Abstract
Product lines are an established framework for software design. They are specified by special diagrams called feature models. For formal analysis, the latter are usually encoded by propositional theories with Boolean semantics. We discuss a major deficiency of this semantics, and show that it can be fixed by considering that a product is an instantiation process rather than its final result. We call intermediate states of this process partial products, and argue that what a feature model M really defines is a poset of partial products called a partial product line, PPL(M). We argue that such PPLs can be viewed as special partial product Kripke structures (ppKS) specifiable by a suitable version of CTL (partial product CTL or ppCTL). We show that any feature model M is representable by a ppCTL theory \(\varPhi (M)\) such that for any ppKs K, \(K\models \varPhi (M)\) iff \(K = PPL (M)\); hence, \(\varPhi (M)\) is a sound and complete representation of the feature model.
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- 1.
With a more flexible view of product assembly, some possible interleavings could be prohibited and some allowed. (We owe this idea to an anonymous reviewer.) Then we need to add a suitable annotating mechanism to the fm formalism.
- 2.
We cannot declare that features \(\mathsf {en}\) and \(\mathsf {ge}\) are mutually exclusive and write \(\{\mathsf {en}{\wedge }\mathsf {ge}\rightarrow \bot \}\) as down the lattice they are combined in the product \(\{{\mathsf {c}},\mathsf {en},\mathsf {ele},\mathsf {ge}\}\).
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Acknowledgement
We are grateful to Krzysztof Czarnecki for several fruitful discussions of the subject, particularly, of the staged configuration approach. Thanks also go to anonymous reviewers for stimulating criticism and, specifically, for references to DCR-graphs and PL-CCS work. Financial support was provided by Automotive Partnership Canada via the NECSIS project.
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Appendix
Appendix
The list of abbreviations used in the paper:
Abbreviation | Meaning | Abbreviation | Meaning |
ccc | crosscutting constraint | \(\mathsf {BL}\) | Boolean logic |
ppKS | partial product Kripke structure | \(\mathsf {ppCTL}\) | partial product CTL |
fm | feature model | FM | feature modeling |
pl | product line | \(\mathsf {I2C}\) | instantiate-to-completion |
ppl | partial product line | \(\mathsf {ML}\) | Modal Logic |
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Diskin, Z., Safilian, A., Maibaum, T., Ben-David, S. (2015). Modeling Product Lines with Kripke Structures and Modal Logic. In: Leucker, M., Rueda, C., Valencia, F. (eds) Theoretical Aspects of Computing - ICTAC 2015. ICTAC 2015. Lecture Notes in Computer Science(), vol 9399. Springer, Cham. https://doi.org/10.1007/978-3-319-25150-9_12
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