Abstract A process is called computable if it can be modelled by a transition system that has a recursive structure—implying finite branching. The equivalence relation between transition systems considered is strong bisimulation equivalence. The transition systems studied in this paper can be associated to processes specified in common specification languages such as CCS, LOTOS, ACP and PSF. As a means for defining transition systems up to bisimulation equivalence, the specification language μ CRL is used. Two simple fragments of, μ CRL are singled out, yielding universal expressivity with respect to recursive and primitive recursive transition systems. For both these domains the following properties are classified in the arithmetical hierarchy: bisimilarity, perpetuity (both ∏ 1 0 ), regularity (having a bisimilar, finite representation, Σ 2 0 ), acyclic regularity (Σ 1 0 ), and deadlock freedom (distinguishing deadlock from successful termination, ∏ 1 0 ). Finally, it is shown that in the domain of primitive recursive transition systems over a fixed, finite label set, a genuine hierarchy in bisimilarity can be defined by the complexity of the witnessing relations, which extends r.e. bisimilarity. Hence, primitive recursive transition systems already form an interesting class.