Revision #5 Authors: Or Meir, Avi Wigderson

Accepted on: 25th December 2018 14:11

Downloads: 47

Keywords:

Consider a random sequence of $n$ bits that has entropy at least $n-k$, where $k\ll n$. A commonly used observation is that an average coordinate of this random sequence is close to being uniformly distributed, that is, the coordinate “looks random”. In this work, we prove a stronger result that says, roughly, that the average coordinate looks random to an adversary that is allowed to query $\approx\frac{n}{k}$ other coordinates of the sequence, even if the adversary is non-deterministic. This setting generalizes decision trees and certificates for Boolean functions.

As an application of this result, we prove a new result on depth-$3$ circuits, which recovers as a direct corollary the known lower bounds for the parity and majority functions, as well as a lower bound on sensitive functions due to Boppana (IPL 63(5), 1997). An interesting feature of this proof is that it works in the framework of Karchmer and Wigderson (SIDMA 3(2), 1990), and in particular it is a “top-down” proof (Hastad, Jukna and Pudlak, Computational Complexity 5(2), 1995). Finally, it yields a new kind of a random restriction lemma for non-product distributions, which may be of independent interest.

Journal version

Revision #4 Authors: Or Meir, Avi Wigderson

Accepted on: 28th June 2018 11:40

Downloads: 113

Keywords:

Consider a random sequence of $n$ bits that has entropy at least $n-k$, where $k\ll n$. A commonly used observation is that an average coordinate of this random sequence is close to being uniformly distributed, that is, the coordinate “looks random”. In this work, we prove a stronger result that says, roughly, that the average coordinate looks random to an adversary that is allowed to query $\approx\frac{n}{k}$ other coordinates of the sequence, even if the adversary is non-deterministic. This setting generalizes decision trees and certificates for Boolean functions.

As an application of this result, we prove a new result on depth-$3$ circuits, which recovers as a direct corollary the known lower bounds for the parity and majority functions, as well as a lower bound on sensitive functions due to Boppana (IPL 63(5), 1997). An interesting feature of this proof is that it works in the framework of Karchmer and Wigderson (SIDMA 3(2), 1990), and in particular it is a “top-down” proof (Hastad, Jukna and Pudlak, Computational Complexity 5(2), 1995). Finally, it yields a new kind of a random restriction lemma for non-product distributions, which may be of independent interest.

Added discussion of the connection to the satisfiability coding lemma.

Revision #3 Authors: Or Meir, Avi Wigderson

Accepted on: 17th January 2018 14:50

Downloads: 199

Keywords:

Consider a random sequence of $n$ bits that has entropy at least $n-k$, where $k\ll n$. A commonly used observation is that an average coordinate of this random sequence is close to being uniformly distributed, that is, the coordinate “looks random”. In this work, we prove a stronger result that says, roughly, that the average coordinate looks random to an adversary that is allowed to query $\approx\frac{n}{k}$ other coordinates of the sequence, even if the adversary is non-deterministic. This setting generalizes decision trees and certificates for Boolean functions.

As an application of this result, we prove a new result on depth-$3$ circuits, which recovers as a direct corollary the known lower bounds for the parity and majority functions, as well as a lower bound on sensitive functions due to Boppana (IPL 63(5), 1997). An interesting feature of this proof is that it works in the framework of Karchmer and Wigderson (SIDMA 3(2), 1990), and in particular it is a “top-down” proof (Hastad, Jukna and Pudlak, Computational Complexity 5(2), 1995). Finally, it yields a new kind of a random restriction lemma for non-product distributions, which may be of independent interest.

Revision #2 Authors: Or Meir, Avi Wigderson

Accepted on: 15th January 2018 17:09

Downloads: 177

Keywords:

Fixed some typos, and added discussion of a recent follow-up work.

Revision #1 Authors: Or Meir, Avi Wigderson

Accepted on: 13th November 2017 18:49

Downloads: 192

Keywords:

Fixed a small error in the introduction, and clarified the connection between Proposition 1.7 and Claim 3.2.

TR17-149 Authors: Or Meir, Avi Wigderson

Publication: 8th October 2017 07:51

Downloads: 454

Keywords: