Related papers: A finite alternation result for reversible boolean…
Let $ACC \circ THR$ be the class of constant-depth circuits comprised of AND, OR, and MOD$m$ gates (for some constant $m > 1$), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen…
The regression depth of a hyperplane with respect to a set of n points in R^d is the minimum number of points the hyperplane must pass through in a rotation to vertical. We generalize hyperplane regression depth to k-flats for any k between…
Recently, deep residual networks have been successfully applied in many computer vision and natural language processing tasks, pushing the state-of-the-art performance with deeper and wider architectures. In this work, we interpret deep…
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…
Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit $d^{O(1)}$-variate and degree $d$ polynomial $P_{d}\in VNP$ such that if any depth four circuit $C$ of bounded formal degree $d$ which computes a…
We present necessary and sufficient conditions for a Boolean function to be a negabent function for both even and odd number of variables, which demonstrate the relationship between negabent functions and bent functions. By using these…
We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal…
Reversible computation is one of the most promising emerging technologies of the future. The usage of reversible circuits in computing devices can lead to a significantly lower power consumption. In this paper we study reversible logic…
Let $U_{k,N}$ denote the Boolean function which takes as input $k$ strings of $N$ bits each, representing $k$ numbers $a^{(1)},\dots,a^{(k)}$ in $\{0,1,\dots,2^{N}-1\}$, and outputs 1 if and only if $a^{(1)} + \cdots + a^{(k)} \geq 2^N.$…
In this paper, we consider bounded width circuits and nondeterministic circuits in three somewhat new directions. In the first part of this paper, we mainly consider bounded width circuits. The main purpose of this part is to prove that…
We study pseudorandomness properties of permutations on $\{0,1\}^n$ computed by random circuits made from reversible $3$-bit gates (permutations on $\{0,1\}^3$). Our main result is that a random circuit of depth $n \cdot \tilde{O}(k^2)$,…
We investigate shift-invariant transformations, also known as rotation-symmetric vectorial Boolean functions, on $n$ bits that are induced from Boolean functions on $k$ bits, for $k\leq n$. We consider such transformations that are not…
The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically…
Classification of Non-linear Boolean functions is a long-standing problem in the area of theoretical computer science. In this paper, effort has been made to achieve a systematic classification of all n-variable Boolean functions, where…
It is known that a positive Boolean function f depending on n variables has at least n + 1 extremal points, i.e. minimal ones and maximal zeros. We show that f has exactly n + 1 extremal points if and only if it is linear read-once. The…
In his 2003 paper "Towards an algebraic theory of Boolean circuits", Lafont notes that the class of reversible circuits over a set of k truth values is finitely generated when k is odd. He cites a private communication for the proof. The…
Finite blocklength and second-order (dispersion) results are presented for the arbitrarily-varying channel (AVC), a classical model wherein an adversary can transmit arbitrary signals into the channel. A novel finite blocklength…
We continue the study of the approximate $k$-wise independence of random reversible circuits as permutations of $\{\pm1\}^n$. Our main result is the first construction of a natural class of random reversible circuits with a sublinear-in-$n$…
Motivated by DNA storage in living organisms, and by known biological mutation processes, we study the reverse-complement string-duplication system. We fully classify the conditions under which the system has full expressiveness, for all…
A novel fast recursive coding technique is proposed. It operates with only integer values not longer 8 bits and is multiplication free. Recursion the algorithm is based on indirectly provides rather effective coding of symbols for very…