CS6840 - Advanced Complexity Theory

Meetings  

• Theme 1 : Probabilistic Proof Systems - 15 meetings

  • Meeting 01 : Thu, Feb 09, 11:00 am-11:50 am

  Interactive Proofs - introduction and examples. Protocol for GNI.

  References	Du-Ko Section 10.1, Example 10.1 Section 8.1.1, Lemma 8.4
  Exercises	Get-to-the-mindblock question : We made a crucial assumption that the prover cannot see the verifier's randomness? What if the prover can see the random bits too?
  Reading	<a href="http://www.cs.princeton.edu/courses/archive/spr07/cos522/BabaiEmail.pdf">Email and the Unexpected Power of Interaction</a> - Lazlo Babai.

  Interactive Proofs - introduction and examples. Protocol for GNI.
  Scribe : Vamsi Krishna   (to be edited)

  Scribe	:
  Key	:

  References	:	Du-Ko Section 10.1, Example 10.1 Section 8.1.1, Lemma 8.4
  Exercises	:	Get-to-the-mindblock question : We made a crucial assumption that the prover cannot see the verifier's randomness? What if the prover can see the random bits too?
  Reading	:	Email and the Unexpected Power of Interaction - Lazlo Babai.

  • Meeting 02 : Mon, Feb 13, 08:00 am-08:50 am

  Proof of correctness of GNI protocol, Historical Aspects of IP, P^#P, and final proof. The interactive protocol for the permanent (outline).

  References	<a href="http://www.cs.princeton.edu/courses/archive/spr07/cos522/BabaiEmail.pdf">Email and the Unexpected Power of Interaction</a> - Lazlo Babai. <br> Du-Ko Book, Chapter10, Example 10.3
  Exercises	Again, do we need the random bits to be private?
  Reading	<a href="http://www.cs.princeton.edu/courses/archive/spr07/cos522/BabaiEmail.pdf">Email and the Unexpected Power of Interaction</a> - Lazlo Babai.

  Proof of correctness of GNI protocol, Historical Aspects of IP, P^#P, and final proof. The interactive protocol for the permanent (outline).
  Scribe : Anup Joshi   (to be edited)

  Scribe	:
  Key	:

  References	:	Email and the Unexpected Power of Interaction - Lazlo Babai. Du-Ko Book, Chapter10, Example 10.3
  Exercises	:	Again, do we need the random bits to be private?
  Reading	:	Email and the Unexpected Power of Interaction - Lazlo Babai.

  • Meeting 03 : Tue, Feb 14, 12:00 pm-12:50 pm

  Proof of LFKN Protocol. Protocol for #SAT.

  References	<a href="http://www.cs.princeton.edu/courses/archive/spr07/cos522/BabaiEmail.pdf">Email and the Unexpected Power of Interaction</a> - Lazlo Babai. <br> Du-Ko Book, Chapter10
  Exercises
  Reading

  Proof of LFKN Protocol. Protocol for #SAT.
  Scribe : T Devanathan   (rough draft; better one coming soon)

  Scribe	:
  Key	:

  References

  :

  Email and the Unexpected Power of Interaction - Lazlo Babai. Du-Ko Book, Chapter10

  • Meeting 04 : Thu, Feb 16, 11:00 am-11:50 am

  Proof of #SAT protocol, Arithmetization of quantified expressions. Reproving that PH is contained in IP. Quantified Boolean Formulae for PSPACE.

  References	<a href="http://www.cs.princeton.edu/courses/archive/spr07/cos522/BabaiEmail.pdf">Email and the Unexpected Power of Interaction</a> - Lazlo Babai. <br>
  Exercises
  Reading

  Proof of #SAT protocol, Arithmetization of quantified expressions. Reproving that PH is contained in IP. Quantified Boolean Formulae for PSPACE.
  Scribe : Sivaramakrishnan   (to be edited)

  Scribe	:
  Key	:

  References

  :

  Email and the Unexpected Power of Interaction - Lazlo Babai.

  • Meeting 05 : Fri, Feb 17, 10:00 am-10:50 am

  Shamir's Interactive Protocol for PSPACE.

  References	<a href="http://www.cs.princeton.edu/courses/archive/spr07/cos522/BabaiEmail.pdf">Email and the Unexpected Power of Interaction</a> - Lazlo Babai. <br> For chinese remaindering - Supplementary Lecture B - in Kozen's Theory of Computation Book.<br> Kozen's Book also has a lecture on PSPACE is contained in IP.
  Exercises	Prove (or read up) Chinese remaindering theorem.
  Reading	<a href="http://people.cs.uchicago.edu/~fortnow/papers/mip2.pdf">Nondeterministic Exponential Time has two prover interactive protocols</a> - Babai, Fortnow, Lund.

  Shamir's Interactive Protocol for PSPACE.
  Scribe : Sivaramakrishnan   (to be edited)

  Scribe	:
  Key	:

  References	:	Email and the Unexpected Power of Interaction - Lazlo Babai. For chinese remaindering - Supplementary Lecture B - in Kozen's Theory of Computation Book. Kozen's Book also has a lecture on PSPACE is contained in IP.
  Exercises	:	Prove (or read up) Chinese remaindering theorem.
  Reading	:	Nondeterministic Exponential Time has two prover interactive protocols - Babai, Fortnow, Lund.

  • Meeting 06 : Sat, Feb 18, 10:00 am-10:50 am

  IP is contained in PSPACE. Computing, in PSPACE, acceptance probability (of the verifier) against the maximizing prover.

  References	Lecture in Kozen's Theory of Computation Textbook.
  Exercises
  Reading	<a href="./CS6840/IPinPSPACE.pdf">An algorithmic view</a> - notes written by Balagopal.

  IP is contained in PSPACE. Computing, in PSPACE, acceptance probability (of the verifier) against the maximizing prover.
  Scribe : Abdulla Anam  (notes being written up)

  Scribe	:
  Key	:

  References	:	Lecture in Kozen's Theory of Computation Textbook.
  Reading	:	An algorithmic view - notes written by Balagopal.

  • Meeting 07 : Sat, Feb 18, 11:00 am-11:50 am

  Probabilistically Checkable Proofs, Basic definitions. GAP3SAT problem.

