Equational Characterization of Covariant-Contravariant Simulation and Conformance Simulation Semantics

Covariant-contravariant simulation and conformance simulation generalize plain simulation and try to capture the fact that it is not always the case that “the larger the number of behaviors, the better”. We have previously studied their logical characterizations and in this paper we present the axiomatizations of the preorders defined by the new simulation relations and their induced equivalences. The interest of our results lies in the fact that the axiomatizations help us to know the new simulations better, understanding in particular the role of the contravariant characteristics and their interplay with the covariant ones; moreover, the axiomatizations provide us with a powerful tool to (algebraically) prove results of the corresponding semantics. But we also consider our results interesting from a metatheoretical point of view: the fact that the covariant-contravariant simulation equivalence is indeed ground axiomatizable when there is no action that exhibits both a covariant and a contravariant behaviour, but becomes non-axiomatizable whenever we have together actions of that kind and either covariant or contravariant actions, offers us a new subtle example of the narrow border separating axiomatizable and non-axiomatizable semantics. We expect that by studying these examples we will be able to develop a general theory separating axiomatizable and non-axiomatizable semantics.

💡 Research Summary

The paper introduces two novel behavioral relations that extend the classic notion of simulation: covariant‑contravariant simulation (CCS) and conformance simulation (CS). Both aim to capture the intuition that “more behaviours are not always better” by distinguishing between actions that are optional (covariant) and those that are mandatory (contravariant).

Technical definitions.
The authors start from a basic process algebra (BCCSP) and partition the set of action labels into two disjoint subsets: C (covariant) and D (contravariant). A relation R is a covariant‑contravariant simulation if, for every pair (p,q)∈R:

• If p can perform a covariant action a∈C and reach p′, then q must be able to perform the same a and reach some q′ with (p′,q′)∈R (the usual forward simulation condition).
• If q can perform a contravariant action b∈D and reach q′, then p must also be able to perform b and reach some p′ with (p′,q′)∈R (a backward condition).

Conformance simulation is a specialization of CCS in which all D‑actions are regarded as required: a process q conforms to p only if every required action of q is also present in p, while optional (C) actions may be added freely.

Logical characterisation.
The paper revisits earlier work on Hennessy‑Milner logic (HML) and shows that CCS and CS can be captured by a modal logic that mixes possibility modalities ⟨a⟩ (for a∈C) with necessity modalities