Input languages of answer set solvers are based on the mathe-matically simple concept of a stable model. But many useful constructs available in these languages, including local vari-ables, conditional literals, and aggregates, cannot be easily explained in terms of stable models in the sense of the origi-nal definition of this concept and its straightforward general-izations. 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 refer-ences 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 trans-formations 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 optimiza-tion methods may be more formally studied. 1

Abstract. Recent work has shown that the infinitary logic of here-and-there is closely related to the input language of the ASP grounder gringo. A formal system axiomatizing that logic exists, but a proof in that system may include infinitely many formulas. In this note, we define a correspondence between the validity of infinitary formulas in the logic of here-and-there and the provability of formulas in some finite deductive systems. On the basis of this correspondence, we can use finite proofs to justify the validity of infinitary formulas. 1