Coherent modal transition systems refinement

Modal Transition Systems (MTS) are a well-known formalism that extend Labelled Transition Systems (LTS) with the possibility of specifying necessary and permitted behaviour. Coherent MTS (CMTS) have been introduced to model Software Product Lines (SPL) based on a correspondence between the necessary and permitted modalities of MTS transitions and their associated actions, and the core and optional features of SPL. In this paper, we address open problems of the coherent fragment of MTS and introduce the notions of refinement and thorough refinement of CMTS. Most notably, we prove that refinement and thorough refinement coincide for CMTS, while it is known that this is not the case for MTS. We also define (thorough) equivalence and strong bisimilarity of both MTS and CMTS. We show their relations and, in particular, we prove that also strong bisimilarity and equivalence coincide for CMTS, whereas they do not for MTS. Finally, we extend our investigation to CMTS equipped with Constraints (MTSC), originally introduced to express alternative behaviour, and we prove that novel notions of refinement and strong thorough refinement coincide for MTSC, and so do their extensions to strong (thorough) equivalence and strong bisimilarity.