The Family of Stable Models

The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a so-called knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, sP — it is the well-founded model. There is also a dual largest stable model, S P , which has not been considered before. There is another ordering based on degree of truth. Taking the meet and the join, in the truth ordering, of the two extreme stable models sP and S k P just mentioned, yields the alternating fixed points of [29], denoted sP and S t P here. From s t P and S t P in turn, sP and S k P can be produced again, using the meet and join of the knowledge ordering. All stable models are bounded by these four valuations. Further, the methods of proof apply not just to logic programs considered classically, but to logic programs over any bilattice meeting certain conditions, and thus apply in a vast range of settings. The methods of proof are largely algebraic.