A Constructive semantic characterization of aggregates in answer set programming

Abstract

This technical note describes a monotone and continuous fixpoint operator to compute the answer sets of programs with aggregates. The fixpoint operator relies on the notion of aggregate solution. Under certain conditions, this operator behaves identically to the three-valued immediate consequence operator Φ aggr P for aggregate programs, independently proposed in Pelov (2004) and Pelov et al. (2004). This operator allows us to closely tie the computational complexity of the answer set checking and answer sets existence problems to the cost of checking a solution of the aggregates in the program. Finally, we relate the semantics described by the operator to other proposals for logic programming with aggregates.

Fixpoint semantics and optimization of recursive Datalog programs with aggregates

“…However it used different lattices for different aggregates and thus incurred in the problems described in (Van Gelder 1993). Therefore, the search for general solutions has motivated several approaches that have addressed the problem using powerful semantics, such as answer-set semantics, that however require higher levels of computational complexity (Pelov et al 2007;Son and Pontelli 2007;Swift and Warren 2010;Faber et al 2011;Gelfond and Zhang 2014).…”

On Reductive Semantics of Aggregates in Answer Set Programming

“…Son and Pontelli [5] presented two equivalent definitions of aggregates, one in terms of "unfolding" into nested expressions, and the other in terms of "conditional satisfaction." The latter notion was simplified by Son, Pontelli and Tu [3].…”

“…On the other hand, a few reasonable "reductive" semantics of aggregates were also developed. In [4] and [5], the authors defined translations of aggregates into nested expressions, which are somewhat complex than the naive approach above. In [6], instead of translating into nested expressions, Ferraris proposed to identify an aggregate with conjunctions of implications under his extension of the answer set semantics for arbitrary propositional formulas.…”

Computing Weighted Solutions in Answer Set Programming

“…Then we can compute at least (resp. most) w-weighted solutions by computing answer sets for the ASP program P ∪ W ∪ C. In some cases, we do not have to define the weight of a solution explicitly; we can use aggregates (e.g., sum, count, times) to compute the weight, in the sense of [9,6,10]. Some other problems require an explicit definition of the weight of a solution.…”