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 : Counting vs Decision**- 10 meetings- Meeting 01 : Mon, Jan 12, 10:00 am-10:50 am
- Review the reduction from SAT to HAMCYCLE Problem.
- Calculate the exact gadget length that you need for the reduction to #CYCLE problem.
- Meeting 02 : Tue, Jan 13, 09:00 am-09:50 am
- Meeting 03 : Wed, Jan 14, 08:00 am-08:50 am
- Review the reduction between SAT to VC that we have done in the previous course and check if it is parsimonious. If not, can you make parsimonious.
- Meeting 04 : Tue, Jan 20, 09:00 am-09:50 am
- Meeting 05 : Wed, Jan 21, 08:00 am-08:50 am
- Meeting 06 : Fri, Jan 23, 12:00 pm-12:50 pm
- Meeting 07 : Tue, Jan 27, 09:00 am-09:50 am
- Meeting 08 : Wed, Jan 28, 08:00 am-08:50 am
- Meeting 09 : Mon, Feb 02, 10:00 am-10:50 am
- Meeting 10 : Tue, Feb 03, 09:00 am-09:50 am

Administrative Announcements. Counting Complexity. Computational Model. The class FP. Clique vs #Clique Problem. Cycle vs #Cycle Problem. If #CYCLE is in FP, then P = NP. References Arora-Barak Textbook chapter on Complexity of Counting - section 17.1 Exercises <ul> <li>Review the reduction from SAT to HAMCYCLE Problem.</li> <li>Calculate the exact gadget length that you need for the reduction to #CYCLE problem. </li> </ul> Reading Administrative Announcements. Counting Complexity. Computational Model. The class FP. Clique vs #Clique Problem. Cycle vs #Cycle Problem. If #CYCLE is in FP, then P = NP.References : Arora-Barak Textbook chapter on Complexity of Counting - section 17.1 Exercises : The Class #P. #P = FP if and only if PP = P. The bits and value of the #P function. Counting as a generalization of NP, BPP, RP, PP and Parity P. References Arora-Barak Textbook chapter on Complexity of Counting - section 17.2 Exercises Question to think about : how do we develop a theory for integer valued functions which produces negative values too? <br> A reading related to the same : <a href="http://dx.doi.org/10.1006/inco.1996.0080">Gap Definable Counting Classes</a> Reading <ul> <li><a href="www.cse.buffalo.edu/~regan/papers/pdf/MPJrev.pdf">Power of the middle bit of a #P function.</a></li> </ul> The Class #P. #P = FP if and only if PP = P. The bits and value of the #P function. Counting as a generalization of NP, BPP, RP, PP and Parity P.References : Arora-Barak Textbook chapter on Complexity of Counting - section 17.2 Exercises : Question to think about : how do we develop a theory for integer valued functions which produces negative values too?

