Abstract In two earlier papers we derived congruence formats with regard to transition system specifications for weak semantics on the basis of a decomposition method for modal formulas. The idea is that a congruence format for a semantics must ensure that the formulas in the modal characterisation of this semantics are always decomposed into formulas that are again in this modal characterisation. The stability and divergence requirements that are imposed on many of the known weak semantics have so far been outside the realm of this method. Stability refers to the absence of a τ-transition. We show, using the decomposition method, how congruence formats can be relaxed for weak semantics that are stability-respecting. This relaxation for instance brings the priority operator within the range of the stability-respecting branching bisimulation format. Divergence, which refers to the presence of an infinite sequence of τ-transitions, escapes the inductive decomposition method. We circumvent this problem by proving that a congruence format for a stability-respecting weak semantics is also a congruence format for its divergence-preserving counterpart.
We present a general class of process languages for rooted eager bisimulation preorder and prove its congruence result. Also, we present classes of process languages for the rooted versions of several other weak preorders. The process languages we propose are defined by the Ordered SOS method which combines the traditional SOS approach, based on transition rules, with the ordering on transition rules. This new feature specifies the order of application of rules when deriving transitions of process terms. We support and illustrate our results with examples of process operators and process languages which benefit from the OSOS formulation. Earlier we presented a method to decompose modal formulas for processes with the internal action τ; congruence formats for branching and η-bisimilarity were derived on the basis of this decomposition method. The idea is that a congruence format for a semantics must ensure that formulas in the modal characterisation of this semantics are always decomposed into formulas in this modal characterisation. Here the decomposition method is enhanced to deal with modal characterisations that contain a modality 〈e〉〈a〉p, to derive congruence formats for delay and weak bisimilarity. Earlier we presented a method to decompose modal formulas for processes with the internal action τ , and congruence formats for branching and η -bisimilarity were derived on the basis of this decomposition method. The idea is that a congruence format for a semantics must ensure that the formulas in the modal characterisation of this semantics are always decomposed into formulas that are again in this modal characterisation. In this follow-up paper the decomposition method is enhanced to deal with modal characterisations that contain a modality 〈ϵ〉〈a〉φ 〈 ϵ 〉 〈 a 〉 φ , to derive congruence formats for delay and weak bisimilarity.