  Among the myriad of desirable properties discussed in the context of
forgetting in Answer Set Programming (ASP), strong persistence naturally
captures its essence. Recently, it has been shown that it is not always
possible to forget a set of atoms from a program while obeying this property,
and a precise criterion regarding what can be forgotten has been presented,
accompanied by a class of forgetting operators that return the correct result
when forgetting is possible.
  However, it is an open question what to do when we have to forget a set of
atoms, but cannot without violating this property. In this paper, we address
this issue and investigate three natural alternatives to forget when forgetting
without violating strong persistence is not possible, which turn out to
correspond to the different possible relaxations of the characterization of
strong persistence. Additionally, we discuss their preferable usage, shed light
on the relation between forgetting and notions of relativized equivalence
established earlier in the context of ASP, and present a detailed study on
their computational complexity.