  References	<a href="http://www.cse.cuhk.edu.hk/~andrejb/csc5170/notes/10L12.pdf">Lecture 12 from Andrej Bogdanov's course</a>.
  Exercises
  Reading	<a href="http://www.cs.washington.edu/education/courses/cse533/05au/pcp-history.pdf">The history of PCP theorem</a> by Ryan Odonnel

  Probabilistically Checkable Proofs, Basic definitions. GAP3SAT problem.
  Scribe : Rahul CS   (to be edited)

  Scribe	:
  Key	:

  References	:	Lecture 12 from Andrej Bogdanov's course.
  Reading	:	The history of PCP theorem by Ryan Odonnel

  • Meeting 08 : Mon, Feb 20, 08:00 am-08:50 am

  qCSP, L is in PCP(O(log n), q) if and only if L reduces to qCSP. <br> Reduction from qCSP to GAPSAT.

  References	<a href="http://www.cse.cuhk.edu.hk/~andrejb/csc5170/notes/10L12.pdf">Lecture 12 from Andrej Bogdanov's course</a>.
  Exercises
  Reading	<a href="http://www.cs.washington.edu/education/courses/cse533/05au/pcp-history.pdf">The history of PCP theorem</a> by Ryan Odonnel

  qCSP, L is in PCP(O(log n), q) if and only if L reduces to qCSP.
  Reduction from qCSP to GAPSAT.
  Scribe : Sunil K S   (to be edited)

  Scribe	:
  Key	:

  References	:	Lecture 12 from Andrej Bogdanov's course.
  Reading	:	The history of PCP theorem by Ryan Odonnel

  • Meeting 09 : Tue, Feb 21, 12:00 pm-12:50 pm

  Inapproximability of Independent set Problem. GAPCSP to GAPIS<br> PCP for LIN, Attempts, Proof in the long-code form. Need of linearity testing.

  References	<a href="http://www.cse.cuhk.edu.hk/~andrejb/csc5170/notes/10L12.pdf">Lecture 12 from Andrej Bogdanov's course</a>.
  Exercises
  Reading	<a href="http://www.cs.washington.edu/education/courses/cse533/05au/pcp-history.pdf">The history of PCP theorem</a> by Ryan Odonnel

  Inapproximability of Independent set Problem. GAPCSP to GAPIS
  PCP for LIN, Attempts, Proof in the long-code form. Need of linearity testing.
  Scribe : Balagopal  

  Scribe	:
  Key	:

  References	:	Lecture 12 from Andrej Bogdanov's course.
  Reading	:	The history of PCP theorem by Ryan Odonnel

  • Meeting 10 : Thu, Feb 23, 11:00 am-11:50 am

  Linearity testing. Local decoding. The proof of PCP for LIN.

  References	<a href="http://www.cse.cuhk.edu.hk/~andrejb/csc5170/notes/10L12.pdf">Lecture 12 from Andrej Bogdanov's course</a>.
  Exercises
  Reading

  Linearity testing. Local decoding. The proof of PCP for LIN.
  Scribe : Balagopal  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  Lecture 12 from Andrej Bogdanov's course.

  • Meeting 11 : Fri, Feb 24, 10:00 am-10:50 am

  Proof of Linearity Testing

  References	<a href="http://www.cse.cuhk.edu.hk/~andrejb/csc5170/notes/10L12.pdf">Lecture 12 from Andrej Bogdanov's course</a>.
  Exercises
  Reading	<a href="http://www.cs.washington.edu/education/courses/cse533/05au/pcp-history.pdf">The history of PCP theorem</a> by Ryan Odonnel

  Proof of Linearity Testing
  Scribe : Jayalal Sarma  (notes being written up)

  Scribe	:
  Key	:

  References	:	Lecture 12 from Andrej Bogdanov's course.
  Reading	:	The history of PCP theorem by Ryan Odonnel

  • Meeting 12 : Mon, Feb 27, 08:00 am-08:50 am

  Generalization to Quadratic Programs.

  References	<a href="http://www.cse.cuhk.edu.hk/~andrejb/csc5170/notes/10L12.pdf">Lecture 12 from Andrej Bogdanov's course</a>.
  Exercises
  Reading	<a href="http://www.cs.washington.edu/education/courses/cse533/05au/pcp-history.pdf">The history of PCP theorem</a> by Ryan Odonnel

  Generalization to Quadratic Programs.
  Scribe : Jayalal Sarma  (notes being written up)

  Scribe	:
  Key	:

  References	:	Lecture 12 from Andrej Bogdanov's course.
  Reading	:	The history of PCP theorem by Ryan Odonnel

  • Meeting 13 : Tue, Feb 28, 12:00 pm-12:50 pm

  Dinur's Proof of PCP theorem. The proof outline. Query reduction step.

  References	<a href="http://www.cse.cuhk.edu.hk/~andrejb/csc5170/notes/10L13.pdf">Lecture 13 from Andrej Bogdanov's course</a>.
  Exercises
  Reading	<a href="http://www.cs.washington.edu/education/courses/cse533/05au/pcp-history.pdf">The history of PCP theorem</a> by Ryan Odonnel

  Dinur's Proof of PCP theorem. The proof outline. Query reduction step.
  Scribe : Anup Joshi   (to be edited)

  Scribe	:
  Key	:

  References	:	Lecture 13 from Andrej Bogdanov's course.
  Reading	:	The history of PCP theorem by Ryan Odonnel

  • Meeting 14 : Fri, Mar 02, 10:00 am-10:50 am

  Expander Graphs, Degree reduction

  References	<a href="http://www.cse.cuhk.edu.hk/~andrejb/csc5170/notes/10L13.pdf">Lecture 13 from Andrej Bogdanov's course</a>.
  Exercises
  Reading	<a href="http://www.cs.washington.edu/education/courses/cse533/05au/pcp-history.pdf">The history of PCP theorem</a> by Ryan Odonnel

  Expander Graphs, Degree reduction
  Scribe : Vamsi Krishna   (to be edited)

  Scribe	:
  Key	:

  References	:	Lecture 13 from Andrej Bogdanov's course.
  Reading	:	The history of PCP theorem by Ryan Odonnel

  • Meeting 15 : Sat, Mar 03, 02:00 pm-03:30 pm

  Gap Amplification, Alphabet reduction

  References	<a href="http://www.cse.cuhk.edu.hk/~andrejb/csc5170/notes/10L13.pdf">Lecture 13 from Andrej Bogdanov's course</a>. As a side product, we mentioned in class how the ideas we discussed could also be useful in testing reachability using a simple randomized algorithm. Here is references related to it. <br> Read section 21.1 for the eigen-value connections. <br> Read section 21.4 for the Reingold's log-space algorithm. <br>
  Exercises
  Reading	<a href="http://www.cs.washington.edu/education/courses/cse533/05au/pcp-history.pdf">The history of PCP theorem</a> by Ryan Odonnel <br> Here are some related papers to expanders and ideas we discussed in class while describing the intuition behind Dinur's proof:<br> <a href="www.wisdom.weizmann.ac.il/~reingold/publications/zigzag.ps">Construction of explicit expanders through a combinatorial approach</a>.<br> <a href="www.wisdom.weizmann.ac.il/~reingold/publications/sl.ps"> Testing reachability in undirected graph can be done in log-space.</a><br> <a href="http://brahma.tcs.tifr.res.in/~prahladh/teaching/05spring/">Expander Graphs in Computer Science</a> - Course by Prahladh Harsha.

