Fundamenta Informaticae - Volume 108, issue 3-4

Authors: Ciucci, Davide | Yao, Yiyu

Article Type: Other

DOI: 10.3233/FI-2011-418

Citation: Fundamenta Informaticae, vol. 108, no. 3-4, pp. i-i, 2011

Authors: Wolski, Marcin

Article Type: Research Article

Abstract: The paper studies complete and incomplete information systems, i.e. basic structures of rough set theory (RST), from the standpoint of monadic Boolean algebras (MBAs). In the first part we recall a simple characterisation of RST as a theory of MBAs. In the second part the paper discusses approximation operators obtained from incomplete information systems. In this case we lose a direct link to MBAs and thus our aim is to obtain an MBA which will additionally be interesting from the perspective of RST. In the third part we consider a simple application of approximation operators based on this MBA.

DOI: 10.3233/FI-2011-419

Citation: Fundamenta Informaticae, vol. 108, no. 3-4, pp. 181-196, 2011

Authors: Cattaneo, Gianpiero

Article Type: Research Article

Abstract: The approach to rough set theory is investigated from the foundational point of view, starting from the formal analysis of the mathematical structures generated by equivalence relations (the standard Pawlak approach to complete information systems) and then by tolerance or similarity relations (the approach to roughness by incomplete information systems). Making use of these approaches the “minimal” meta–theoretical principles which allow one to treat approximations in rough set theory are discussed and formulated in explicit way, in particular the conditions which can assure what can be rightly considered as lower and upper approximations of some approximable set. The hierarchy of …approximation spaces generated in this way on a universe is then discussed. Show more

DOI: 10.3233/FI-2011-420

Citation: Fundamenta Informaticae, vol. 108, no. 3-4, pp. 197-221, 2011

Authors: Couso, Inés | Dubois, Didier

Article Type: Research Article

Abstract: Rough sets are often induced by descriptions of objects based on the precise observations of an insufficient number of attributes. In this paper, we study generalizations of rough sets to incomplete information systems, involving imprecise observations of attributes. The precise role of covering-based approximations of sets that extend the standard rough sets in the presence of incomplete information about attribute values is described. In this setting, a covering encodes a set of possible partitions of the set of objects. A natural semantics of two possible generalisations of rough sets to the case of a covering (or a non transitive tolerance …relation) is laid bare. It is shown that uncertainty due to granularity of the description of sets by attributes and uncertainty due to incomplete information are superposed, whereby upper and lower approximations themselves (in Pawlak's sense) become ill-known, each being bracketed by two nested sets. The notion of measure of accuracy is extended to the incomplete information setting, and the generalization of this construct to fuzzy attribute mappings is outlined. Show more

Keywords: Rough sets, Possibility theory, Covering, fuzzy sets

DOI: 10.3233/FI-2011-421

Citation: Fundamenta Informaticae, vol. 108, no. 3-4, pp. 223-247, 2011

Authors: Yao, Yiyu

Article Type: Research Article

Abstract: Probabilistic rough set models are quantitative generalizations of the classical and qualitative Pawlak model by considering degrees of overlap between equivalence classes and a set to be approximated. The extensive studies, however, have not sufficiently addressed some semantic issues in a probabilistic rough set model. This paper examines two fundamental semantics-related questions. One is the interpretation and determination of the required parameters, i.e., thresholds on probabilities, for defining the probabilistic lower and upper approximations. The other is the interpretation of rules derived from the probabilistic positive, boundary and negative regions. We show that the two questions can be answered within …the framework of a decision-theoretic rough set model. Parameters for defining probabilistic rough sets are interpreted and determined in terms of loss functions based on the well established Bayesian decision procedure. Rules constructed from the three regions are associated with different actions and decisions, which immediately leads to the notion of three-way decision rules. A positive rule makes a decision of acceptance, a negative rule makes a decision of rejection, and a boundary rules makes a decision of deferment. The three-way decisions are, again, interpreted based on the loss functions. Show more

Keywords: Decision-theoretic rough sets, probabilistic rough sets, semantics, three-way decisions

