Abstract
We show that any ASP aggregate interpreted under Gelfond and Zhang’s (GZ) semantics can be replaced (under strong equivalence) by a propositional formula. Restricted to the original GZ syntax, the resulting formula is reducible to a disjunction of conjunctions of literals but the formulation is still applicable even when the syntax is extended to allow for arbitrary formulas (including nested aggregates) in the condition. Once GZ-aggregates are represented as formulas, we establish a formal comparison (in terms of the logic of Here-and-There) to Ferraris’ (F) aggregates, which are defined by a different formula translation involving nested implications. In particular, we prove that if we replace an F-aggregate by a GZ-aggregate in a rule head, we do not lose answer sets (although more can be gained). This extends the previously known result that the opposite happens in rule bodies, i.e., replacing a GZ-aggregate by an F-aggregate in the body may yield more answer sets. Finally, we characterise a class of aggregates for which GZ- and F-semantics coincide.
Partially supported by grants GPC 2016/035 (Xunta de Galicia, Spain), TIN 2013-42149-P (MINECO, Spain), and SCHA 550/9 (DFG, Germany).
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Notes
- 1.
Note that, for arbitrary languages and semantics, this definition is stronger than usual, as it refers to any subformula replacement and not just conjunctions of formulas, as usual. For instance, \(\varphi \equiv _s\psi \) also implies \(\{\varphi \otimes \alpha \} \equiv _s\{\psi \otimes \alpha \}\) for any binary operator \(\otimes \) in our language. When \(\mathcal{L}\) is a logical language and \(\equiv _s\) amounts to equivalence in HT (or any logic with substitution of equivalents) the distinction becomes irrelevant.
- 2.
An extension to a full first-order language is under development.
- 3.
Ferraris actually uses \(\varphi _i=w\) rather than \(\mathbf {c}\!:\!\varphi \), but this is not a substantial difference, assuming w is the first element in tuple \(\mathbf {c}\).
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Cabalar, P., Fandinno, J., Schaub, T., Schellhorn, S. (2017). Gelfond-Zhang Aggregates as Propositional Formulas. In: Balduccini, M., Janhunen, T. (eds) Logic Programming and Nonmonotonic Reasoning. LPNMR 2017. Lecture Notes in Computer Science(), vol 10377. Springer, Cham. https://doi.org/10.1007/978-3-319-61660-5_12
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