A reading related to the same : Gap Definable Counting ClassesReading : Notion of #P-completeness.Attempt 1 : Parsimonious reductions. #SAT is #P-complete. #IND-SET is #P-complete. Shortcomings of the theory. General oracle query model of #P-completeness. Transitivity of reductions. References Arora-Barak Textbook chapter on Complexity of Counting - section 17.3 Exercises <ul> <li>Review the reduction between SAT to VC that we have done in the previous course and check if it is parsimonious. If not, can you make parsimonious.</li> </ul> Reading Notion of #P-completeness.Attempt 1 : Parsimonious reductions. #SAT is #P-complete. #IND-SET is #P-complete. Shortcomings of the theory. General oracle query model of #P-completeness. Transitivity of reductions.References : Arora-Barak Textbook chapter on Complexity of Counting - section 17.3 Exercises : The permanent and the determinant functions. Combinatorial interpretation of permanent of matrix with 0-1 entries. Counting perfect matchings. References Arora-Barak Textbook chapter on Complexity of Counting - section 17.3 Exercises Reading The permanent and the determinant functions. Combinatorial interpretation of permanent of matrix with 0-1 entries. Counting perfect matchings.References : Arora-Barak Textbook chapter on Complexity of Counting - section 17.3 Permanent if integer matrices. Combinatorial interpretation in terms of cycle covers. The upper bound of FP^#P. Permanent over {-1,0,1} matrices is #P-complete. Proof idea, Gadget construction. References Kristoffer Hansen's <a href="http://www.cs.au.dk/~arnsfelt/CT10/scribenotes/lecture10.pdf">Lecture Notes</a> Exercises Reading Permanent if integer matrices. Combinatorial interpretation in terms of cycle covers. The upper bound of FP^#P. Permanent over {-1,0,1} matrices is #P-complete. Proof idea, Gadget construction.References : Kristoffer Hansen's Lecture Notes Reduction from PERM(-1,0,1) to PERM(0,1,...,n), and then to PERM(0,1). References Kristoffer Hansen's <a href="http://www.cs.au.dk/~arnsfelt/CT10/scribenotes/lecture10.pdf">Lecture Notes</a> Exercises Reading Reduction from PERM(-1,0,1) to PERM(0,1,...,n), and then to PERM(0,1).References : Kristoffer Hansen's Lecture Notes Toda's Theorem. The Roadmap. U-SAT : Valiant-Vazirani Reduction. References Arora-Barak Textbook. Exercises Reading Toda's Theorem. The Roadmap. U-SAT : Valiant-Vazirani Reduction.References : Arora-Barak Textbook. Hash families, Pairwise independent hashing. Proof of the uniqueness lemma. Reduction to Parity SAT. Amplification of success probability. References <a href="http://www.cs.umd.edu/~jkatz/complexity/f11/lecture24.pdf">Jonathan Katz's Lecture notes</a> Exercises Reading Hash families, Pairwise independent hashing. Proof of the uniqueness lemma. Reduction to Parity SAT. Amplification of success probability.References : Jonathan Katz's Lecture notes Generalizations to show that PH BP-reduces to Parity-SAT. References <a href="http://www.cs.umd.edu/~jkatz/complexity/f11/lecture24.pdf">Jonathan Katz's Lecture notes</a> Exercises Reading Generalizations to show that PH BP-reduces to Parity-SAT.References : Jonathan Katz's Lecture notes Any language that BP-reduces to Parity-SAT can be decided with a counting oracle. Proof idea. Toda's polynomials. Details of the proof. References <a href="http://www.cs.umd.edu/~jkatz/complexity/f11/lecture24.pdf">Jonathan Katz's Lecture notes</a><br> Arora Barak's Textbook. Exercises Reading Any language that BP-reduces to Parity-SAT can be decided with a counting oracle. Proof idea. Toda's polynomials. Details of the proof.References : Jonathan Katz's Lecture notes

Arora Barak's Textbook.**Theme 2 : Interactive Proof Systems**- 9 meetings- Meeting 11 : Wed, Feb 04, 08:00 am-08:50 am
- Meeting 12 : Fri, Feb 06, 12:00 pm-12:50 pm
- Meeting 13 : Sat, Feb 07, 11:00 am-01:00 pm
- MA is in Sigma_2.
- Completeness for MA can be improved to 1.
- MA is contained in AM.
- Completeness for AM can be improved to 1.
- AM is contained in Pi_2.
- If CoNP is in AM, PH collapses to AM.
- Meeting 14 : Mon, Feb 09, 10:00 am-10:50 am
- Meeting 15 : Tue, Feb 10, 09:00 am-09:50 am
- Meeting 16 : Wed, Feb 11, 08:00 am-08:50 am
- Meeting 17 : Fri, Feb 13, 12:00 pm-12:50 pm
- Meeting 18 : Mon, Feb 16, 10:00 am-10:50 am
- Meeting 19 : Wed, Feb 18, 08:00 am-08:50 am

Interactive Proof Systems. Deterministic, Randomized. <br> Power of Interaction 1 : Interactive Proof System for GNI. References Arora Barak Textbook Chapter on Interactive Proofs. First few sections. Exercises Reading Interactive Proof Systems. Deterministic, Randomized.

