Term models, in which a process is represented as an expression in a system description language, labelled event structures, Petri nets, process graphs, and many other models of concurrency can be regarded as labelled transition spaces. Namely, there are canonical definitions of the transition relations between two terms, event structures, nets, etc. For these models the definitions of runs and their visible content from the previous section apply, and hence it is determined what it means for a semantic equivalence on such a model to respect branching time. Unless causality is taken to be part of the observable content of runs, in which case could be modelled as a mixed ordering (cf. DEGANO, DE NICOLA & MONTANARI ) and the coarsest branching time equivalence would be history preserving branching bisimulation equivalence.
Some models however, such as the failures model of BROOKES, HOARE & ROSCOE , can not be regarded as labelled transition spaces. In these cases it is often difficult to associate with a process a prefix-ordered set of runs. However, usually such a model can be canonically interpreted as the homomorphic image of a labelled transition space. In that case the model can be said to respect branching time iff the canonical homomorphism does not map two processes which have a different branching structure to the same element. Similarly, an equivalence on such a model respects branching time iff no two images of processes with a different branching structure are equivalent.