Improving completeness in multi-adjoint logic computations via general reductants

Fuzzy extensions of logic programming often require the notion of reductant to ensure completeness when working with some lattices modeling the concept of truth degree beyond the simpler case of true and false. Initially introduced in the context of generalized annotated logic programming, some adaptations of this theoretical tool have been proposed for the more recent and flexible framework of multi-adjoint logic programming. To the best of our knowledge, all of them suffer the important problem of usually requiring an infinite set of reductants (one for each ground atom) for being added to a given program in order to guarantee its completeness. The main goal of this paper is the introduction of a generalized notion of reductant, called G-reductant, which only depends on (a finite number of) predicate symbols instead of ground atoms (whose number is always infinite for programs considering at least a non constant function symbol in their signature). More exactly, given a predicate p/n in the signature of a fuzzy program p, we generate just a single G-reductant with head p(X
₁
, ... , X
_n
) (being X
₁
, ... , X
_n
different variable symbols) which covers all the possible calls to p in a completely safe way. Since the number of G-reductants is finite for programs with a finite number of predicates, our notion can be really applied in practice in contrast with older versions of reductants which are only applicable at a non-practical, but purely theoretical level.