# Theory of Computing 2019 2020

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## General Information

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

Homework 1, deadline: 4 October, before the lecture
Homework 2, deadline: 8 November, before the lecture
Homework 3, deadline: 6 December, before the lecture

## 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 Kushilevich: communication complexity, 1997 Problem list 11
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. 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. Lower bound for randomized 1-shot communication complexity of set disjointness, see Roughgarden's lecture notes). Problem list 13

For interested students: lecture notes on quantum computation

## Office hours

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