Theory of Computing 2018 2019 — различия между версиями

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[http://www.mi.ras.ru/~podolskii/files/computability1819/grading.pdf Grading]
 
[http://www.mi.ras.ru/~podolskii/files/computability1819/grading.pdf Grading]
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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.
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[https://docs.google.com/spreadsheets/d/1SnGKycG4m42VtKTq3Yb6rF5e0756GboLjIBO3GthpmE/edit?usp=sharing Homework results]
  
 
== Dates and Deadlines ==
 
== Dates and Deadlines ==
  
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Homework 1, deadline: 2 October
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Homework 2, deadline: 6 November
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Homework 3, deadline: 4 December
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Before the lecture.
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== Colloquium ==
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Date and time: December 10, 16:40<br>
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Room: 400 <br>
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[http://www.mi.ras.ru/~podolskii/files/computability1819/col.pdf Program]
  
 
== Course Materials ==
 
== Course Materials ==
  
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<span style="color:red">New</span> [http://www.mi.ras.ru/~podolskii/files/computability1819/exam171222.pdf Last year exam]
  
 
{| class="wikitable"
 
{| class="wikitable"
Строка 17: Строка 36:
 
  || 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. || [http://www.mi.ras.ru/~podolskii/files/computability1819/prob_2.pdf Problem list 2 ]
 
  || 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. || [http://www.mi.ras.ru/~podolskii/files/computability1819/prob_2.pdf Problem list 2 ]
 
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  || 18/9 || NP-completeness: Circuit-SAT, 3-SAT, NAE-3-SAT, IND-SET || [https://www.dropbox.com/s/8nisqdaia715ib0/prob_3.pdf?dl=0 Problem list 3]
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  || 18/9 || NP-completeness: Circuit-SAT, 3-SAT, NAE-3-SAT, IND-SET || [http://www.mi.ras.ru/~podolskii/files/computability1819/prob_3.pdf Problem list 3]
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[http://www.mi.ras.ru/~podolskii/files/computability/prob_2.pdf Problem list 2]
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  || NP-completeness: NAE-3-SAT, Exactly-1-3-SAT, IND-SET, Subset-SUM, 3-COLORING.  || [http://www.mi.ras.ru/~podolskii/files/computability/prob_3.pdf Problem list 3]
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  || 25/9 || NP-completeness: Subset-SUM, 3COLORING || [http://www.mi.ras.ru/~podolskii/files/computability1819/prob_4.pdf Problem list 4]
 
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  || 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) (additional material). Interpretation of PSPACE in terms of games. || [http://www.mi.ras.ru/~podolskii/files/computability/prob_4.pdf Problem list 4]
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  || 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). || [http://www.mi.ras.ru/~podolskii/files/computability1819/prob_5.pdf Problem list 5]
 
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  || 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.
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  || 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) || [http://www.mi.ras.ru/~podolskii/files/computability1819/prob_6.pdf Problem list 6]
|| [http://www.mi.ras.ru/~podolskii/files/computability/prob_5.pdf Problem list 5]
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  || Definition of Sigma_2 and Pi_2. BPP is in Sigma_2 and Pi_2. 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.
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  || 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.
  || [http://www.mi.ras.ru/~podolskii/files/computability/prob_6.pdf Problem list 6] <span style="color:red">Updated: 07.10.17</span>
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  || [https://www.dropbox.com/s/adl3acqeqebctbk/prob_7.pdf?dl=0 Problem list 7]
 
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  || Communication protocols. Functions EQ, GT, DISJ, IP. Fooling sets. Combinatorial rectangles. Rectangle size lower bound. Rank lower bound. Non-deterministic complexity. Communication complexity classes P, NP, coNP, intersection of NP and coNP.
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  || 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.
  || [http://www.mi.ras.ru/~podolskii/files/computability/prob_7.pdf Problem list 7]
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  || [https://www.dropbox.com/s/n18684gtl9kd9i9/prob_8.pdf?dl=0 Problem list 8]
 
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  || D(f)=O(N^0(f)N^1(f)). Randomized communication complexity, definitions. R(EQ)=O(1). N^1(f) vs. R^1(f). Newman's theorem, formulation.
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  || 6/11 || Circuit complexity: directed reachability in AC1, NC0 is trivial, polynomial size Boolean formulas equal NC_1, relations with L and NL. [https://www.dropbox.com/s/txpwymgx1s7f4id/09lect.pdf?dl=0 Lecture notes]
  || [http://www.mi.ras.ru/~podolskii/files/computability/prob_8.pdf Problem list 8]
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  || [http://www.mi.ras.ru/~podolskii/files/computability1819/prob_9.pdf Problem list 9]
 
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  || Proof of Newman's theorem, distributional complexity and the characterization of public coin communication complexity, the discrepancy method.
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  || 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. [http://theory.stanford.edu/~tim/w15/l/l1.pdf Roughgarden's lecture notes]
  || [http://www.mi.ras.ru/~podolskii/files/computability/prob_9.pdf Problem list 9]
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  || [https://www.dropbox.com/s/z6qygibdhoioo4g/prob_10.pdf?dl=0 Problem list 10]  
 
