We present a complete logic for reasoning with functional dependencies (FDs) with semantics defined over classes of commutative integral partially ordered monoids and complete residuated lattices. The dependencies allow us to express stronger relationships between attribute values than the ordinary FDs. In our setting, the dependencies not only express that certain values are determined by others but also express that similar values of attributes imply similar values of other attributes. We show complete axiomatization using a system of Armstrong-like rules, comment on related computational issues, and the relational vs. propositional semantics of the dependencies. A logic for reasoning with similarity-preserving dependencies is proposed.Its semantics is defined using commutative integral partially ordered monoids.The logic is complete and decidable.A closure-like algorithm for non-contracting sets of dependencies is presented.Two kinds of semantics are discussed: a propositional and a relational one.
First, let us note that there exists a vast amount of papers on fuzzy functional dependencies'', often with questionable technical quality, which combine (in various ways) the concepts of fuzzy sets and functional dependencies in order to formalize vague dependencies between attributes. While this idea is tempting and close to what we present here, our objection is that most of these papers are purely definitional or just experimental and are not interested in the underlying logic in the narrow sense of it (i.e., in logic as a study of consequence). From one viewpoint this is not surprising since a number of papers in this category predate the beginning of systematic formalization of various types of fuzzy logics which appeared in the late 90's, see @cite_21 as a standard reference and a historical overview. One of the most influential early approaches that enjoyed interest in the database community is @cite_12 , further papers dealing with fuzzy functional dependencies and related phenomena include @cite_29 @cite_17 @cite_33 . Since our paper is not a survey, we do not write further details on such approaches and refer interested readers to @cite_39 where they can find further comments.
@cite_33 @cite_29 @cite_21 @cite_39 @cite_12 @cite_17
2074622901 2072960837 2110182592 2139040790 2017978889 1991924287
Abstract This paper deals with relational databases which are extended in the sense that fuzzily known values are allowed for attributes. Precise as well as partial (imprecise, uncertain) knowledge concerning the value of the attributes are represented by means of [0,1]-valued possibility distributions in Zadeh's sense. Thus, we have to manipulate ordinary relations on Cartesian products of sets of fuzzy subsets rather than fuzzy relations. Besides, vague queries whose contents are also represented by possibility distributions can be taken into account. The basic operations of relational algebra, union, intersection, Cartesian product, projection, and selection are extended in order to deal with partial information and vague queries. Approximate equalities and inequalities modeled by fuzzy relations can also be taken into account in the selection operation. Then, the main features of a query language based on the extended relational algebra are presented. An illustrative example is provided. This approach, which enables a very general treatment of relational databases with fuzzy attribute values, makes an extensive use of dual possibility and necessity measures. A structure for representing inexact information in the form of a relational database is presented. The structure differs from ordinary relational databases in two important respects: Components of tuples need not be single values and a similarity relation is required for each domain set of the database. Two critical properties possessed by ordinary relational databases are proven to exist in the fuzzy relational structure. These properties are (1) no two tuples have identical interpretations, and (2) each relational operation has a unique result. Preface. 1. Preliminaries. 2. Many-Valued Propositional Calculi. 3. Lukasiewicz Propositional Logic. 4. Product Logic, Godel Logic. 5. Many-Valued Predicate Logics. 6. Complexity and Undecidability. 7. On Approximate Inference. 8. Generalized Quantifiers and Modalities. 9. Miscellanea. 10. Historical Remarks. References. Index. The article deals with Codd's relational model of data and its fuzzy logic extensions. Our main purpose is to examine, from the point of view of fuzzy logic in the narrow sense, some of the extensions proposed in the literature and the relationships between them. We argue that fuzzy logic in the narrow sense is important for the fuzzy logic extensions because it provides conceptual and methodological foundations, clarity and simplicity. We present several comparative observations as well as new technical results. This paper deals with the application of fuzzy logic in a relational database environment with the objective of capturing more meaning of the data. It is shown that with suitable interpretations for the fuzzy membership functions, a fuzzy relational data model can be used to represent ambiguities in data values as well as impreciseness in the association among them. Relational operators for fuzzy relations have been studied, and applicability of fuzzy logic in capturing integrity constraints has been investigated. By introducing a fuzzy resemblance measure EQUAL for comparing domain values, the definition of classical functional dependency has been generalized to fuzzy functional dependency (ffd). The implication problem of ffds has been examined and a set of sound and complete inference axioms has been proposed. Next, the problem of lossless join decomposition of fuzzy relations for a given set of fuzzy functional dependencies is investigated. It is proved that with a suitable restriction on EQUAL, the design theory of a classical relational database with functional dependencies can be extended to fuzzy relations satisfying fuzzy functional dependencies. The need to incorporate and treat information given in fuzzy terms in Relational Databases has concentrated a great effort in the last years. This article focuses on the treatment of functional dependencies (f.d.) between attributes of a relation scheme. We review other approaches to this problem and present some of its missfunctions concerning intuitive properties a fuzzy extension of f.d. should verify. Then we introduce a fuzzy extension of this concept to overcome the previous anomalous behaviors and study its properties. of primary interest is the completeness of our fuzzy version of Armstrong axioms in order to derive all the fuzzy functional dependencies logically implied by a set of f.f.d. just using these axioms. © 1994 John Wiley & Sons, Inc.