Stable Models for Infinitary Formulas with Extensional Atoms

The definition of stable models for propositional formulas with infinite conjunctions and disjunctions can be used to describe the semantics of answer set programming languages. In this note, we enhance that definition by introducing a distinction between intensional and extensional atoms. The symmetric splitting theorem for first-order formulas is then extended to infinitary formulas and used to reason about infinitary definitions. This note is under consideration for publication in Theory and Practice of Logic Programming.

On the Semantics of Gringo

Input languages of answer set solvers are based on the mathematically simple concept of a stable model. But many useful constructs available in these languages, including local variables, conditional literals, and aggregates, cannot be easily explained in terms of stable models in the sense of the original definition of this concept and its straightforward generalizations. Manuals written by designers of answer set solvers usually explain such constructs using examples and informal comments that appeal to the user's intuition, without references to any precise semantics. We propose to approach the problem of defining the semantics of gringo programs by translating them into the language of infinitary propositional formulas. This semantics allows us to study equivalent transformations of gringo programs using natural deduction in infinitary propositional logic.