A general logic program (abbreviated to "program" hereafter) is a set of roles that have both positive and negative subgoals. It is common to view a deductive database as a general logic program consisting of rules (IDB) slttmg above elementary relations (EDB, facts). It is desirable to associate one Herbrand model with a program and think of that model as the "meaning of the program, " or Its "declarative semantics. " Ideally, queries directed to the program would be answered in accordance with this model. Recent research indicates that some programs do not have a "satisfactory" total model; for such programs, the question of an appropriate partial model arises. Unfounded sets and well-founded partial models are introduced and the well-founded semantics of a program are defined to be its well-founded partial model. If the well-founded partial model is m fact a total model. it is called the well-founded model. It n shown that the class of programs possessing a total well-founded model properly includes previously studied classes of "stratified" and "locally stratified" programs, The method in this paper is also compared with other proposals in the literature, including Clark's "program completion, " Fitting's and Kunen's 3-vahred interpretations of it, and the "stable models" of Gelfond and Lifschitz.A preliminary version of this paper was presented at the Seventh ACM Symposium on Principles of Database Systems, 1988 (published as VANGELDER, A., Ross, K. A., AND SCHLIPF, J S. Unfounded sets and well-founded semantics for general logic programs.

A general logic program (abbreviated to "prog-ram hereafter) I a eet of ruler that have both positive and negative subgoals It 1s common to view a deductive database as a general logic program conslating of rulea (IDB) slttmg above elementary relatlone (EDB, facts)It 18 deelrable to aseoclate one Herbrand model with a program and think of that model ae the =meanmg of the program," or Its "declarative semantlce." Ideally, queries dlrected to the program would be anewered in accordance with this model We mtroduce unbunded sets and well-founded pa&al modeb, and define the well-founded eemantlcs of a program to be Its wellfounded partial model If the well-founded partial model 18 m fact a model, we call it the well-founded model, and eay the program IU 'well-behaved n We show that the class of well-behaved programs properly Includes previously studied classes of %tratifiedD and "locally stratified' programs Gelfond and kfschits have propoeed a definltlon of 'unique stable model" for general logic programs We show that a program hae a umque stable model d It has a well-founded model, in which case they are the same We discuss why the convene M not true

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The notions of recursively saturated and resplendent models grew out of the study of admissible sets with urelements and admissible fragments of Lω1ω, but, when applied to ordinary first order model theory, give us new tools for research and exposition. We will discuss their history in §3.The notion of saturated model has proven to be important in model theory. Its most important property for applications is that if , are saturated and of the same cardinality then = iff ≅ . See, e.g., Chang-Keisler [3]. The main drawback is that saturated models exist only under unusual assumptions of set theory. For example, if 2κ = κ+ then every theory T of L has a saturated model of power κ+. (Similarly, if κ is strongly inaccessible, then every T has a saturated model of power κ.) On the other hand, a theory T like Peano arithmetic, with types, cannot have a saturated model in any power κ with ω ≤ κ ≤ .One method for circumventing these problems of existence (or rather non-existence) is the use of “special” models (cf. [3]). If κ = Σλ<κ2λ, κ < ω, then every theory T of L has a special model of power κ. Such cardinals are large and, themselves, rather special. There are definite aesthetic objections to the use of these large, singular models to prove results about first order logic.