  The formalism of generalized probabilistic theories (GPTs) was originally
developed as a way to characterize the landscape of conceivable physical
theories. Thus, the GPT describing a given physical theory necessarily includes
all physically possible processes. We here consider the question of how to
provide a GPT-like characterization of a particular experimental setup within a
given physical theory. We show that the resulting characterization is not
generally a GPT in and of itself-rather, it is described by a more general
mathematical object that we introduce and term an accessible GPT fragment. We
then introduce an equivalence relation, termed cone equivalence, between
accessible GPT fragments (and, as a special case, between standard GPTs). We
give a number of examples of experimental scenarios that are best described
using accessible GPT fragments, and where moreover cone-equivalence arises
naturally. We then prove that an accessible GPT fragment admits of a classical
explanation if and only if every other fragment that is cone-equivalent to it
also admits of a classical explanation. Finally, we leverage this result to
prove several fundamental results regarding the experimental requirements for
witnessing the failure of generalized noncontextuality. In particular, we prove
that neither incompatibility among measurements nor the assumption of freedom
of choice is necessary for witnessing failures of generalized noncontextuality,
and, moreover, that such failures can be witnessed even using arbitrarily
inefficient detectors.
