The Semantics of Gringo and Infinitary Propositional Formulas

ABSTRACT. 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, so that the properties of these programs can be more precisely characterized. In this way, we aim to create a foundation on which important issues such as the correctness of gringo programs and optimization methods may be more formally studied.