This paper deals with distributed procedures, caused by negation as failure through a network, where general logic programs are distributed so that they communicate with each other in terms of negation as failure inquiries and responses, but not in terms of derivations of SLD resolutions. The common variables as channels in share for distributed programs are not treated, but negation as failure validated in the whole network is the object for communications of distributed programs. We can define the semantics for the distributed programs in a network. At the same time, we have distributed proof procedures for distributed programs, by means of negation as failure to be implemented through the network, where the soundness of the procedure is guaranteed by the defined semantics.

Negation as failure is realized to be combined with SLD resolution for general logic programs, where the combined resolution is called an SLDNF resolution. In this paper, we introduce narrowing and infinite failure to SLDNF resolution for general logic programs with equations. The combination of SLDNF resolution with narrowing and infinite failure is called an SLDNFN resolution. In Shepherdson (1992), equation theory is combined with SLDNF resolution so that the soundness may be guaranteed with respect to Clark's completion. Generalizing the method of Yamamoto (1987) for definite clause sets with equations, we formally define a least fixpoint semantics, which is an extension of Fitting (1985) and Kunen (1987) semantics, and which includes the pair of success and failure sets defined by the SLDNFN resolution. The relationship between the fixpoint semantics and the pair of sets is regarded as an extension of the relationships for general logic programs as in Marriott and et al. (1992) and in Yamasaki (1996). Instead of generalizing Clark's completion for SLDNFN resolution, we establish, as a model for general logic programs with equations, an extended well-founded model so that the SLDNFN resolution is sound and complete for non-floundering queries with respect to the extended well-founded model.