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Attribute-based variability in feature models

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Abstract

Extended feature models enable the expression of complex cross-tree constraints involving feature attributes. The inclusion of attributes in cross-tree relations not only enriches the constraints, but also engenders an extended type of variability that involves attributes. In this article, we elaborate on the effects of this new variability type on feature models. We start by analyzing the nature of the variability involving attributes and extend the definitions of the configuration and the product to suit the emerging requirements. Next, we propose classifications for the features, configurations, and products to identify and formalize the ramifications that arise due to the new type of variability. Then, we provide a semantic foundation grounded on constraint satisfaction for our proposal. We introduce an ordering relation between configurations and show that the set of all the configurations represented by a feature model forms a semilattice. This is followed by a demonstration of how the feature model analyses will be affected using illustrative examples selected from existing and novel analysis operations. Finally, we summarize our experiences, gained from a commercial research and development project that employs an extended feature model.

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Authors and Affiliations

  1. Department of Computer Engineering, Middle East Technical University, Ankara, Turkey

    Ahmet Serkan Karataş & Halit Oğuztüzün

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  1. Ahmet Serkan Karataş
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  2. Halit Oğuztüzün
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Correspondence to Ahmet Serkan Karataş.

Appendix 1

Appendix 1

Proof of Proposition 1

We first start by observing that the subset relation is also reflexive (i.e., X ⊆ X), antisymmetric (i.e., X ⊆ Y and Y ⊆ X imply X = Y), and transitive (i.e., X ⊆ Y and Y ⊆ Z imply X ⊆ Z), where X, Y, and Z are sets. We define the configuration ordering using the subset relation between the sets of included features and the sets of excluded features of two valid configurations; thus, it quickly follows that the configuration ordering relation is a partial order, as follows.

Let C1 = (I1, E1), C2 = (I2, E2) and C3 = (I3, E3) be three valid configurations with respect to M. The configuration ordering relation is

  • (reflexive) C1 ≤ C1, since I1 ⊆ I1 and E1 ⊆ E1.

  • (antisymmetric) C1 ≤ C2 and C2 ≤ C1 imply C1 = C2, since I1 ⊆ I2 and I2 ⊆ I1 imply I1 = I2, and E1 ⊆ E2 and E2 ⊆ E1 imply E1 = E2

  • (transitive) C1 ≤ C2 and C2 ≤ C3 imply C1 ≤ C3, since I1 ⊆ I2 and I2 ⊆ I3 imply I1 ⊆ I3, and E1 ⊆ E2 and E2 ⊆ E3 imply E1 ⊆ E3 □

Proof of Proposition 2

We first start by observing that Cinf ≤ C1 (since I1 ∩ I2 ⊆ I1 and E1 ∩ E2 ⊆ E1) and Cinf ≤ C2 (since I1 ∩ I2 ⊆ I2 and E1 ∩ E2 ⊆ E2). Thus, Cinf is a lower bound of {C1, C2}.

Then, we show that Cinf is the greatest lower bound of {C1, C2}. Assume that there exists another configuration C = (I, E), where C ≠ Cinf, such that C is a lower bound of {C1, C2} and Cinf ≤ C. Due to the assumption C ≠ Cinf, at least one of the following two conditions must be true: (i) I ≠ I1 ∩ I2, (ii) E ≠ E1 ∩ E2.

We will first examine (i). Since we have assumed that Cinf ≤ C, then, due to the definition of the configuration ordering relation, it must be the case that I1 ∩ I2 ⊆ I. If (i) is true, then there must be at least one feature f such that f is a member of I, but not I1 ∩ I2. f is not a member of I1 ∩ I2 if and only if f ∉ I1 or f ∉ I2. However, since we assumed that C is a lower bound of {C1, C2}, it must be the case that C ≤ C1 and C ≤ C2. If C ≤ C1, then I ⊆ I1; thus, every member of I, including f, must also be a member of I1; hence, f ∉ I1 cannot be true. For similar reasons, f ∉ I2 cannot also be true. Therefore, it is not possible to find such an f. Consequently, (i) cannot be true.

