Comp 531 Advanced Theory of Computation




Hamed Hatami



Lianna Hambardzumyan 



Tuesday-Thursday 11:35am - 12:55pm (Maybe Eventually McConnell Engineering Building 304). 


course outline

Office hours: 


Tuesday-Thursday 14:00 - 15:00 on zoom


Homeworks 1 and 2 and 3 are posted on mycourses.

Lecture notes 

Here are the lecture notes. They will be updated as we go.


The presentations are going to be as scheduled. The write-ups are due End of April


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. 

Complexity Background: I will start the first lecture assuming that everybody is familiar with Turing Machines, Undecidability, P, NP, CoNP, EXP, reductions, NP-completeness to the extend that they are covered in Comp 330. If you feel that you have forgotten some of these topics, it is useful to quickly review them before the first lecture.

Probabilistic method background: Some of the proofs in this course use probabilistic arguments. I will assume that you are familiar with elementary discrete probability theory: Discrete Random variables, Expected value, Variance, Linearity of expectation. If you feel rusty about these topics, I suggest reading the first two (or even better: four) chapters of the excellent book, the probabilistic method by Alon and Spencer.

Linear programming background: We might come across some arguments based on the linear programming duality. If you have not seen this topic in Comp 360, you can do some preparation ahead of time by reading these two short lecture notes Linear programming, and Linear programming duality.


Homework (80% = 4 x 20%). There will be four homework assignments. The due dates are going to be announced. The homework 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.

Group project. 20% 

Late homeworks can be submitted until 96 hours after the deadline. There will be a penalty of -10 (out of 100) points for a two-day delay, and -20 points for four-day delays on late homeworks unless a valid reason is provided by the student. Some personal circumstances for which accommodation may be warranted include, but are not limited to: Student illness (mental/physical), Family/partner illness, Death in the immediate family or of a person with whom the student has a similarly close relationship, Religious Observances, Pregnancy, Delivery of a child, Parenting issues. 

The following are reasons for which an extension request will normally NOT be granted: Employment reasons, Travel/vacation/social plans, Airline flights and schedules, Other assignments and exams due on or about the due date. 

Class participation: Although not a formal component of the course grade, active participation can effect your grade in a positive way. 

Tentative plan 

Other Resources 

I will provide links to notes from other similar courses.

Course Schedule



Recommended Reading

0: Jan 7

No class. The first lecture is going to be on the 12th.

Chapters 1, 2, 3 of Math and Computation. 

1: Jan 12

P vs NP. Time and Space Hierarchy Theorems. 

2: Jan 14

Polynomial Hierarchy. 

3: Jan 19

Space complexity 

4: Jan 21 

L, NL, Savitch's theorem. 

5: Jan 26

Immerman-Szelepcsenyi theorem (NL=CoNL) 

6: Jan 28

Larnder's Theorem 

7: Feb 2


8: Feb 4

AC, NC classes and their relation to NL, L. 

Finishing the notes from lecture 10 


Alternating Circuits, Switching Lemma, Razborov-Smolensky 



Razborov's lower-bound for monotone circuit size of clique.




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.