On implicational bases of closure systems with unique critical sets

We show that every optimum basis of a finite closure system, in D.Maier's sense, is also right-side optimum, which is a parameter of a minimum CNF representation of a Horn Boolean function. New parameters for the size of the binary part are also established. We introduce a K-basis of a general closure system, which is a refinement of the canonical basis of Duquenne and Guigues, and discuss a polynomial algorithm to obtain it. We study closure systems with the unique criticals and some of its subclasses, where the K-basis is unique. A further refinement in the form of the E-basis is possible for closure systems without D-cycles. There is a polynomial algorithm to recognize the D-relation from a K-basis. Thus, closure systems without D-cycles can be effectively recognized. While E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete problem.Comment: Presented on International Symposium of Artificial Intelligence and Mathematics (ISAIM-2012), Ft. Lauderdale, FL, USA Results are included into plenary talk on conference Universal Algebra and Lattice Theory, June 2012, Szeged, Hungary 29 pages and 2 figure

Ordered direct implicational basis of a finite closure system

Closure system on a finite set is a unifying concept in logic programming, relational data bases and knowledge systems. It can also be presented in the terms of finite lattices, and the tools of economic description of a finite lattice have long existed in lattice theory. We present this approach by describing the so-called D-basis and introducing the concept of ordered direct basis of an implicational system. A direct basis of a closure operator, or an implicational system, is a set of implications that allows one to compute the closure of an arbitrary set by a single iteration. This property is preserved by the D-basis at the cost of following a prescribed order in which implications will be attended. In particular, using an ordered direct basis allows to optimize the forward chaining procedure in logic programming that uses the Horn fragment of propositional logic. One can extract the D-basis from any direct unit basis S in time polynomial in the size of S, and it takes only linear time of the cardinality of the D-basis to put it into a proper order. We produce examples of closure systems on a 6-element set, for which the canonical basis of Duquenne and Guigues is not ordered direct.Comment: 25 pages, 10 figures; presented at AMS conference, TACL-2011,ISAIM-2012 and at RUTCOR semina

Ordered direct implication basis of a finite closure system

Closure system on a nite set is a unifying concept in logic programming, relational data bases and knowledge systems. It can also be presented in the terms of nite lattices, and the tools of economic description of a nite lattice have long existed in lattice theory. We present this approach by describing the so-called D-basis and introducing the concept of ordered direct basis of an implicational system. A direct basis of a closure operator, or an implicational system, is a set of implications that allows one to compute the closure of an arbitrary set by a single iteration. This property is preserved by the D-basis at the cost of following a prescribed order in which implications will be attended. In particular, using an ordered direct basis allows to optimize the forward chaining procedure in logic programming that uses the Horn fragment of propositional logic. One can extract the D-basis from any direct unit basis in time polynomial in the size s( ), and it takes only linear time of the cardinality of the D-basis to put it into a proper order. We produce examples of closure systems on a 6-element set, for which the canonical basis of Duquenne and Guigues is not ordered direc

On implicational bases of closure system with unique critical sets

We show that every optimum basis of a nite closure system, in D. Maier's sense, is also right-side optimum, which is a parameter of a minimum CNF representation of a Horn Boolean function. New parameters for the size of the binary part are also established. We introduce the K-basis of a general closure system, which is a re nement of the canonical basis of V. Duquenne and J.L. Guigues, and discuss a polynomial algorithm to obtain it. We study closure systems with unique critical sets, and some subclasses of these where the K-basis is unique. A further re nement in the form of the E-basis is possible for closure systems without D-cycles. There is a polynomial algorithm to recognize the D-relation from a K-basis. Thus, closure systems without D-cycles can be e ectively recognized. While the E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete proble

On implicational bases of closure system with unique critical sets

We show that every optimum basis of a nite closure system, in D. Maier's sense, is also right-side optimum, which is a parameter of a minimum CNF representation of a Horn Boolean function. New parameters for the size of the binary part are also established. We introduce the K-basis of a general closure system, which is a re nement of the canonical basis of V. Duquenne and J.L. Guigues, and discuss a polynomial algorithm to obtain it. We study closure systems with unique critical sets, and some subclasses of these where the K-basis is unique. A further re nement in the form of the E-basis is possible for closure systems without D-cycles. There is a polynomial algorithm to recognize the D-relation from a K-basis. Thus, closure systems without D-cycles can be e ectively recognized. While the E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete proble

