1. Happy 60 th B’day Mike

2. Lower bounds, anyone? Avi Wigderson Institute for Advanced Study

3. Lower bounds & Randomness & Expanders

4. P = NP ?

5. Astronomical Language Seismic Genomic NeuroVisual Internet WeatherLHC Stock Market Correlations ClusteringRegularitiesIrregularities Essential parameters TranslationPrediction What is going on? Removing noise Low dim surface HMM Bayesian network Decision tree Generative grammar Statistical mechanics SVDSamplingAnnealing Genetic algor Neural network Boosting Gradient descent Occam’s razor Dimension reduction Big DATA

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

7. P = NP ? PH = PSPACE ? [BGS]  A P A  NP A (diagonalization is useless) ?  A PH A  PSPACE A ? Mike’s dictionary Oracle machines  Circuit comp.  Set theory PH A ~ AC 0 ~ Finite Borel hierarchy PSPACE A ~ NC 1 ~ Borel sets [Sipser] New, “more combinatorial” proof [Furst-Saxe-Sipser,Ajtai] Parity  AC 0 [Yao, Hastad]  A PH A  PSPACE A Switching Lemma, Restrictions Random 

8. NL = L ? Mike’s dictionary Comp classes  Finite automata NL ~ polysize 2NFA L ~ polysize 2DFA [Sipser]  n  language S n such that - S n is accepted by an O(n)-state 2NFA -S n requires 2 n -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 ) or BPP = P if Explicit extractors exist X Hardness vs. Randomness

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?] Affine expander? [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. NC 1 vs. P - 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! g f f gofgof

14. The KRW conjecture [KRW conj]: D(gof) ≈ D(g)+D(f) [KRW]: Conjecture implies P ≠ NC 1. [KRW]: Conjecture holds for monotone circuits [Cor]: mP ≠ mNC 1. [Grigni-Sipser]: mL ≠ mNC 1. Natural proof Barrier doesn’t Seem to apply

15. KRW program Universal Relations: g ≤ U m, f ≤ U n, gof < U m oU n [EIRS, HW]: D(Um o Un) ≈ D(Um) + D(Un) [GMWW’13]:  g D(g o Un) ≈ D(g) + D(Un) [Open]:  g,f D(g o f) ≈ D(g) + D(f) [Open]:  f D(Um o f) ≈ D(Um) + D(f )

16. Happy 60 th b’day Mike!!!