On Extension of Dependency and Consistency Degrees of Two Knowledges Represented by Covering
18 Dec 2008-Vol. 9, pp 351-364
TL;DR: In this paper dependency degrees of two knowledges have been considered in both the cases and a measure of consistency and inconsistency of knowledging are discussed.
Abstract: Knowledge of an agent depends on the granulation procedure adopted by the agent. The knowledge granules may form a partition of the universe or a covering. In this paper dependency degrees of two knowledges have been considered in both the cases. A measure of consistency and inconsistency of knowledges are also discussed. This paper is a continuation of our earlier work [3].
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15 Dec 2009
TL;DR: This paper deals with a survey of some aspects of covering based approaches to rough set theory and their implication lattices.
Abstract: This paper deals with a survey of some aspects of covering based approaches to rough set theory and their implication lattices.
68 citations
01 Jan 2011
TL;DR: This paper consists of an extensive survey of various generalized approaches to the lower and upper approximation of a set, the two approximations being first defined by Pawlak while introducing rough set theory.
Abstract: This paper consists of an extensive survey of various generalized approaches to the lower and upper approximations of a set, the two approximations being first defined by Pawlak while introducing rough set theory. Particularly, relational, covering based and operator based approaches are considered. Categorization of various approaches in terms of implication lattices is shown. Significance of this categorization in rough logics is briefly mentioned.
24 citations
01 Jan 2013
TL;DR: Semantic frameworks for dealing with concepts of relative consistency of knowledge, correspondences between evolvents of knowledges and problems of conflict representation and resolution are introduced and developed in this research paper.
Abstract: Pawlak had introduced a concept of knowledge as a state of relative exactness in classical rough set theory (RST) [30]. From a theory of knowledge and application perspective, it is of much interest to study concepts of relative consistency of knowledge, correspondences between evolvents of knowledges and problems of conflict representation and resolution. Semantic frameworks for dealing with these are introduced and developed in this research paper by the present author. New measures that deal with different levels of contamination are also proposed. Further, it is shown that the algebraic semantics are computationally very amenable. The proposed semantics would also be of interest for multi-agent systems, dynamic spaces and collections of general approximation spaces. Part of the literature on related areas is also critically surveyed.
23 citations
TL;DR: This paper uses an example in evidence-based medicine to illustrate how to use Pawlak’s rough membership function on numerically characterizing decisions under the circumstances where data are complete, and presents theoretical backgrounds for these covering-based rough membership functions.
18 citations
TL;DR: In this article, the authors generalize the definition of consistency for incomplete data sets using rough set theory and discuss two types of missing attribute values: lost values and "do not care" conditions.
15 citations
References
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Book•
31 Oct 1991
TL;DR: Theoretical Foundations.
Abstract: I. Theoretical Foundations.- 1. Knowledge.- 1.1. Introduction.- 1.2. Knowledge and Classification.- 1.3. Knowledge Base.- 1.4. Equivalence, Generalization and Specialization of Knowledge.- Summary.- Exercises.- References.- 2. Imprecise Categories, Approximations and Rough Sets.- 2.1. Introduction.- 2.2. Rough Sets.- 2.3. Approximations of Set.- 2.4. Properties of Approximations.- 2.5. Approximations and Membership Relation.- 2.6. Numerical Characterization of Imprecision.- 2.7. Topological Characterization of Imprecision.- 2.8. Approximation of Classifications.- 2.9. Rough Equality of Sets.- 2.10. Rough Inclusion of Sets.- Summary.- Exercises.- References.- 3. Reduction of Knowledge.- 3.1. Introduction.- 3.2. Reduct and Core of Knowledge.- 3.3. Relative Reduct and Relative Core of Knowledge.- 3.4. Reduction of Categories.- 3.5. Relative Reduct and Core of Categories.- Summary.- Exercises.- References.- 4. Dependencies in Knowledge Base.- 4.1. Introduction.- 4.2. Dependency of Knowledge.- 4.3. Partial Dependency of Knowledge.- Summary.- Exercises.- References.- 5. Knowledge Representation.- 5.1. Introduction.- 5.2. Examples.- 5.3. Formal Definition.- 5.4. Significance of Attributes.- 5.5. Discernibility Matrix.- Summary.- Exercises.- References.- 6. Decision Tables.- 6.1. Introduction.- 6.2. Formal Definition and Some Properties.- 6.3. Simplification of Decision Tables.- Summary.- Exercises.- References.- 7. Reasoning about Knowledge.- 7.1. Introduction.- 7.2. Language of Decision Logic.- 7.3. Semantics of Decision Logic Language.- 7.4. Deduction in Decision Logic.- 7.5. Normal Forms.- 7.6. Decision Rules and Decision Algorithms.- 7.7. Truth and Indiscernibility.- 7.8. Dependency of Attributes.- 7.9. Reduction of Consistent Algorithms.- 7.10. Reduction of Inconsistent Algorithms.- 7.11. Reduction of Decision Rules.- 7.12. Minimization of Decision Algorithms.- Summary.- Exercises.- References.- II. Applications.- 8. Decision Making.- 8.1. Introduction.- 8.2. Optician's Decisions Table.- 8.3. Simplification of Decision Table.- 8.4. Decision Algorithm.- 8.5. The Case of Incomplete Information.- Summary.- Exercises.- References.- 9. Data Analysis.- 9.1. Introduction.- 9.2. Decision Table as Protocol of Observations.- 9.3. Derivation of Control Algorithms from Observation.- 9.4. Another Approach.- 9.5. The Case of Inconsistent Data.- Summary.- Exercises.- References.- 10. Dissimilarity Analysis.- 10.1. Introduction.- 10.2. The Middle East Situation.- 10.3. Beauty Contest.- 10.4. Pattern Recognition.- 10.5. Buying a Car.- Summary.- Exercises.- References.- 11. Switching Circuits.- 11.1. Introduction.- 11.2. Minimization of Partially Defined Switching Functions.- 11.3. Multiple-Output Switching Functions.- Summary.- Exercises.- References.- 12. Machine Learning.- 12.1. Introduction.- 12.2. Learning From Examples.- 12.3. The Case of an Imperfect Teacher.- 12.4. Inductive Learning.- Summary.- Exercises.- References.
