Dedekind–MacNeille completion and Cartesian product of multi-adjoint lattices

Abstract

Fuzzy logic programming tries to introduce fuzzy logic into logic programming in order to provide new generation computer languages which incorporate comfortable programming resources for helping the development of applications where uncertainty could play an important role. In this sense, the mathematical concept of multi-adjoint lattice has been successfully exploited into the so-called Multi-Adjoint Logic Programming approach, MALP in brief, for modelling flexible notions of truth-degrees beyond the simpler case of true and false. In this paper, we focus on two relevant mathematical concepts for this kind of domains useful for evaluating MALP. On the one side, we adapt the classical notion of Dedekind-MacNeille completion in order to relax some usual hypothesis on such kind of ordered sets, and next we study the advantages of generating multi-adjoint lattices as the Cartesian product of previous ones. On the practical side, we show that the formal mechanisms described before have direct correspondences with interesting debugging tasks into the 'Fuzzy Logic Programming Environment for Research', FLOPER in brief, developed in our research group.

Fuzzy Logic Programming in Action with <i>FLOPER</i>

“…This section presents different methods to gain expressiveness when designing a MALP program by manipulating its associated lattice. The main technique consists on agglutinating several lattices in order to obtain the Cartesian product of them, which is also a multi-adjoint lattice [24]. Once this is done, a new functionality emerges for obtaining declarative traces (when evaluating goals) at a very low cost.…”

Beyond multi-adjoint logic programming

“…Focusing on this setting, in [1,17,18,21,[43][44][45] we describe some fundamental and practical issues regarding the design of declarative languages, programming environments and real-world applications with a fuzzy taste 1 developed in our research group during the last years. In essence, our research on MALP follows the fruitful path initiated by Lee [26], where the classical inference mechanism of selective linear definite clause resolution (SLD-resolution) used in LP, has been replaced by a fuzzy variant able to handle partial truth and uncertainty in a comfortable way.…”