Pruning Processes and a New Characterization of Convex Geometries

We provide a new characterization of convex geometries via a multivariate version of an identity that was originally proved by Maneva, Mossel and Wainwright for certain combinatorial objects arising in the context of the k-SAT problem. We thus highlight the connection between various characterizations of convex geometries and a family of removal processes studied in the literature on random structures.Comment: 14 pages, 3 figures; the exposition has changed significantly from previous versio

On the Satisfiability Threshold and Clustering of Solutions of Random 3-SAT Formulas

We study the structure of satisfying assignments of a random 3-SAT formula. In particular, we show that a random formula of density 4.453 or higher almost surely has no non-trivial "core" assignments. Core assignments are certain partial assignments that can be extended to satisfying assignments, and have been studied recently in connection with the Survey Propagation heuristic for random SAT. Their existence implies the presence of clusters of solutions, and they have been shown to exist with high probability below the satisfiability threshold for k-SAT with k>8, by Achlioptas and Ricci-Tersenghi, STOC 2006. Our result implies that either this does not hold for 3-SAT or the threshold density for satisfiability in 3-SAT lies below 4.453. The main technical tool that we use is a novel simple application of the first moment method

P versus NP: el problema estrella de la matemàtica de la computació

El problema «P versus NP» és un dels set Problemes del Mil.lenni de l?Institut Clay de Matemàtiques, la solució del qual estaria premiada amb un milió de dòlars. En aquest article presentem de manera divulgativa el problema i els seus orígens, donant pel camí exemples de problemes computacionals de diferents nivells de dificultat, alguns algoritmes no trivials, la definició de màquina de Turing ?el model matemàtic d?ordinador? i el concepte de reducció polinòmica entre problemes. La part més avançada de l?article presenta una demostració del teorema de Razborov sobre circuits monòtons de l?any 1985 que resol un cas especial de la conjectura. També donem una traducció al català d?una carta de Gödel a Von Neumann de l?any 1956 que es va descobrir l?any 1988 i que es pot considerar com la primera formulació per escrit del problema «P versus NP».The problem ?P versus NP? is one of the seven Millennium Prize Problems of the Clay Mathematics Institute, whose solution would be awarded one million dollars. In this article we present in an informal manner the problem and its origin, giving along the way examples of computational problems of diferent levels of hardness, some non-trivial algorithms, the definition of Turing machine ? the mathematical model of a computer ? and the concept of polynomial reduction between problems. The most advanced part of the article presents a proof of Razborov?s 1985 theorem for monotone circuits, which solves a special case of the conjecture. We also give a translation into Catalan of a letter from Gödel to Von Neumann from 1956 which was discovered in 1988 and can be considered the first written formulation of the problem ?P versus NP?

Analysis of the use of automatic judges in computer programming classes in vocational education

The goal of this thesis is to analyse the use of automatic judges in the teaching of Computer Programming in Vocational Training programs, in a sample of secondary schools in Catalonia and Spain. We review the academic literature on automatic judges and describe their history and the different services that are currently available. Through surveys to teachers and students from Vocational Training centres, we have collected information about the tools in use. We classify twelve of them according to features that we found to be relevant to the teachers we surveyed. Using the collected data, we study the reasons in favour and against automatic judges, as well as the level of familiarity of active teachers with those tools. We identify some discrepancies between teacher's expectations and actual experiences of teachers who have adopted them. We use students' responses to derive statistics about their satisfaction, and to detect differences in the attitude of those who use judges in class and those who don't with respect to the subject of programming. In order to detect any effects on learning, we use three Bebras challenges to evaluate the competence of algorithmic and computational thinking. No difference in the ability of students to complete these challenges has been observed, but a difference in how willing they are to work on them has been detected. The effect is contrary to the one originally hypothesised, which leads to some interesting questions for further study

P versus NP: el problema estrella de la matemàtica de la computació

El problema «P versus NP» és un dels set Problemes del Mil . lenni de l'Institut Clay de Matemàtiques, la solució del qual estaria premiada amb un milió de dòlars. En aquest article presentem de manera divulgativa el problema i els seus orígens, donant pel camí exemples de problemes computacionals de diferents nivells de dificultat, alguns algoritmes no trivials, la definició de màquina de Turing el model matemàtic d'ordinador i el concepte de reducció polinòmica entre problemes. La part més avançada de l'article presenta una demostració del teorema de Razborov sobre circuits monòtons de l'any 1985 que resol un cas especial de la conjectura. També donem una traducció al català d'una carta de Gödel a Von Neumann de l'any 1956 que es va descobrir l'any 1988 i que es pot considerar com la primera formulació per escrit del problema «P versus NP»

A Hike in the Phases of the 1-in-3 Satisfiability

We summarise our results for the random $\epsilon$--1-in-3 satisfiability problem, where $\epsilon$ is a probability of negation of the variable. We employ both rigorous and heuristic methods to describe the SAT/UNSAT and Hard/Easy transitions.Comment: 2 pages, introductory level, proceed. for the Les Houches Session LXXXV 2006 on Complex System