A logic-based approach to deal with implicational systems and direct bases

El tratamiento de la información y el conocimiento es uno de los muchos campos en los que confluyen los métodos matemáticos y computacionales. Una de las áreas donde encontramos de forma clara esta concurrencia es en el Análisis de Conceptos Formales, donde los métodos de almacenamiento, descubrimiento, análisis y manipulación del conocimiento descansan sobre las sólidas bases del Álgebra y de la Lógica. En el Análisis de Conceptos Formales la información se representa en tablas binarias en las que se relacionan objetos con sus atributos. Dichas tablas, denominadas contextos formales, son el repositorio de datos del que se extrae el conocimiento mediante la utilización de técnicas algebraicas. Este conocimiento se puede representar de diversas formas, entre ellas se encuentran los conjuntos de implicaciones. Una de las principales ventajas de usar sistemas de implicaciones para representar el conocimiento es que admiten un tratamiento sintáctico por medio de la lógica, segundo pilar matemático en el que se sustenta la tesis. La mejor alternativa de cara al razonamiento automático viene de mano de la Lógica de Simplificación. El conjunto de axiomas y reglas de inferencias de esta lógica lleva directamente a un conjunto de equivalencias que permiten eliminar redundancias en los sistemas de implicaciones. La extracción de sistemas de implicaciones, y su posterior tratamiento y manipulación, constituyen un tema de actualidad en la comunidad del Análisis de Conceptos Formales. Los conjuntos de implicaciones extraídos pueden contener gran cantidad de información redundante, por lo que el estudio de propiedades que permitan caracterizar conjuntos equivalentes de implicaciones con menor redundancia o sin ella, se erige como uno de los retos más importantes. Sin embargo, como sucede en otras áreas, en algunas ocasiones puede ser interesante almacenar cierta clase de información redundante en función del uso posterior que se le pretenda dar. Sobresale pues, entre los temas de interés del área, el problema de la búsqueda de representaciones canónicas de sistemas de implicaciones que, satisfaciendo ciertas propiedades, permitan compilar todo el conocimiento extraído del contexto formal. Estas representaciones canónicas para los sistemas de implicaciones suelen recibir el nombre de `bases'. En esta tesis ponemos nuestra atención en un grupo de bases conocidas como `bases directas', que son aquellas que permiten calcular el cierre de cualquier conjunto en un único recorrido del sistema de implicaciones. Los objetivos generales de la tesis son dos: - El estudio de las bases directas en Análisis de Conceptos Formales clásico con la finalidad de obtener algoritmos eficientes para calcular dichas bases. Para ello analizamos las definiciones que aparecen en la bibliografía (base directa-optimal y D-base) y proponemos una alternativa (base dicótoma directa), así como métodos para su cálculo. - Establecer las bases para la extensión de estos resultados al Análisis de Conceptos Triádicos, en particular, introducir una lógica que permita el razonamiento automático sobre implicaciones en esta extensión. Se presentan dos lógicas: CAIL y CAISL. La primera permite caracterizar la semántica de las implicaciones y la segunda el razonamiento automático

Discovery of the D-basis in binary tables based on hypergraph dualization

Discovery of (strong) association rules, or implications, is an important task in data management, and it nds application in arti cial intelligence, data mining and the semantic web. We introduce a novel approach for the discovery of a speci c set of implications, called the D-basis, that provides a representation for a reduced binary table, based on the structure of its Galois lattice. At the core of the method are the D-relation de ned in the lattice theory framework, and the hypergraph dualization algorithm that allows us to e ectively produce the set of transversals for a given Sperner hypergraph. The latter algorithm, rst developed by specialists from Rutgers Center for Operations Research, has already found numerous applications in solving optimization problems in data base theory, arti cial intelligence and game theory. One application of the method is for analysis of gene expression data related to a particular phenotypic variable, and some initial testing is done for the data provided by the University of Hawaii Cancer Cente

Discovery of the D-basis in binary tables based on hypergraph dualization

Discovery of (strong) association rules, or implications, is an important task in data management, and it nds application in arti cial intelligence, data mining and the semantic web. We introduce a novel approach for the discovery of a speci c set of implications, called the D-basis, that provides a representation for a reduced binary table, based on the structure of its Galois lattice. At the core of the method are the D-relation de ned in the lattice theory framework, and the hypergraph dualization algorithm that allows us to e ectively produce the set of transversals for a given Sperner hypergraph. The latter algorithm, rst developed by specialists from Rutgers Center for Operations Research, has already found numerous applications in solving optimization problems in data base theory, arti cial intelligence and game theory. One application of the method is for analysis of gene expression data related to a particular phenotypic variable, and some initial testing is done for the data provided by the University of Hawaii Cancer Cente

Efficient Algorithms on the Family Associated to an Implicational System

International audienceAn implication system (IS) on a finite set S is a set of rules called Σ -implications of the kind A →_Σ B, with A,B ⊆ S. A subset X ⊆ S satisfies A →_Σ B when ''A ⊆ X implies B ⊆ X'' holds, so ISs can be used to describe constraints on sets of elements, such as dependency or causality. ISs are formally closely linked to the well known notions of closure operators and Moore families. This paper focuses on their algorithmic aspects. A number of problems issued from an IS Σ (e.g. is it minimal, is a given implication entailed by the system) can be reduced to the computation of closures φ _Σ (X), where φ _Σ is the closure operator associated to Σ . We propose a new approach to compute such closures, based on the characterization of the direct-optimal IS Σ _do which has the following properties: \beginenumerate ıtemit is equivalent to Σ ıtemφ _Σ _do(X) (thus φ _Σ (X)) can be computed by a single scanning of Σ _do-implications ıtemit is of minimal size with respect to ISs satisfying 1. and 2. \endenumerate We give algorithms that compute Σ _do, and from Σ _do closures φ _Σ (X) and the Moore family associated to φ _Σ