  Gap Amplification, Alphabet reduction
  Scribe : Vamsi Krishna   (to be edited)

  Scribe	:
  Key	:

  References	:	Lecture 13 from Andrej Bogdanov's course. As a side product, we mentioned in class how the ideas we discussed could also be useful in testing reachability using a simple randomized algorithm. Here is references related to it. Read section 21.1 for the eigen-value connections. Read section 21.4 for the Reingold's log-space algorithm.
  Reading	:	The history of PCP theorem by Ryan Odonnel Here are some related papers to expanders and ideas we discussed in class while describing the intuition behind Dinur's proof: Construction of explicit expanders through a combinatorial approach. Testing reachability in undirected graph can be done in log-space. Expander Graphs in Computer Science - Course by Prahladh Harsha.

• Theme 2 : Circuits, Lower Bounds & Derandomization - 24 meetings

  • Meeting 16 : Mon, Mar 05, 08:00 am-08:50 am

  Boolean Circuit Model of Computation. Circuits, Gates and Basis functions, <br> <a href="http://en.wikipedia.org/wiki/Emil_Leon_Post">Emil Post</a>'s characterization (1941) of a complete basis.

  References	Lecture 1 in Uri Zwick's course. <br> pdf file sent as an email to the class mailing list.
  Exercises	Work out the full detailed proof of Post's theorem developing on the ideas that we discussed.
  Reading

  Boolean Circuit Model of Computation. Circuits, Gates and Basis functions,
  Emil Post's characterization (1941) of a complete basis.
  Scribe : Rahul CS   (to be edited)

  Scribe	:
  Key	:

  References	:	Lecture 1 in Uri Zwick's course. pdf file sent as an email to the class mailing list.
  Exercises	:	Work out the full detailed proof of Post's theorem developing on the ideas that we discussed.

  • Meeting 17 : Tue, Mar 06, 12:00 pm-12:50 pm

  Shannon's counting argument, Lupanov's construction.

  References
  Exercises
  Reading

  Shannon's counting argument, Lupanov's construction.
  Scribe : Dinesh K  

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 18 : Fri, Mar 09, 10:00 am-10:50 am

  P/poly = functions computed by polynomial sized circuits. CVP is P-complete under log-space reductions.

  References
  Exercises
  Reading

  P/poly = functions computed by polynomial sized circuits. CVP is P-complete under log-space reductions.
  Scribe : Prashant Vasudevan   (to be edited)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 19 : Mon, Mar 12, 08:00 am-08:50 am

  Uniformity, Log-space Uniformity. P is computed by uniform polysize circuits. Review of parameters and questions. Circuit Lower bound Problem.

  References
  Exercises
  Reading

  Uniformity, Log-space Uniformity. P is computed by uniform polysize circuits. Review of parameters and questions. Circuit Lower bound Problem.
  Scribe : Princy Lunawat   (to be edited)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 20 : Tue, Mar 13, 12:00 pm-12:50 pm

  Circuit Lower Bound Problem. Trivial size and depth lower bounds, current frontiers. <br> Upper Bounds, simple functions and their circuits, PARITY, ADD. <br> The class NC

  References
  Exercises
  Reading

  Circuit Lower Bound Problem. Trivial size and depth lower bounds, current frontiers.
  Upper Bounds, simple functions and their circuits, PARITY, ADD.
  The class NC
  Scribe : Rahul CS   (to be edited)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 21 : Thu, Mar 15, 11:00 am-11:50 am

  The class AC-hierarchy. Interleaving with NC hierarchy. Relationship between space complexity classes and uniform circuit complexity classes.

  References	None
  Exercises	We constructed a circuit family (for AC^1 upper bound) for the languages in NL using Savitch's theorem. Is the family uniform? What is the best uniformity machine that you can think of?
  Reading

  The class AC-hierarchy. Interleaving with NC hierarchy. Relationship between space complexity classes and uniform circuit complexity classes.
  Scribe : Balagopal   (to be edited)

  Scribe	:
  Key	:

  References	:	None
  Exercises	:	We constructed a circuit family (for AC^1 upper bound) for the languages in NL using Savitch's theorem. Is the family uniform? What is the best uniformity machine that you can think of?

  • Meeting 22 : Fri, Mar 16, 10:00 am-10:50 am

  Zooming in to NC^1. Adding n n-bit numbers. Thresholds, Bit count. NC^1 upper bounds. Constant depth reductions among these problems.

  References	<a href="http://www.cs.au.dk/~arnsfelt/CT08/scribenotes/lecture8.pdf">Kristoffer Hansen's Lecture Notes</a>.
  Exercises
  Reading

  Zooming in to NC^1. Adding n n-bit numbers. Thresholds, Bit count. NC^1 upper bounds. Constant depth reductions among these problems.
  Scribe : Nilkamal Adak  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  Kristoffer Hansen's Lecture Notes.

  • Meeting 23 : Mon, Mar 19, 08:00 am-08:50 am

  Computing symmetric functions using Threshold gates. The complexity classes TC^0 and AC^0[2]. The class ACC^0.

  References
  Exercises
  Reading

  Computing symmetric functions using Threshold gates. The complexity classes TC^0 and AC^0[2]. The class ACC^0.
  Scribe : Sivaramakrishnan   (to be edited)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 24 : Tue, Mar 20, 12:00 pm-12:50 pm

  Formulas, BF = NC^1. Formula size lower bounds, overview and ideas.

