Related papers: On Boolean Functions with Low Polynomial Degree an…
The sensitivity of a Boolean function f is the maximum over all inputs x, of the number of sensitive coordinates of x. The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-s Boolean…
Sensitivity, block sensitivity and certificate complexity are basic complexity measures of Boolean functions. The famous sensitivity conjecture claims that sensitivity is polynomially related to block sensitivity. However, it has been…
We generalize and extend the ideas in a recent paper of Chiarelli, Hatami and Saks to prove new bounds on the number of relevant variables for boolean functions in terms of a variety of complexity measures. Our approach unifies and refines…
Sensitivity conjecture is a longstanding and fundamental open problem in the area of complexity measures of Boolean functions and decision tree complexity. The conjecture postulates that the maximum sensitivity of a Boolean function is…
Sensitivity, certificate complexity and block sensitivity are widely used Boolean function complexity measures. A longstanding open problem, proposed by Nisan and Szegedy, is whether sensitivity and block sensitivity are polynomially…
We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a…
Nisan and Szegedy (CC 1994) showed that any Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ that depends on all its input variables, when represented as a real-valued multivariate polynomial $P(x_1,\ldots,x_n)$, has degree at least $\log…
The sensitivity conjecture of Nisan and Szegedy [CC '94] asks whether for any Boolean function $f$, the maximum sensitivity $s(f)$, is polynomially related to its block sensitivity $bs(f)$, and hence to other major complexity measures.…
A natural measure of smoothness of a Boolean function is its sensitivity (the largest number of Hamming neighbors of a point which differ from it in function value). The structure of smooth or equivalently low-sensitivity functions is still…
Boolean matching is significant to digital integrated circuits design. An exhaustive method for Boolean matching is computationally expensive even for functions with only a few variables, because the time complexity of such an algorithm for…
A number of complexity measures for Boolean functions have previously been introduced. These include (1) sensitivity, (2) block sensitivity, (3) witness complexity, (4) subcube partition complexity and (5) algorithmic complexity. Each of…
In this paper we construct a cyclically invariant Boolean function whose sensitivity is $\Theta(n^{1/3})$. This result answers two previously published questions. Tur\'an (1984) asked if any Boolean function, invariant under some transitive…
In this paper we study the separation between two complexity measures: the degree of a Boolean function as a polynomial over the reals and its block sensitivity. We show that separation between these two measures can be improved from $…
Empirical evidence has revealed that biological regulatory systems are controlled by high-level coordination between topology and Boolean rules. In this study, we study the joint effects of degree and Boolean functions on the stability of…
Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing binary minimal codes from sets, Boolean functions and vectorial Boolean…
Propagation criterion of degree $l$ and order $k$ ($PC(l)$ of order $k$) and resiliency of vectorial Boolean functions are important for cryptographic purpose (see [1, 2, 3,6, 7,8,10,11,16]. Kurosawa, Stoh [8] and Carlet [1] gave a…
We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of polynomial threshold functions. More specifically, for a Boolean function f on n variables equal to the sign of a real, multivariate polynomial…
Sensitivity \cite{CD82,CDR86} and block sensitivity \cite{Nisan91} are two important complexity measures of Boolean functions. A longstanding open problem in decision tree complexity, the "Sensitivity versus Block Sensitivity" question,…
A Boolean function on n variables is q-resilient if for any subset of at most q variables, the function is very likely to be determined by a uniformly random assignment to the remaining n-q variables; in other words, no coalition of at most…
Polynomial representations of Boolean functions over various rings such as $\mathbb{Z}$ and $\mathbb{Z}_m$ have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of fields including…