Power of Interaction 1 : Interactive Proof System for GNI.References : Arora Barak Textbook Chapter on Interactive Proofs. First few sections. Power of Interaction 2: Making BPP one sided error using interaction. References Jonathan Katz's <a href="http://www.cs.umd.edu/~jkatz/complexity/f11/lecture16.pdf">lecture notes</a>. Exercises Reading Power of Interaction 2: Making BPP one sided error using interaction.References : Jonathan Katz's lecture notes. (Compensatory class for Jan 16 & 19) <br> Public Coin Protocols. Classes AM and MA. Structural theorems:<ul> <li>MA is in Sigma_2.</li> <li>Completeness for MA can be improved to 1.</li> <li>MA is contained in AM.</li> <li>Completeness for AM can be improved to 1.</li> <li>AM is contained in Pi_2.</li> <li>If CoNP is in AM, PH collapses to AM.</li> </ul> References Jonathan Katz's <a href="http://www.cs.umd.edu/~jkatz/complexity/f11/lecture16.pdf">lecture notes</a>. <br> Exercises Reading <a href="http://pages.cs.wisc.edu/~sfdiehl/ma-full.pdf">Lowerbounds for switching Arthur and Merlin</a>. (Compensatory class for Jan 16 & 19)

Public Coin Protocols. Classes AM and MA. Structural theorems:References : Jonathan Katz's lecture notes. Reading : Lowerbounds for switching Arthur and Merlin. Set lower bound protocol.Public Coin Protocol for GNI. GNI is in AM. Set lower bound protocol.<br> Hence if GI is NP-complete, PH = Sigma_2. References <a href="http://www.cs.umd.edu/~jkatz/complexity/f11/lecture17.pdf">Jonathan Katz's Lecture Notes</a> Exercises Reading <a href="http://crypto.cs.mcgill.ca/~crepeau/COMP647/2007/TOPIC01/goldwasser-Sipser.pdf">IP = AM : Paper by Goldwasser and Sipser</a> Set lower bound protocol.Public Coin Protocol for GNI. GNI is in AM. Set lower bound protocol.

Hence if GI is NP-complete, PH = Sigma_2.References : Jonathan Katz's Lecture Notes Reading : IP = AM : Paper by Goldwasser and Sipser Protocol for permanent. LFKN protocol and analysis. Protocol for #SAT. Generalization to prove IP = PSPACE. References Exercises Reading Protocol for permanent. LFKN protocol and analysis. Protocol for #SAT. Generalization to prove IP = PSPACE.References : None Arithmetization of QBF. Properties of Arithmetization. Mindblocks with respect to the protocol #SAT protocol. References Exercises Reading Arithmetization of QBF. Properties of Arithmetization. Mindblocks with respect to the protocol #SAT protocol.References : None Degree Reduction using Quantifier introduction. Examples. References Exercises Reading Degree Reduction using Quantifier introduction. Examples.References : None Chinese remaindering. Coefficient size reduction. Completing Shamir's Proof of PSPACE is contained in IP<br> IP is contained in NEXP. <br> IP is contained in PSPACE. References Exercises Reading Chinese remaindering. Coefficient size reduction. Completing Shamir's Proof of PSPACE is contained in IP

IP is contained in NEXP.

