Fundamenta Informaticae - Volume 27, issue 2-3

Authors: Skowron, Andrzej | Ziarko, Wojciech

Article Type: Other

DOI: 10.3233/FI-1996-272316

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 101-102, 1996

Authors: Pawlak, Zdzislaw

Article Type: Research Article

Abstract: The paper explores the concepts of approximate relations and functions in the framework of the theory of rough sets. The difficulties with the application of the idea of rough relation to general rough function definition are discussed. The definition of rough function for the domain of real numbers is introduced and its properties are investigated in detail including the generalization of the standard notion of function continuity known in the theory of real functions.

DOI: 10.3233/FI-1996-272301

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 103-108, 1996

Authors: Bryniarski, Edward

Article Type: Research Article

Abstract: In the paper we present a formal description of rough sets within the framework of the generalized set theory, which is interpreted in the set approximation theory. The rough sets are interpreted as approximations, which are defined by means of the Pawlak's rough sets.

DOI: 10.3233/FI-1996-272302

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 109-136, 1996

Authors: Lin, T.Y. | Liu, Qing

Article Type: Research Article

Abstract: Earlier the authors have shown that rough sets can be characterized by six topological properties. In this paper, a new formal logic system based on such axioms is proposed. It will be called First-Order Logic for Rough Approximation or simply Rough Logic. The axiom schemas of rough logic turn out to be the same as those of the modal logic S5 . In other words, topological and modal logic considerations led to the same conclusion. So rough logic must have captured the intrinsic meaning of approximate reasoning However, their interpretations are different. To reflect the differences in semantics, possible worlds …are renamed as observable worlds. Each observable world represents a different rough observation of the actual world. Rough logic also provides a frame work for approximation. It integrates imperfect observations (observable worlds) into an approximation of actual world. Any good approximation theory should have a convergence theorem-its details are deferred to next paper. A sample theorem is as follows: If there is a convergent sequence of rough observations (namely, equivalence relations), then the corresponding rough models converge to the Tarskian model of first-order classical logic. Show more

DOI: 10.3233/FI-1996-272303

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 137-153, 1996

Authors: Katzberg, Jack David | Ziarko, Wojciech

Article Type: Research Article

Abstract: We present a generalization of the original idea of rough sets as introduced by Pawlak. The generalization, called the Variable Precision Rough Sets Model with Asymmetric Bounds, is aimed at modeling decision situations characterized by uncertain information expressed in terms of probability distributions estimated form frequency distributions observed in empirical data. The model presented is a direct extension of the previous concept, the Variable Precision Rough Sets Model. The properties of the extended model are investigated and compared to the original model. Also, a real life problem of identifying the factors which most affect the likelihoods of specified events in …the steel industry is discussed in the context of this theory. Show more

DOI: 10.3233/FI-1996-272304

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 155-168, 1996

Authors: Kent, Robert E.

Article Type: Research Article

Abstract: The theory introduced, presented and developed in this paper, is concerned with Rough Concept Analysis. This theory is a synthesis of the theory of Rough Sets pioneered by Zdzislaw Pawlak [10] with the theory of Formal Concept Analysis pioneered by Rudolf Wille [11]. The central notion in this paper of a rough formal concept combines in a natural fashion the two notions of rough set and formal concept — to use a slogan: “rough set + formal concept = rough formal concept”. This paper is an extension of the paper [5] presented at the international workshop on Rough Sets and …Knowledge Discovery (RSKD'93). A related paper [8] using distributed constraints provides a synthesis of the two important data modeling techniques: conceptual scaling of Formal Concept Analysis, and Entity-Relationship database modeling. A follow-up paper [9] will extend rough concept analysis from formal contexts to distributed constraints. Show more

DOI: 10.3233/FI-1996-272305

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 169-181, 1996

Authors: Kryszkiewicz, Marzena | Rybinski, Henryk

Article Type: Research Article

Abstract: A set-theoretical approach to finding reducts of composed information systems is presented. It is shown how the search space can be represented in form of a pair of boundaries. It is also shown, how reducts of composing information systems can be used to reduce the search space of the composed system. Presented solutions are implied directly from the properties of composed monotonic Boolean functions.

DOI: 10.3233/FI-1996-272306

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 183-195, 1996

Authors: Moshkov, Mikhail

Article Type: Research Article

Abstract: We investigate decision trees for decision tables. We present upper and lower bounds on the minimal decision tree depth. Some bounds are expressed by parameters of decision rule systems constructed for decision tables.

Keywords: decision table, decision rule, decision tree, depth, bounds

DOI: 10.3233/FI-1996-272307

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 197-203, 1996

Authors: Pagliani, Piero

Article Type: Research Article

Abstract: Any Rough Sets System induced by an Approximation Space can be given several logic-algebraic interpretations. In this paper a Rough Sets System is investigated as a finite semi-simple Nelson algebra whose structure is inherently described using the properties of the underlying Approximation Space. Moreover some of the most characterizing features of Rough Sets Systems are derived from this interpretation in logic-algebraic terms. Particularly the logic-algebraic structure given to the Rough Sets System, qua a Nelson algebra is equipped by a weak negation and a strong negation, and, since it is a finite distributive lattice, it can be regarded also as …a Heyting algebra equipped by its own pseudocomplementation. Moreover the weak Nelson negation reveals to be a dual pseudocomplementation in the lattice of Rough Sets. In this way we are able, for instance, to recover the well-known fact that Rough Sets Systems are double Stone algebras, and to exploit both their properties and the general properties of Nelson algebras in order to analyse the notions of ”definable set” and ”rough top (bottom) equality” in Approximation Spaces. Show more

DOI: 10.3233/FI-1996-272308

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 205-219, 1996

Authors: Pomykala, Janusz | De Haas, Erik

Article Type: Research Article

Abstract: In this article we present a framework in which we can reason about damaged or lost information. As a basis of reasoning about information we use the notion of ’information system’ of Pawlak [1]. Furthermore this article presents introductory considerations for analyzing information systems using tools of category theory.