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  || Randomized communication complexity of IP. Streaming algorithms. Finding the majority element. Deciding whether there is a most frequent element is hard. One-sided probabilistic complexity of disjointness.
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  || 20/11 || Communication protocols. Functions EQ, GT, DISJ, IP. Fooling sets. Combinatorial rectangles. Rectangle size lower bound. Rank lower bound. Book: [http://infotheorytcs.wikischolars.columbia.edu/file/view/Noam+Nisan+and+Eyal+Kushilevitz+-+Communication+Complexity.pdf Nisan Kushilevich: communication complexity, 1997]
  || [http://www.mi.ras.ru/~podolskii/files/computability/prob_10.pdf Problem list 10] <span style="color:red">Updated: 22.11.17</span>
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  || [https://www.dropbox.com/s/08slc776301dhfk/prob_11.pdf?dl=0 Problem list 11]
 
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  || Property testing: definitions, testing of halfplanes, sorted listed, connectedness of graphs, testing of linearity.  [https://www.dropbox.com/s/r35jr22fb3lfo3k/propTest.pdf?dl=0 Lecture notes.] Version 25.11.17
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  || 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.
  || [https://www.dropbox.com/s/uzh9dur6aiy88dl/prob_11.pdf?dl=0 Problem list 11] <span style="color:red">Updated: 22.11.17</span>
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  || [https://www.dropbox.com/s/3e1as6lok0h1oxw/prob_12.pdf?dl=0 Problem list 12]
 
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  || Property testing: connectedness of graphs (cont.) and testing of monotonicity. (See notes from the previous lecture.)
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  || 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 [http://theory.stanford.edu/~tim/w15/l/l2.pdf lecture notes]).
  || [https://www.dropbox.com/s/y2xujzeftsg6zlr/prob_12.pdf?dl=0 Problem list 12]  
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  || [https://www.dropbox.com/s/nsa2tped7qp39ch/prob_13.pdf?dl=0 Problem list 13]
 
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  || Property testing: lower bounds for monotonicity (see Sect 4 [http://theory.stanford.edu/~tim/w15/l/l8.pdf here]) and k-linearity using communication complexity. Approximation algorithms for some NP-complete problems (see seminar).
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  || 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. || [https://www.dropbox.com/s/cifs60rf6vpfn3i/prob_14.pdf?dl=0 Problem list 14]  
|| [https://www.dropbox.com/s/5xsxqv098y5wmx0/prob_13.pdf?dl=0 Problem list 13]
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  || The class PCP: definition, basic properties, and relation to with the MAX-Clique approximation problem. Beginning of the proof that NP is a subset of PCP(poly(n), 1). See Dexter Kozen, "Introduction to the theory of computation", lectures 18-20 (or see the book of Arora and Barak, chapter 11).  
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  || 18/12 || Questions from students about exercises and homework. Kolmogorov complexity, symmetry of information. [Optional lecture, not for the exams.] || [https://www.dropbox.com/s/b3g60wwwbqpmotm/prob_15.pdf?dl=0 Problem list 15]
|| [https://www.dropbox.com/s/gbg30ajwes45ne8/prob_14.pdf?dl=0 Problem list 14]
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|| NP is a subset of PCP(poly(n),1) [continued]. Solutions of some extra problems.
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|| No problem list.
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In the first lectures, we follow Sipser's book [https://theswissbay.ch/pdf/Book/Introduction%20to%20the%20theory%20of%20computation_third%20edition%20-%20Michael%20Sipser.pdf Introduction to the theory of computation], chapters 7-9.
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During the first module, we follow Sipser's book [https://theswissbay.ch/pdf/Book/Introduction%20to%20the%20theory%20of%20computation_third%20edition%20-%20Michael%20Sipser.pdf Introduction to the theory of computation], chapters 7-9.
 
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== Office hours ==
 
== Office hours ==
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| <center>1</center> || Vladimir Podolskii, room&nbsp;621 ||  || 18:00&ndash;19:00 || 16:40&ndash;18:00  ||  ||   
 
| <center>1</center> || Vladimir Podolskii, room&nbsp;621 ||  || 18:00&ndash;19:00 || 16:40&ndash;18:00  ||  ||   
 
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| <center>2</center> || Bruno Bauwens, room&nbsp;620 || 16:40&ndash;18:00 || 15:00&ndash;18:00 || ||  ||
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| <center>2</center> || Bruno Bauwens, room&nbsp;620 || 16:40&ndash;19:00 || 15:00&ndash;18:00 || ||  ||
 
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Текущая версия на 21:38, 18 декабря 2018

General Information

Grading

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.

Homework results

Dates and Deadlines

Homework 1, deadline: 2 October

Homework 2, deadline: 6 November

Homework 3, deadline: 4 December

Before the lecture.

Colloquium

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

Course Materials

New Last year exam

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. [Optional lecture, not for the exams.] Problem list 15

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