IP is contained in PSPACE.References : None Multiprover interactive Proofs. Outline of MIP = NEXP proof. From MIP to PCP. References Exercises Reading Multiprover interactive Proofs. Outline of MIP = NEXP proof. From MIP to PCP.References : None **Theme 3 : PCPs and Inapproximability**- 12 meetings- Meeting 20 : Fri, Feb 20, 12:00 pm-12:50 pm
- Meeting 21 : Mon, Feb 23, 10:00 am-10:50 am
- Meeting 22 : Tue, Feb 24, 09:00 am-09:50 am
- Meeting 23 : Wed, Feb 25, 08:00 am-08:50 am
- Meeting 24 : Fri, Feb 27, 12:00 pm-12:50 pm
- Meeting 25 : Mon, Mar 02, 12:00 pm-12:50 pm
- Meeting 26 : Tue, Mar 03, 09:00 am-09:50 am
- Meeting 27 : Wed, Mar 04, 08:00 am-08:50 am
- Meeting 28 : Mon, Mar 09, 10:00 am-10:50 am
- Meeting 29 : Tue, Mar 10, 09:00 am-09:50 am
- Meeting 30 : Wed, Mar 11, 08:00 am-08:50 am
- Meeting 31 : Fri, Mar 13, 12:00 pm-12:50 pm

The setting of Inapproximability. Independent Set Problem. GapIS. Connection between GapIS problem and inapproximability of IS. GapSAT and q-GapCSP. <br> From PCP theorem to hardness of q-GapCSP. References Exercises Reading Course by Prahladh Harsha : <a href="http://www.tcs.tifr.res.in/~prahladh/teaching/2009-10/limits/">Limits of approximation algorithms : PCPs and Unique Games - Spring Semester (2009-10)</a> The setting of Inapproximability. Independent Set Problem. GapIS. Connection between GapIS problem and inapproximability of IS. GapSAT and q-GapCSP.

From PCP theorem to hardness of q-GapCSP.References : None Reading : Course by Prahladh Harsha : Limits of approximation algorithms : PCPs and Unique Games - Spring Semester (2009-10) L is in PCP(O(log n),q) if and only if L reduces to q-GapCSP. Statement of PCP theorem. References Exercises Reading Course by Prahladh Harsha : <a href="http://www.tcs.tifr.res.in/~prahladh/teaching/2009-10/limits/">Limits of approximation algorithms : PCPs and Unique Games - Spring Semester (2009-10)</a> L is in PCP(O(log n),q) if and only if L reduces to q-GapCSP. Statement of PCP theorem.References : None Reading : Course by Prahladh Harsha : Limits of approximation algorithms : PCPs and Unique Games - Spring Semester (2009-10) Hardness of GapSAT. Hardness of GapIS. Constant factor inapproximability for Independent Set problem. <br> History of PCP theorem. A weaker version first. PCP for LIN. References Exercises Reading Course by Prahladh Harsha : <a href="http://www.tcs.tifr.res.in/~prahladh/teaching/2009-10/limits/">Limits of approximation algorithms : PCPs and Unique Games - Spring Semester (2009-10)</a> Hardness of GapSAT. Hardness of GapIS. Constant factor inapproximability for Independent Set problem.

History of PCP theorem. A weaker version first. PCP for LIN.References : None Reading : Course by Prahladh Harsha : Limits of approximation algorithms : PCPs and Unique Games - Spring Semester (2009-10) Weak PCP for LIN. Random vector testing. Allowing exponential sized proofs. Need for linearity testing. BLR Test for linearity. References Exercises Reading Weak PCP for LIN. Random vector testing. Allowing exponential sized proofs. Need for linearity testing. BLR Test for linearity.References : None Reading : BLR test. Closeness to a Linear Function. Proof of Linearity testing. References Exercises Reading BLR test. Closeness to a Linear Function. Proof of Linearity testing.References : None Reading : Proof of correctness of linearity testing using combinatorial arguments. Soundness for PCP for LIN. <br> Generalizing to setting of Qudratic Equations. Proof of consistency. References Exercises Reading Proof of correctness of linearity testing using combinatorial arguments. Soundness for PCP for LIN.

