Related papers: On the number of inequivalent monotone Boolean fun…

In this paper, the author presents algorithms that allow determining the number of fixed points in permutations of a set of monotone Boolean functions. Then, using Burnside's lemma, the author determines the number of inequivalent monotone…

Monotone Boolean functions (MBFs) are Boolean functions $f: {0,1}^n \rightarrow {0,1}$ satisfying the monotonicity condition $x \leq y \Rightarrow f(x) \leq f(y)$ for any $x,y \in {0,1}^n$. The number of MBFs in n variables is known as the…

We show that the problem of counting the number of $n$-variable unate functions reduces to the problem of counting the number of $n$-variable monotone functions. Using recently obtained results on $n$-variable monotone functions, we obtain…

We present a few algorithms and methods to count fixes of permutations acting on monotone Boolean functions. Some of these methods was used by Pawelski \cite{P} to compute the number of inequivalent monotone Boolean functions with 8…

Let $D_n$ denote the set of monotone Boolean functions with $n$ variables. Elements of $D_n$ can be represented as strings of bits of length $2^n$. Two elements of $D_0$ are represented as 0 and 1 and any element $g\in D_n$, with $n>0$, is…

We study the number of queries needed to identify a monotone Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$. A query consists of a 0-1-sequence, and the answer is the value of $f$ on that sequence. It is well-known that the number of…

Every simple game is a monotone Boolean function. For the other direction we just have to exclude the two constant functions. The enumeration of monotone Boolean functions with distinguishable variables is also known as the Dedekind's…

Let $V_n$ be the number of equivalence classes of invertible maps from $\{0,1\}^n$ to $\{0,1\}^n$, under action of permutation of variables on domain and range. So far, the values $V_n$ have been known for $n\le 6$. This paper describes the…

The only known example of an almost perfect nonlinear (APN) permutation in even dimension was obtained by applying CCZ-equivalence to a specific quadratic APN function. Motivated by this result, there have been numerous recent attempts to…

This paper establishes several upper and lower estimates for the maximal number of the connected components of the solution sets of monotone affine vector variational inequalities. Our results give a partial solution to Question~2 in [N.D.…

A Boolean function of n bits is balanced if it takes the value 1 with probability 1/2. We exhibit a balanced Boolean function with a randomized evaluation procedure (with probability 0 of making a mistake) so that on uniformly random…

In this report, we show that all n-variable Boolean function can be represented as polynomial threshold functions (PTF) with at most $0.75 \times 2^n$ non-zero integer coefficients and give an upper bound on the absolute value of these…

We prove a lower bound of $\Omega(n^{1/2 - c})$, for all $c>0$, on the query complexity of (two-sided error) non-adaptive algorithms for testing whether an $n$-variable Boolean function is monotone versus constant-far from monotone. This…

We prove two conjectures on correlation inequalities for functions that are linear combinations of unimodal Boolean monotone nondecreasing functions

We study the most-informative Boolean function conjecture using a differential equation approach. This leads to a formulation of a functional inequality on finite-dimensional random variables. We also develop a similar inequality in the…

Any monotone Boolean circuit computing the $n$-dimensional Boolean convolution requires at least $n^2$ and-gates. This precisely matches the obvious upper bound.

A boolean function $f(x_1,...,x_n)$ is \textit{weakly symmetric} if it is invariant under a transitive permutation group on its variables. A boolean function $f(x_1,...,x_n)$ is \textit{elusive} if we have to check all $x_1$,..., $x_n$ to…

The 93 minions of Boolean functions stable under left composition with the clone of self-dual monotone functions are described. As an easy consequence, all $(C_1,C_2)$-stable classes of Boolean functions are determined for an arbitrary…

A simple way to generate a Boolean function is to take the sign of a real polynomial in $n$ variables. Such Boolean functions are called polynomial threshold functions. How many low-degree polynomial threshold functions are there? The…

Let $\Psi_{k,n}$ denote the number of inequivalent binary self-orthogonal $[n,k]$ codes. We present a method which allows us to compute $\Psi_{k,n}$ explicitly for a moderate $k$ and an arbitrary $n$. Included in this paper are explicit…