7.11.2000 | - An exponential gap between randomized
and deterministic communication complexity.
D(f) \ge log(Rank(M_f)), D(EQ) \ge n. R(EQ)=O(log (n/epsilon)). - Defining NC, RNC. |
14.11.2000 | - Recognizing a perfect matching in
RNC.
- The isolating lemma. - Finding a perfect matching in RNC. - RNC vs. P, the problem of whether a determinant of a matrix of variables is identically zero. |
21.11.2000 | - Space classes, L,RL,NL. Complexity
map of space, time and NC classes. USTCON is in RL.
- Defining expanders (with expansion and second eigenvalue), dispersers and extractors. |
28.11.2000 | - Markov, Chebychev, Chernoff.
- Deterministic amplification. 1) Chernoff 2) Pair wise independence, Chebyshev. 3) The KPS generator. The mixing lemma. 4) The AKS generator. Random walks on expander graphs. |
05.12.2000 | - GG, LPS (no proof).
- Cayley graphs, characters of commutative groups, the Fourier transform. - Back to the deterministic amplification problem: 1) One sided case, with dispersers. 2) Two sided case, with extractors. |
12.12.2000 | - Error correcting codes. Hadamard. Reed Solomon.
Concatenation codes. Binary codes with distance almost half
have good list decoding properties.
- Defining Trevisan's extractor. - Guy Kindler gave a nice intuition, Thanks! - An attempt to explain the proof. |
19.12.2000 | Back to proving Trevisan's extractor.
Distinguishes, next bit predictors, Yao's trick. Luca's proof. A discussion of the parameters. Lower bound, non explicit constructions, Luca's parameters. |
26.12.2000 | Weak Design. The method of conditional expectations.
BPP in Sigma_2, using extractors and amplification. MA is in AM. If NP in BPP then PH=Sigma_2=BPP . BPP is in P/poly. |
02.01.2001 | promise-RP=promise-P ==> promise-BPP=promise-P
PRG, derandomization of BPP using PRG HSH, derandomization of RP using HSP. Derandomization of BPP using HSG. The NW pseudo random generator, with average case hardness assumption. |
09.01.2001 | List Decoding of RS codes.
Random self-reducibilty. Hardness amplification 1: If the permanent is hard, there is no quick algorithm that is right on most inputs. Reed-Muller codes. Algorithm for list decoding Reed-Muller codes. |
16.01.2001 | List decoding Reed-Muller codes.
List decoding Reed-Muller concatenated with Hadamard codes. Hardness amplification of functions in EXP (PSPACE). NW+Hardness amplification. If EXP not in P|poly then BPP in i.o.-SUBEXP. |
23.01.2001 | If PSPACE not in NC|poly RNC in i.o. SUB-linear space.
Derandomization under uniform assumptions. Either E=BPP or PRG against uniform. The class Learn. The class HUER. Natural proofs - I |
30.01.2001 | Natural proofs - II.
INW PRG against RL. Universal sequences. An explicit universal sequence of length n^O(log n). |