Generalizing to setting of Qudratic Equations. Proof of consistency.References : None Reading : Soundness Analysis of the weak PCP for QE.<br> Dinur's proof of PCP theorem. The strategy. Constraint graph of a 2-query CSP. Outline of the proof. Query reduction. References Exercises Reading Soundness Analysis of the weak PCP for QE.

Dinur's proof of PCP theorem. The strategy. Constraint graph of a 2-query CSP. Outline of the proof. Query reduction.References : None Reading : Proof of Query reduction at the expense of alphabet size and soundness. References Exercises Reading Proof of Query reduction at the expense of alphabet size and soundness.References : None Reading : Degree reduction and detailed proof of upper bound for increase in the soundness. Role of expanders. References Exercises Reading Degree reduction and detailed proof of upper bound for increase in the soundness. Role of expanders.References : None Reading : Expanderization. Review of second largest eigen value. Soundness Amplification (construction and overview). References Exercises Reading Expanderization. Review of second largest eigen value. Soundness Amplification (construction and overview).References : None Reading : Alphabet Reduction construction and proof sketch. References Exercises Reading Alphabet Reduction construction and proof sketch.References : None Reading : Intution and proof sketch for soundness amplification. References Exercises Reading Intution and proof sketch for soundness amplification.References : None Reading : **Theme 4 : Boolean Circuit Complexity**- 25 meetings- Meeting 32 : Mon, Mar 16, 10:00 am-10:50 am
- Class Notes
- Zwick's Notes (sent to the mailing list) is the reference for Post's theorem.
- Meeting 33 : Tue, Mar 17, 09:00 am-09:50 am
- Previous Year's Class Notes
- Jukna's Textbook Section 1.4
- Meeting 34 : Fri, Mar 20, 12:00 pm-12:50 pm
- Jukna's Textbook Section 1.6
- Meeting 35 : Sat, Mar 21, 10:00 am-11:30 am
- Class Notes
- Meeting 36 : Mon, Mar 23, 10:00 am-10:50 am
- Class Notes
- Meeting 37 : Tue, Mar 24, 09:00 am-09:50 am
- Lecture Notes from V.Vinay (sent to the mailing list)
- Meeting 38 : Wed, Mar 25, 08:00 am-08:50 am
- Class Notes
- Meeting 39 : Fri, Mar 27, 12:00 pm-12:50 pm
- Class Notes
- Jukna's Textbook Section 6.1
- Meeting 40 : Sat, Mar 28, 10:00 am-11:30 am
- Jukna's Textbook Section 6.2, 6.3
- Meeting 41 : Mon, Mar 30, 10:00 am-10:50 am
- Jukna's Textbook Section 6.2, 6.3
- Meeting 42 : Tue, Mar 31, 09:00 am-09:50 am
- Jukna's Textbook Section 6.2, 6.3, 6.4
- Meeting 43 : Mon, Apr 06, 10:00 am-10:50 am
- Switching Lemma Primer - Paul Beame.
- Meeting 44 : Tue, Apr 07, 09:00 am-09:50 am
- Switching Lemma Primer - Paul Beame.
- Meeting 45 : Wed, Apr 08, 08:00 am-08:50 am
- Switching Lemma Primer - Paul Beame.
- Meeting 46 : Fri, Apr 10, 12:00 pm-12:50 pm
- Switching Lemma Primer - Paul Beame.
- Meeting 47 : Mon, Apr 13, 10:00 am-10:50 am
- Jukna's Textbook section 12.3
- Meeting 48 : Tue, Apr 14, 09:00 am-09:50 am
- Jukna's Textbook section 2.6, 12.3 and 12.4
- Meeting 49 : Wed, Apr 15, 08:00 am-08:50 am
- Lectures Notes by Ramprasad Saptarishi in Arvind's course
- Meeting 50 : Fri, Apr 17, 12:00 pm-12:50 pm
- Lectures Notes by Ramprasad Saptarishi in Arvind's course
- Meeting 51 : Mon, Apr 20, 10:00 am-10:50 am
- Meeting 52 : Tue, Apr 21, 09:00 am-09:50 am
- Meeting 53 : Wed, Apr 22, 08:00 am-08:50 am
- Meeting 54 : Fri, Apr 24, 12:00 pm-12:50 pm
- Meeting 55 : Mon, Apr 27, 10:00 am-10:50 am
- Meeting 56 : Tue, Apr 28, 09:00 am-09:50 am

