Related papers: Which Boolean Functions are Most Informative?
This paper develops upper and lower bounds for the probability of Boolean functions by treating multiple occurrences of variables as independent and assigning them new individual probabilities. We call this approach dissociation and give an…
Consider a large Boolean network with a feed forward structure. Given a probability distribution on the inputs, can one find, possibly small, collections of input nodes that determine the states of most other nodes in the network? To answer…
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…
It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction…
The Boolean function implicit in the famous Dayenu song, sung at the Passover meal, is expressed in full conjunctive normal form, and it is proved that if there are n miracles the number of truth-vectors satisfying it is $2^n -(n+1)$.
A novel multinomial theorem for commutative idempotents is shown to lead to new results about the moments, central moments, factorial moments, and their generating functions for any random variable $X = \sum_{i} Y_i $ expressible as a sum…
We describe several families of efficiently implementable Boolean functions achieving provable trade-offs between resiliency, nonlinearity, and algebraic immunity. In particular, the following statement holds for each of the function…
We obtain new separation results for the two-party external information complexity of boolean functions. The external information complexity of a function $f(x,y)$ is the minimum amount of information a two-party protocol computing $f$ must…
Typical properties of computing circuits composed of noisy logical gates are studied using the statistical physics methodology. A growth model that gives rise to typical random Boolean functions is mapped onto a layered Ising spin system,…
Determining the maximal separation between sensitivity and block sensitivity of Boolean functions is of interest for computational complexity theory. We construct a sequence of Boolean functions with bs(f) = 1/2 s(f)^2 + 1/2 s(f). The best…
In this paper, we prove that most of the boolean functions, $f : \{-1,1\}^n \rightarrow \{-1,1\}$ satisfy the Fourier Entropy Influence (FEI) Conjecture due to Friedgut and Kalai (Proc. AMS'96). The conjecture says that the Entropy of a…
We consider the transmission of nonexponentially many messages through a binary symmetric channel with noiseless feedback. We obtain an upper bound for the best decoding error exponent. Combined with the corresponding known lower bound,…
We associate to each Boolean function a polynomial whose evaluations represents the distances from all possible Boolean affine functions. Both determining the coefficients of this polynomial from the truth table of the Boolean function and…
Information bottleneck (IB) is a technique for extracting information in one random variable $X$ that is relevant for predicting another random variable $Y$. IB works by encoding $X$ in a compressed "bottleneck" random variable $M$ from…
A collaborative distributed binary decision problem is considered. Two statisticians are required to declare the correct probability measure of two jointly distributed memoryless process, denoted by $X^n=(X_1,\dots,X_n)$ and…
We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x)=1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a…
We improve results of Kahn, Kalai, and Linial from the late 80s on the existence of influential large coalitions for Boolean functions, and we give counterexamples to conjectures (of Benny Chor and others) also from the late 80s, by…
We study nondeterministic communication complexity and related concepts (fooling sets, fractional covering number) of random functions $f\colon X\times Y \to \{0,1\}$ where each value is chosen to be 1 independently with probability…
The equivalence of the characteristic function approach and the probabilistic approach to monotone and boolean convolutions is proven for non-compactly supported probability measures. A probabilistically motivated definition of the…
Let $f:\{-1,1\}^{n}\rightarrow \{-1,1\}$ be a Boolean valued function having total degree $d$. Then a conjecture due to Servedio and Gopalan asserts that $\sum_{i=1}^{n}\widehat{f}(i)\leq \sum_{j=1}^{d}\widehat{\text{Maj}}_{d}(j)$ where…