Related papers: Unsatisfiable (k,(4*2^k/k))-CNF formulas

We solve the boson normal ordering problem for F[(a*)^r a^s], with r,s positive integers, where a* and a are boson creation and annihilation operators satisfying [a,a*]=1. That is, we provide exact and explicit expressions for the normal…

For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…

We study the typical case properties of the 1-in-3 satisfiability problem, the boolean satisfaction problem where a clause is satisfied by exactly one literal, in an enlarged random ensemble parametrized by average connectivity and…

We prove a version of Gauss's Lemma. It recursively constructs polynomials {c_k} for k=0,1,...,m+n, in Z[a_i,A_i,b_j,B_j] for i=0,...,m, and j=0,1,...,n, having degree at most (m+n choose m) in each of the four variable sets, such that…

The main result is a direct proof of the implication $(LVKF_{k,3})\Rightarrow( LT_{3k-1,3})$ below. Consider the following statements: ($LVKF_{1,3}$) From any 11 points in $ \mathbb{R}^{3}$ one can choose 3 pairwise disjoint triples whose…

Hitting formulas have been studied in many different contexts at least since [Iwama,89]. A hitting formula is a set of Boolean clauses such that any two of them cannot be simultaneously falsified. [Peitl,Szeider,05] conjectured that hitting…

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$ and let $r$ denote the largest degree of an irreducible factor of $F$ over the rationals. Let $k$ be an integer with $k \geq 2$…

An asymptotic formula is proved for the k-fold divisor function averaged over homogeneous polynomials of degree k in k-1 variables coming from incomplete norm forms.

A regular language is $k$-piecewise testable if it is a finite boolean combination of languages of the form $\Sigma^* a_1 \Sigma^* \cdots \Sigma^* a_n \Sigma^*$, where $a_i\in\Sigma$ and $0\le n \le k$. Given a DFA $A$ and $k\ge 0$, it is…

Decomposable Negation Normal Forms (DNNFs) are Boolean circuits in negation normal form where the subcircuits leading into each AND gate are defined on disjoint sets of variables. We prove a strongly exponential lower bound on the size of…

We introduce a multi-sorted stratified syllogistic, called 4LQSR, admitting variables of four sorts and a restricted form of quantification over variables of the first three sorts, and prove that it has a solvable satisfiability problem by…

In many decision-making processes, one may prefer multiple solutions to a single solution, which allows us to choose an appropriate solution from the set of promising solutions that are found by algorithms. Given this, finding a set of…

Given a Boolean function f, the quantity ess(f) denotes the largest set of assignments that falsify f, no two of which falsify a common implicate of f. Although ess(f)$ is clearly a lower bound on cnf_size(f) (the minimum number of clauses…

The topic of this paper is the Finiteness Conjecture for minimally unsatisfiable clause-sets (MUs), stating that for each fixed deficiency (number of clauses minus number of variables) there are only finitely many patterns, given a certain…

We ask, for which $n$ does there exists a $k$, $1 \leq k < n$ and $(k,n)=1$, so that $k/n$ has a continued fraction whose partial quotients are bounded in average by a constant $B$? This question is intimately connected with several other…

In sphere of research of discrete optimization algorithms efficiency the important place occupies a method of polynomial reducibility of some problems to others with use of special purpose components. In this paper a novel method of compact…

Dependency Quantified Boolean Formulas (DQBF) generalize QBF by explicitly specifying which universal variables each existential variable depends on, instead of relying on a linear quantifier order. The satisfiability problem of DQBF is…

We construct simple, explicit matrices with columns having unit $\ell^2$ norm and discrepancy approaching $1 + \sqrt{2} \approx 2.414$. This number gives a lower bound, the strongest known as far as we are aware, on the constant appearing…

Firewall policies are an important line of defence in cybersecurity, specifying which packets are allowed to pass through a network and which are not. These firewall policies are made up of a list of interacting rules. In practice, firewall…

We evaluate in closed form several alternating infinite series involving the binomial coefficients $C(4n,2n)$ and $C(4n+2,2n+1)$ in the denominator. One of our results generalizes an identity that was obtained experimentally by Sprugnoli in…