Boolean Circuit Model of Computation. Why circuits? Connections to parallel algorithm design. The relevant parameters of the circuit model - Size, depth, fanin. What is allowed as a gate? Basis and completeness. Post's characterization of a complete basis (statement). References <ul> <li>Class Notes</li> <li>Zwick's Notes (sent to the mailing list) is the reference for Post's theorem.</li> </ul> Exercises Reading Boolean Circuit Model of Computation. Why circuits? Connections to parallel algorithm design. The relevant parameters of the circuit model - Size, depth, fanin. What is allowed as a gate? Basis and completeness. Post's characterization of a complete basis (statement).References : PSIZE = P/poly. Circuit Lower Bound Problem. Shannon's counting argument. Lupanov's construction (sketch) References <ul> <li>Previous Year's Class Notes</li> <li>Jukna's Textbook Section 1.4</li> </ul> Exercises Reading PSIZE = P/poly. Circuit Lower Bound Problem. Shannon's counting argument. Lupanov's construction (sketch)References : Gate Elimination Argument for Circuit Lower Bounds. Lower bounds for Parity and Threshold. References <ul> <li>Jukna's Textbook Section 1.6</li> </ul> Exercises Reading Gate Elimination Argument for Circuit Lower Bounds. Lower bounds for Parity and Threshold.References : Uniformity, Log-space Uniformity. P is computed by uniform polysize circuits. <br> Parity and Addition Functions. NC^1 and AC^0. Uniform NC^1 is contained in L. References <ul> <li>Class Notes</li> </ul> Exercises Reading Uniformity, Log-space Uniformity. P is computed by uniform polysize circuits.

