Equivalences in Answer-Set Programming by Countermodels in the Logic of Here-and-There

In Answer-Set Programming different notions of equivalence, such as the prominent notions of strong and uniform equivalence, have been studied and characterized by various selections of models in the logic of Here-and-There (HT). For uniform equivalence however, correct characterizations in terms of HT-models can only be obtained for finite theories, respectively programs. In this paper, we show that a selection of countermodels in HT captures uniform equivalence also for infinite theories. This result is turned into coherent characterizations of the different notions of equivalence by countermodels, as well as by a mixture of HT-models and countermodels (so-called equivalence interpretations), which are lifted to first-order theories under a very general semantics given in terms of a quantified version of HT. We show that countermodels exhibit expedient properties like a simplified treatment of extended signatures, and provide further results for non-ground logic programs. In particular, uniform equivalence coincides under open and ordinary answer-set semantics, and for finite non-ground programs under these semantics, also the usual characterization of uniform equivalence in terms of maximal and total HT-models of the grounding is correct, even for infinite domains, when corresponding ground programs are infinite.