Theory of Computing 2019 2020 — различия между версиями

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[https://www.dropbox.com/s/suzbqo0k37blw2b/grading.pdf?dl=0 Grading]
 
[https://www.dropbox.com/s/suzbqo0k37blw2b/grading.pdf?dl=0 Grading]
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[https://docs.google.com/spreadsheets/d/12KPgGxfFTZNrBse0JPnh1Jdd4cFpJBGCD_37YszaJ8I/edit?usp=sharing Homework results]
  
 
== Dates and Deadlines ==
 
== Dates and Deadlines ==
  
Homework 1, deadline: 4 October, before the lecture
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Homework 1, deadline: 4 October, before the lecture <br>
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Homework 2, deadline: 8 November, before the lecture <br>
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Homework 3, deadline: 8 December, <span style="color:red">2 days extension</span>
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== Colloquium ==
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Date and time: December 13, 15:10<br>
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Room:  R405<br>
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[https://www.dropbox.com/s/rbjd1hsuhtbmvzq/col.pdf?dl=0 Program] <span style="color:red">Updated: 11.12.19 (question 7 of part 2 is removed)</span>
  
 
== Course Materials ==
 
== Course Materials ==
  
In the first several lecture we follow Sipser's book "Introduction to the theory of computation"
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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:<br>
 
If you need some background in math, consider these two sourses:<br>
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! Date !! Summary !! Problem list
 
! Date !! Summary !! Problem list
 
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  || 06.09 || Turing machines, multitape Turing machines, connection between them. Examples. Time and space complexity. Complexity classes P, PSPACE, EXP. || [https://www.dropbox.com/s/lxoenwqohopsoca/prob_1.pdf?dl=0 Problem list 1 ]
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  || 06.09 || Turing machines, multitape Turing machines, connection between them. Examples. Time and space complexity. Complexity classes P, PSPACE, EXP. || [https://www.dropbox.com/s/lxoenwqohopsoca/prob_1.pdf?dl=0 Problem list 1 ] <span style="color:red">Updated: 07.09.19</span> 
<!---|-
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|| 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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|-
 
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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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  || 13.09 || Universal Turing machine. Space hierarchy theorem. Space constructable functions. || [https://www.dropbox.com/s/vs07m4iaepi64ao/prob_2.pdf?dl=0 Problem list 2]
 
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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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  || 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. || [https://www.dropbox.com/s/3yu3xtdgigugkaw/prob_3.pdf?dl=0 Problem list 3 ]
 
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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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  || 27.09 || Circuit complexity. Examples. All functions are computed by circuits. Existence of functions with exponential circuit complexity. P is in P/poly. || [https://www.dropbox.com/s/zn5n3gihxrb1gko/prob_4.pdf?dl=0 Problem list 4]
 
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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]
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  || 04.10 || NP-completeness: Circuit-SAT, 3-SAT, IND-SET, BIN-INT-PROG || [https://www.dropbox.com/s/gfdnexki0dfj3g0/prob_5.pdf?dl=0 Problem list 5]
 
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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.
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  || 11.10 || NP-completeness: Subset-SUM, 3COLORING. coNP, completeness of CIRC-TAUT || [https://www.dropbox.com/s/qa2t7bm7lmw6rbe/prob_6.pdf?dl=0 Problem list 6]
  || [https://www.dropbox.com/s/adl3acqeqebctbk/prob_7.pdf?dl=0 Problem list 7]
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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.
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  || 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. || [https://www.dropbox.com/s/nllomqsrocqcjcv/prob_7.pdf?dl=0 Problem list 7]
|| [https://www.dropbox.com/s/n18684gtl9kd9i9/prob_8.pdf?dl=0 Problem list 8]
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|-
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|| 01.11 || PSPACE-completeness of formula game and generalized geography. Oracle computation definitions. There exists a language A for which P^A = NP^A. || [https://www.dropbox.com/s/e5ccevcu9ig23lu/prob_8.pdf?dl=0 Problem list 8]
 
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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]
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  || 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. || [https://www.dropbox.com/s/36ov36op8wvkdf7/prob_9.pdf?dl=0 Problem list 9]
|| [http://www.mi.ras.ru/~podolskii/files/computability1819/prob_9.pdf Problem list 9]
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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]  
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  || 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. [http://theory.stanford.edu/~tim/w15/l/l1.pdf Roughgarden's lecture notes]  
  || [https://www.dropbox.com/s/z6qygibdhoioo4g/prob_10.pdf?dl=0 Problem list 10]  
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  || [https://www.dropbox.com/s/nbc2m3gama625po/prob_10.pdf?dl=0 Problem list 10]  
 
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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]
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  || 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 [https://epdf.pub/communication-complexity.html download]  
  || [https://www.dropbox.com/s/08slc776301dhfk/prob_11.pdf?dl=0 Problem list 11]
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  || [https://www.dropbox.com/s/zn0nn6rzxvyjtwd/prob_11.pdf?dl=0 Problem list 11] <span style="color:red">Updated HW-problem: 22.11.19</span>
 
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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.
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  || 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)
  || [https://www.dropbox.com/s/3e1as6lok0h1oxw/prob_12.pdf?dl=0 Problem list 12]
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  || [https://www.dropbox.com/s/km4wkungoh0qplt/prob_12.pdf?dl=0 Problem list 12]
 
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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]).  
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  || 6/12 || Probabilistic versus deterministic complexity. Newman's theorem. Space-time tradeoffs for Turing machines. See Nisan Kushilevich chapters 3 and 12.  
|| [https://www.dropbox.com/s/nsa2tped7qp39ch/prob_13.pdf?dl=0 Problem list 13]
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|| [https://www.dropbox.com/s/daj8me2wtt7b5eg/prob_13.pdf?dl=0 Problem list 13]
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|-
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|| 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 [http://www.mi-ras.ru/~podolskii/files/computability1819/exam171222.pdf the exam of 2 years ago].
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||
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<!--
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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]). </span>
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|-
 
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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]  
 
  || 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]  
 
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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]--->
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  ||--->
 
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For interested students: [https://homepages.cwi.nl/~rdewolf/qcnotes.pdf lecture notes on quantum computation]
  
 
== Office hours ==
 
== Office hours ==
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|  Vladimir Podolskii, room&nbsp;S830 ||  ||  ||  ||  ||   
 
|  Vladimir Podolskii, room&nbsp;S830 ||  ||  ||  ||  ||   
 
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|  Bruno Bauwens, room&nbsp;S834 ||  ||  || ||  ||
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|  Bruno Bauwens, room&nbsp;S834 ||  ||  || 14-18h ||  || 18-19h
 
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Текущая версия на 18:09, 18 декабря 2019

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