7,826 citations
"On Extension of Dependency and Cons..." refers background in this paper
...Novotn´ y and Pawlak defined a dependency degree between two knowledges given by two partitions on a set [6,7,8, 9 ]....
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...In propositions 1,2 and 3, we enlist some elementary, often trivial, properties of dependency degree some of them being newly exercised but most of which are present in [6, 9 ]....
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...It is interestingly observed that the properties of partial dependency that were developed in [3,6, 9 ] hold good in the general case of covering based approximation system....
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TL;DR: This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.
Abstract: Rough set theory, introduced by Zdzislaw Pawlak in the early 1980s [11, 12], is a new mathematical tool to deal with vagueness and uncertainty. This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.
7,185 citations
Book•
01 Jan 1995TL;DR: Fuzzy Sets and Fuzzy Logic is a true magnum opus; it addresses practically every significant topic in the broad expanse of the union of fuzzy set theory and fuzzy logic.
Abstract: Fuzzy Sets and Fuzzy Logic is a true magnum opus. An enlargement of Fuzzy Sets, Uncertainty,
and Information—an earlier work of Professor Klir and Tina Folger—Fuzzy Sets and Fuzzy Logic
addresses practically every significant topic in the broad expanse of the union of fuzzy set theory
and fuzzy logic. To me Fuzzy Sets and Fuzzy Logic is a remarkable achievement; it covers its vast
territory with impeccable authority, deep insight and a meticulous attention to detail.
To view Fuzzy Sets and Fuzzy Logic in a proper perspective, it is necessary to clarify a point
of semantics which relates to the meanings of fuzzy sets and fuzzy logic.
A frequent source of misunderstanding fias to do with the interpretation of fuzzy logic. The
problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow
sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization
of classical multivalued logics. But in a wider sense, fuzzy logic, FL^ is almost synonymous
with the theory of fuzzy sets. In this context, what is important to recognize is that: (a) FLW is much
broader than FLn and subsumes FLn as one of its branches; (b) the agenda of FLn is very different
from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is
usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FLW
In Fuzzy Sets and Fuzzy Logic, fuzzy logic is interpreted in a sense that is close to FLW. However,
to avoid misunderstanding, the title refers to both fuzzy sets and fuzzy logic.
Underlying the organization of Fuzzy Sets and Fuzzy Logic is a fundamental fact, namely,
that any field X and any theory Y can be fuzzified by replacing the concept of a crisp set in X and Y
by that of a fuzzy set. In application to basic fields such as arithmetic, topology, graph theory, probability
theory and logic, fuzzification leads to fuzzy arithmetic, fuzzy topology, fuzzy graph theory,
fuzzy probability theory and FLn. Similarly, hi application to applied fields such as neural network
theory, stability theory, pattern recognition and mathematical programming, fuzzification leads to
fuzzy neural network theory, fuzzy stability theory, fuzzy pattern recognition and fuzzy mathematical
programming. What is gained through fuzzification is greater generality, higher expressive
power, an enhanced ability to model real-world problems and, most importantly, a methodology for
exploiting the tolerance for imprecision—a methodology which serves to achieve tractability,
7,131 citations
"On Extension of Dependency and Cons..." refers background in this paper
...These are conjunction operators used extensively and are in some sense the basic t-norms [ 4 ]....
[...]
TL;DR: In tolerance approximation spaces the lower and upper set approximations are defined and 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.
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
955 citations
"On Extension of Dependency and Cons..." refers background or methods in this paper
...Definition 9. [ 10 ] Let X be a subset of U . Then the lower and upper approximations are defined as follows :...
[...]
...This observation gives rise to the study of Rough Set Theory based on coverings instead of partitions [2,10,11, 13 ,14,15,16]....
[...]
...Now, in the covering based approximation systems lower and upper approximations of a set are defined in at least five different ways [ 10 ]....
[...]
...This observation gives rise to the study of Rough Set Theory based on coverings instead of partitions [2, 10 ,11,13,14,15,16]....
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TL;DR: The RSFDGrC 2013 was the 14th International Conference on Distributed Sensor Networks for Computer Science (RSFDG-2013) as mentioned in this paper, held in Halifax, NS, Canada, October 11-14, 2013.
Abstract: 14th International Conference, RSFDGrC 2013, Halifax, NS, Canada, October 11-14, 2013. Proceedings - Part of the Lecture Notes in Computer Science book series
535 citations