DOI: 10.3233/FI-1996-272309

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 221-227, 1996

Authors: Slowiński, Roman | Stefanowski, Jerzy

Article Type: Research Article

Abstract: Rough set theory refers to classification of objects described by well-defined values of qualitative and quantitative attributes. The values of attributes defined for each pair [object, attribute], called descriptors, are assumed to be unique and precise. In practice, however, these attribute values may be neither unique nor precise, i.e. they can be uncertain. We are distinguishing four types of uncertainty affecting values of attributes: uncertain discretization of quantitative attributes, imprecision of values of numerical attributes, unknown (missing) values of attributes, multiple values possible for one pair [object, attribute]. We propose a special way of modelling the first three types of …uncertainty using fuzzy sets, which boils them down to the fourth type, called shortly, multiple descriptors. Thus, the generalization of the rough set approach consists in handling the case of multiple descriptors for both condition and decision attributes. The generalization preserves all characteristic features of the rough set approach while enabling reasoning about uncertain data. This capacity is illustrated by a simple example. Show more

Keywords: Rough sets, uncertain descriptors, fuzzy sets, approximate reasoning

DOI: 10.3233/FI-1996-272310

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 229-243, 1996

Authors: Skowron, Andrzej | Stepaniuk, Jaroslaw

Article Type: Research Article

Abstract: We generalize the notion of an approximation space introduced in [8]. In tolerance approximation spaces we define the lower and upper set approximations. We investigate some attribute reduction problems for tolerance approximation spaces determined by tolerance information systems. The tolerance relation defined by the so called uncertainty function or the positive region of a given partition of objects have been chosen as invariants in the attribute reduction process. We obtain the solutions of the reduction problems by applying boolean reasoning [1]. The solutions are represented by tolerance reducts and relative tolerance reducts.

DOI: 10.3233/FI-1996-272311

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 245-253, 1996

Authors: Skowron, Andrzej | Polkowski, Lech

Article Type: Research Article

Abstract: We propose a method called analytical morphology for data filtering. The method was created on the basis of some ideas of rough set theory and mathematical morphology. Mathematical morphology makes an essential use of geometric structure of objects while the aim of our method is to provide tools for data filtering when there is no directly available geometric structure in the data set.

DOI: 10.3233/FI-1996-272312

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 255-271, 1996

Authors: Tsumoto, Shusaku | Tanaka, Hiroshi

Article Type: Research Article

Abstract: In order to acquire knowledge from databases, there have been proposed several methods of inductive learning, such as ID3 family and AQ family. These methods are applied to discover meaningful knowledge from large databases, and their usefulness is ensured. However, since there has been no formal approach proposed to treat these methods, efficiency of each method is only compared empirically. In this paper, we introduce matroid theory and rough sets to construct a common framework for empirical machine learning methods which induce the combination of attribute-value pairs from databases. Combination of the concepts of rough sets and matroid theory gives …us an excellent set-theoretical framework and enables us to understand the differences and the similarities between these methods clearly. In this paper, we compare three classical methods, AQ, Pawlak's Consistent Rules and ID3. The results show that there exist the differences in algebraic structure between the former two and the latter and that this causes the differences between AQ and ID3. Show more

DOI: 10.3233/FI-1996-272313

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 273-288, 1996

Authors: Yao, Y.Y. | Li, Xining

Article Type: Research Article

Abstract: In the rough-set model, a set is represented by a pair of ordinary sets called the lower and upper approximations. In the interval-set model, a pair of sets is referred to as the lower and upper bounds which define a family of sets. A significant difference between these models lies in the definition and interpretation of their extended set-theoretic operators. The operators in the rough-set model are not truth-functional, while the operators in the interval-set model are truth-functional. Within the framework of possible-worlds analysis, we show that the rough-set model corresponds to the modal logic system S5 , while the …interval-set model corresponds to Kleene's three-valued logic system K3 . It is argued that these two models extend set theory in the same manner as the logic systems S5 and K3 extend standard propositional logic. Their relationships to probabilistic reasoning are also examined. Show more

DOI: 10.3233/FI-1996-272314

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 289-298, 1996

Authors: Zytkow, Jan M.

Article Type: Research Article

Abstract: We define the problem of empirical search for knowledge by interaction with a setup experiment, and we present a solution implemented in the FAHRENHEIT discovery system. FAHRENHEIT autonomously explores multi-dimensional empirical spaces of numerical parameters, making experiments, generalizing them into empirical equations, finding the scope of applications for each equation, and setting new discovery goals, until it reaches the empirically complete theory. It turns out that a small number of generic goals and a small number of data structures, when combined recursively, can lead to complex discovery processes and to the discovery of complex theories. We present FAHRENHEIT's knowledge representation …and the ways in which the discovery mechanism interacts with the emerging knowledge. Brief descriptions of several real-world applications demonstrate the system's discovery potential. Show more

DOI: 10.3233/FI-1996-272315

Citation: Fundamenta Informaticae, vol. 27, no. 2-3, pp. 299-318, 1996