Theory of Computation 2021
General Information
Classes: Tuesdays, 14:40–17:40.
Dates and Deadlines
Homework 1, deadline: October 5, 14:00
Homework 2, deadline: November 2, 14:00
Homework 3, deadline: December 7, 14:00
Colloquium: December 7, 14:40–17:40
Course Materials
The main reference is Sipser's book "Introduction to the theory of computation" Chapters 3, 7–10.
If you need some background in math, consider these two sources:
Lecture notes: Discrete Mathematics, L. Lovasz, K. Vesztergombi
Лекции по дискретной математике (черновик учебника, in Russian)
| Date/Movie | Summary | Problem list |
|---|---|---|
| 07.09 | Turing machines, multitape Turing machines, connection between them. Universal Turing machine. Examples. Time and space complexity. Complexity classes P, PSPACE, EXP. | |
| 14.09 | Time and space hierarchy theorem. Time and space constructible functions. | |
| 21.09 | Circuit complexity. Examples. All functions are computed by circuits. Existence of functions with exponential circuit complexity. P is in P/poly. | |
| 29.09 | Circuit complexity. Examples. All functions are computed by circuits. Existence of functions with exponential circuit complexity. P is in P/poly. | Problem list 4 |
| 06.10 | NP-completeness: Circuit-SAT, 3-SAT, IND-SET, BIN-INT-PROG, Clique, Vertex-Cover, Exactly 1-3-SAT, NAE-3-SAT | Problem list 5 |
| 13.10 | NP-completeness: Subset-SUM, 3COLORING. coNP, completeness of CIRC-TAUT | Problem list 6 |
| 27.10 | Space complexity. Classes L, NL, PSPACE and NPSPACE. Directed Reachability is in SPACE(log2 n). Configuration graph. Inclusions between time and space classes. TQBF problem, its PSPACE-completeness. PSPACE = NPSPACE. NSPACE(s(n)) is in SPACE(s(n)2) for space constructible s. | Problem list 7 |
| 03.11 | L ⊆ P. NL ⊆ P. Log-space reductions and NL-completeness. Path is NL-complete and, probably, outside L. Class coNL. NL = coNL. Generalised Geography and Formula Game are PSPACE-complete. | Problem list 8 |
| 10.11 | Oracle computation definitions. There exists an oracle A for which PA = NPA. There is an oracle B such that PB is not equal to NPB. | Problem list 9 |
| 17.11 | Probabilistic computation. Probabilistic machines, the class BPP, invariance of the definition BPP for different thresholds, RP, coRP, PP, ZPP. BPP is in P/poly. | Problem list 10 |
| 24.11 | Streaming algorithms: finding the majority element, computation of the moment F_2 in logarithmic space. Roughgarden's lecture notes | Problem list 11 |
| 01.12 | Lower-bound for exact and probabilistic computation of F_0 using one-shot communication complexity. Communication protocols. Functions EQ, GT, DISJ. Fooling sets. Combinatorial rectangles. Book: Kushilevitz and Nisan, Communication Complexity, 1997 download | Problem list 12 |
| 08.12 | Questions from students P vs NP vs PSPACE | Problem list 13 |
| 15-16.12 | Colloquium. |
For interested students, we give a few lectures about parameterized complexity. We follow the book Parameterized algorithms by Cygan, Marek, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Vol. 4, no. 8. Cham: Springer, 2015.
Office hours
| Person | Monday | Tuesday | Wednesday | Thursday | Friday |
|---|---|---|---|---|---|
| Sergei Obiedkov, Zoom | 16:30–18:00 | 16:30–18:00 | |||
| Bruno Bauwens, Zoom (email in advance) | 14h-18h | 16h15-19h |
