Theory of Computing 2019 2020

General Information

Classes: Fridays, 15:10-18:00, R406

Grading

Homework results

Dates and Deadlines

Homework 1, deadline: 4 October, before the lecture
Homework 2, deadline: 8 November, before the lecture
Homework 3, deadline: 8 December, 2 days extension

Colloquium

Date and time: December 13, 15:10
Room: R405
Program Updated: 11.12.19 (question 7 of part 2 is removed)

Course Materials

In the first 10 lectures, we follow Sipser's book "Introduction to the theory of computation" Chapters 7, 8, 9 (not Theorem 9.15), and Section 10.2.

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

Date	Summary	Problem list
06.09	Turing machines, multitape Turing machines, connection between them. Examples. Time and space complexity. Complexity classes P, PSPACE, EXP.	Problem list 1 Updated: 07.09.19
13.09	Universal Turing machine. Space hierarchy theorem. Space constructable functions.	Problem list 2
20.09	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
27.09	Circuit complexity. Examples. All functions are computed by circuits. Existence of functions with exponential circuit complexity. P is in P/poly.	Problem list 4
04.10	NP-completeness: Circuit-SAT, 3-SAT, IND-SET, BIN-INT-PROG	Problem list 5
11.10	NP-completeness: Subset-SUM, 3COLORING. coNP, completeness of CIRC-TAUT	Problem list 6
18.10	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.	Problem list 7
01.11	PSPACE-completeness of formula game and generalized geography. Oracle computation definitions. There exists a language A for which P^A = NP^A.	Problem list 8
08.11	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.	Problem list 9
15/11	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	Problem list 10
22/11	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	Problem list 11 Updated HW-problem: 22.11.19
29/11	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)	Problem list 12
6/12	Probabilistic versus deterministic complexity. Newman's theorem. Space-time tradeoffs for Turing machines. See Nisan Kushilevich chapters 3 and 12.	Problem list 13
20/12	Questions from students about exercises and homework. Poly time reductions on graphs and NP-completeness of Hamiltonion graphs. (If interested, one-shot complexity of the disjointness problem, this is not for the exam.) Solving the exam of 2 years ago.

For interested students: lecture notes on quantum computation

Office hours

Person	Monday	Tuesday	Wednesday	Thursday	Friday
Vladimir Podolskii, room S830
Bruno Bauwens, room S834			14-18h		18-19h