Axiomatization of Non-Recursive Aggregates in First-Order Answer Set Programming

This paper contributes to the development of theoretical foundations of answer set programming. Groundbreaking work on the SM operator by Ferraris, Lee, and Lifschitz proposed a definition/semantics for logic (answer set) programs based on a syntactic transformation similar to parallel circumscription. That definition radically differed from its predecessors by using classical (second-order) logic and avoiding reference to either grounding or fixpoints. Yet, the work lacked the formalization of crucial and commonly used answer set programming language constructs called aggregates. In this paper, we present a characterization of logic programs with aggregates based on a many-sorted generalization of the SM operator. This characterization introduces new function symbols for aggregate operations and aggregate elements, whose meaning can be fixed by adding appropriate axioms to the result of the SM transformation. We prove that our characterization coincides with the ASP-Core-2 semantics for logic programs and, if we allow non-positive recursion through aggregates, it coincides with the semantics of the answer set solver CLINGO.