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.

Keywords:

• lattice
• completion
• Cartesian product
• fuzzy (multi-adjoint) logic programming
• declarative debugging

Keywords:

• 06B23
• 03E04
• 68N17
• 90C70
• 97P50

Notes

^† Extended version of a previous work entitled ‘Dedekind–MacNeille Completion, Multi-adjoint Lattices’, presented in the ‘11th International Conference on Mathematical Methods in Science, Engineering, CMMSE 2011’.

We assume familiarity with pure Logic Programming and its most popular language Prolog 15.

Visit, for instance, http://dectau.uclm.es/fuzzyXPath/ where we provide a fuzzy extension of the standard XPath language for querying XML documents in a flexible way.

This German mathematician was student of Gauss in Göttingen and nowadays is considered one of the founders of modern algebra.

From http://dectau.uclm.es/floper/ it is possible to freely download the system, as well as to consult more detailed information about its current state and capabilities.

By definition, an element x∈P is an upper bound of A if a≤x for all a∈A.

By definition, an element x∈P is an lower bound of A if x≤a for all a∈A.

That is, a pair (A,B) of subsets of P such that and .

Then, it is a bounded lattice, i.e. it has bottom and top elements, denoted by ⊥ and ⊤, respectively.

By definition of least upper bound and since is a down-set.