  Answer Set Programming (ASP) is a prominent rule-based language for knowledge
representation and reasoning with roots in logic programming and non-monotonic
reasoning. The aim to capture the essence of removing (ir)relevant details in
ASP programs led to the investigation of different notions, from strong
persistence (SP) forgetting, to faithful abstractions, and, recently, strong
simplifications, where the latter two can be seen as relaxed and strengthened
notions of forgetting, respectively. Although it was observed that these
notions are related, especially given that they have characterizations through
the semantics for strong equivalence, it remained unclear whether they can be
brought together. In this work, we bridge this gap by introducing a novel
relativized equivalence notion, which is a relaxation of the recent
simplification notion, that is able to capture all related notions from the
literature. We provide necessary and sufficient conditions for relativized
simplifiability, which shows that the challenging part is for when the context
programs do not contain all the atoms to remove. We then introduce an operator
that combines projection and a relaxation of (SP)-forgetting to obtain the
relativized simplifications. We furthermore present complexity results that
complete the overall picture.
