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

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 $\{0, 1\}^n$

Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for $f(x \wedge y)$ and $f(x\oplus y)$ with monotone functions $f$, where $\wedge$ and $\oplus$ are bit-wise AND and XOR, respectively. In this paper, we extend these results to functions $f$ which alternate values for a relatively small number of times on any monotone path from $0^n$ to $1^n$. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers

Low-Sensitivity Functions from Unambiguous Certificates

We provide new query complexity separations against sensitivity for total Boolean functions: a power $3$ separation between deterministic (and even randomized or quantum) query complexity and sensitivity, and a power $2.22$ separation between certificate complexity and sensitivity. We get these separations by using a new connection between sensitivity and a seemingly unrelated measure called one-sided unambiguous certificate complexity ($UC_{min}$). We also show that $UC_{min}$ is lower-bounded by fractional block sensitivity, which means we cannot use these techniques to get a super-quadratic separation between $bs(f)$ and $s(f)$. We also provide a quadratic separation between the tree-sensitivity and decision tree complexity of Boolean functions, disproving a conjecture of Gopalan, Servedio, Tal, and Wigderson (CCC 2016). Along the way, we give a power $1.22$ separation between certificate complexity and one-sided unambiguous certificate complexity, improving the power $1.128$ separation due to G\"o\"os (FOCS 2015). As a consequence, we obtain an improved $\Omega(\log^{1.22} n)$ lower-bound on the co-nondeterministic communication complexity of the Clique vs. Independent Set problem.Comment: 25 pages. This version expands the results and adds Pooya Hatami and Avishay Tal as author

On the Sensitivity Conjecture

The sensitivity of a Boolean function f:{0,1}^n -> {0,1} is the maximal number of neighbors a point in the Boolean hypercube has with different f-value. Roughly speaking, the block sensitivity allows to flip a set of bits (called a block) rather than just one bit, in order to change the value of f. The sensitivity conjecture, posed by Nisan and Szegedy (CC, 1994), states that the block sensitivity, bs(f), is at most polynomial in the sensitivity, s(f), for any Boolean function f. A positive answer to the conjecture will have many consequences, as the block sensitivity is polynomially related to many other complexity measures such as the certificate complexity, the decision tree complexity and the degree. The conjecture is far from being understood, as there is an exponential gap between the known upper and lower bounds relating bs(f) and s(f). We continue a line of work started by Kenyon and Kutin (Inf. Comput., 2004), studying the l-block sensitivity, bs_l(f), where l bounds the size of sensitive blocks. While for bs_2(f) the picture is well understood with almost matching upper and lower bounds, for bs_3(f) it is not. We show that any development in understanding bs_3(f) in terms of s(f) will have great implications on the original question. Namely, we show that either bs(f) is at most sub-exponential in s(f) (which improves the state of the art upper bounds) or that bs_3(f) >= s(f){3-epsilon} for some Boolean functions (which improves the state of the art separations). We generalize the question of bs(f) versus s(f) to bounded functions f:{0,1}^n -> [0,1] and show an analog result to that of Kenyon and Kutin: bs_l(f) = O(s(f))^l. Surprisingly, in this case, the bounds are close to being tight. In particular, we construct a bounded function f:{0,1}^n -> [0, 1] with bs(f) n/log(n) and s(f) = O(log(n)), a clear counterexample to the sensitivity conjecture for bounded functions. Finally, we give a new super-quadratic separation between sensitivity and decision tree complexity by constructing Boolean functions with DT(f) >= s(f)^{2.115}. Prior to this work, only quadratic separations, DT(f) = s(f)^2, were known