Theory of Computation 2021 — различия между версиями

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  || 12.10 || Cook–Levin theorem. ||
 
  || 12.10 || Cook–Levin theorem. ||
 
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  || [https://zoom.us/rec/share/3LrMJXXTJDAYF2lzH0r33hv336AWBjtufpF6yKdidy4ic6uFvsYEpK7BlZEKFNyV.xYkSb0P6NBjVRsao?startTime=1603793959000 27.10] || Space complexity. Classes L, NL, PSPACE and NPSPACE. Directed Reachability is in SPACE(log<sup>2</sup> ''n''). Configuration graph. Inclusions between time and space classes. TQBF problem, its PSPACE-completeness. PSPACE = NPSPACE. NSPACE(''s''(''n'')) is in SPACE(''s''(''n'')<sup>2</sup>) for space constructible ''s''. || [https://www.dropbox.com/s/hlubofza612y6cj/prob_07.pdf?dl=0 Problem list 7]
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  || 26.10 || Space complexity. ||
 
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|| [https://zoom.us/rec/share/14xX_cDOu2Gr2fpFFxfmJKEclKNoDr57XCq2gyGqSGmsgilZdrTwyZihBHz6XRmO.OfTjHf0GXEA9Et4H 03.11] || [https://www.dropbox.com/s/5lt1z4qxhksgs15/WCM_def5d.mp4?dl=0 L ⊆ P]. [https://www.dropbox.com/s/rtizfkx2m2ebyec/NLsubsetP.png?dl=0 NL ⊆ P]. [https://www.dropbox.com/s/9jtajtadc8vrgby/reductions.mp4?dl=0 Log-space reductions and NL-completeness]. [https://www.dropbox.com/s/ca86r3h2fvm3q8o/WCM_09830.mp4?dl=0 Path is NL-][https://www.dropbox.com/s/3k5fld05zisqgrp/WCM_df84e.mp4?dl=0 complete] and, [https://www.dropbox.com/s/ko2zmvk31jqm0ma/WCM_03d13.mp4?dl=0 probably, outside L]. [https://www.dropbox.com/s/mhqz2n52hab37zw/WCM_695bc.mp4?dl=0 Class coNL]. [https://www.dropbox.com/s/ul739s45r3palqi/WCM_7ddbe.mp4?dl=0 NL] [https://www.dropbox.com/s/0tqcxf9uzpev711/WCM_65c40.mp4?dl=0 =] [https://www.dropbox.com/s/ecdy6tno88a1ffs/WCM_63463.mp4?dl=0 coNL]. [https://www.dropbox.com/s/hilscsxiicgr6ho/WCM_3c921.mp4?dl=0 Generalised Geography] and [https://www.dropbox.com/s/4tbu304rw7vje4q/WCM_81f5b.mp4?dl=0 Formula Game] are [https://www.dropbox.com/s/xrlz1u4ho2z1u5x/WCM_bc82a.mp4?dl=0 PSPACE-complete]. || [https://www.dropbox.com/s/2i4hj8ilp4exqr2/prob_08.pdf?dl=0 Problem list 8]
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|| 02.11 || Oracle computation definitions. There exists an oracle ''A'' for which P<sup>''A''</sup> = NP<sup>''A''</sup>. There is an oracle B such that P<sup>''B''</sup> is not equal to NP<sup>''B''</sup>. ||
 
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  || [https://zoom.us/rec/share/XqpUIS7G8bB2TjD4l0XZuGtpLOKdI5ICuyPA3PyMfaxSB0dsjO0x-U-9Yn0QAOcb.nZQA6cqTBv7ytmiP 10.11] || Oracle computation definitions. There exists an oracle ''A'' for which P<sup>''A''</sup> = NP<sup>''A''</sup>. There is an oracle ''B'' such that P<sup>''B''</sup> is not equal to NP<sup>''B''</sup>. || [https://www.dropbox.com/s/8h3a98h938162gz/prob_09.pdf?dl=0 Problem list 9]
 
  || [https://zoom.us/rec/share/XqpUIS7G8bB2TjD4l0XZuGtpLOKdI5ICuyPA3PyMfaxSB0dsjO0x-U-9Yn0QAOcb.nZQA6cqTBv7ytmiP 10.11] || Oracle computation definitions. There exists an oracle ''A'' for which P<sup>''A''</sup> = NP<sup>''A''</sup>. There is an oracle ''B'' such that P<sup>''B''</sup> is not equal to NP<sup>''B''</sup>. || [https://www.dropbox.com/s/8h3a98h938162gz/prob_09.pdf?dl=0 Problem list 9]

Версия 15:35, 25 августа 2021

General Information

Classes: Tuesdays, 14:40–17:40.

Dates and Deadlines

Homework 1, deadline: October 5, 14:00
Homework 2, deadline: November 2, 14:00
Homework 3, deadline: December 7, 14:00
Colloquium: December 7, 14:40–17:40

Course Materials

The main reference is Sipser's book "Introduction to the theory of computation" Chapters 3, 7–10.

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


Date/Movie Summary Problem list
07.09 Turing machines, multitape Turing machines, connection between them. Universal Turing machine. Examples. Time and space complexity. Complexity classes P, PSPACE, EXP.
14.09 Time and space hierarchy theorem. Time and space constructible functions.
21.09 Circuit complexity. Examples. All functions are computed by circuits. Existence of functions with exponential circuit complexity. P is in P/poly.
28.09 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 4
05.10 Proving NP-hardness by reduction from an NP-complete problem. Examples of NP-complete problems.
12.10 Cook–Levin theorem.
26.10 Space complexity.
02.11 Oracle computation definitions. There exists an oracle A for which PA = NPA. There is an oracle B such that PB is not equal to NPB.
10.11 Oracle computation definitions. There exists an oracle A for which PA = NPA. There is an oracle B such that PB is not equal to NPB. Problem list 9
17.11 Probabilistic computation. Probabilistic machines, the class BPP, invariance of the definition BPP for different thresholds, RP, coRP, PP, ZPP. BPP is in P/poly. Problem list 10
24.11 Streaming algorithms: finding the majority element, computation of the moment F_2 in logarithmic space. Roughgarden's lecture notes Problem list 11
01.12 Lower-bound for exact and probabilistic computation of F_0 using one-shot communication complexity. Communication protocols. Functions EQ, GT, DISJ. Fooling sets. Combinatorial rectangles. Book: Kushilevitz and Nisan, Communication Complexity, 1997 download Problem list 12
08.12 Questions from students P vs NP vs PSPACE Problem list 13
15-16.12 Colloquium.

For interested students, we give a few lectures about parameterized complexity. We follow the book Parameterized algorithms by Cygan, Marek, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Vol. 4, no. 8. Cham: Springer, 2015.

Office hours

Person Monday Tuesday Wednesday Thursday Friday
Sergei Obiedkov, Zoom 16:30–18:00 16:30–18:00
Bruno Bauwens, Zoom (email in advance) 14h-18h 16h15-19h
Источник — «http://wiki.cs.hse.ru/index.php?title=Theory_of_Computation_2021&oldid=56641»