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Happy 60 th B’day Mike

Happy 60 th B’day Mike. Lower bounds, anyone? Avi Wigderson Institute for Advanced Study. Lower bounds & Randomness & Expanders. P = NP ?. Gradient descent. Neural network. Genetic algor. Dimension reduction. Occam’s razor. Removing noise. Generative grammar. Annealing.

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Happy 60 th B’day Mike

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  1. Happy 60th B’day Mike

  2. Lower bounds, anyone?Avi WigdersonInstitute for Advanced Study

  3. Lower bounds&Randomness&Expanders

  4. P = NP ?

  5. Gradient descent Neural network Genetic algor Dimension reduction Occam’s razor Removing noise Generative grammar Annealing Low dim surface HMM Bayesian network Statistical mechanics Boosting SVD Sampling Decision tree Stock Market Essential parameters What is going on? Irregularities Regularities Clustering Correlations LHC Weather Internet Visual Neuro Genomic Seismic Language Translation Prediction Big DATA Astronomical

  6. NP = coNP? Mike’s dictionary: Comput. Complexity  Set Theory Polynomial ~ Countable Exponential ~ Uncountable NPcoNP PolysizeNondet DNF PolysizeNondet CNF Countable Nondet DNF Countable Nondet CNF Analytic coAnalytic [Sipser] New, “more combinatorial” proof  Topological approach

  7. P = NP ? PH = PSPACE ? [BGS] APANPA (diagonalization is useless) ? APHAPSPACEA ? Mike’s dictionary Oracle machines  Circuit comp. Set theory PHA~ AC0 ~ Finite Borel hierarchy PSPACEA ~ NC1 ~ Borelsets [Sipser] New, “more combinatorial” proof [Furst-Saxe-Sipser,Ajtai] Parity  AC0 [Yao, Hastad] APHAPSPACEA •  •  •  Switching Lemma, Restrictions • Random

  8. NL = L ? • Mike’s dictionary • Comp classes  Finite automata • NL~ polysize 2NFA • L ~ polysize 2DFA • [Sipser] nlanguage Snsuch that • - Snis accepted by an O(n)-state 2NFA • Snrequires 2n-state (sweeping) 2DFA • REGULAR = 2DFA = 2NFA = 2PFA* • [Open] 2AMFA* = REGULAR ? • [CHPW] True if 2AMFA* = co2AMFA* *Polytime

  9. Time vs. Space [HPV] Time(t)Space(t/log t) [Open] Time(t)Space(t.99) ? Randomness vs. Determinism [Open] BPP = P ? [Sipser] either Time(t)Space(t.99) orBPP =P Hardness vs. Randomness if Explicit extractors exist X

  10. Utilizing Expanders [Sipser] Expanders T(t) S(t.99) or BPP = P [Karp-Pippenger-Sipser] Deterministic amplification [Sipser-Spielman] Expander codes ( [Gallager, Tanner] ) [Spielman] linear time encoding and decoding good codes [Sipser?] Affineexpander? [Klawe]Impossibility!

  11. Hashing in Comput. Complexity [Sipser] BPPPH [Gacs, Lautemann] [Goldwasser-Sipser] PublicCoinIP = PrivateCoinsIP

  12. Randomness & Lower bounds Probabilistic method (AC0) Natural proofs

  13. - Can sequential computation be parallelized? - Are formulas weaker than circuits? Composition g:{0,1}m{0,1} f:{0,1}n{0,1} gof:{0,1}mn{0,1} D(gof) ≤ D(g)+D(f), L(gof) ≤ L(g) L(f) [Karchmer-Raz-Wigderson Conj] This is tight! NC1 vs. P gof g f f

  14. [KRWconj]: D(gof) ≈ D(g)+D(f) [KRW]: Conjecture implies P ≠ NC1. [KRW]: Conjecture holds for monotone circuits [Cor]:mP ≠ mNC1. [Grigni-Sipser]:mL≠ mNC1. Natural proof Barrier doesn’t Seem to apply The KRW conjecture

  15. Universal Relations: g ≤ Um, f ≤ Un, gof < UmoUn [EIRS, HW]: D(Um o Un) ≈ D(Um) + D(Un) [GMWW’13]: g D(go Un) ≈ D(g) + D(Un) [Open]: g,f D(g o f) ≈ D(g) + D(f) [Open]: fD(Umo f) ≈ D(Um) + D(f) KRW program

  16. Happy 60thb’dayMike!!!

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