  Compositionality of denotational semantics is an important concern in
programming semantics. Mathematical operational semantics in the sense of Turi
and Plotkin guarantees compositionality, but seen from the point of view of
stateful computation it applies only to very fine-grained equivalences that
essentially assume unrestricted interference by the environment between any two
statements. We introduce the more restrictive stateful SOS rule format for
stateful languages. We show that compositionality of two more coarse-grained
semantics, respectively given by assuming read-only interference or no
interference between steps, remains an undecidable property even for stateful
SOS. However, further restricting the rule format in a manner inspired by the
cool GSOS formats of Bloom and van Glabbeek, we obtain the streamlined and cool
stateful SOS formats, which respectively guarantee compositionality of the two
more abstract equivalences.
