Tight Fourier Tails for AC 0 Circuits

1. Tight Fourier Tails for AC0 Circuits Avishay Tal (IAS) CCC ’2017

2. Bounded Depth Circuits • variables • gates (size of the circuit) • depth • alternating gates

3. Brief History [Ajtai’83, Furst-Saxe-Sipser’84, Yao’85]: Parity is not in AC0 [Håstad ’86]:any depth-circuit computing parity is of size at least . Result is tight: there exists acircuit of size and depth computing Parity Challenge: Give an explicit function with better lower bounds. Really good lower bounds will imply lower bounds for NC1 & log-space.

4. Brief History [Linial-Mansour-Nisan’89]:Bounded depth circuits are well-approximated in L2 by low degree polynomials. Theorem: Let.Then, of s.t. [Håstad ’12]: anymay agree with Parityon at mostof the inputs. [Imagaliazzo-Matthews-Paturi’12]:… [Håstad ’12]and [IMP’12]results are tight!

5. Discrete Fourier Analysis 101 For functions define inner-product as The characters for form an orthonormal basis. Hence, any function has a uniqueexpansion called the Fourier expansion. The Fourier coefficientsare real numbers given by Plancherel’s Identity: Parseval’s Identity: If is Boolean, i.e., , then

6. Discrete Fourier Analysis 101 The Fourier transform of a Boolean function naturally defines a distribution over sets : Denote by Denote by The probability to sample from equals .

7. Tails and Low-Degree Approximation Equivalence Let . The truncated Fourier expansion of at level is a degree polynomial defined by By Parseval: . By Parseval: this is the best L2-approx. of among degree polys. has a degree- L2-approximation with error iff

9. Comparison of Results in Fourier language LMN’89 decay Boppana’97 Our Result decay Håstad’01 Lower Bound decay Håstad’12 IMP’12

10. Comparison of Results in Polynomial Language If can be computed by a circuit with size and depth , then can be -approximated in L2 by polynomials of degree:

11. Main Theorem If can be computed by a circuit of size and depth ,then Alternatively, can be -approximated in L2 by a polynomial of degree . • A significant improvement for . • Tight (for any )

12. Applications to Pseudo-randomness A distribution over is pseudorandom for crkts of class if A pseudo-random generator (PRG) for is a function such that is pseudorandom for . F F PRG

13. Summary of Applications

14. Why should we care? Why are we not satisfied by decay in tails and want decay? Motivating question: give a Fourier analytical proof that Majority cannot be approximated by AC0circuits. (Other proofs: [Smolensky’93, O’Donnell-Wimmer’07])

15. Different Notions of Fourier Concentration Let be a Boolean function and a parameter. TFAE: • for all k: • for all k: • for all p, k: . and they imply Exp. Small Fourier Tails Fourier Moments “Switching Lemma”

16. Majority is not approximated by AC0 Problem: both MAJ and AC0are concentrated on lower levels of the Fourier spectrum. Idea: Recall .  on the k’th level, ’s Fourier mass is concentrated on only coefs out of all the coefs. Since MAJis symmetric, it spreads its Fourier weight equally within each layer: every coefficient in the k’th level is at most .

17. Majority is not approximated by AC0 Using Plancherel: For : For : 

18. Open Question Which distributions fool AC0? [Aaronson’10, Fefferman-Shaltiel-Umans-Viola’12] Can you find a distribution which is pseudorandom for AC0 but not pseudorandom for log-time quantum algorithms? F F  an oracle separation between BQP from PH

19. Exponentially Small Fourier Tails Definition:has ESFT(t) if for all : Several interesting classes of functions have ESFT(t) • CNFs/DNFs of width-[Håstad’86, LMN’89] • Formulas of size [Reichardt’11] • Read-Once Formulas[Impagliazzo-Kabanets’14] • Circuits of size and depth • Functions with max-sensitivity [Gopalan-Servedio-T-Wigderson’16]:

20. Thank You!