
8 (10/05) Probabilistic complexity classes: BPP Chapter 9
9 (10/13) The class PP, P/Poly Chapter 9
10 (10/17) P/Poly Chapter 10
11 (10/19) NC and AC, switching lemma Chapter 11
12 (10/24) Lower bounds against AC^0, Influences Chapter 11
13 (10/26) XOR gates, Razborov-Smolensky Chapter 12
14 (10/31) Razborov-Smolensky, start of monotone circuits Chapter 13
15 (11/02) Monotone circuit lower bound for clique Chapter 13
16 (11/07) Natural Proofs Chapter 14
17 (11/09) Natural Proofs + Fourier analysis Chapter 15
18 (11/14) Fourier analysis of Abelian groups Chapter 15
19 (11/16) LMN: PAC learning of AC0 under uniform distribution Chapter 16




This is a rigorous course with an emphasis on mathematical proofs. This course is a continuation of Comp 330 and as a result it requires Comp 330 as a prerequisite. You must be comfortable with basic concepts from logic, linear algebra, Turing Machines, and you must be able to read and write precise mathematical statements.



Homework (80% = 4 x 20%). There will be four homework assignments. The due dates are going to be announced. The homework and the exams will be graded based on correctness rather than effort alone. Each assignment will be posted on the course web page. Your grades will be posted on mycourses.

Late homeworks can be submitted for valid reasons.

Group project. 20%


Other Resources


Academic honesty. McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see for more information). Most importantly, work submitted for this course must represent your own efforts. Copying assignments or tests from any source, completely or partially, allowing others to copy your work, will not be tolerated, and they will be reported to disciplinary office.

Submission of written work in French. In accord [sic] with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded.