Translational Expressiveness. Comparing Process Calculi using Encodings

ion is not of much use without operational correspondence. Because of the various possibilities two choose these two equivalences, it is often not possible to compare different full abstraction results, which is a major drawback of this criterion. 3.2. 3. Operational Correspondence Intuitively, operational correspondence requires preservation and reflection of executions. Again, it consists of a completeness a soundness part. The completeness condition, also called adequacy, requires that for all source term steps S 7−→S S′ or source term executions S Z=⇒S S′ there is one emulating execution in the target language such that J S K Z =⇒T T J S′ K, where T ⊆ PT × PT is some equivalences on the target language. Note that there is no difference in the consideration of single source term steps or source term executions. Intuitively, the completeness condition requires that any source term execution is emulated by the target term modulo some equivalence T . Again, completeness is usually the easiest part. For the soundness condition we basically find two formulations. The stricter formulation requires that for all executions of the target J S K Z =⇒T T there exists some execution of the source S Z=⇒S S′ such that J S′ K T T . Intuitively, soundness requires that whatever J S K can do is a translation of some behaviour of S modulo T [FL10]. The weaker formulation requires that for all executions of the target J S K Z =⇒T T there exists some execution of the source S Z=⇒S S′ and some execution of the target T Z=⇒T T ′ such that J S′ K T T ′. Intuitively, it states that any execution of the target is some part of the emulation of an execution in the source modulo T [Par08, Gor10b]. The main difference is that the later formulation allows for intermediate or partial commitment states, i.e., for states that do not need to be related directly to the states of the respective source term but that have to belong to some emulation of a source term step. In this sense, an intermediate state results from the partial emulation of a source term step. We discuss this issue in Section 6.3. 1. Again different variants of operational correspondence may arise from different requirements on the assumed equivalence T on the target language. Note that [Nes96, NP00] present operational correspondence without the equivalence, i.e., require J S K Z=⇒T J S′ K whenever S 7−→S S′ and J S K Z =⇒T T implies S Z=⇒S S′ for some S′ such that T Z=⇒T J S′ K, which again leads to a stricter formulation than above. They also present a stricter variant of the soundness part—J S K 7−→T T implies S 7−→S S′ for some S′ such that T T J S′ K—and state that only prompt encodings can satisfy this stricter variant. Moreover, in [FL10] labelled steps are considered instead of a reduction semantics under the assumption that there exists a mapping ·̂ from the labels of the source term into the labels of the target term. Hence, the resulting requirement—J S K λ̂ =⇒ T J S′ K whenever S λ =⇒ S′ and J S K λ =⇒ T implies S λ ′ =⇒ S′ for some λ′, S′ such that J S′ K T T and λ̂′ = λ—can be considered as stricter than the above variant of operational correspondence, because also observables have to be preserved and reflected in some sense. In fact, without this strengthening to labelled semantics, operational correspondence