Fuzzy Functional and Multivalued Dependencies for Frank’s Class of Additive Generators

In this paper we consider all possible dependencies that can be built upon similarity-based fuzzy relations, that is, fuzzy functional and fuzzy multivalued dependencies. Motivated by the fact that the classical obtaining of new dependencies via inference rules may be tedious and uncertain, we replace it by the automated one, where the key role is played by the resolution principle techniques and the fuzzy formulas in place of fuzzy dependencies. We prove that some fuzzy multivalued dependency is actively correct with respect to given fuzzy relation instance if and only if the corresponding fuzzy formula is in line with the attached interpretation. Additionally, we require the tuples of the instance to be conformant (up to some extent) on the leading set of attributes. The equivalence as well as the conclusion are generalized to sets of attributes. The research is conducted by representing the attributes and fuzzy dependencies in the form of fuzzy formulas, and the application of fuzzy implication operators derived from carefully selected Frank’s classes of additive generators

On Some Applications of h-generated Fuzzy Implications

In this paper we apply the h-generated fuzzy implications to prove a number of results which are of fundamental importance to the theory of fuzzy and vague functional and multivalued dependencies defined on given scheme. Our research is motivated by the fact that some analogous results already hold true for the families of f- and g-generated fuzzy implications, and the fact that these three collections of implications share many similar mutual properties. While some of the aforementioned implications are introduced in order to be applied in approximate reasoning, the results derived in this paper represent the main tool in the process of automation and are also used to complement the resolution principle. More precisely, the main result of this research states that the fact that some fuzzy (vague) relation instance r, |r| = 2, satisfies some fuzzy (vague) functional or fuzzy (vague) multivalued dependency c /∈ C (under assumption that r satisfies some set C of fuzzy (vague) functional and fuzzy (vague) multivalued dependencies), yields that the fuzzy formula attached to c is valid whenever all of the fuzzy formulas attached to the elements of C are valid. What is more important is that the opposite claim is also proven. Its importance stems from the fact that the verification by hand, which means purely theoretical verification, that C implies c is not required anymore. Now, in order to prove that some C yields some c, it is enough to make the use of the resolution principle, and automatically verify whether or not the set of the attached fuzzy formulas yields the fuzzy formula attached to c. In the case of affirmative answer, the desired dependency follows. The research conducted in this paper represent a natural generalization of our previous research since it includes and considers both, fuzzy and vague theories.