Counting self-dual monotone Boolean functions

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 represented as a concatenation $g_0\cdot g_1$, where $g_0, g_1\in D_{n-1}$ and $g_0\le g_1$. For each $x\in D_n$, we have dual $x^*\in D_n$ which is obtained by reversing and negating all bits. An element $x\in D_n$ is self-dual if $x=x^*$. Let $\lambda_n$ denote the cardinality of the set of all self-dual monotone Boolean functions of $n$ variables. The value $\lambda_n$ is also known as the $n$-th Hosten-Morris number. In this paper, we derive several algorithms for counting self-dual monotone Boolean functions and confirm the known result that $\lambda_9$ equals 423,295,099,074,735,261,880

Self-duality of bounded monotone boolean functions and related problems

AbstractIn this paper we examine the problem of determining the self-duality of a monotone boolean function in disjunctive normal form (DNF). We show that the self-duality of monotone boolean functions with n disjuncts such that each disjunct has at most k literals can be determined in O(2k2k2n) time. This implies an O(n2logn) algorithm for determining the self-duality of logn-DNF functions. We also consider the version where any two disjuncts have at most c literals in common. For this case we give an O(n4(c+1)) algorithm for determining self-duality

Counting and enumerating aggregate classifiers

peer reviewedaudience: researcherWe propose a generic model for the "weighted voting" aggregation step performed by several methods in supervised classification. Further, we construct an algorithm to count the number of distinct aggregate classifiers that arise in this model. When there are only two classes in the classification problem, we show that a class of functions that arises from aggregate classifiers coincides with the class of self-dual positive threshold Boolean functions

Improving the availability of mutual exclusion Systems on Incomplete Networks

We model a distributed system by a graph G=(V, E), where V represents the set of processes and E the set of bidirectional communication links between two processes. G may not be complete. A popular (distributed) mutual exclusion algorithm on G uses a coterie C(⊆2v), which is a nonempty set of nonempty subsets of V (called quorums) such that, for any two quorums P, Q ∈ C, 1) P∩Q ≠φ and 2) P￠Q hold. The availability is the probability that the algorithm tolerates process and/or link failures, given the probabilities that a process and a link, respectively, are operational. The availability depends on the coterie used in the algorithm. This paper proposes a method to improve the availability by transforming a given coterie

Transversal Merge Operation : A Nondominated Coterie Construction method for distributed mutual exclusion

A coterie is a set of subsets (called quorums) of the processes in a distributed system such that any two quorums intersect with each other and is mainly used to solve the mutual exclusion problem in a quorum-based algorithm. The choice of a coterie sensitively affects the performance of the algorithm and it is known that nondominated (ND) coteries achieve good performance in terms of criteria such as availability and load. On the other hand, grid coteries have some other attractive features : 1) A quorum size is small, which implies a low message complexity, and 2) a quorum is constructible on the fly, which benefits a low space complexity. However, they are not ND coteries unfortunately. To construct ND coteries having the favorite features of grid coteries, we introduce the transversal merge operation that transforms a dominated coterie into an ND coterie and apply it to grid coteries. We call the constructed ND coteries ND grid coteries. These ND grid coteries have availability higher than the original ones, inheriting the above desirable features from them. To demonstrate this fact, we then investigate their quorum size, load, and availability, and propose a dynamic quorum construction algorithm for an ND grid coterie

System support for object replication in distributed systems

Distributed systems are composed of a collection of cooperating but failure prone system components. The number of components in such systems is often large and, despite low probabilities of any particular component failing, the likelihood that there will be at least a small number of failures within the system at a given time is high. Therefore, distributed systems must be able to withstand partial failures. By being resilient to partial failures, a distributed system becomes more able to offer a dependable service and therefore more useful. Replication is a well known technique used to mask partial failures and increase reliability in distributed computer systems. However, replication management requires sophisticated distributed control algorithms, and is therefore a labour intensive and error prone task. Furthermore, replication is in most cases employed due to applications' non-functional requirements for reliability, as dependability is generally an orthogonal issue to the problem domain of the application. If system level support for replication is provided, the application developer can devote more effort to application specific issues. Distributed systems are inherently more complex than centralised systems. Encapsulation and abstraction of components and services can be of paramount importance in managing their complexity. The use of object oriented techniques and languages, providing support for encapsulation and abstraction, has made development of distributed systems more manageable. In systems where applications are being developed using object-oriented techniques, system support mechanisms must recognise this, and provide support for the object-oriented approach. The architecture presented exploits object-oriented techniques to improve transparency and to reduce the application programmer involvement required to use the replication mechanisms. This dissertation describes an approach to implementing system support for object replication, which is distinct from other approaches such as replicated objects in that objects are not specially designed for replication. Additionally, object replication, in contrast to data replication, is a function-shipping approach and deals with the replication of both operations and data. Object replication is complicated by objects' encapsulation of local state and the arbitrary interaction patterns that may exist among objects. Although fully transparent object replication has not been achieved, my thesis is that partial system support for replication of program-level objects is practicable and assists the development of certain classes of reliable distributed applications. I demonstrate the usefulness of this approach by describing a prototype implementation and showing how it supports the development of an example toy application. To increase their flexibility, the system support mechanisms described are tailorable. The approach adopted in this work is to provide partial support for object replication, relying on some assistance from the application developer to supply application dependent functionality within particular collators for dealing with processing of results from object replicas. Care is taken to make the programming model as simple and concise as possible