  References
  Exercises
  Reading

  Formulas, BF = NC^1. Formula size lower bounds, overview and ideas.
  Scribe : Sivaramakrishnan   (to be edited)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 25 : Thu, Mar 22, 11:00 am-11:50 am

  Formula size lower bounds : Neciporuk's Method.

  References
  Exercises
  Reading

  Formula size lower bounds : Neciporuk's Method.
  Scribe : Anup Joshi  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 26 : Mon, Mar 26, 08:00 am-08:50 am

  Neciprouk's method. Applications. Random Restrictions. The history.

  References	<a href="http://lovelace.thi.informatik.unifrankfurt.de/~jukna/EC_Book/Topics/lect01.ps">Boolean Formulas - The classics</a>.
  Exercises
  Reading

  Neciprouk's method. Applications. Random Restrictions. The history.
  Scribe : Sajin Koroth  

  Scribe	:
  Key	:

  References

  :

  Boolean Formulas - The classics.

  • Meeting 27 : Tue, Mar 27, 12:00 pm-12:50 pm

  Sabbotavskaya's Lower bound using Random Restrictions. Lower Bounds for Parity.

  References	<a href="http://lovelace.thi.informatik.uni-frankfurt.de/~jukna/EC_Book/Topics/lect01.ps">Formula Lower bounds - The classics</a><br> <a href="http://logic.pdmi.ras.ru/~kulikov/formulaminicourse/lecture4-slides.pdf">Slides</a> - Alexander Kulikov.
  Exercises
  Reading

  Sabbotavskaya's Lower bound using Random Restrictions. Lower Bounds for Parity.
  Scribe : Sajin Koroth  

  Scribe	:
  Key	:

  References

  :

  Formula Lower bounds - The classics Slides - Alexander Kulikov.

  • Meeting 28 : Thu, Mar 29, 11:00 am-11:50 am

  Exponential size lower bounds for constant depth circuits computing parity. The proof assuming the switching lemma.

  References	Section 6.1 of Paul Beame's Survey : <a href="http://www.cs.washington.edu/homes/beame/papers/primer.ps">Switching Lemma Primer</a>
  Exercises
  Reading

  Exponential size lower bounds for constant depth circuits computing parity. The proof assuming the switching lemma.
  Scribe : Dinesh K  

  Scribe	:
  Key	:

  References

  :

  Section 6.1 of Paul Beame's Survey : Switching Lemma Primer

  • Meeting 29 : Fri, Mar 30, 10:00 am-10:50 am

  Proof of the switching Lemma.

  References	Section 2 of Paul Beame's Survey : <a href="http://www.cs.washington.edu/homes/beame/papers/primer.ps">Switching Lemma Primer</a>
  Exercises
  Reading	Other versions of switching lemma in the survey : <a href="http://www.cs.washington.edu/homes/beame/papers/primer.ps">Switching Lemma Primer</a>

  Proof of the switching Lemma.
  Scribe : Abdulla Anam  (notes being written up)

  Scribe	:
  Key	:

  References	:	Section 2 of Paul Beame's Survey : Switching Lemma Primer
  Reading	:	Other versions of switching lemma in the survey : Switching Lemma Primer

  • Meeting 30 : Sat, Mar 31, 11:00 am-12:00 pm

  Proof of the switching Lemma.

  References
  Exercises
  Reading

  Proof of the switching Lemma.
  Scribe : Prashant Vasudevan  

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 31 : Mon, Apr 02, 08:00 am-08:50 am

  Representation of Boolean functions by polynomials. The strategy of the proof.

  References
  Exercises
  Reading

  Representation of Boolean functions by polynomials. The strategy of the proof.
  Scribe : Nilkamal Adak  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 32 : Tue, Apr 03, 10:00 am-10:50 am

  Razborov-Smolensky lower bound for PARITY.

  References
  Exercises
  Reading

  Razborov-Smolensky lower bound for PARITY.
  Scribe : Abdulla Anam  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 33 : Tue, Apr 10, 12:00 pm-12:50 pm

  Monotone functions and circuits. CLIQUE requires exponential size for any monotone circuit computing it. The overall strategy. Clique Indicators and Approximators. Positive and negative inputs.

  References
  Exercises
  Reading

  Monotone functions and circuits. CLIQUE requires exponential size for any monotone circuit computing it. The overall strategy. Clique Indicators and Approximators. Positive and negative inputs.
  Scribe : Dinesh K  

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 34 : Thu, Apr 12, 11:00 am-11:50 am

  Sunflower Lemma. Approximation procedure for AND and OR gates. Estimating the errors of the approximator at the root. Size lower bounds. Proof of the sunflower lemma. <br> Lecture extended to a session in the afternoon.

  References
  Exercises
  Reading

  Sunflower Lemma. Approximation procedure for AND and OR gates. Estimating the errors of the approximator at the root. Size lower bounds. Proof of the sunflower lemma.
  Lecture extended to a session in the afternoon.
  Scribe : Princy Lunawat  

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 35 : Fri, Apr 13, 10:00 am-10:50 am

  Branching Programs and Skew circuits. From space bounded algorithms to Branching Programs.

  References
  Exercises
  Reading

  Branching Programs and Skew circuits. From space bounded algorithms to Branching Programs.
  Scribe : T Devanathan  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 36 : Mon, Apr 16, 08:00 am-08:50 am

  Width of the Branching Programs. Programs over groups. Permutations. Conjugates, Cycle conjugacy lemma.

  References
  Exercises
  Reading

  Width of the Branching Programs. Programs over groups. Permutations. Conjugates, Cycle conjugacy lemma.
  Scribe : Prashant Vasudevan  

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 37 : Tue, Apr 17, 12:00 pm-12:50 pm

  From Programs over permutation groups to Branching Programs. Simulating NC^1 circuits as programs over permutation groups. The "commutator" solution for AND gate. Non-solvability of the group for simulation.

  References	<ul> <li><a href="http://users.eecs.northwestern.edu/~fortnow/classes/f10/395-Complexity/Lecture15.pdf">Lecture Notes from Lance Fortnow's course</a></li> </ul>
  Exercises	Explore the proof that S_5 is the smallest non-solvable symmetric group.
  Reading

  From Programs over permutation groups to Branching Programs. Simulating NC^1 circuits as programs over permutation groups. The "commutator" solution for AND gate. Non-solvability of the group for simulation.
  Scribe : Abdulla Anam  (notes being written up)

  Scribe	:
  Key	:

  References	:	Lecture Notes from Lance Fortnow's course
  Exercises	:	Explore the proof that S_5 is the smallest non-solvable symmetric group.

