Quantifying over Optimum Answer Sets

Answer Set Programming with Quantifiers (ASP(Q)) has been introduced to provide a natural extension of ASP modeling to problems in the polynomial hierarchy (PH). However, ASP(Q) lacks a method for encoding in an elegant and compact way problems requiring a polynomial number of calls to an oracle in $\Sigma_n^p$ (that is, problems in $\Delta_{n+1}^p$). Such problems include, in particular, optimization problems. In this paper we propose an extension of ASP(Q), in which component programs may contain weak constraints. Weak constraints can be used both for expressing local optimization within quantified component programs and for modeling global optimization criteria. We showcase the modeling capabilities of the new formalism through various application scenarios. Further, we study its computational properties obtaining complexity results and unveiling non-obvious characteristics of ASP(Q) programs with weak constraints.

Beyond NP: Quantifying over Answer Sets

Answer Set Programming (ASP) is a logic programming paradigm featuring a purely declarative language with comparatively high modeling capabilities. Indeed, ASP can model problems in NP in a compact and elegant way. However, modeling problems beyond NP with ASP is known to be complicated, on the one hand, and limited to problems in {\Sigma}^P_2 on the other. Inspired by the way Quantified Boolean Formulas extend SAT formulas to model problems beyond NP, we propose an extension of ASP that introduces quantifiers over stable models of programs. We name the new language ASP with Quantifiers (ASP(Q)). In the paper we identify computational properties of ASP(Q); we highlight its modeling capabilities by reporting natural encodings of several complex problems with applications in artificial intelligence and number theory; and we compare ASP(Q) with related languages. Arguably, ASP(Q) allows one to model problems in the Polynomial Hierarchy in a direct way, providing an elegant expansion of ASP beyond the class NP. Under consideration for acceptance in TPLP.