Kira V. Adaricheva

Ordered direct implicational basis of a finite closure system

The closure system on a finite set is a unifying concept in logic programming, relational databases 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 defining the D-basis and introducing the concept of an 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 s(@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.

Representing finite convex geometries by relatively convex sets

A closure system with the anti-exchange axiom is called a convex geometry. One geometry is called a sub-geometry of the other if its closed sets form a sublattice in the lattice of closed sets of the other. We prove that convex geometries of relatively convex sets in n-dimensional vector space and their finite sub-geometries satisfy the n-Carousel Rule, which is the strengthening of the n-Caratheodory property. We also find another property, that is similar to the simplex partition property and independent of 2-Carousel Rule, which holds in sub-geometries of 2-dimensional geometries of relatively convex sets.

On implicational bases of closure systems with unique critical sets

We present results inspired by the study of closure systems with unique critical sets. Many of these results, however, are of a more general nature. Among those is the statement that every optimum basis of a finite closure system, in D. Maiers sense, is also right-side optimum. New parameters for the size of the binary part of a closure system are established. We introduce the K-basis of a closure system, which is a refinement of the canonical basis of V. Duquenne and J.L. Guigues, and discuss a polynomial algorithm to obtain it. The main part of the paper is devoted to closure systems with unique critical sets, and some subclasses of these 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. Consequently, closure systems without D-cycles can be effectively recognized. While the E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete problem.

Note on the description of join-distributive lattices by permutations

Let L be a join-distributive lattice with length n and width(JiL) ≤ k. There are two ways to describe L by k − 1 permutations acting on an n-element set: a combinatorial way given by P.H. Edelman and R. E. Jamison in 1985 and a recent lattice theoretical way of the second author. We prove that these two approaches are equivalent. Also, we characterize join-distributive lattices by trajectories.

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 finds application in artificial intelligence, data mining and the semantic web. We introduce a novel approach for the discovery of a specific 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 defined in a lattice theoretic framework and the hypergraph dualization algorithm that allows us to effectively produce the set of transversals for a given Sperner hypergraph. The latter algorithm, first developed by specialists from Rutgers Center for Operations Research, has already found numerous applications in solving optimization problems in data base theory, artificial 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 data provided by the University of Hawaii Cancer Center.

Realization of abstract convex geometries by point configurations

The Edelman-Jamison problem is to characterize those abstract convex geometries that are representable by a set of points in the plane. We show that some natural modification of the Edelman-Jamison problem is equivalent to the well known NP-hard order type problem. The relation to the realizability of oriented matroids is clarified.

Measuring the Implications of the D-Basis in Analysis of Data in Biomedical Studies

We introduce the parameter of relevance of an attribute of a binary table to another attribute of the same table, computed with respect to an implicational basis of a closure system associated with the table. This enables a ranking of all attributes, by relevance parameter to the same fixed attribute, and, as a consequence, reveals the implications of the basis most relevant to this attribute. As an application of this new metric, we test the algorithm for D-basis extraction presented in Adaricheva and Nation [1] on biomedical data related to the survival groups of patients with particular types of cancer. Each test case requires a specialized approach in converting the real-valued data into binary data and careful analysis of the transformed data in a multi-disciplinary environment of cross-field collaboration.

On Scattered Convex Geometries

A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of algebraic convex geometries we describe when the collection of closed sets is order scattered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice Ω(η), that does not appear among minimal obstructions to order-scattered algebraic modular lattices, plays a prominent role in convex geometries case. The connection to topological scatteredness is established in convex geometries of relatively convex sets.

Optimum basis of finite convex geometry

Convex geometries form a subclass of closure systems with unique criticals, or

Bases of Closure Systems

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