  • Meeting 38 : Thu, Apr 19, 11:00 am-11:50 am

  Derandomization Problem, Pseudo random generators. Conditional derandomization results.

  References	<a href="">Markus Blaser's Lecture notes on Derandomization</a>
  Exercises
  Reading

  Derandomization Problem, Pseudo random generators. Conditional derandomization results.
  Scribe : Nilkamal Adak  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  Markus Blaser's Lecture notes on Derandomization

  • Meeting 39 : Fri, Apr 20, 09:30 am-11:00 am

  Nisan-Wigderson generator, Proof of pseudorandomness, Construction of NW-designs.

  References
  Exercises
  Reading

  Nisan-Wigderson generator, Proof of pseudorandomness, Construction of NW-designs.
  Scribe : Vamsi Krishna  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

• Theme 3 : Course Project Presentations - 13 meetings

  • Meeting 40 : Fri, Mar 30, 02:00 pm-03:00 pm

  <a href="CS6840/2012/project/PLT-VK-slides.pdf">Polylogarithmic Threshold is in AC^0</a> - (Presenter : Vamsi Krishna)<br> We saw in class that Th(n,k) has an AC^0 circuit when k is a constant. A stronger result can be proved using hash-functions : Even when k is polylogarithmic in n, the function has an AC^0 circuit. Contrast it with the fact that majority (which is a threshold function with k=n/2) is not in AC^0.

  References
  Exercises
  Reading

  Polylogarithmic Threshold is in AC^0 - (Presenter : Vamsi Krishna)
  We saw in class that Th(n,k) has an AC^0 circuit when k is a constant. A stronger result can be proved using hash-functions : Even when k is polylogarithmic in n, the function has an AC^0 circuit. Contrast it with the fact that majority (which is a threshold function with k=n/2) is not in AC^0.
  Scribe : Vamsi Krishna  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 41 : Sat, Mar 31, 02:00 pm-03:00 pm

  <a href="CS6840/2012/project/IC-SRK-slides.pdf">Infeasibility of Instance Compression</a>: - (Presenter : Sivaramakrishnan) <br> The aim here is to explore a connection between parameterized complexity and classical complexity theory. One of the central concept in parameterized complexity is that of Kernelization. A closely related notion is instance compression. A language L in NP is instance compressible if there is a polynomial-time computable function f and a set A such that for each instance x of L, f(x) is of size polynomial in the witness size of x, and f reduces L to A. One of the interesting results here is due to <a href="http://www.eccc.uni-trier.de/report/2007/096/">Fortnow and Santhanam</a> which says SAT is incompressible unless NP is contained in CoNP/poly. </li>

  References
  Exercises
  Reading

  Infeasibility of Instance Compression: - (Presenter : Sivaramakrishnan)
  The aim here is to explore a connection between parameterized complexity and classical complexity theory. One of the central concept in parameterized complexity is that of Kernelization. A closely related notion is instance compression. A language L in NP is instance compressible if there is a polynomial-time computable function f and a set A such that for each instance x of L, f(x) is of size polynomial in the witness size of x, and f reduces L to A. One of the interesting results here is due to Fortnow and Santhanam which says SAT is incompressible unless NP is contained in CoNP/poly.
  Scribe : Prashant Vasudevan  

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 42 : Sat, Mar 31, 03:00 pm-04:00 pm

  <a href="CS6840/2012/project/PP-AA-slides.pdf">Structural Properties of PP:</a> - (Presenter : Abdulla Anam)<br> We saw in class that the PP vs P question is equivalent to FP vs #P question. This project is to explore some structural properites of the class PP using the power of "polynomials" again. The reference is: <a href="http://www.sciencedirect.com/science/article/pii/S0022000085710173">PP is closed under intersection</a> by Richard Beigel, Nick Reingold, and Daniel Spielman. (STOC 1991, JCSS 1995)

  References
  Exercises
  Reading

  Structural Properties of PP: - (Presenter : Abdulla Anam)
  We saw in class that the PP vs P question is equivalent to FP vs #P question. This project is to explore some structural properites of the class PP using the power of "polynomials" again. The reference is: PP is closed under intersection by Richard Beigel, Nick Reingold, and Daniel Spielman. (STOC 1991, JCSS 1995)
  Scribe : Nilkamal Adak  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 43 : Sat, Mar 31, 04:00 pm-05:00 pm

  <a href="./CS6840/2012/project/Reach-PL-slides.pdf">Space Complexity of Reachability</a>: (Presenter : Princy Lunawat)<br> This project aims to explore some very recent series of works in the space complexity of directed graph reachability problem. It has close connections to the NL vs L problem and its possibly simpler variants of the same. The most recent one is available <a href="http://www.cse.unl.edu/~vinod/papers/lmpd.pdf">here</a>.

  References
  Exercises
  Reading

  Space Complexity of Reachability: (Presenter : Princy Lunawat)
  This project aims to explore some very recent series of works in the space complexity of directed graph reachability problem. It has close connections to the NL vs L problem and its possibly simpler variants of the same. The most recent one is available here.
  Scribe : Dinesh K  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 44 : Mon, Apr 09, 05:30 pm-06:30 pm

  <a href="CS6840/2012/project/UGC-AJ-slides.pdf">Unique Games Conjecture and Hardness of Approx.</a> - (Presenter : Anup Joshi) <br> The aim of this project is get an introduction to Unique Games Conjecture (UGC) and explore the associated inapproximability results. <a href="http://repository.upenn.edu/cis_reports/123/">Here is a survey</a>.

  References
  Exercises
  Reading

  Unique Games Conjecture and Hardness of Approx. - (Presenter : Anup Joshi)
  The aim of this project is get an introduction to Unique Games Conjecture (UGC) and explore the associated inapproximability results. Here is a survey.
  Scribe : T Devanathan  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 45 : Tue, Apr 10, 04:00 pm-05:00 pm

  <a href="CS6840/2012/project/RPP-DT-slides.pdf">Remote Point Problem and Circuit Lower Bounds</a>: (Presenter : Devanathan) <br> This project is to investigate the connection between the problem of proving explicit lower bounds against a special class of circuits (constant depth circuits with help functions) and obtaining upper bounds for a combinatorial-algebraic problem called the "Remote Point Problem". The paper is <a href="http://arxiv.org/abs/0911.4337">here</a>.

