Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > DETAIL:

### Revision(s):

Revision #2 to TR14-077 | 7th June 2016 14:00

#### Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma

Revision #2
Authors: Andris Ambainis, Jevgenijs Vihrovs
Accepted on: 7th June 2016 14:00
Keywords:

Abstract:

We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving lower bounds on sensitivity of Boolean functions, with applications to the theory of parallel computing and the "sensitivity vs. block sensitivity" conjecture.

In this paper, we take a deeper look at the size of such sets and their structure. We show an unexpected "gap theorem": if $S\subseteq\{0, 1\}^n$ has sensitivity $s$, then we either have $|S|=2^{n-s}$ or $|S|\geq \frac{3}{2} 2^{n-s}$. This is shown via classifying such sets into sets that can be constructed from low-sensitivity subsets of $\{0, 1\}^{n'}$ for $n'<n$ and irreducible sets which cannot be constructed in such a way and then proving a lower bound on the size of irreducible sets.

This provides new insights into the structure of low sensitivity subsets of the Boolean hypercube $\{0, 1\}^n$.

Changes to previous version:

- Updated proof of Theorem 2.
- Updated reference to "Sensitivity versus certificate complexity of Boolean functions" by A. Ambainis, K. Pr?sis, and J. Vihrovs.

Revision #1 to TR14-077 | 10th June 2014 19:10

#### Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma

Revision #1
Authors: Andris Ambainis, Jevgenijs Vihrovs
Accepted on: 10th June 2014 19:11
Keywords:

Abstract:

We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving lower bounds on sensitivity of Boolean functions, with applications to the theory of parallel computing and the "sensitivity vs. block sensitivity" conjecture.

In this paper, we take a deeper look at the size of such sets and their structure. We show an unexpected "gap theorem": if $S\subseteq\{0, 1\}^n$ has sensitivity $s$, then we either have $|S|=2^{n-s}$ or $|S|\geq \frac{3}{2} 2^{n-s}$. This is shown via classifying such sets into sets that can be constructed from low-sensitivity subsets of $\{0, 1\}^{n'}$ for $n'<n$ and irreducible sets which cannot be constructed in such a way and then proving a lower bound on the size of irreducible sets.

This provides new insights into the structure of low sensitivity subsets of the Boolean hypercube $\{0, 1\}^n$.

Changes to previous version:

Added reference to "Boolean functions with small spectral norm." by B. Green and T. Sanders.

### Paper:

TR14-077 | 2nd June 2014 14:07

#### Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma

TR14-077
Authors: Andris Ambainis, Jevgenijs Vihrovs
Publication: 3rd June 2014 12:15
Keywords:

Abstract:

We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving lower bounds on sensitivity of Boolean functions, with applications to the theory of parallel computing and the "sensitivity vs. block sensitivity" conjecture.

In this paper, we take a deeper look at the size of such sets and their structure. We show an unexpected "gap theorem": if $S\subseteq\{0, 1\}^n$ has sensitivity $s$, then we either have $|S|=2^{n-s}$ or $|S|\geq \frac{3}{2} 2^{n-s}$. This is shown via classifying such sets into sets that can be constructed from low-sensitivity subsets of $\{0, 1\}^{n'}$ for $n'<n$ and irreducible sets which cannot be constructed in such a way and then proving a lower bound on the size of irreducible sets.

This provides new insights into the structure of low sensitivity subsets of the Boolean hypercube $\{0, 1\}^n$.

ISSN 1433-8092 | Imprint