相关论文: The number of 3-SAT functions

Secure multi-party computation using a physical deck of cards, often called card-based cryptography, has been extensively studied during the past decade. Card-based protocols to compute various Boolean functions have been developed. As each…

Three algorithms are presented that determine the existence of satisfying assignments for 3SAT Boolean satisfiability expressions. One algorithm is presented for determining an instance of a satisfying assignment, where such exists. The…

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…

Since the early 2000s physicists have developed an ingenious but non-rigorous formalism called the cavity method to put forward precise conjectures on phase transitions in random problems [Mezard, Parisi, Zecchina: Science 2002]. The cavity…

Boolean satisfiability [1] (k-SAT) is one of the most studied optimization problems, as an efficient (that is, polynomial-time) solution to k-SAT (for $k\geq 3$) implies efficient solutions to a large number of hard optimization problems…

We consider multifold convolutions of a combinatorial sequence $(a_n)_{n=0}^{\infty}$: namely, for each $k \in \N$ the $k$-fold convolution is $\mathcal{M}^{(k)}_n(\boldsymbol{a}) = \sum_{j_1+\dots+j_k=n} a_{j_1} \cdots a_{j_k}$. Let $C_n$…

Consider a non-negative sequence $c_n = h(n) \cdot n^{\alpha-1} \cdot \rho^{-n}$, where $h$ is slowly varying, $\alpha>0$, $0<\rho<1$ and $n\in\mathbb{N}$. We investigate the coefficients of $G(x,y) = \prod_{k\ge1}(1-x^ky)^{-c_k}$, which is…

We show that Monotone 3-Sat remains NP-complete if (i) each clause contains exactly three distinct variables, (ii) each clause is unique, i.e., there are no duplicates of the same clause, and (iii), amongst the clauses, each variable…

We consider simplified, monotone versions of Not-All-Equal 3-Sat and 3-Sat, variants of the famous Satisfiability Problem where each clause is made up of exactly three distinct literals. We show that Not-All-Equal 3-Sat remains NP-complete…

It is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers $(n_1,\ldots,n_R)$ by a system of quadratic forms $Q_1,\ldots, Q_R$ in $k$ variables, as long as $k$ is…

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$…

We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous counting algorithm for the permissive version…

Recently, quantitative versions of the Gibbard-Satterthwaite theorem were proven for $k=3$ alternatives by Friedgut, Kalai, Keller and Nisan and for neutral functions on $k \geq 4$ alternatives by Isaksson, Kindler and Mossel. We prove a…

The Boolean Satisfiability (SAT) problem is the canonical NP-complete problem and is fundamental to computer science, with a wide array of applications in planning, verification, and theorem proving. Developing and evaluating practical SAT…

Let $k \ge 3$ be a fixed integer. We exactly determine the asymptotic distribution of $\ln Z_k(G(n,m))$, where $Z_k(G(n,m))$ is the number of $k$-colourings of the random graph $G(n,m)$. A crucial observation to this aim is that the…

The Sun polynomials $g_n(x)$ are defined by \begin{align*} g_n(x)=\sum_{k=0}^n{n\choose k}^2{2k\choose k}x^k. \end{align*} We prove that, for any positive integer $n$, there hold \begin{align*} &\frac{1}{n}\sum_{k=0}^{n-1}(4k+3)g_k(x)…

The random $k$-SAT problem serves as a model that represents the 'typical' $k$-SAT instances. This model is thought to undergo a phase transition as the clause density changes, and it is believed that the random $k$-SAT problem is primarily…

Here we examine the number of ways to partition an integer $n$ into $k$th powers when $n$ is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion…

We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$…

We verify the 3-dimensional Glassey conjecture on asymptotically flat manifolds $(R^{1+3}, g)$, where the metric $g$ is certain small space-time perturbation of the flat metric, as well as the nontrapping asymptotically Euclidean manifolds.…