Ultimate Well-Founded and Stable Semantics for Logic Programs with Aggregates

In [3] we investigate the problem of defining a well-founded and stable semantics for programs with aggregates. Our work is based on Approximation Theory [1] which is a general algebraic framework for approximating non-monotone operators on a complete lattice L by approximating operators on the bi-lattice L. The theory identifies basic properties of the approximating operators and gives a method to construct a stable operator and their associated fixpoints KripkeKleene (KK), well-founded (WF), and set of stable fixpoints. The authors then show that the different fixpoints enjoy similar properties and relationships as the corresponding semantics in standard logic programming. For example the WF fixpoint is more precise than the KK fixpoint and also approximates every stable fixpoint. Moreover, stable fixpoints are always minimal fixpoints. In a more recent paper [2] the authors study the concept of precision of an approximating operator. They show that more precise approximating operators have more precise KK and WF fixpoints and a larger number of stable fixpoints. Moreover, there exists a most-precise approximating operator which they call the ultimate approximation. In contrast with other approximating operators which have to be defined explicitly, the construction of the ultimate approximating operator is entirely algebraic and depends only on the original non-monotone operator. This makes it particularly well-suited for our purposes because defining a TP operator for programs with aggregates is relatively straightforward. Another advantage of the ultimate approximation is that in cases where TP is monotone the ultimate well-founded model will be 2-valued and will coincide with the least fixpoint of TP . This is not the case for the standard well-founded semantics. For example in the standard well-founded model of the program: