Multiset Theories of Cardinality-based Feature Diagrams
  Software product line engineering is a very common method for designing complex software systems. Feature modeling is the most common approach to specify product lines. The main part of a feature model is a special tree of features called a feature diagram. Cardinality-based feature diagrams provide the most expressive tool among the current feature diagram languages. The most common characterization of the semantics of a cardinality-based diagram is the set of flat multisets over features satisfying the constraints. However, this semantics provides a poor abstract view of the diagram. We address this problem by proposing another multiset theory for the cardinality-based feature diagram, called the hierarchical theory of the diagram. We show that this semantics captures all information of the diagram so that one can retrieve the diagram from its hierarchical semantics. We also characterize sets of multisets, which can provide a hierarchical semantics of some diagrams. 
Aliakbar Safilian and Tom Maibaum
cs.SE