Theory of computing, AMI

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General Information

Classes: Tuesdays, 13h00-16h00, for the room see

First lecture September 8.


Dates and Deadlines

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

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 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.
29.09 Circuit complexity. Examples. All functions are computed by circuits. Existence of functions with exponential circuit complexity. P is in P/poly.
06.10 NP-completeness: Circuit-SAT, 3-SAT, IND-SET, BIN-INT-PROG
13.10 NP-completeness: Subset-SUM, 3COLORING. coNP, completeness of CIRC-TAUT
20.10 Space complexity. Classes L, NL, PSPACE and NPSPACE. Directed Reachability is in SPACE(log^2 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 constructable s.
03.11 PSPACE-completeness of formula game and generalized geography. Oracle computation definitions. There exists a language A for which P^A = NP^A.
10.11 There is an oracle B such that P^B is not equal to NP^B. Probabilistic computation. Probabilistic machines, the class BPP, invariance of the definition BPP for different thresholds, RP, coRP, PP, ZPP.
17.11 Streaming algorithms: finding the majority element, computation of the moment F_2 in logarithmic space, lower-bound for exact and probabilistic computation of F_0 using one-shot communication complexity. Roughgarden's lecture notes
24.11 Communication protocols. Functions EQ, GT, DISJ, IP. Fooling sets. Combinatorial rectangles. Rectangle size lower bound. Rank lower bound. Book: Nisan and Kushilevich: communication complexity, 1997 download
01.12 Nondeterministic communication complexity. D(f) < O(N^0(f) N^1(f)). Deterministic complexity vs number of leafs in a protocol tree. Randomized communication complexity: definitions. Functions EQ, GT, MCE. Newman's theorem (only the statement)
08.12 Probabilistic versus deterministic complexity. Newman's theorem. Space-time tradeoffs for Turing machines. See Nisan Kushilevich chapters 3 and 12.
22.12 Questions from students about exercises and homework. Poly time reductions on graphs and NP-completeness of Hamiltonion graphs. Solving the exam of 2 years ago.

Office hours

Person Monday Tuesday Wednesday Thursday Friday
Sergei Obiedkov, room T915 16:30–18:00
Bruno Bauwens, room S834