Federico Infusino

Micro and macro models of granular computing induced by the indiscernibility relation

In rough set theory (RST), and more generally in granular computing on information tables (GRC-IT), a central tool is the Pawlaks indiscernibility relation between objects of a universe set with respect to a fixed attribute subset. Let us observe that Pawlaks relation induces in a natural way an equivalence relation ź on the attribute power set that identifies two attribute subsets yielding the same indiscernibility partition. We call indistinguishability relation of a given information table I the equivalence relation ź, that can be considered as a kind of global indiscernibility. In this paper we investigate the mathematical foundations of indistinguishability relation through the introduction of two new structures that are, respectively, a complete lattice and an abstract simplicial complex. We show that these structures can be studied at both a micro granular and a macro granular level and that are naturally related to the core and the reducts of I . We first discuss the role of these structures in GrC-IT by providing some interpretations, then we prove several mathematical results concerning the fundamental properties of such structures.

Rough Set Theory Applied to Simple Undirected Graphs

The incidence matrix of a simple undirected graph is used as an information table. Then, rough set notions are applied to it: approximations, membership function, positive region and discernibility matrix. The particular cases of complete and bipartite graphs are analyzed. The symmetry induced in graphs by the indiscernibility relation is studied and a new concept of generalized discernibility matrix is introduced.

The granular partition lattice of an information table

In this paper we study the lattice of all indiscernibility partitions induced from attribute subsets of a knowledge representation system (information table in the finite case). This lattice, that we here call granular partition lattice, is a very well studied order structure in granular computing and data base theory and it provides a complete hierarchical classification of the knowledge obtained from all possible choices of attribute subsets. We show that it has a lattice structure also in the infinite case and we provide several isomorphic characterizations for this lattice. We discuss the potentiality of this order structure from both a micro-granular and a macro-granular perspective. Furthermore, the sub-poset of all the indiscernibility closures needed to determine when an arbitrary partition is an indiscernibility one is studied. Finally, we show the monotonic behaviour of the granular partition lattice with respect to entropy of partitions and attribute dependency in decision tables.

Generalizations of rough set tools inspired by graph theory

We introduce and study new generalizations of some rough set tools. Namely, the extended core, the generalized discernibility function, the discernibility space and the maximum partitioner. All these concepts where firstly introduced during the application of rough set theory to graphs, here we show that they have an interesting and useful interpretation also in the general setting. Indeed, among other results, we prove that reducts can be computed in incremental polynomial time, we give some conditions in order that a partition coincides with an indiscernibility partition of a given information table and we give the conditions such that a discernibility matrix corresponds to an information table.

Knowledge pairing systems in granular computing

In Pawlaks theory of information systems, one has a collection of objects and knows the values of any object with respect a certain class of properties, usually called attributes. An implicit assumption of this theory is that the difference between the nature of objects and attributes is well outlined. In our paper we generalize the concept of information system by analyzing the case in which there is no a priori distinction in the nature of objects and attributes and, hence, both the interpretations are admissible. We call the structure arising in the previous context knowledge pairing system. We study the indiscernibility relations induced in both the admissible interpretations by means of up-down operators, in such a way to have a direct analogy with the extent and intent operators used in Formal Context Analysis. In particular, we investigate three models of knowledge pairing systems arising from real contexts and modeled respectively by graphs, digraphs and hypergraphs. We show the real convenience to use the notion of knowledge pairing system focusing on interpretation of this structure and discussing the two admissible perspectives obtained by avoiding the difference between the roles of objects and attributes.

Preclusivity and Simple Graphs

The adjacency relation of a simple undirected graph is a preclusive (irreflexive and symmetric) relation. Hence, it originates a preclusive space enabling us to define the lower and upper preclusive approximations of graphs and two orthogonality graphs. Further, the possibility of defining the similarity lower and upper approximations and the sufficiency operator on graphs will be investigated, with particular attention to complete and bipartite graphs. All these mappings will be put in relation with Formal Concept Analysis and the theory of opposition.

Preclusivity and Simple Graphs: The n –cycle and n –path Cases

Two classes of graphs, the n–cycles and n–paths, are interpreted as preclusivity spaces. In this way, it is possible to define two pairs of approximations on them: one based on a preclusive relation and another one based on a similarity relation. Further, two relations can be defined among the set of vertices and they define two different graphs, which are here studied.

Dependency structures for decision tables

Abstract In this paper we study decision tables from a more general and abstract outlook. We focus our attention on consistency and inconsistency of decision tables. The starting point of our analysis is the remark that an inconsistent table has different local degrees of consistency depending on how an object and a condition attribute subset are chosen. When X and A run respectively over the object and the condition attribute set, we describe the interrelations of local consistencies by means of two set operators. They enable us to generalize the classical Pawlaks attribute dependency function . The operatorial standpoint correlates the study of decision tables in RST to the classical mathematical theories investigated through functional operators. In this perspective, we are also interested in finding which condition attribute subsets preserve the local positive region. This is the main reason to introduce the notions of local positive essentials and local positive reducts . These attribute subset families, in general, do not satisfy the properties of their counterparts in information table theory. Hence, in order to extend these results to decision tables, we can follow two different approaches: to define a subclass of decision tables in which they hold or to change the nature of the hypergraph induced by the decision discernibility matrix.

Granular computing on information tables: Families of subsets and operators

Abstract In this work we use the granular computing paradigm to study specific types of families of subsets, operators and families of ordered pairs of sets of attributes which are naturally induced by information tables. In an unifying perspective, by means of some representation results, we connect the study of finite closure systems, matroids and finite lattice theory in the scope of the more general notion of attribute dependency based on information tables. For a fixed finite set Ω and for a corresponding information table J having attribute set Ω, the fundamental tool we use to proceed in our investigation is the equivalence relation ≈ J on the power set P ( Ω ) which identifies two any sets of attributes inducing the same indiscernibility relation on the object set of J . We interpret the attribute dependency as a preorder ≥ J on P ( Ω ) whose induced equivalence relation coincides with ≈ J . Then we investigate in detail the links between the preorder ≥ J , a closure system and an abstract simplicial complex on Ω naturally induced by ≈ J and specific families of ordered pairs of sets of attributes on Ω.

Dependency and accuracy measures for directed graphs

Abstract In this paper we use finite directed graphs (digraphs) as mathematical models to study two basic notions widely analyzed in granular computing: the attribute dependency and the approximation accuracy. To be more specific, at first we interpret any digraph as a Boolean information table, next we study the approximation accuracy for three fundamentals digraph families: the directed path, the directed cycle and the transitive tournament. We also introduce a new global average for the attribute dependency in any information table and we determine such number for any directed path. For the transitive tournament we provide a lower bound.