  References
  Exercises
  Reading

  Remote Point Problem and Circuit Lower Bounds: (Presenter : Devanathan)
  This project is to investigate the connection between the problem of proving explicit lower bounds against a special class of circuits (constant depth circuits with help functions) and obtaining upper bounds for a combinatorial-algebraic problem called the "Remote Point Problem". The paper is here.
  Scribe : Anup Joshi  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 46 : Tue, Apr 10, 05:00 pm-06:00 pm

  <a href="CS6840/2012/project/MonotoneDepthLB-PV-slides.pdf">Monotone circuit depth lower bounds</a>: (Presenter : Prashant Vasudevan)<br> This project is to explore a connection between communication complexity and circuit lower bounds against monotone circuits. The sample references are : <a href="http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/KW88/KW88.pdf">Monotone Circuits for Connectivity Requires Super-logarithmic Depth</a> - Karchmer, Wigderson (SIDM 1990).<br> <a href="http://lovelace.thi.informatik.uni-frankfurt.de/~jukna/EC_Book/Topics/lect07.ps">Communication Complexity and Monotone Depth</a> - an expository article by Stasys Jukna.

  References
  Exercises
  Reading

  Monotone circuit depth lower bounds: (Presenter : Prashant Vasudevan)
  This project is to explore a connection between communication complexity and circuit lower bounds against monotone circuits. The sample references are : Monotone Circuits for Connectivity Requires Super-logarithmic Depth - Karchmer, Wigderson (SIDM 1990).
  Communication Complexity and Monotone Depth - an expository article by Stasys Jukna.
  Scribe : Sivaramakrishnan  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 47 : Wed, Apr 11, 04:30 pm-05:30 pm

  <a href="CS6840/2012/project/Evasiveness-RahulCS.pdf">Evasive Boolean Functions</a> - (Presenter : Rahul CS)<br> A Boolean function is said to be evasive if every decision tree is of depth at least n. This project explores the property of evasiveness of Boolean functions. <a href="http://www.cs.uiuc.edu/~jeffe/teaching/497/05-evasive.pdf">Here</a> is a specific reference.</li>

  References
  Exercises
  Reading

  Evasive Boolean Functions - (Presenter : Rahul CS)
  A Boolean function is said to be evasive if every decision tree is of depth at least n. This project explores the property of evasiveness of Boolean functions. Here is a specific reference.
  Scribe : Rahul CS  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 48 : Fri, Apr 13, 05:00 pm-06:00 pm

  <a href="./CS6840/2012/project/Regular-Sunil-slides.pdf">Circuit complexity of regular languages</a> : (Presenter - Sunil K S.)<br> This project explores the circuit complexity of regular languages. Here is <a href="http://www.math.cas.cz/preprint/pre-67.pdf">an article</a> for reference.

  References
  Exercises
  Reading

  Circuit complexity of regular languages : (Presenter - Sunil K S.)
  This project explores the circuit complexity of regular languages. Here is an article for reference.
  Scribe : Sunil K S  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 49 : Mon, Apr 16, 05:00 pm-06:00 pm

  <a href="./CS6840/2012/project/AM-Sajin-slides.pdf">Lower Bounds Against Circuits with Limited Negations</a>: (Presenter : Sajin Koroth)<br> The aim is to understand approaches towards proving lower bounds against circuits with limited number of negations. <br> <a href="http://epubs.siam.org/sicomp/resource/1/smjcat/v35/i1/p201_s1">A superpolynomial Lower Bound for a Circuit Computing the Clique Function with at most (1/6)log log n Negation Gates</a> - Kazuyuki Amano and Akira Maruoka

  References
  Exercises
  Reading

  Lower Bounds Against Circuits with Limited Negations: (Presenter : Sajin Koroth)
  The aim is to understand approaches towards proving lower bounds against circuits with limited number of negations.
  A superpolynomial Lower Bound for a Circuit Computing the Clique Function with at most (1/6)log log n Negation Gates - Kazuyuki Amano and Akira Maruoka
  Scribe : Sajin Koroth  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 50 : Tue, Apr 17, 04:00 pm-05:00 pm

  <a href="./CS6840/2012/project/Self-Reducibility-B.pdf">Amplifying Lower bounds by Self-reducibility</a> - (Presenter : Balagopal)<br> This project aims to explore an exciting recent development in the circuit lower bounds against TC^0 and NC^1 which showed some surprises using simple properties like self-reducibility. <br> <a href="http://www.eccc.uni-trier.de/report/2008/038/">Amplifying Lower Bounds by Means of Self-Reducibility</a> - Eric Allender, Michal Koucky.<br> <a href="http://ftp.cs.rutgers.edu/pub/allender/cie.plenary.pdf">New Surprises from Self-reducibility</a> - Eric Allender.

  References
  Exercises
  Reading

  Amplifying Lower bounds by Self-reducibility - (Presenter : Balagopal)
  This project aims to explore an exciting recent development in the circuit lower bounds against TC^0 and NC^1 which showed some surprises using simple properties like self-reducibility.
  Amplifying Lower Bounds by Means of Self-Reducibility - Eric Allender, Michal Koucky.
  New Surprises from Self-reducibility - Eric Allender.
  Scribe : Balagopal  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 51 : Wed, Apr 18, 04:30 pm-05:30 pm

  <a href="./CS6840/2012/project/PolgLogIndFoolsAC0-Dinesh-slides.pdf">Pseudorandomness against Constant Depth Circuits</a> - (Presenter : Dinesh K.) <br> This project is to explore the power of poly-sized AC^0 circuits in distinguishing dependent distributions from the uniform one. Here is a <a href="http://www.cs.toronto.edu/~mbraverm/FoolAC0v5.pdf">recent paper</a> due to Mark Braverman.

  References
  Exercises
  Reading

  Pseudorandomness against Constant Depth Circuits - (Presenter : Dinesh K.)
  This project is to explore the power of poly-sized AC^0 circuits in distinguishing dependent distributions from the uniform one. Here is a recent paper due to Mark Braverman.
  Scribe : Princy Lunawat  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 52 : Thu, Apr 19, 05:00 pm-06:00 pm

  <a href="./CS6840/2012/project/HardnessExtraction-NA.pdf">Hardness Extractors.</a> - (Presenter : Nilkamal Adak) <br> This project looks at general approaches of improving computational hardness. A hardness extractor takes in a Boolean function as input, as well as an advice string, and outputs a Boolean function defined on a smaller number of bits which has close to maximum hardness. The project will explore positive and negative results about these objects. A relevant paper is <a href="http://www.cs.utoronto.ca/~bureshop/eccc_condense.pdf">here</a>.

