# Theory of Computing 2018 2019

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## General Information

Send your home assignments only in pdf format to the teacher assistant (Andrey Storozhenko) by email (storozhenkoaa [at] yandex.ru) with the following subject: "[Computing, Name Surname, HWx]". You can also submit them in person before the deadline.

Before the lecture.

## Colloquium

Date and time: December 10, 16:40
Room: 400
Program

## Course Materials

Date Summary Problem list
4/9 Time and space hierarchy theorems (see also Sipser Section 9.1) Problem list 1
11/9 Complexity class NP. Examples. Inclusions between P, NP and EXP. Non-deterministic TMs. Another definition of NP. Polynomial reductions, their properties. NP-hardness and NP-completeness, their properties. Problem list 2
18/9 NP-completeness: Circuit-SAT, 3-SAT, NAE-3-SAT, IND-SET Problem list 3
25/9 NP-completeness: Subset-SUM, 3COLORING Problem list 4
2/10 Space complexity. Classes PSPACE and NPSPACE. Configuration graph. Inclusions between time and space classes. TQBF problem, its PSPACE-completeness. PSPACE = NPSPACE. NSPACE(s(n)) is in SPACE(s(n)^2). Problem list 5
9/10 Classes L and NL. Examples. Log-space reductions, their properties. REACHABILITY is NL-complete. NL is equal to coNL (proof is not included in the exams) Problem list 6
16/10 Interpretation of PSPACE in terms of games. Probabilistic computation. Probabilistic machines, the class BPP, prime testing and Carmichael numbers, invariance of the definition BPP for different thresholds, RP, coRP, PP, ZPP. BPP is in P/poly. Problem list 7
30/10 Computations with oracles, its simple properties. There are oracles A and B such that P^A is equal to NP^A and P^B is not equal to NP^B. Problem list 8
6/11 Circuit complexity: directed reachability in AC1, NC0 is trivial, polynomial size Boolean formulas equal NC_1, relations with L and NL. Lecture notes Problem list 9
13/11 Streaming algorithms: finding the majority element, computation of moments F_0 and F_2 in logarithmic space, lower-bounds using one-shot communication complexity. Roughgarden's lecture notes Problem list 10
20/11 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
27/11 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
4/12 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
11/12 Linear programming is in NP, NP-completeness of Hamiltonian path, TQBF as a game, PSPACE-completeness of generalized geography. Various other NP-complete problems. Problem list 14
18/12 Questions from students about exercises and homework. Kolmogorov complexity, symmetry of information, information distance. [Optional lecture, not for the exams.

During the first module, we follow Sipser's book Introduction to the theory of computation, chapters 7-9.

## Office hours

Person Monday Tuesday Wednesday Thursday Friday
1
Vladimir Podolskii, room 621 18:00–19:00 16:40–18:00
2
Bruno Bauwens, room 620 16:40–19:00 15:00–18:00