- Meeting 16 : Tue, Feb 07, 11:00 am-11:50 am
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Boolean Circuits as a model of computation. PARITY is a motivating example. Connections with Parallel computation. Parameters of interest. Trivial circuits for any Boolean functions. Post's characterization of Basis.
Languages in P has polynomial size circuits.
- Meeting 17 : Wed, Feb 08, 10:00 am-10:50 am
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Can PSIZE be contained in P. Undecidable problem in PSIZE. Uniformity in circuits. P is equal to P-uniform PSIZE. Logspace uniformity. P/poly is exactly PSIZE (PSIZE will not be used from now on).
Circuit Lower Bound Problem : Can CLIQUE be computed by polynomial size circuits at all?
- Meeting 18 : Wed, Feb 08, 03:30 pm-04:30 pm
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Are there hard functions at all? Shannon's counting argument and non-explicit exponential lower bound. Lupanov's upper bound almost matching Shannon's lower bound.
- Meeting 19 : Thu, Feb 09, 08:00 am-08:50 am
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Example problems. PARITY has O(n) size, O(n) depth bounded fanin circuits. Improving it to O(log n) depth. ADD(n,2) function. O(n) depth circuit. Constant depth circuit of O(n^3) size using Carry Lookahead adder. Comparison of ADD(n,2) and PARITY. Classes AC and NC - and the hierarchy and the containments.
- Meeting 20 : Tue, Feb 14, 11:00 am-11:50 am
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Review of the NC hierarchy. PARITY is in NC^1. ADD(n,2) is in AC^0. Uniform NC^1 is contained in L. NL is a subset of AC^1. Class SAC^1 and interleaving with the hierarchy.
- Meeting 21 : Wed, Feb 15, 10:00 am-10:50 am
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Two restriction in circuits. Poly Size Skew Circuits characterize NL.
- Meeting 22 : Wed, Feb 15, 03:30 pm-04:30 pm
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Restriction on Fanout - formulas. Boolean Formulas. NC^1 = BF. Brent's construction.
- Meeting 23 : Thu, Feb 16, 08:00 am-08:50 am
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The world between NC^1 and AC^0.
ADD(n,n), MULT(n, 2), MULT(n,n), BCOUNT, MAJ, Th, Mod-k
Offman's construction : ADD(n,n) is in NC^1
MULT(n,2), BCOUNT, Threshold is in NC^1. Symmetric functions.
- Meeting 24 : Tue, Feb 21, 11:00 am-11:50 am
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The number of symmetric functions. All Symmetric functions are in NC^1. Exact Threshold Function using Threshold.
Constant depth reductions. Threshold reduces to BCOUNT. Overview of ADD(n,n) reduces to BCOUNT.
- Meeting 25 : Wed, Feb 22, 10:00 am-10:50 am
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Details of ADD(n,n) reduces to BCOUNT. Threshold reduces to MAJ. Completeness for NC^1. The class TC^0 to study majority. ACC^0. The hierarchy of ACC, TC, NC, AC and the interleaving.
- Meeting 26 : Wed, Feb 22, 03:30 pm-04:30 pm
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Polylog threshold is in AC^0. The overview and building towards the proof. Recall the hash families.
- Meeting 27 : Thu, Feb 23, 08:00 am-08:50 am
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Correctness proof and the final AC^0 circuit for polylog threshold.