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

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.

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.

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.

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

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.

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

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.