Attribute-based variability in feature models

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.

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 C_inf ≤ C₁ (since I₁ ∩ I₂ ⊆ I₁ and E₁ ∩ E₂ ⊆ E₁) and C_inf ≤ C₂ (since I₁ ∩ I₂ ⊆ I₂ and E₁ ∩ E₂ ⊆ E₂). Thus, C_inf is a lower bound of {C₁, C₂}.

Then, we show that C_inf is the greatest lower bound of {C₁, C₂}. Assume that there exists another configuration C = (I, E), where C ≠ C_inf, such that C is a lower bound of {C₁, C₂} and C_inf ≤ C. Due to the assumption C ≠ C_inf, 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 C_inf ≤ 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 C_inf = (I₁ ∩ I₂, E₁ ∩ E₂) is the infimum of {C₁, C₂} (i.e., C_inf = 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 C_inf ≤ C₁ (since I₁ ∩ I₂ ⊆ I₁, E₁ ∩ E₂ ⊆ E₁, and A_1-1 ⊆ A_1-1 ∪ A_2-1, …, A_1-n ⊆ A_1-n ∪ A_2-n). Similarly, C_inf ≤ C₂ is also true. Thus, C_inf is a lower bound of {C₁, C₂}.

Then, we show that C_inf is the greatest lower bound of {C₁, C₂}. Assume that there exists another configuration C = (I, E, Δ), where C ≠ C_inf, such that C is a lower bound of {C₁, C₂} and C_inf ≤ C. Due to the assumption C ≠ C_inf, at least one of the following three conditions must be true: (i) I ≠ I₁ ∩ I₂, (ii) E ≠ E₁ ∩ E₂, (iii) Δ ≠ Δ_inf. 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, …, a_n ∈ A_n}. Since we have assumed that C_inf ≤ C, then, due to the definition of the extended configuration ordering relation, it must be the case that A₁ ⊆ A_1-1 ∪ A_2-1, …, A_n ⊆ A_1-n ∪ A_2-n. If (iii) is true, then there must be at least one value v such that v is a member of A_1-x ∪ A_2-x, but not A_x, where 1 ≤ x ≤ n. v is a member of A_1-x ∪ A_2-x if and only if v ∈ A_1-x or v ∈ A_2-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 A_1-x ⊆ A_x; thus, a value that is not a member of A_x cannot be a member of A_1-x; hence, v ∈ A_1-x cannot be true. For similar reasons, v ∈ A_2-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 C_inf is the infimum of {C₁, C₂} (i.e., C_inf = 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, Δ_inf), where Δ₁ = {a₁ ∈ A_1-1, …, a_n ∈ A_1-n, …}, Δ₂ = {a₁ ∈ A_2-1, …, a_n ∈ A_2-n, …}, and Δ_inf = {a₁ ∈ A_1-1 ∪ A_2-1, …, a_n ∈ A_1-n ∪ A_2-n}. Therefore, S is a meet semilattice. □