Scale-Free Random SAT Instances

We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known that many industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. This is not possible, in general, with classical randomly generated instances. We provide a different generation model of SAT instances, called \emph{scale-free random SAT instances}. It is based on the use of a non-uniform probability distribution $P(i)\sim i^{-\beta}$ to select variable $i$, where $\beta$ is a parameter of the model. This results into formulas where the number of occurrences $k$ of variables follows a power-law distribution $P(k)\sim k^{-\delta}$ where $\delta = 1 + 1/\beta$. This property has been observed in most real-world SAT instances. For $\beta=0$, our model extends classical random SAT instances. We prove the existence of a SAT-UNSAT phase transition phenomenon for scale-free random 2-SAT instances with $\beta<1/2$ when the clause/variable ratio is $m/n=\frac{1-2\beta}{(1-\beta)^2}$. We also prove that scale-free random k-SAT instances are unsatisfiable with high probability when the number of clauses exceeds $\omega(n^{(1-\beta)k})$. %This implies that the SAT/UNSAT phase transition phenomena vanishes when $\beta>1-1/k$, and formulas are unsatisfiable due to a small core of clauses. The proof of this result suggests that, when $\beta>1-1/k$, the unsatisfiability of most formulas may be due to small cores of clauses. Finally, we show how this model will allow us to generate random instances similar to industrial instances, of interest for testing purposes

An Improved Separation of Regular Resolution from Pool Resolution and Clause Learning

We prove that the graph tautology principles of Alekhnovich, Johannsen, Pitassi and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses and without degenerate resolution inferences. We also prove that these graph tautology principles can be refuted by polynomial size DPLL proofs with clause learning, even when restricted to greedy, unit-propagating DPLL search

Degree complexity for a modified pigeonhole principle

We consider a modification of the pigeonhole principle, M P H P, introduced by Goerdt in [7]. M P H P is defined over n pigeons and log n holes, and more than one pigeon can go into a hole (according to some rules). Using a technique of Razborov [9] and simplified by Impagliazzo, Pudlak and Sgall [8], we prove that any Polynomial Calculus refutation of a set of polynomials encoding the M P H P, requires degree Omega(log n). We also prove a simple Lemma giving a simulation of Resolution by Polynomial Calculus. Using this lemma, and a Resolution upper bound by Goerdt [7], we obtain that the degree lower bound is tight. Our lower bound establishes the optimality of the tree-like Resolution simulation by the Polynomial Calculus given in [6]

Community Structure in Industrial SAT Instances

Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental process. It is believed that these techniques exploit the underlying structure of industrial instances. However, there are few works trying to exactly characterize the main features of this structure. The research community on complex networks has developed techniques of analysis and algorithms to study real-world graphs that can be used by the SAT community. Recently, there have been some attempts to analyze the structure of industrial SAT instances in terms of complex networks, with the aim of explaining the success of SAT solving techniques, and possibly improving them. In this paper, inspired by the results on complex networks, we study the community structure, or modularity, of industrial SAT instances. In a graph with clear community structure, or high modularity, we can find a partition of its nodes into communities such that most edges connect variables of the same community. In our analysis, we represent SAT instances as graphs, and we show that most application benchmarks are characterized by a high modularity. On the contrary, random SAT instances are closer to the classical Erd\"os-R\'enyi random graph model, where no structure can be observed. We also analyze how this structure evolves by the effects of the execution of a CDCL SAT solver. In particular, we use the community structure to detect that new clauses learned by the solver during the search contribute to destroy the original structure of the formula. This is, learned clauses tend to contain variables of distinct communities