  References
  Exercises
  Reading

  Hardness Extractors. - (Presenter : Nilkamal Adak)
  This project looks at general approaches of improving computational hardness. A hardness extractor takes in a Boolean function as input, as well as an advice string, and outputs a Boolean function defined on a smaller number of bits which has close to maximum hardness. The project will explore positive and negative results about these objects. A relevant paper is here.
  Scribe : Abdulla Anam  (notes being written up)

  Scribe	:
  Key	:

  References

  :

  None

• Theme 4 : Between P and PSPACE - 17 meetings

  • Meeting 53 : Tue, Jan 03, 12:00 pm-12:50 pm

  Why this course? <br> Course outline, expected course activities, grading, scribing, course project.

  References
  Exercises
  Reading

  Why this course?
  Course outline, expected course activities, grading, scribing, course project.
  

  References

  :

  None

  • Meeting 54 : Thu, Jan 05, 11:00 am-11:50 am

  Barriers to overcome : the story from the 70s. Relativisation, Baker-Gill-Solovoy theorem.

  References	Section 3.2.2 of Arora-Barak Textbook.
  Exercises	What is the complexity the oracle languages that we constructed?
  Reading	<a href="http://www.wisdom.weizmann.ac.il/~oded/PS/roh.ps">Random Oracle Hypothesis</a> - the question of P vs NP relative to a random oracle.

  Barriers to overcome : the story from the 70s. Relativisation, Baker-Gill-Solovoy theorem.
  

  References	:	Section 3.2.2 of Arora-Barak Textbook.
  Exercises	:	What is the complexity the oracle languages that we constructed?
  Reading	:	Random Oracle Hypothesis - the question of P vs NP relative to a random oracle.

  • Meeting 55 : Fri, Jan 06, 10:00 am-10:50 am

  Quest for structure in counting problems. Problem definitions, #SAT, counting versions of "easy" decision problems can be hard. #CYCLE problem.

  References	Chapter on Counting Complexity in Arora Barak Textbook
  Exercises	Is getting an approximation to the number of cycles an easier problem?
  Reading

  Quest for structure in counting problems. Problem definitions, #SAT, counting versions of "easy" decision problems can be hard. #CYCLE problem.
  

  References	:	Chapter on Counting Complexity in Arora Barak Textbook
  Exercises	:	Is getting an approximation to the number of cycles an easier problem?

  • Meeting 56 : Tue, Jan 10, 12:00 pm-12:50 pm

  Counting complexity classes FP, FPSPACE. Non-determinism and counting, the class #P, Basic containments.

  References	Arora-Barak Textbook - Chapter on Counting Complexity.
  Exercises	Attempt to define the class FNP.
  Reading

  Counting complexity classes FP, FPSPACE. Non-determinism and counting, the class #P, Basic containments.
  Scribe : Dinesh K  

  Scribe	:
  Key	:

  References	:	Arora-Barak Textbook - Chapter on Counting Complexity.
  Exercises	:	Attempt to define the class FNP.

  • Meeting 57 : Thu, Jan 12, 11:00 am-11:50 am

  FP vs #P question, a counter part in the decision world. The class PP. PP vs P is equivalent to FP vs #P.

  References	Chapter on counting problems in Arora-Barak texbook.
  Exercises	Use a similar strategy to show P^#P = P^PP.
  Reading

  FP vs #P question, a counter part in the decision world. The class PP. PP vs P is equivalent to FP vs #P.
  Scribe : Prasun Kumar  

  Scribe	:
  Key	:

  References	:	Chapter on counting problems in Arora-Barak texbook.
  Exercises	:	Use a similar strategy to show P^#P = P^PP.

  • Meeting 58 : Fri, Jan 13, 10:00 am-10:50 am

  Reductions in the counting world. Parsimonious reductions. #P-hardness. #SAT is #P-complete. Permanent and determinant. Counting number of perfect matchings in bipartite graphs.

  References	Counting Complexity Chapter in DK Book.
  Exercises	Work out the details that Cook-Levin reduction showing the SAT is NP-complete preserves the number of certificates. That is the number of satisfying assignments of the formula produced is precisely the number of accepting paths of the NP-machine that we start with.<br> Check if SAT to 3SAT us parsimonious.<br> Check if 3SAT to INDSET is parsimonious.
  Reading

  Reductions in the counting world. Parsimonious reductions. #P-hardness. #SAT is #P-complete. Permanent and determinant. Counting number of perfect matchings in bipartite graphs.
  Scribe : Sunil K S  

  Scribe	:
  Key	:

  References	:	Counting Complexity Chapter in DK Book.
  Exercises	:	Work out the details that Cook-Levin reduction showing the SAT is NP-complete preserves the number of certificates. That is the number of satisfying assignments of the formula produced is precisely the number of accepting paths of the NP-machine that we start with. Check if SAT to 3SAT us parsimonious. Check if 3SAT to INDSET is parsimonious.

  • Meeting 59 : Mon, Jan 16, 08:00 am-08:50 am

  A combinatorial interpretation of permanent of integer matrices, using cycle covers.

  References	Counting Complexity Chapter in DK textbook.
  Exercises	Show that computing the Permanent over integer matrices is in FP^#P
  Reading

  A combinatorial interpretation of permanent of integer matrices, using cycle covers.
  Scribe : Dinesh K  

  Scribe	:
  Key	:

  References	:	Counting Complexity Chapter in DK textbook.
  Exercises	:	Show that computing the Permanent over integer matrices is in FP^#P

  • Meeting 60 : Tue, Jan 17, 12:00 pm-12:50 pm

  Permanent is #P-complete, Counting the Number of Perfect Matchings is #P-complete. Valiant's gadgets and constructions and complete proof.

