Meetings

Click on the theme item for the meeting plan for that theme.Click on the meeting item for references, exercises, and additional reading related to it.

**Theme 1 : Probabilistic Proof Systems**- 15 meetings- Meeting 01 : Thu, Feb 09, 11:00 am-11:50 am
- Meeting 02 : Mon, Feb 13, 08:00 am-08:50 am
- Meeting 03 : Tue, Feb 14, 12:00 pm-12:50 pm
- Meeting 04 : Thu, Feb 16, 11:00 am-11:50 am
- Meeting 05 : Fri, Feb 17, 10:00 am-10:50 am
- Meeting 06 : Sat, Feb 18, 10:00 am-10:50 am
- Meeting 07 : Sat, Feb 18, 11:00 am-11:50 am
- Meeting 08 : Mon, Feb 20, 08:00 am-08:50 am
- Meeting 09 : Tue, Feb 21, 12:00 pm-12:50 pm
- Meeting 10 : Thu, Feb 23, 11:00 am-11:50 am
- Meeting 11 : Fri, Feb 24, 10:00 am-10:50 am
- Meeting 12 : Mon, Feb 27, 08:00 am-08:50 am
- Meeting 13 : Tue, Feb 28, 12:00 pm-12:50 pm
- Meeting 14 : Fri, Mar 02, 10:00 am-10:50 am
- Meeting 15 : Sat, Mar 03, 02:00 pm-03:30 pm

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

Du-Ko Book, Chapter10Proof 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 References : Email and the Unexpected Power of Interaction - Lazlo Babai. 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. 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. 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. 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 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 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 References : Lecture 12 from Andrej Bogdanov's course. Reading : The history of PCP theorem by Ryan Odonnel Linearity testing. Local decoding. The proof of PCP for LIN. References 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. Proof of Linearity Testing References Exercises Reading 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 Generalization to Quadratic Programs. References Exercises Reading 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 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 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 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 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 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
- Meeting 17 : Tue, Mar 06, 12:00 pm-12:50 pm
- Meeting 18 : Fri, Mar 09, 10:00 am-10:50 am
- Meeting 19 : Mon, Mar 12, 08:00 am-08:50 am
- Meeting 20 : Tue, Mar 13, 12:00 pm-12:50 pm
- Meeting 21 : Thu, Mar 15, 11:00 am-11:50 am
- Meeting 22 : Fri, Mar 16, 10:00 am-10:50 am
- Meeting 23 : Mon, Mar 19, 08:00 am-08:50 am
- Meeting 24 : Tue, Mar 20, 12:00 pm-12:50 pm
- Meeting 25 : Thu, Mar 22, 11:00 am-11:50 am
- Meeting 26 : Mon, Mar 26, 08:00 am-08:50 am
- Meeting 27 : Tue, Mar 27, 12:00 pm-12:50 pm
- Meeting 28 : Thu, Mar 29, 11:00 am-11:50 am
- Meeting 29 : Fri, Mar 30, 10:00 am-10:50 am
- Meeting 30 : Sat, Mar 31, 11:00 am-12:00 pm
- Meeting 31 : Mon, Apr 02, 08:00 am-08:50 am
- Meeting 32 : Tue, Apr 03, 10:00 am-10:50 am
- Meeting 33 : Tue, Apr 10, 12:00 pm-12:50 pm
- Meeting 34 : Thu, Apr 12, 11:00 am-11:50 am
- Meeting 35 : Fri, Apr 13, 10:00 am-10:50 am
- Meeting 36 : Mon, Apr 16, 08:00 am-08:50 am
- Meeting 37 : Tue, Apr 17, 12:00 pm-12:50 pm
- Meeting 38 : Thu, Apr 19, 11:00 am-11:50 am
- Meeting 39 : Fri, Apr 20, 09:30 am-11:00 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. Shannon's counting argument, Lupanov's construction. References Exercises Reading References : None P/poly = functions computed by polynomial sized circuits. CVP is P-complete under log-space reductions. References Exercises Reading References : None Uniformity, Log-space Uniformity. P is computed by uniform polysize circuits. Review of parameters and questions. Circuit Lower bound Problem. References Exercises Reading References : None 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 References : None 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 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? 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. 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 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 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 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 References : Boolean Formulas - The classics. 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 References : Formula Lower bounds - The classics