DOI: 10.3233/FI-2011-422

Citation: Fundamenta Informaticae, vol. 108, no. 3-4, pp. 249-265, 2011

Authors: Herbert, Joseph P. | Yao, JingTao

Article Type: Research Article

Abstract: This article investigates the Game-theoretic Rough Set (GTRS) model and its capability of analyzing a major decision problem evident in existing probabilistic rough set models. A major challenge in the application of probabilistic rough set models is their inability to formulate a method of decreasing the size of the boundary region through further explorations of the data. To decrease the size of this region, objects must be moved to either the positive or negative regions. Game theory allows a solution to this decision problem by having the regions compete or cooperate with each other in order to find which is …best fit to be selected for the move. There are two approaches discussed in this article. First, the region parameters that define the minimum conditional probabilities for region inclusion can either compete or cooperate in order to increase their size. The second approach is formulated by having classification approximation measures compete against each other. We formulate a learning method using the GTRS model that repeatedly analyzes payoff tables created from approximation measures and modified conditional risk strategies to calculate parameter values. Show more

Keywords: rough sets, set approximation, game theory, decision-theoretic model, variable precision model, probabilistic rough set model, game-theoretic model

DOI: 10.3233/FI-2011-423

Citation: Fundamenta Informaticae, vol. 108, no. 3-4, pp. 267-286, 2011

Authors: Ciucci, Davide

Article Type: Research Article

Abstract: The term orthopair is introduced to group under a unique definition different ways used to denote the same concept. Some orthopair models dealing with uncertainty are analyzed both from a mathematical and semantical point of view, outlining similarities and differences among them. Finally, lattice operations on orthopairs are studied and a survey on algebraic structures is provided.

DOI: 10.3233/FI-2011-424

Citation: Fundamenta Informaticae, vol. 108, no. 3-4, pp. 287-304, 2011

Authors: Diker, Murat

Article Type: Research Article

Abstract: In this paper, we consider the Alexandroff topology for texture spaces. We prove that there exists a one-to-one correspondence between the Alexandroff ditopologies, and the reflexive and transitive direlations on a given texture. Using textural fuzzy direlations on a fuzzy lattice, we obtain a fuzzy rough set algebra where the inverse fuzzy relation and inverse fuzzy corelation are the upper approximation and lower approximation, respectively. In a special case, this gives us the fuzzy rough sets which are calculated with respect to the min-t norm introduced by S. Gottwald, or the fuzzy rough sets which are considered by D. Pei.

Keywords: Alexandroff topology, approximation operator, direlation, fuzzy rough set algebra, rough set algebra, textural fuzzy direlation, textural rough set algebra, texture space

DOI: 10.3233/FI-2011-425

Citation: Fundamenta Informaticae, vol. 108, no. 3-4, pp. 305-336, 2011

Authors: Wu, Wei-Zhi

Article Type: Research Article

Abstract: In this paper, a general framework for the study of fuzzy rough approximation operators determined by a triangular norm in infinite universes of discourse is investigated. Lower and upper approximations of fuzzy sets with respect to a fuzzy approximation space in infinite universes of discourse are first introduced. Essential properties of various types of T-fuzzy rough approximation operators are then examined. An operator-oriented characterization of fuzzy rough sets is also proposed, that is, T-fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations …which produce the same operators. A comparative study of T-fuzzy rough set algebras with some other mathematical structures are presented. It is proved that there exists a one-to-one correspondence between the set of all reflexive and T-transitive fuzzy approximation spaces and the set of all fuzzy Alexandrov spaces such that the lower and upper T-fuzzy rough approximation operators are, respectively, the fuzzy interior and closure operators. It is also shown that a reflexive fuzzy approximation space induces a measurable space such that the family of definable fuzzy sets in the fuzzy approximation space forms the fuzzy σ-algebra of the measurable space. Finally, it is explored that the fuzzy belief functions in the Dempster-Shafer of evidence can be interpreted by the T-fuzzy rough approximation operators in the rough set theory, that is, for any fuzzy belief structure there must exist a probability fuzzy approximation space such that the derived probabilities of the lower and upper approximations of a fuzzy set are, respectively, the T-fuzzy belief and plausibility degrees of the fuzzy set in the given fuzzy belief structure. Show more

Keywords: Approximation operators, fuzzy belief functions, fuzzy measurability sets, fuzzy rough sets, fuzzy rough set algebras, fuzzy topological spaces, rough sets, triangular norms

DOI: 10.3233/FI-2011-426

Citation: Fundamenta Informaticae, vol. 108, no. 3-4, pp. 337-369, 2011