Related papers: Sensitivity Conjecture and Log-rank Conjecture for…
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…
We compare several complexity measures for self-mappings of finite fields. In particular, we show that Carlitz rank and additive index cannot be small simultaneously up to trivial exceptions. That is, these two measures detect cryptographic…
The 2013 logic blog has focussed on the following: 1. Higher randomness. Among others, the Borel complexity of $\Pi^1_1$ randomness and higher weak 2 randomness is determined. 2. Reverse mathematics and its relationship to randomness. For…
The sensitivity conjecture which claims that the sensitivity complexity is polynomially related to block sensitivity complexity, is one of the most important and challenging problem in decision tree complexity theory. Despite of a lot of…
Comon's conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen's conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor…
The aim of this work is to provide a unified framework for ordinal representations of uncertainty lying at the crosswords between possibility and probability theories. Such confidence relations between events are commonly found in monotonic…
A Boolean function $f({\vec x})$ is sensitive to bit $x_i$ if there is at least one input vector $\vec x$ and one bit $x_i$ in $\vec x$, such that changing $x_i$ changes $f$. A function has sensitivity $s$ if among all input vectors, the…
The probabilistic satisfiability of a logical expression is a fundamental concept known as the partition function in statistical physics and field theory, an evaluation of a related graph's Tutte polynomial in mathematics, and the…
This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and…
Boolean functions are important primitives in different domains of cryptology, complexity and coding theory. In this paper, we connect the tools from cryptology and complexity theory in the domain of Boolean functions with low polynomial…
A logconcave likelihood is as important to proper statistical inference as a convex cost function is important to variational optimization. Quantization is often disregarded when writing likelihood models, ignoring the limitations of the…
This note is an attempt to unconditionally prove the existence of weak one way functions (OWF). Starting from a provably intractable decision problem $L_D$ (whose existence is nonconstructively assured from the well-known discrete…
We prove two new results about the randomized query complexity of composed functions. First, we show that the randomized composition conjecture is false: there are families of partial Boolean functions $f$ and $g$ such that $R(f\circ g)\ll…
We introduce a framework to consider transport problems for integer-valued random variables. We introduce weighting coefficients which allow us to characterize transport problems in a gradient flow setting, and form the basis of our…
In this paper we introduce a new method to produce lower bounds for the Waring rank of symmetric tensors. We also introduce the notion of $e$-computability and we use it to prove that Strassen's Conjecture holds in infinitely many new…
Let $(X,d)$ be a compact metric space and $f:X \to X$ be a self-map. The compact dynamical system $(X,f)$ is called sensitive or sensitivity depends on initial conditions, if there is a positive constant $\delta$ such that in each non-empty…
Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y)=f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for…
Based on a sequence of numerical computations, a conjecture is presented regarding the class of functions $H(x;a)=\exp(a)-(1+a/x)^x$, and the open problem of determining the values of $a$ for which the functions are completely monotonic…
The minimum number of NOT gates in a Boolean circuit computing a Boolean function is called the inversion complexity of the function. In 1957, A. A. Markov determined the inversion complexity of every Boolean function and proved that…
In this paper, we present a comprehensive study of the monotonicity and log-concavity of the generalized Marcum and Nuttall Q-functions. More precisely, a simple probabilistic method is firstly given to prove the monotonicity of these two…