On the Stable Model Semantics of First-Order Formulas with Aggregates

The original definition of a stable model has been generalized to logic programs with aggregates. On the other hand, it was extended to first-order formulas using a syntactic transformation SM, similar to circumscription. In a recent paper, Lee and Meng combined these two ideas in a single framework by defining the operator SM for generalized formulas that may include, besides the usual syntactic features of first-order logic, symbols used in dlv-style aggregate expressions. In this note, we make the syntax proposed by Lee and Meng more uniform by allowing aggregate symbols to be used at any stage of the recursive process of building a formula from atoms, along with propositional connectives and quantifiers. We also generalize several useful properties of SM to formulas with aggregates, and investigate the relationship between the Lee-Meng semantics of aggregates and the semantics of counting adopted in the language RASPL-1.