Structural and computational aspects of simple and influence games

Simple games are a fundamental class of cooperative games. They have a huge relevance in several areas of computer science, social sciences and discrete applied mathematics. The algorithmic and computational complexity aspects of simple games have been gaining notoriety in the recent years. In this thesis we review different computational problems related to properties, parameters, and solution concepts of simple games. We consider different forms of representation of simple games, regular games and weighted games, and we analyze the computational complexity required to transform a game from one representation to another. We also analyze the complexity of several open problems under different forms of representation. In this scenario, we prove that the problem of deciding whether a simple game in minimal winning form is decisive (a problem that is associated to the duality problem of hypergraphs and monotone Boolean functions) can be solved in quasi-polynomial time, and that this problem can be polynomially reduced to the same problem but restricted to regular games in shift-minimal winning form. We also prove that the problem of deciding wheter a regular game is strong in shift-minimal winning form is coNP-complete. Further, for the width, one of the parameters of simple games, we prove that for simple games in minimal winning form it can be computed in polynomial time. Regardless of the form of representation, we also analyze counting and enumeration problems for several subfamilies of these games. We also introduce influence games, which are a new approach to study simple games based on a model of spread of influence in a social network, where influence spreads according to the linear threshold model. We show that influence games capture the whole class of simple games. Moreover, we study for influence games the complexity of the problems related to parameters, properties and solution concepts considered for simple games. We consider extremal cases with respect to demand of influence, and we show that, for these subfamilies, several problems become polynomial. We finish with some applications inspired on influence games. The first set of results concerns to the definition of collective choice models. For mediation systems, several of the problems of properties mentioned above are polynomial-time solvable. For influence systems, we prove that computing the satisfaction (a measure equivalent to the Rae index and similar to the Banzhaf value) is hard unless we consider some restrictions in the model. For OLFM systems, a generalization of OLF systems (van den Brink et al. 2011, 2012) we provide an axiomatization of satisfaction. The second set of results concerns to social network analysis. We define new centrality measures of social networks that we compare on real networks with some classical centrality measures.Los juegos simples son una clase fundamental de juegos cooperativos, que tiene una enorme relevancia en diversas áreas de ciencias de la computación, ciencias sociales y matemáticas discretas aplicadas. En los últimos años, los distintos aspectos algorítmicos y de complejidad computacional de los juegos simples ha ido ganando notoriedad. En esta tesis revisamos los distintos problemas computacionales relacionados con propiedades, parámetros y conceptos de solución de juegos simples. Primero consideramos distintas formas de representación de juegos simples, juegos regulares y juegos de mayoría ponderada, y estudiamos la complejidad computacional requerida para transformar un juego desde una representación a otra. También analizamos la complejidad de varios problemas abiertos bajo diferentes formas de representación. En este sentido, demostramos que el problema de decidir si un juego simple en forma ganadora minimal es decisivo (un problema asociado al problema de dualidad de hipergrafos y funciones booleanas monótonas) puede resolverse en tiempo cuasi-polinomial, y que este problema puede reducirse polinomialmente al mismo problema pero restringido a juegos regulares en forma ganadora shift-minimal. También demostramos que el problema de decidir si un juego regular en forma ganadora shift-minimal es fuerte (strong) es coNP-completo. Adicionalmente, para juegos simples en forma ganadora minimal demostramos que el parámetro de anchura (width) puede computarse en tiempo polinomial. Independientemente de la forma de representación, también estudiamos problemas de enumeración y conteo para varias subfamilias de juegos simples. Luego introducimos los juegos de influencia, un nuevo enfoque para estudiar juegos simples basado en un modelo de dispersión de influencia en redes sociales, donde la influencia se dispersa de acuerdo con el modelo de umbral lineal (linear threshold model). Demostramos que los juegos de influencia abarcan la totalidad de la clase de los juegos simples. Para estos juegos también estudiamos la complejidad de los problemas relacionados con parámetros, propiedades y conceptos de solución considerados para los juegos simples. Además consideramos casos extremos con respecto a la demanda de influencia, y probamos que para ciertas subfamilias, varios de estos problemas se vuelven polinomiales. Finalmente estudiamos algunas aplicaciones inspiradas en los juegos de influencia. El primer conjunto de estos resultados tiene que ver con la definición de modelos de decisión colectiva. Para sistemas de mediación, varios de los problemas de propiedades mencionados anteriormente son polinomialmente resolubles. Para los sistemas de influencia, demostramos que computar la satisfacción (una medida equivalente al índice de Rae y similar al valor de Banzhaf) es difícil a menos que consideremos algunas restricciones en el modelo. Para los sistemas OLFM, una generalización de los sistemas OLF (van den Brink et al. 2011, 2012) proporcionamos una axiomatización para la medida de satisfacción. El segundo conjunto de resultados se refiere al análisis de redes sociales, y en particular con la definición de nuevas medidas de centralidad de redes sociales, que comparamos en redes reales con otras medidas de centralidad clásica