Pablo Cordero

SLFD Logic: Elimination of Data Redundancy in Knowledge Representation

In this paper, we propose the use of formal techniques on Software Engineering in two directions: 1)We present, within the general framework of lattice theory, the analysis of relational databases. To do that, we characterize the concept of f-family (Armstrong relations) by means of a new concept which we call non-deterministic ideal operator. This characterization allows us to formalize database redundancy in a more significant way than it was thought of in the literature. 2) We introduce the Substitution Logic SLFD for functional dependencies that will allows us the design of automatic transformations of data models to remove redundancy.

Closure via functional dependence simplification

In this paper, a method for computing the closure of a set of attributes according to a specification of functional dependencies of the relational model is described. The main feature of this method is that it computes the closure using solely the inference system of the SL FD logic. For the first time, logic is used in the design of automated deduction methods to solve the closure problem. The strong link between the SL FD logic and the closure algorithm is presented and an SL FD simplification paradigm emerges as the key element of our method. In addition, the soundness and completeness of the closure algorithm are shown. Our method has linear complexity, as the classical closure algorithms, and it has all the advantages provided by the use of logic. We have empirically compared our algorithm with the Diederich and Milton classical algorithm. This experiment reveals the best behaviour of our method which shows a significant improvement in the average speed.

A New Algebraic Tool for Automatic Theorem Provers

The concepts of implicates and implicants are widely used in several fields of “Automated Reasoning”. Particularly, our research group has developed several techniques that allow us to reduce efficiently the size of the input, and therefore the complexity of the problem. These techniques are based on obtaining and using implicit information that is collected in terms of unitary implicates and implicants. Thus, we require efficient algorithms to calculate them. In classical propositional logic it is easy to obtain efficient algorithms to calculate the set of unitary implicants and implicates of a formula. In temporal logics, contrary to what we see in classical propositional logic, these sets may contain infinitely many members. Thus, in order to calculate them in an efficient way, we have to base the calculation on the theoretical study of how these sets behave. Such a study reveals the need to make a generalization of Lattice Theory, which is very important in “Computational Algebra”. In this paper we introduce the multisemilattice structure as a generalization of the semilattice structure. Such a structure is proposed as a particular type of poset. Subsequently, we offer an equivalent algebraic characterization based on non-deterministic operators and with a weakly associative property. We also show that from the structure of multisemilattice we can obtain an algebraic characterization of the multilattice structure. This paper concludes by showing the relevance of the multisemilattice structure in the design of algorithms aimed at calculating unitary implicates and implicants in temporal logics. Concretely, we show that it is possible to design efficient algorithms to calculate the unitary implicants/implicates only if the unitary formulae set has the multisemilattice structure.

Fuzzy Logic, Soft Computing, and Applications

We survey on the theoretical and practical developments of the theory of fuzzy logic and soft computing. Specifically, we briefly review the history and main milestones of fuzzy logic (in the wide sense), the more recent development of soft computing, and finalise by presenting a panoramic view of applications: from the most abstract to the most practical ones.

On galois connections and soft computing

After recalling the different interpretations usually assigned to the term Galois connection, both in the crisp and in the fuzzy case, we survey on several of their applications in Computer Science and specifically, in Soft Computing.

On the definition of suitable orderings to generate adjunctions over an unstructured codomain

Given a mappingf:A->B from a (pre-)ordered set A into an unstructured set B, we study the problem of defining a suitable (pre-)ordering relation on B such that there exists a mapping g:B->A such that the pair of mappings (f,g) forms an adjunction between (pre-)ordered sets. The necessary and sufficient conditions obtained are then expressed in terms of closure operators and closure systems.

Generalizations of lattices via non-deterministic operators

Benado (Cehoslovak. Mat. Z. 79(4) (1954) 105-129) and later Hansen (Discrete Math. 33(1) (1981) 99-101) have offered an algebraic characterization of multilattice (i.e., a poset where every pair of elements satisfies that any upper bound is greater than or equal to a minimal upper bound, and also satisfies the dual property). To that end, they introduce two algebraic operators that are a generalization of the operators @? and @? in a lattice. However, in Martinez et al. (Math. Comput. Sci. Eng. (2001) 238-248), we give the only algebraic characterization of the multisemilattice structure that exists in the literature. Moreover, this characterization allows us to give a more adequate characterization of the multilattice structure. The main advantage of our algebraic characterizations is that they are natural generalizations of the semilattice and lattice structures. It is well-known that in the lattice theory we can use indistinctly pairs of elements or finite subsets to characterize them. However, this is not true when we work with multilattices. For this reason in this paper we introduce two new structures from the ordered point of view, called universal multisemilattice and universal multilattice, and we propose an equivalent algebraic characterization for them. These new structures are generalizations, on one hand, of semilattice and lattice and, on the other hand, of multisemilattice and multilattice, respectively. The algebraic characterizations have the same advantages as the two introduced by us in Martinez et al. The most important purpose of this paper is to deepen the theoretical study of universal multisemilattices and universal multilattices.

A Complete Logic for Fuzzy Functional Dependencies over Domains with Similarity Relations

An axiomatic system for fuzzy functional dependencies is introduced. The main novelty of the system is that it is not based on the transitivity rule like all the others, but it is built around a simplification rule which allows the removal of redundancy. The axiomatic system presented here is shown to be sound and complete.

On residuation in multilattices: Filters, congruences, and homomorphisms

Continuing with our general study of algebraic hyperstructures, we focus on the residuated operation in the framework of multilattices. Firstly, we recall the existing relation between filters, homomorphisms and congruences in the framework of multilattices; then, introduce the notion of residuated multilattice and further study the notion of filter, which has to be suitably modified so that the results in the first section are conveniently preserved also in the residuated case.

Knowledge discovery in social networks by using a logic-based treatment of implications

This work can be seen as a contribution to the area of social network analysis. By considering Formal Concept Analysis (FCA) as the underlying formalizing tool, we use logic-based techniques in order to offer novel solutions to identify users influence in a social network. We propose the use of the Simplification Logic SL FD for attribute implications as the core of an automated method to build a structure containing the complete set of influences among users.