Similarly, it is easy to show that (ii) cannot be true as well. Thus, our assumption on the existence of such a C fails. Therefore, we conclude that Cinf = (I1 ∩ I2, E1 ∩ E2) is the infimum of {C1, C2} (i.e., Cinf = C1 ∧ C2). □

Proof of Proposition 3

Let the set of all valid configurations represented by M be S. Since M is not void, it represents at least one valid product; thus, S will be non-empty. It follows from Proposition 1 that S is equipped with the partial order relation ≤ ; thus, it is a partially ordered set. It follows from Proposition 2 that for any two valid configurations C1 = (I1, E1) and C2 = (I2, E2) such that C1, C2 ∈ S, C1 ∧ C2 exists and is equal to (I1 ∩ I2, E1 ∩ E2). Therefore, S is a meet semilattice. □

Proof of Proposition 4

We first start by observing that Cinf ≤ C1 (since I1 ∩ I2 ⊆ I1, E1 ∩ E2 ⊆ E1, and A1-1 ⊆ A1-1 ∪ A2-1, …, A1-n ⊆ A1-n ∪ A2-n). Similarly, Cinf ≤ C2 is also true. Thus, Cinf is a lower bound of {C1, C2}.

Then, we show that Cinf is the greatest lower bound of {C1, C2}. Assume that there exists another configuration C = (I, E, Δ), where C ≠ Cinf, such that C is a lower bound of {C1, C2} and Cinf ≤ C. Due to the assumption C ≠ Cinf, at least one of the following three conditions must be true: (i) I ≠ I1 ∩ I2, (ii) E ≠ E1 ∩ E2, (iii) Δ ≠ Δinf. The proofs that show (i) and (ii) cannot be true can be directly borrowed from the proof of Proposition 2. Therefore, we will focus on (iii).

Let Δ = {a1 ∈ A1, …, an ∈ An}. Since we have assumed that Cinf ≤ C, then, due to the definition of the extended configuration ordering relation, it must be the case that A1 ⊆ A1-1 ∪ A2-1, …, An ⊆ A1-n ∪ A2-n. If (iii) is true, then there must be at least one value v such that v is a member of A1-x ∪ A2-x, but not Ax, where 1 ≤ x ≤ n. v is a member of A1-x ∪ A2-x if and only if v ∈ A1-x or v ∈ A2-x. However, since we assumed that C is a lower bound of {C1, C2}, it must be the case that C ≤ C1 and C ≤ C2. If C ≤ C1, then A1-x ⊆ Ax; thus, a value that is not a member of Ax cannot be a member of A1-x; hence, v ∈ A1-x cannot be true. For similar reasons, v ∈ A2-x cannot also be true. Therefore, it is not possible to find such a v and (iii) cannot be true.

Thus, our assumption on the existence of such a C fails. Therefore, we conclude that Cinf is the infimum of {C1, C2} (i.e., Cinf = C1 ∧ C2). □

Proof of Proposition 5

Let the set of all valid configurations represented by M be S. Since M is not void, it represents at least one valid product; thus, S will be non-empty. As discussed above, S is equipped with the partial order relation ≤ ; thus, it is a partially ordered set. It follows from Proposition 4 that for any two configurations C1 = (I1, E1, Δ1) and C2 = (I2, E2, Δ2) such that C1, C2 ∈ S, C1 ∧ C2 exists and is equal to (I1 ∩ I2, E1 ∩ E2, Δinf), where Δ1 = {a1 ∈ A1-1, …, an ∈ A1-n, …}, Δ2 = {a1 ∈ A2-1, …, an ∈ A2-n, …}, and Δinf = {a1 ∈ A1-1 ∪ A2-1, …, an ∈ A1-n ∪ A2-n}. Therefore, S is a meet semilattice. □

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Karataş, A.S., Oğuztüzün, H. Attribute-based variability in feature models. Requirements Eng 21, 185–208 (2016). https://doi.org/10.1007/s00766-014-0216-9

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