On Equivalences Between Fuzzy Dependencies and Fuzzy Formulas’ Satisfiability for Yager’s Fuzzy Implication Operator
In this paper we consider fuzzy functional and fuzzy multivalued dependencies introduced by Sozat and Yazici. We appropriately relate these dependencies to fuzzy formulas. In particular, we relate any subset of the universal set of attributes to fuzzy conjunction of its attributes. Thus, being in the form of implication between such subsets, we naturally relate a fuzzy dependency to fuzzy implication between corresponding fuzzy conjunctions. In this paper we choose standard min, max as well as Yager’s fuzzy implication operator for definitions of fuzzy conjunction, fuzzy disjunction and fuzzy implication, respectively. If any two-element fuzzy relation instance on a given scheme, known to satisfy some set of fuzzy functional and fuzzy multivalued dependencies, satisfies some fuzzy functional or fuzzy multivalued dependency f which is not member of the given set of fuzzy dependencies, then, we prove that satisfiability of the related set of fuzzy formulas yields satisfiability of the fuzzy formula related to f and vice versa. A methodology behind the proofs of our results is mainly based on an application of definitions of the introduced fuzzy logic operators. Our results can be verified for various choices of fuzzy logic operators however.