Parity and Addition Functions. NC^1 and AC^0. Uniform NC^1 is contained in L.References : The class AC-hierarchy. Interleaving with NC hierarchy. NL is contained in uniform-AC^1. References <ul> <li>Class Notes</li> </ul> Exercises Reading The class AC-hierarchy. Interleaving with NC hierarchy. NL is contained in uniform-AC^1.References : Back to NC^1, Adding n, n-bit numbers. Trivial AC^1 upper bound. Offman's technique and the NC^1 upper bound. Majority in NC^1. Constant depth reductions. Zooming in to NC^1. Review of constant depth reductions. Six Problems : ADD(n,n). MULT(2,n), Th(n,k), BCOUNT, MAJ. NC^1 upper bounds. Constant depth reductions among these problems. Motivation and definition of the class TC^0. References <ul> <li>Lecture Notes from V.Vinay (sent to the mailing list)</li> </ul> Exercises Reading Back to NC^1, Adding n, n-bit numbers. Trivial AC^1 upper bound. Offman's technique and the NC^1 upper bound. Majority in NC^1. Constant depth reductions. Zooming in to NC^1. Review of constant depth reductions. Six Problems : ADD(n,n). MULT(2,n), Th(n,k), BCOUNT, MAJ. NC^1 upper bounds. Constant depth reductions among these problems. Motivation and definition of the class TC^0.References : Symmetric Functions are in TC^0. Motivation and definition of the class ACC^0. The hierarchy interleaving with NC and AC hierarchies. References <ul> <li>Class Notes</li> </ul> Exercises Reading Symmetric Functions are in TC^0. Motivation and definition of the class ACC^0. The hierarchy interleaving with NC and AC hierarchies.References : Formulas, BF = NC^1. Formula size lower bounds, References <li>Class Notes</li> Exercises Reading <li>Jukna's Textbook Section 6.1</li> Formulas, BF = NC^1. Formula size lower bounds,References : Reading : Nechiporuk's Method. Lower bound for Indirect Access function. Proof of Nechiporuk's Theorem. Nechiporuk's "hardness embedding" technique. Sabbotavskaya's Lower bound using Restrictions. Lower Bounds for Parity. References <li>Jukna's Textbook Section 6.2, 6.3</li> Exercises Reading Nechiporuk's Method. Lower bound for Indirect Access function. Proof of Nechiporuk's Theorem. Nechiporuk's "hardness embedding" technique. Sabbotavskaya's Lower bound using Restrictions. Lower Bounds for Parity.References : Extensions of Sabotavskaya's argument to random restrictions. References <li>Jukna's Textbook Section 6.2, 6.3</li> Exercises Reading Extensions of Sabotavskaya's argument to random restrictions.References : Andreev's Lower Bound. References <li>Jukna's Textbook Section 6.2, 6.3, 6.4</li> Exercises Reading Andreev's Lower Bound.References : Constant depth circuits cannot compute parity in polysize. Overview of the proof. Trivial cases like AND, OR. First nontrivial case - DNF. References <li><a href="http://homes.cs.washington.edu/~beame/papers/primer.ps">Switching Lemma Primer</a> - Paul Beame.</li> Exercises Reading Constant depth circuits cannot compute parity in polysize. Overview of the proof. Trivial cases like AND, OR. First nontrivial case - DNF.References : Inductive Argument. Exponential size lower bounds for constant depth circuits computing parity. References <li><a href="http://homes.cs.washington.edu/~beame/papers/primer.ps">Switching Lemma Primer</a> - Paul Beame.</li> Exercises Reading Inductive Argument. Exponential size lower bounds for constant depth circuits computing parity.References : Review of the proof. Proof of Switching Lemma. References <li><a href="http://homes.cs.washington.edu/~beame/papers/primer.ps">Switching Lemma Primer</a> - Paul Beame.</li> Exercises Reading Review of the proof. Proof of Switching Lemma.References : Details of Proof of switching lemma. References Exercises Reading Details of Proof of switching lemma.References : Can Majority be computed by Constant depth circuits of poly-size using PARITY, AND and OR. <br> Main Technique : Approximation of Boolean functions by polynomials. Outline of the argument. References <li>Jukna's Textbook section 12.3</li> Exercises Reading Can Majority be computed by Constant depth circuits of poly-size using PARITY, AND and OR.

Main Technique : Approximation of Boolean functions by polynomials. Outline of the argument.References : OR-approximation Lemma over F_q. Lower Bounds for the degree of polynomials approximating Threshold functions well. Majority is not in AC^o[2].<br> Degree upper bounds for the polynomials approximating circuits over MOD-3, AND, OR using OR-approximation lemma for q=3. Lower bounds for the degree for approximating parity using polynomials over F_3. Razborov Smolensky theorem - PARITY is not in AC^0[3]. References <li>Jukna's Textbook section 2.6, 12.3 and 12.4</li> Exercises Reading OR-approximation Lemma over F_q. Lower Bounds for the degree of polynomials approximating Threshold functions well. Majority is not in AC^o[2].