  References	<a href="www.cs.princeton.edu/theory/complexity/countchap.pdf">Chapter on counting comlexity</a> in Arora-Barak Textbook.<br> <a href="www.cs.au.dk/~arnsfelt/CT10/scribenotes/lecture10.pdf">Lecture 10</a> from Kristoffer Hanses's course.
  Exercises	Think about "why is the reduction not parsimonious?". Remember that if the reduction is parsimonious we have SAT is in P. We discussed this briefly in the class. But get the complete details right.
  Reading	<a href="http://www.cse.sc.edu/~fenner/papers/cc-and-q.pdf">Counting Complexity and Quantum Computation</a> - A survey (for those who are taking the Quantum computing course)

  Permanent is #P-complete, Counting the Number of Perfect Matchings is #P-complete. Valiant's gadgets and constructions and complete proof.
  Scribe : T Devanathan   (to be edited)

  Scribe	:
  Key	:

  References	:	Chapter on counting comlexity in Arora-Barak Textbook. Lecture 10 from Kristoffer Hanses's course.
  Exercises	:	Think about "why is the reduction not parsimonious?". Remember that if the reduction is parsimonious we have SAT is in P. We discussed this briefly in the class. But get the complete details right.
  Reading	:	Counting Complexity and Quantum Computation - A survey (for those who are taking the Quantum computing course)

  • Meeting 61 : Tue, Jan 24, 12:00 pm-12:50 pm

  Schwartz-Zippel Lemma, BPP, Error reduction. Amplification Lemma.

  References
  Exercises	Calculate precisely the value of k that you would choose if the error probability that you want is less than 2^{q(n)} for some polynomial k? Is k a polynomial in n? <br> More interesting question. Is NP contained in BPP? Why would not the same proof that NP is contained in PP work? Or at least, could that strategy be saved?
  Reading	<a href="http://eccc.hpi-web.de/report/2010/096/">An Alternative Proof of Schwartz-Zippel Lemma</a> - Dana Moshkovitz

  Schwartz-Zippel Lemma, BPP, Error reduction. Amplification Lemma.
  Scribe : Sunil K S  

  Scribe	:
  Key	:

  References	:	None
  Exercises	:	Calculate precisely the value of k that you would choose if the error probability that you want is less than 2^{q(n)} for some polynomial k? Is k a polynomial in n? More interesting question. Is NP contained in BPP? Why would not the same proof that NP is contained in PP work? Or at least, could that strategy be saved?
  Reading	:	An Alternative Proof of Schwartz-Zippel Lemma - Dana Moshkovitz

  • Meeting 62 : Wed, Jan 25, 11:00 am-11:50 am

  Derandomizing BPP, trivial one, Quantifier Based - BPP is in Sigma^2.

  References	Du-Ko Book <br> Kozen's Book on Theory of Computation Lecture on BPP is in Sigma^2
  Exercises	Prove that Parity map is one-one.
  Reading

  Derandomizing BPP, trivial one, Quantifier Based - BPP is in Sigma^2.
  Scribe : Princy Lunawat  

  Scribe	:
  Key	:

  References	:	Du-Ko Book Kozen's Book on Theory of Computation Lecture on BPP is in Sigma^2
  Exercises	:	Prove that Parity map is one-one.

  • Meeting 63 : Sat, Jan 28, 02:00 pm-02:50 pm

  A strange consequence of amplification result on BPP. Advice complexity classes. The class P/poly. BPP is contained in P/poly.

  References	Du-Ko Book, section 8.6 and 6.6
  Exercises
  Reading

  A strange consequence of amplification result on BPP. Advice complexity classes. The class P/poly. BPP is contained in P/poly.
  Scribe : Sajin Koroth  

  Scribe	:
  Key	:

  References

  :

  Du-Ko Book, section 8.6 and 6.6

  • Meeting 64 : Sat, Jan 28, 03:00 pm-03:50 pm

  Complete problem for Sigma_k. Self reduction property of SAT.

  References
  Exercises
  Reading

  Complete problem for Sigma_k. Self reduction property of SAT.
  Scribe : Sajin Koroth  

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 65 : Mon, Jan 30, 08:00 am-08:50 am

  Karp-Lipton-Sipser Theorem : If NP is contained in P/poly then PH collapses to Sigma^2.

  References	Secton 6.2 in the end of Du-Ko textbook. <br> (Problem Set spoiler warning.)
  Exercises	Try this - if you have not seen it before: Does NP=P does imply a PH collapse? Converse? <br> We discussed examples of problems in Sigma_2 in class. Are there any in Sigma_3 that you can think of? How about an arbitrary level of the PH?
  Reading

  Karp-Lipton-Sipser Theorem : If NP is contained in P/poly then PH collapses to Sigma^2.
  Scribe : Anup Joshi  

  Scribe	:
  Key	:

  References	:	Secton 6.2 in the end of Du-Ko textbook. (Problem Set spoiler warning.)
  Exercises	:	Try this - if you have not seen it before: Does NP=P does imply a PH collapse? Converse? We discussed examples of problems in Sigma_2 in class. Are there any in Sigma_3 that you can think of? How about an arbitrary level of the PH?

  • Meeting 66 : Tue, Jan 31, 12:00 pm-12:50 pm

  Back to ParityP, PP. , BP, exists, ParityP as operators on complexity classes. Toda's theorem and the interpretation. Proof strategy. Statement of Valiant Vazirani Lemma.

  References	No reference.
  Exercises	Show that Parity as an operator, operated on the class parity P remains within the same class.
  Reading	<a href="http://www.cse.buffalo.edu/~regan/papers/pdf/MPJrev.pdf">The Power of the Middle Bit of a #P Function</a>

  Back to ParityP, PP. , BP, exists, ParityP as operators on complexity classes. Toda's theorem and the interpretation. Proof strategy. Statement of Valiant Vazirani Lemma.
  Scribe : Nilkamal Adak   (to be edited)

  Scribe	:
  Key	:

  References	:	No reference.
  Exercises	:	Show that Parity as an operator, operated on the class parity P remains within the same class.
  Reading	:	The Power of the Middle Bit of a #P Function

  • Meeting 67 : Thu, Feb 02, 11:00 am-11:50 am

  Valiant-Vazirani Lemma

  References
  Exercises
  Reading

  Valiant-Vazirani Lemma
  Scribe : Rahul CS  

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 68 : Mon, Feb 06, 08:00 am-08:50 am

  Amplification. Extensions to PH.

  References
  Exercises
  Reading

  Amplification. Extensions to PH.
  Scribe : Princy Lunawat   (to be edited)

  Scribe	:
  Key	:

  References

  :

  None

  • Meeting 69 : Tue, Feb 07, 12:00 pm-12:50 pm

  Proof of Toda's Theorem.

  References
  Exercises
  Reading

  Proof of Toda's Theorem.
  Scribe : Sunil K S   (rough draft; better one coming soon)

  Scribe	:
  Key	:

  References

  :

  None