Non-deterministic ideal operators: An adequate tool for formalization in Data Bases

Abstract

In this paper, we propose the application of formal methods to Software Engineering. The most used data model is the relational model and we present, within the general framework of lattice theory, this analysis of functional dependencies. For this reason, we characterize the concept of f -family by means of a new concept which we call non-deterministic ideal operator (nd.ideal-o). The study of nd.ideal-o.s allows us to obtain results about functional dependencies as trivial particularizations, to clarify the semantics of the functional dependencies and to progress in their efficient use, and to extend the concept of schema. Moreover, the algebraic characterization of the concept of Key of a schema allows us to propose new formal definitions in the lattice framework for classical normal forms in relation schemata. We give a formal definition of the normal forms for functional dependencies more frequently used in the bibliography: the second normal form (2FN), the third normal form(3FN) and Boyce-Codd's normal form (FNBC).

Ideal non-deterministic operators as a formal framework to reduce the key finding problem

“…We denote by N do(L) the set of unary non-deterministic operations in a non-empty set L. The unary case is the trivial one that corresponds with binary relations in L. However, using the idea of ndo and its notation, several concepts and results are not only based on theoretical results, but they also become easier to use. In [4], we showed that this approach allows us to directly prove results about FDs, to clarify the semantics of the FDs and to propose new formal definitions in the lattice framework for normal forms.…”

“…In [4] we showed that this concept formalizes the set of dependencies satisfied by a relation. That is, by considering L = 2 A , the set of all FDs that a relation R satisfies is an nd-ideal-o in L, and vice versa, and for all F ∈ N dIo(L) there exists a relation R whose set of satisfied FDs is F .…”

“…In [4], a formalization of these concepts in the framework of the lattice theory is introduced. It is based on the concept of non-deterministic operation (also named multioperation, multivalued operation, polyoperation or hyperoperation, depending on the authors) that has been successfully used in the literature to solve several algebraic and computational problems.…”

“…The scheme pruning transformation that is proposed in this work has linear cost in the worst case and is directly based on the theoretical results concerning the lattice theory and FDs [4].…”

On minimal sets of graded attribute implications

“…(a) Alternative graph-based proof systems [28] as well as automated provers based on simplification equivalences as in [9] are also available. Let us also note that in addition to Theorem 1 which provides a syntactic characterization only for formulas which are semantically entailed to degree 1, the logic of graded attribute implications is also complete in the graded style (Pavelka-style (5) corresponds to the incidence data semantics we have mentioned in the introduction.…”