Comp 531 Advanced Theory of Computation

Instructor   Hamed Hatami
TA's   Teaching assistant: Ran Tao (ran.tao6@mail.mcgill...)
Lecture   Tuesday, Thursday 8:35-9:55 at EGNTR 2120
Course Outline Download
Office hours:   Tuesdays 10:05-10:55. 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.



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


Part I
Lecture Topic Reading
1 (9/2) Computational model, Time and Space classes Chapter 1, Chapter 3 of notes until the beginning of time Hierarchy Theorems.
2 (9/7) Time Hierarchy Theorems, Polynomial Hierarchy. Chapter 3, Chapter 4
3 (9/9) Collapse of Polynomial Hierarchy if P=NP, Space complexity Chapter 5 until the beginning of logspace reductions.
4 (9/14) L and NL, Savitch's Theorem Chapter 6
5 (9/16) NL=coNL, Immerman-Szelepcsenyi's Theorem Chapter 7
6 (9/21) Ladner's Theorem and reletivization Chapter 8 and 9
7 (9/23) PH through Reletivization, and NP vs P Chapter 9
8 (9/28) A crash course in Probabilistic Method I Alon-Spencer Chapter 1
9 (9/30) A crash course in Probabilistic Method II Alon-Spencer Chapter 2
10 (10/5) Randomized Classes I Chapter 10
11 (10/7) Randomized Classes II Chapter 10
Part II
Lecture Topic Reading
12 (10/15) Circuit complexity I Chapter 11
13 (10/19) Circuit complexity II Chapter 11
14 (10/21) AC circuits I Chapter 12
15 (10/26) AC circuits II Chapter 12
16 (10/28) Razborov-Smolensky Chapter 13
17 (11/02) Monotone Circuit Lower-bounds Chapter 14
18 (11/04) Natural Proofs Chapter 15
19 (11/09) Fourier Analysis of Boolean functions Chapter 16


Academic honesty. McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offenses 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.