# Theory of computing, AMI

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## General Information

Classes: Tuesdays, 13:00–16:00, Zoom.

First lecture September 8.

Homework 1, deadline: 6 October, before the lecture
Homework 2, deadline: 3 November, before the lecture
Homework 3, deadline: 8 December, before the lecture

The lecture starts at 13h, it is not allowed to submit after this time. However, the first time you are less than 24 hours late, the homework will still be graded.

## Colloquium

Dates: December 15 and 16. Choose your time
Rules and questions

## Course Materials

In the first 9 lectures, we follow Sipser's book "Introduction to the theory of computation" Chapters 3, 7, 8, 9 (not Theorem 9.15), and Section 10.2.

If you need some background in math, consider these two sourses:
Lecture notes: Discrete Mathematics, L. Lovasz, K. Vesztergombi
Лекции по дискретной математике (черновик учебника, in Russian)

Date/Movie Summary Problem list
08.09 Turing machines, multitape Turing machines, connection between them. Universal Turing machine. Examples. Time and space complexity. Complexity classes P, PSPACE, EXP. Problem list 1
15.09 Time and space hierarchy theorem. Time and space constructible functions. Problem list 2 Update 15.09, problem 2.4
22.09 Complexity class NP. Examples. Inclusions between P, NP and PSPACE. Non-deterministic TMs. Another definition of NP. Polynomial reductions, their properties. NP-hardness and NP-completeness, their properties. Problem list 3 Update 22.09, 3.5 and hint 3.8
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: Nisan and Kushilevich: communication complexity, 1997 download Problem list 12
08.12 Rectangle size lower bound. Rank lower bound. Space-time tradeoffs for Turing machines. See Nisan Kushilevich chapters 3 and 12. Questions from students
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