Polynomial Calculus for MaxSAT

MaxSAT is the problem of finding an assignment satisfying the maximum number of clauses in a CNF formula. We consider a natural generalization of this problem to generic sets of polynomials and propose a weighted version of Polynomial Calculus to address this problem. Weighted Polynomial Calculus is a natural generalization of MaxSAT-Resolution and weighted Resolution that manipulates polynomials with coefficients in a finite field and either weights in ? or ?. We show the soundness and completeness of these systems via an algorithmic procedure. Weighted Polynomial Calculus, with weights in ? and coefficients in ??, is able to prove efficiently that Tseitin formulas on a connected graph are minimally unsatisfiable. Using weights in ?, it also proves efficiently that the Pigeonhole Principle is minimally unsatisfiable

L'Escola com a projecte sociocultural

Abstract not availabl

L'Acumulació de perdigons de plom al Parc Natural del Fondó d'Elx (Alacant): distribució espacial i propostes d'actuació

Presentem els resultats de l'anàlisi de l'acumulació de perdigons de plom als sediments de dues basses del Parc Natural del Fondó d'Elx (Alacant). El plom és un element poc freqüent a la natura però que es pot concentrar als aiguamolls com a conseqüència de l'activitat cinegètica. Aquesta acumulació pot ocasionar que diferents tipus d'organismes ingereixin aquest metall, especialment l'avifauna aquàtica, tot produint-los una greu intoxicació. El Fondó és un sistema aquàtic regulat per l'home on s'ha practicat la caça des de fa molts anys; darrerament s'han trobat en aquest paratge nombrosos exemplars de flamencs (Phoenicopterus ruber) morts per l'emmetzinament amb perdigons de plom. Per tal de corroborar la causa d'aquesta mortalitat, es va procedir a la realització d'aquest estudi. Els resultats obtinguts demostren que s'hi han acumulat grans quantitat de plom al Parc, amb densitats a l'embassament de Llevant de 166,22 perdigons m2, i de 121,57 perdigons m2 a la reserva natural. Es proposa la utilització de méodes de representació especial (Kriging) per localitzar als aiguamolls les zones on l'acumulació és més elevada. També es proposen diferents mesures correctores i possibles solucions específiques que puguin alleugerir el problema.Sediments of two lagoons of the «Parc Natural del Fondo d'Elx» (Alacant, Spain) where analized in order to determine the existing content of lead gunshots accumulated by hunting activity. This can cause poisoning for ingestion in a waste range of organisms, specially in birds. The Fondo is a man made wetland, were hunting has been played for many years. In recent years, many flamingo (Phoenicopterus rubber) individuals have been found dead by lead shot ingestion. The results evidence hight levels of lead accumulation in sediments of the Natural Park. Mean densities are 166.22 shots m2 in the East Pond and 122.57 shots m2 in the Natural Reserve. A method of spatial interpolation (Kriging) is proposed for the location of the higher lead accumulation areas in wetlands. Possible solutions and specific corrective measures are also proposed.Presentamos los resultados del análisis de la acumulac de perdigones de plomo en los sedimentos de dos lagunas del Parque Natural del Hondo de Elche (Alicante). El plomo es un elemento poco frecuente en la naturaleza pero que se puede concentrar en determinados lugares como las zonas húmedas, debido a la actividad cinegética. Esta acumulación puede ocasionar la ingestión del metal por distintos tipos de organismos (sobre todo aves acuáticas), en los que se produce una grave intoxicación. El Hondo es una zona húmeda regulada por el hombre, en la que se ha practicado la caza desde hace muchos años; en estos últimos se han encontrado en este Parque numerosos ejemplares de flamencos (Phoenicopterus ruber) muertos por envenenamiento con perdigones de plomo. Para corroborar la causa de esta mortalidad se procedió a la realización del presente estudio. Los resultados ponen de manifiesto las elevadas densidades de plomo en los sedimentos sumergidos del Parque, siendo las medias obtenidas en el embalse de Levante de 166,22 perdigones m2, y de 121,57 perdigones m2 en la reserva natural. Se proponen técnicas de interpolación espacial (Kriging) para la localización de áreas de zonas húmedas donde la acumulación es mayor. También se proponen distintas medidas correctoras y posibles soluciones que puedan aliviar el problema