Degree upper bounds for the polynomials approximating circuits over MOD-3, AND, OR using OR-approximation lemma for q=3. Lower bounds for the degree for approximating parity using polynomials over F_3. Razborov Smolensky theorem - PARITY is not in AC^0[3].References : Monotone Circuits. Razoborov's lower bound for monotone circuits. Proof strategy. Clique Approximators. Inductively building clique approximators for a monotone circuit. References <li><a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/raz_clique.pdf">Lectures Notes</a> by Ramprasad Saptarishi in Arvind's course</li> Exercises Reading Monotone Circuits. Razoborov's lower bound for monotone circuits. Proof strategy. Clique Approximators. Inductively building clique approximators for a monotone circuit.References : Lower bound on the error estimate for the approximate circuit.The lower bound argument. References <li><a href="http://www.cmi.ac.in/~ramprasad/lecturenotes/raz_clique.pdf">Lectures Notes</a> by Ramprasad Saptarishi in Arvind's course</li> Exercises Reading Lower bound on the error estimate for the approximate circuit.The lower bound argument.References : Upper bound on the error by counting that for each gate. Proof of the Sunflower Lemma. References Exercises Reading Upper bound on the error by counting that for each gate. Proof of the Sunflower Lemma.References : None Branching Programs and Skew circuits. From space bounded algorithms to Branching Programs. Width of the Branching Programs. References Exercises Reading Branching Programs and Skew circuits. From space bounded algorithms to Branching Programs. Width of the Branching Programs.References : None Programs over groups. Permutations. Conjugates, Cycle conjugacy lemma. Barrington's theorem for width-5. Role of Non-solvability of the group. References Exercises Reading Programs over groups. Permutations. Conjugates, Cycle conjugacy lemma. Barrington's theorem for width-5. Role of Non-solvability of the group.References : None Derandomization Problem, Hardness assumptions. Pseudo random generators. References Exercises Reading Derandomization Problem, Hardness assumptions. Pseudo random generators.References : None Nisan-Wigderson generator, NW Design. Proof of pseudorandomness. Hybrid argument. References Exercises Reading Nisan-Wigderson generator, NW Design. Proof of pseudorandomness. Hybrid argument.References : None Arithmetic Circuits. Identity Testing Problem. From Derandomization to Hard Functions - Impagliazzo-Kabanets Theorem. References Exercises Reading Arithmetic Circuits. Identity Testing Problem. From Derandomization to Hard Functions - Impagliazzo-Kabanets Theorem.References : None **Evaluation Meetings**- 9 meetings- Meeting 57 : Mon, Apr 13, 04:00 pm-06:00 pm
- Meeting 58 : Thu, Apr 16, 02:00 pm-05:00 pm
- Meeting 59 : Sat, Apr 18, 11:00 am-01:00 pm
- Meeting 60 : Mon, Apr 20, 04:30 pm-05:30 pm
- Meeting 61 : Thu, Apr 23, 01:30 pm-04:00 pm
- Meeting 62 : Sat, Apr 25, 08:00 am-10:00 am
- Meeting 63 : Sat, Apr 25, 11:00 am-01:00 pm
- Meeting 64 : Sat, Apr 25, 01:30 pm-06:30 pm
- Meeting 65 : Mon, Apr 27, 01:30 pm-05:30 pm

Interim Meeting : Ameya, Aounon, Meenakshi, Vishvajeet References Exercises Reading Interim Meeting : Ameya, Aounon, Meenakshi, VishvajeetReferences : None Interim Meetings : Akshay Gadre, Purnata, Samir, Mitali References Exercises Reading Interim Meetings : Akshay Gadre, Purnata, Samir, MitaliReferences : None Interim Meetings : Ranjan, Aditi, Akshayram, Madhuri References Exercises Reading Interim Meetings : Ranjan, Aditi, Akshayram, MadhuriReferences : None Interim Meetings : Vidhya, Srinivasan References Exercises Reading Interim Meetings : Vidhya, SrinivasanReferences : None Presentations : Ameya, Meenakshi References Exercises Reading Presentations : Ameya, MeenakshiReferences : None Presentations : Purnata, Akshay References Exercises Reading Presentations : Purnata, AkshayReferences : None Presentations : Samir, Ranjan References Exercises Reading Presentations : Samir, RanjanReferences : None Presentations : Mitali, Vishvajeet, Akshayram, Madhuri. References Exercises Reading Presentations : Mitali, Vishvajeet, Akshayram, Madhuri.References : None Presentations : Vidhya, Srinivasan, Aditi, Aounon. References Exercises Reading Presentations : Vidhya, Srinivasan, Aditi, Aounon.References : None