Happy 60th B’day Mike

Lower bounds, anyone?

Avi WigdersonInstitute for Advanced Study

Lower bounds&

Randomness&

Expanders

P = NP ?

Astronomical

Language

Seismic

Genomic

Neuro

Visual

Internet

Weather

LHC

StockMarket

Correlations

Clustering

Regularities

Irregularities

Essentialparameters

Translation

Prediction

What is going on?

Removing noise

Low dim surface

HMM

Bayesiannetwork

Decisiontree

Generativegrammar

Statisticalmechanics

SVD

Sampling

Annealing

Genetic algor

Neuralnetwork

Boosting

Gradientdescent

Occam’srazor

Dimensionreduction

Big DATA

NP = coNP ?Mike’s dictionary: Comput. Complexity Set TheoryPolynomial ~ CountableExponential ~ Uncountable

NP coNPPolysize Nondet DNF Polysize Nondet CNF Countable Nondet DNF Countable Nondet CNFAnalytic coAnalytic

[Sipser] New, “more combinatorial” proof

Topological approach

P = NP ? PH = PSPACE ?

[BGS] A PA NPA (diagonalization is useless)

? A PHA PSPACEA ?

Mike’s dictionaryOracle machines Circuit comp. Set theory

PHA ~ AC0 ~ Finite Borel hierarchy

PSPACEA ~ NC1 ~ Borel sets

[Sipser] New, “more combinatorial” proof

[Furst-Saxe-Sipser,Ajtai] Parity AC0

[Yao, Hastad] A PHA PSPACEA

Switching Lemma, Restrictions

Random

NL = L ?Mike’s dictionaryComp classes Finite automata

NL ~ polysize 2NFA

L ~ polysize 2DFA

[Sipser] n language Sn such that

- Sn is accepted by an O(n)-state 2NFA

- Sn requires 2n-state (sweeping) 2DFA

REGULAR = 2DFA = 2NFA = 2PFA*

[Open] 2AMFA* = REGULAR ?

[CHPW] True if 2AMFA* = co2AMFA*

* Polytime

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

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!

Hashing in Comput. Complexity

[Sipser] BPP PH [Gacs, Lautemann]

[Goldwasser-Sipser] PublicCoinIP = PrivateCoinsIP

Randomness & Lower bounds

Probabilistic method (AC0)

Natural proofs

NC1 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

ff

gof

The KRW conjecture

[KRW conj]: 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 proofBarrier doesn’tSeem to apply

KRW program

Universal Relations: g ≤ Um, f ≤ Un, gof <

UmoUn

[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)

Happy 60th b’day Mike!!!