- Meeting 32 : Mon, Mar 16, 10:00 am-10:50 am
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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 | : |
- Class Notes
- Zwick's Notes (sent to the mailing list) is the reference for Post's theorem.
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- Meeting 33 : Tue, Mar 17, 09:00 am-09:50 am
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Exercises | |
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PSIZE = P/poly. Circuit Lower Bound Problem. Shannon's counting argument. Lupanov's construction (sketch)
References | : |
- Previous Year's Class Notes
- Jukna's Textbook Section 1.4
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- Meeting 34 : Fri, Mar 20, 12:00 pm-12:50 pm
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Exercises | |
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Gate Elimination Argument for Circuit Lower Bounds. Lower bounds for Parity and Threshold.
References | : |
- Jukna's Textbook Section 1.6
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- Meeting 35 : Sat, Mar 21, 10:00 am-11:30 am
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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.
- Meeting 36 : Mon, Mar 23, 10:00 am-10:50 am
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The class AC-hierarchy. Interleaving with NC hierarchy. NL is contained in uniform-AC^1.
- Meeting 37 : Tue, Mar 24, 09:00 am-09:50 am
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Exercises | |
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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 | : |
- Lecture Notes from V.Vinay (sent to the mailing list)
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- Meeting 38 : Wed, Mar 25, 08:00 am-08:50 am
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Symmetric Functions are in TC^0. Motivation and definition of the class ACC^0. The hierarchy interleaving with NC and AC hierarchies.
- Meeting 39 : Fri, Mar 27, 12:00 pm-12:50 pm
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Exercises | |
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Formulas, BF = NC^1. Formula size lower bounds,
References | : | - Class Notes
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Reading | : | - Jukna's Textbook Section 6.1
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- Meeting 40 : Sat, Mar 28, 10:00 am-11:30 am
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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 | : | - Jukna's Textbook Section 6.2, 6.3
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- Meeting 41 : Mon, Mar 30, 10:00 am-10:50 am
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Exercises | |
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Extensions of Sabotavskaya's argument to random restrictions.
References | : | - Jukna's Textbook Section 6.2, 6.3
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- Meeting 42 : Tue, Mar 31, 09:00 am-09:50 am
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Exercises | |
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Andreev's Lower Bound.
References | : | - Jukna's Textbook Section 6.2, 6.3, 6.4
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- Meeting 43 : Mon, Apr 06, 10:00 am-10:50 am
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Exercises | |
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Constant depth circuits cannot compute parity in polysize. Overview of the proof. Trivial cases like AND, OR. First nontrivial case - DNF.
- Meeting 44 : Tue, Apr 07, 09:00 am-09:50 am
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Inductive Argument. Exponential size lower bounds for constant depth circuits computing parity.
- Meeting 45 : Wed, Apr 08, 08:00 am-08:50 am
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Exercises | |
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Review of the proof. Proof of Switching Lemma.
- Meeting 46 : Fri, Apr 10, 12:00 pm-12:50 pm
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Exercises | |
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Details of Proof of switching lemma.
- Meeting 47 : Mon, Apr 13, 10:00 am-10:50 am
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Exercises | |
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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 | : | - Jukna's Textbook section 12.3
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- Meeting 48 : Tue, Apr 14, 09:00 am-09:50 am
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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 | : | - Jukna's Textbook section 2.6, 12.3 and 12.4
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- Meeting 49 : Wed, Apr 15, 08:00 am-08:50 am
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Monotone Circuits. Razoborov's lower bound for monotone circuits. Proof strategy. Clique Approximators. Inductively building clique approximators for a monotone circuit.
- Meeting 50 : Fri, Apr 17, 12:00 pm-12:50 pm
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Lower bound on the error estimate for the approximate circuit.The lower bound argument.
- Meeting 51 : Mon, Apr 20, 10:00 am-10:50 am
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Exercises | |
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Upper bound on the error by counting that for each gate. Proof of the Sunflower Lemma.
- Meeting 52 : Tue, Apr 21, 09:00 am-09:50 am
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Branching Programs and Skew circuits. From space bounded algorithms to Branching Programs. Width of the Branching Programs.
- Meeting 53 : Wed, Apr 22, 08:00 am-08:50 am
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Programs over groups. Permutations. Conjugates, Cycle conjugacy lemma. Barrington's theorem for width-5. Role of Non-solvability of the group.
- Meeting 54 : Fri, Apr 24, 12:00 pm-12:50 pm
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Derandomization Problem, Hardness assumptions. Pseudo random generators.
- Meeting 55 : Mon, Apr 27, 10:00 am-10:50 am
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Nisan-Wigderson generator, NW Design. Proof of pseudorandomness. Hybrid argument.
- Meeting 56 : Tue, Apr 28, 09:00 am-09:50 am
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Arithmetic Circuits. Identity Testing Problem. From Derandomization to Hard Functions - Impagliazzo-Kabanets Theorem.