L= Sets and Systems Elsevier Transitive Fuzzy Dependencies (ii)

In the previous work we accomplished the problem of defining fuzzy dependency as well as projection and join operators which allow us to decompose a database in such a way that this process is fuzzy loss less. The definition of rules-based fuzzy dependency did not satisfy the transitivity axiom, which is crucial to obtaining the completeness of the counterpart of Armstrong axioms for fuzzy dependencies. In this paper we introduce an alternative definition of dependency called restricted dependency, satisfying the transitivity property of inference. All the results obtained with the first definition of dependency are also extended to this case. The price to pay for transitivity is a more restricted definition of fuzzy dependency. (~) 1999 Elsevier Science B.V. All rights reserved. 1. Definition of transitive rules-based fuzzy functional dependencies We assume the same notation and preliminaries introduced in Section 1, Part I. Now, our objective is to extend the notion of fuzzy projection and dependency for single attributes, introduced in Section 2, Part I, to the case of compound attributes, in such a way that transitivity axiom of inference is not lost. The key to maintaining transitivity is to consider, in the definition of fuzzy projection, each attribute separately, i.e., instead of considering those tuples such that their antecedent kernel values are overlapped (definition of RULE operator in Part I) for every Xh EX, we now consider for each value Xh(t), those tuples ti such that Xh(ti),Xh(t) is a sequence of fuzzy values with overlapped kernel values. Let us formalize this process: Definition 1. Let us consider two compound attributes X and Y. We denote by IFX(r) (or simply IF(r) if there is no confusion) a relation which is constructed replacing in r, each Xh(t) by the fuzzy union Vt,~ttl.+h Xh(ti), where [t]xh is the set of tuples given by Definition 11 in Part I. Then, the fuzzy projection relation denoted by IRULEX(r) is constructed merging tuples with equal antecedent values through 1-I x operator