Tight Fourier Tails for AC 0 Circuits Avishay

Tight Fourier Tails for AC 0 Circuits Avishay Tal (IAS)

Bounded Depth Circuits •

Brief History •

Brief History •

Brief History •

Brief Introduction to Discrete Fourier Analysis •

Brief Introduction to Discrete Fourier Analysis •

Tails and Low-Degree Approximation Equivalence •

Comparison of Results in Fourier language • LMN’ 89 Boppana’ 97 Håstad’ 01 Håstad’ 12 IMP’ 12 This Work

Comparison of Results in Polynomial Language • LMN’ 89 Boppana’ 97 Håstad’ 01 This Work

Our Result •

Applications to Pseudorandomness F PRG • F

Summary of Applications

Parity not in AC 0 FF •

Parity not in AC 0 •

Tails for Random Restrictions •

LMN Result

Recall •

Decision Tree

Restriction Tree

Switch-Many Lemma •

Tails for Arbitrary Restrictions •

Tails for Arbitrary Restrictions

Our Result Prob lem

Different Notions of Fourier Concentration •

Exponentially Small Fourier Tails •

Open Questions •

Open Questions • F PRG F

Open Questions 3. (Further) derandomize switching lemma

Open Questions 4. Which distributions fool AC 0? F F In particular, can you find a distribution that log-time quantum algorithms distinguish from the uniform dist. while AC 0 does not.

Open Questions 5. Does exponentially small Fourier tails imply PRGs? Other implications?

Open Questions 1. 2. 3. 4. 5. Better lower bounds for AC 0 Improve PRGs for AC 0 (Further) derandomize the switching lemma Which distributions fool AC 0? Is ESFT(t) enough to derive PRGs? Thank You!