Tommaso Gentile

Simple graphs in granular computing

Given a graph, we interpret its adjacency matrix as an information table. We study this correspondence in two directions. Firstly, on the side of graphs by applying to it standard techniques from granular computing. In this way, we are able to connect automorphisms on graphs to the so-called indiscernibility relation and a particular hypergraph built from the starting graph to core and reducts. On the other hand, new concepts are introduced on graphs that have an interesting correspondence on information tables. In particular, some new topological interpretations of the graph and the concept of extended core are given.

On the connection of hypergraph theory with formal concept analysis and rough set theory

We present a unique framework for connecting different topics: hypergraphs from one side and Formal Concept Analysis and Rough Set Theory from the other. This is done through the formal equivalence among Boolean information tables, formal contexts and hypergraphs. Links with generic (i.e., not Boolean) information tables are established, through so-called nominal scaling. The particular case of k-uniform complete hypergraphs will then be studied. In this framework, we are able to solve typical problems of Rough Set Theory and Formal Concept Analysis using combinatorial techniques. More in detail, we will give a formula to compute the degree of dependency and the partial implication between two sets of attributes, compute the set of reducts and define the structure of the partitions generated by all the definable indiscernibility relations.

Parallel and sequential dynamics of two discrete models of signed integer partitions

Abstract In this paper we complete and generalize some previous results concerning the computing of the sequential and parallel convergent time for two discrete dynamical system of signed integer partitions. We also refine the concept of parallel convergent time for a finite graded partially ordered set (briefly poset) X which is also a discrete dynamical model. To this aim we define the concept of fundamental sequence of X and we compute this sequence in two particularly important cases. In the first case, when X is the finite lattice S ( n , r ) of all the signed integer partitions a r , … , a 1 , b 1 , … , b n - r such that r ⩾ a r ⩾ ⋯ ⩾ a 1 ⩾ 0 ⩾ b 1 ⩾ ⋯ ⩾ b n - r ⩾ - ( n - r ) , where n ⩾ r ⩾ 0 and the unique part that can be repeated is 0. In the second case, when X is the sub-lattice S ( n , d , r ) of all the signed integer partitions of S ( n , r ) having exactly d non-zero parts. The relevance of the previous lattices as discrete dynamical models is related to their link with some unsolved extremal combinatorial sum problems.

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.

Simple Undirected Graphs as Formal Contexts

The adjacency matrix of a graph is interpreted as a formal context. Then, the counterpart of Formal Concept Analysis (FCA) tools are introduced in graph theory. Moreover, a formal context is seen as a Boolean information table, the structure at the basis of Rough Set Theory (RST). Hence, we also apply RST tools to graphs. The peculiarity of the graph case, put in evidence and studied in the paper, is that both FCA and RST are based on a (different) binary relation between objects.

Dominance order on signed integer partitions

Abstract In 1973 Brylawski introduced and studied in detail the dominance partial order on the set Par(m) of all integer partitions of a fixed positive integer m. As it is well known, the dominance order is one of the most important partial orders on the finite set Par(m). Therefore it is very natural to ask how it changes if we allow the summands of an integer partition to take also negative values. In such a case, m can be an arbitrary integer and Par(m) becomes an infinite set. In this paperwe extend the classical dominance order in this more general case. In particular, we consider the resulting lattice Par(m) as an infinite increasing union on n of a sequence of finite lattices O(m, n). The lattice O(m, n) can be considered a generalization of the Brylawski lattice. We study in detail the lattice structure of O(m, n).

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.