Slides - Alexander Kulikov.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 References : Section 6.1 of Paul Beame's Survey : Switching Lemma Primer 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 Proof of the switching Lemma. References Exercises Reading References : None 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 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 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 References : None 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 References : None 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 Width of the Branching Programs. Programs over groups. Permutations. Conjugates, Cycle conjugacy lemma. References Exercises Reading References : None 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 : Exercises : Explore the proof that S_5 is the smallest non-solvable symmetric group. 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 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
- Meeting 41 : Sat, Mar 31, 02:00 pm-03:00 pm
- Meeting 42 : Sat, Mar 31, 03:00 pm-04:00 pm
- Meeting 43 : Sat, Mar 31, 04:00 pm-05:00 pm
- Meeting 44 : Mon, Apr 09, 05:30 pm-06:30 pm
- Meeting 45 : Tue, Apr 10, 04:00 pm-05:00 pm
- Meeting 46 : Tue, Apr 10, 05:00 pm-06:00 pm
- Meeting 47 : Wed, Apr 11, 04:30 pm-05:30 pm
- Meeting 48 : Fri, Apr 13, 05:00 pm-06:00 pm
- Meeting 49 : Mon, Apr 16, 05:00 pm-06:00 pm
- Meeting 50 : Tue, Apr 17, 04:00 pm-05:00 pm
- Meeting 51 : Wed, Apr 18, 04:30 pm-05:30 pm
- Meeting 52 : Thu, Apr 19, 05:00 pm-06:00 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 <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 VasudevanScribe : Key : References : None <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 <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 <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 <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 <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 <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 <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 <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 <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 <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 <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
- Meeting 54 : Thu, Jan 05, 11:00 am-11:50 am
- Meeting 55 : Fri, Jan 06, 10:00 am-10:50 am
- Meeting 56 : Tue, Jan 10, 12:00 pm-12:50 pm
- Meeting 57 : Thu, Jan 12, 11:00 am-11:50 am
- Meeting 58 : Fri, Jan 13, 10:00 am-10:50 am
- Meeting 59 : Mon, Jan 16, 08:00 am-08:50 am
- Meeting 60 : Tue, Jan 17, 12:00 pm-12:50 pm
- Meeting 61 : Tue, Jan 24, 12:00 pm-12:50 pm
- Meeting 62 : Wed, Jan 25, 11:00 am-11:50 am
- Meeting 63 : Sat, Jan 28, 02:00 pm-02:50 pm
- Meeting 64 : Sat, Jan 28, 03:00 pm-03:50 pm
- Meeting 65 : Mon, Jan 30, 08:00 am-08:50 am
- Meeting 66 : Tue, Jan 31, 12:00 pm-12:50 pm
- Meeting 67 : Thu, Feb 02, 11:00 am-11:50 am
- Meeting 68 : Mon, Feb 06, 08:00 am-08:50 am
- Meeting 69 : Tue, Feb 07, 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 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. 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? 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 References : Arora-Barak Textbook - Chapter on Counting Complexity. Exercises : Attempt to define the class FNP. 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 References : Chapter on counting problems in Arora-Barak texbook. Exercises : Use a similar strategy to show P^#P = P^PP. 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 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.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 References : Counting Complexity Chapter in DK textbook. Exercises : Show that computing the Permanent over integer matrices is in FP^#P 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) 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) 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 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 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 References : Du-Ko Book

Kozen's Book on Theory of Computation Lecture on BPP is in Sigma^2Exercises : Prove that Parity map is one-one. 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 References : Du-Ko Book, section 8.6 and 6.6 Complete problem for Sigma_k. Self reduction property of SAT. References Exercises Reading References : None 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 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?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 Valiant-Vazirani Lemma References Exercises Reading References : None Amplification. Extensions to PH. References Exercises Reading Amplification. Extensions to PH.

Scribe : Princy Lunawat (to be edited)Scribe : Key : References : None Proof of Toda's Theorem. References Exercises Reading References : None