Complexity of super-coherence problems in ASP

Adapting techniques from database theory in order to optimize Answer Set Programming
(ASP) systems, and in particular the grounding components of ASP systems, is an important
topic in ASP. In recent years, the Magic Set method has received some interest in this setting,
and a variant of it, called Dynamic Magic Set, has been proposed for ASP. However, this
technique has a caveat, because it is not correct (in the sense of being query-equivalent)
for all ASP programs. In a recent work, a large fragment of ASP programs, referred to as
super-coherent programs, has been identified, for which Dynamic Magic Set is correct. The
fragment contains all programs which possess at least one answer set, no matter which set of
facts is added to them. Two open question remained: How complex is it to determine whether
a given program is super-coherent? Does the restriction to super-coherent programs limit
the problems that can be solved? Especially the first question turned out to be quite difficult
to answer precisely. In this paper, we formally prove that deciding whether a propositional
P
P
program is super-coherent is Π3 -complete in the disjunctive case, while it is Π2 -complete for
normal programs. The hardness proofs are the difficult part in this endeavor: We proceed
by characterizing the reductions by the models and reduct models which the ASP programs
should have, and then provide instantiations that meet the given specifications. Concerning
the second question, we show that all relevant ASP reasoning tasks can be transformed into
tasks over super-coherent programs, although this transformation is more of theoretical than
practical interest.