Modal Transition System Encoding of Featured Transition Systems

Featured transition systems (FTSs) and modal transition systems (MTSs) are two of the most prominent and well-studied formalisms for modeling and analyzing behavioral variability as apparent in software product line engineering. On one hand, it is well-known that for finite behavior FTSs are strictly more expressive than MTSs, essentially due to the inability of MTSs to express logically constrained behavioral variability such as persistently exclusive behaviors. On the other hand, MTSs enjoy many desirable formal properties such as compositionality of semantic refinement and parallel composition. In order to finally consolidate the two formalisms for variability modeling, we establish a rigorous connection between FTSs and MTSs by means of an encoding of one FTS into an equivalent set of multiple MTSs. To this end, we split the structure of an FTS into several MTSs whenever it is necessary to denote exclusive choices that are not expressible in a single MTS. Moreover, extra care is taken when dealing with infinite behaviour: loops may have to be unrolled to accumulate FTS path constraints when encoding them into MTSs. We prove our encoding to be semantic-preserving (i.e., the resulting set of MTSs induces, up to bisimulation, the same set of derivable variants as their FTS counterpart) and to commute with modal refinement. We further give an algorithm to calculate a concise representation of a given FTS as a minimal set of MTSs. Finally, we present experimental results gained from applying a tool implementation of our approach to a collection of case studies.