  Logic programming under the answer-set semantics nowadays deals with numerous
different notions of program equivalence. This is due to the fact that
equivalence for substitution (known as strong equivalence) and ordinary
equivalence are different concepts. The former holds, given programs P and Q,
iff P can be faithfully replaced by Q within any context R, while the latter
holds iff P and Q provide the same output, that is, they have the same answer
sets. Notions in between strong and ordinary equivalence have been introduced
as theoretical tools to compare incomplete programs and are defined by either
restricting the syntactic structure of the considered context programs R or by
bounding the set A of atoms allowed to occur in R (relativized equivalence).For
the latter approach, different A yield properly different equivalence notions,
in general. For the former approach, however, it turned out that any
``reasonable'' syntactic restriction to R coincides with either ordinary,
strong, or uniform equivalence. In this paper, we propose a parameterization
for equivalence notions which takes care of both such kinds of restrictions
simultaneously by bounding, on the one hand, the atoms which are allowed to
occur in the rule heads of the context and, on the other hand, the atoms which
are allowed to occur in the rule bodies of the context. We introduce a general
semantical characterization which includes known ones as SE-models (for strong
equivalence) or UE-models (for uniform equivalence) as special cases.
Moreover,we provide complexity bounds for the problem in question and sketch a
possible implementation method.
  To appear in Theory and Practice of Logic Programming (TPLP).
