Several formalisms can be used to express variability in a product line. Product comparison matrix is a common and simple way to display variability of existing products from a same family, but they lack of formalisation. In this paper, we focus on concept lattices, another alternative already explored in several works to express variability. We first propose a method to translate a description from existing product comparison matrices into a concept lattice using Formal Concept Analysis. Then, we propose an approach to represent the case where a product family is described by other product families with interconnected lattices using Relational Concept Analysis. Because of the combinatorial aspect of these approaches, we evaluate the scalability of the produced structures. We show that a particular structure (AOC-poset) possesses interesting properties for the studies that we envision.
In product line engineering [
Among existing formalisms, the most common is Product Comparison
Matrices (PCMs). PCMs describe product properties in a tabular form. It is a simple
way to display features of products from a same family and to compare them.
However, there is no existing format or good practices to design these PCMs.
Therefore, cells in PCMs lack of formalisation and it is difficult to perform
automatic and efficient processing or analysis on them [
Formal Concept Analysis (FCA) and concept lattices have already been
studied to express variability [
Concept lattices formal and structural aspects make them good candidates to apply automatic or manual processing including product comparison, research by attributes, partial visualisation around points of interest, or decision support. But the exponential growth of the size of concept lattices can make them difficult to exploit. In this paper, we will study the dimensions of these structures and try to find out if depicting product line variability with concept lattices provides structures that can be exploited from a perspective of size. For this, we will build in a first phase concept lattices from existing descriptions. Because there are abundant and focus on both products and features, we will extract descriptions from PCMs. Besides, we can find PCMs that possess a feature whose value domain corresponds to a set of products described by another PCM. It is a special case where a product family is described by another product family. In a second phase, we will model this case by interconnecting concept lattices obtained from the descriptions of these two PCMs using Relational Concept Analysis.
In this paper, we want to answer these questions: How can we represent the variability expressed by a PCM with a concept lattice? To what extent can we model the case where a product family is described by another product family with RCA? Can we efficiently exploit structures obtained with FCA and RCA with regard to their size?
The remainder of this paper is organised as follows. In Section 2, we will review Formal Concept Analysis and propose a way to build a concept lattice from a PCM’s description. In Section 3, we will study how to represent the case where a product family is described by another product family with Relational Concept Analysis. Then, in Section 4, we will apply these two methods on Wikipedia’s PCMs to get an order of magnitude of the obtained structure size. Section 5 discusses related work. Section 6 presents conclusion and future work.
This section proposes an approach to represent with a concept lattice the variability originally expressed in a PCM. 2.1
Formal Concept Analysis [
A formal context is a 3-tuple (O, A, R) where O and A are two sets and R ⊆ O × A a binary relation. Elements from O are called objects and elements from A are called attributes. A pair from R states that the object o possesses the attribute a. Given a formal context K = (O, A, R), a concept is a tuple (E, I) such that E ⊆ O and I ⊆ A. It depicts a maximal set of objects that share a maximal set of common attributes. E = {o ∈ O|∀a ∈ I, (o, a) ∈ R} is the concept’s extent and I = {a ∈ A|∀o ∈ E, (o, a) ∈ R} is the concept’s intent. Given a formal context K = (O, A, R) and two concepts C1 = (E1, I1) and C2 = (E2, I2) from K, C1 ≤ C2 if and only if E1 ⊆ E2 and I2 ⊆ I1. C1 is a subconcept of C2 and C2 is a superconcept of C1. When we provide all the concepts from K with the specialisation order ≤, we obtain a lattice structure called a concept lattice.
We represent intent and extent of a concept in an optimised way by making elements appear only in the concept where they are introduced. Figure 7 represents a concept lattice having simplified intent and extent. We call object-concept and attribute-concept the concepts which introduce respectively at least an object or an attribute. In Figure 7, Concept 7 and Concept 2 introduce neither attributes nor objects: their simplified intent and extent are empty. If they are not necessary, we can choose to ignore these concepts. Attribute-Object-Concept poset (AOC-poset) from a formal context K is the sub-order of (CK ,≤) restricted to object-concepts and attribute-concepts. Figure 4 presents the AOC-poset matching the concept lattice from Figure 7. In our case, interesting properties of concept lattices with regard to variability are preserved in AOC-posets: this smaller structure can be used as an alternative to concept lattices. 2.2
A PCM describes a set of products from a same family with variable features in a tabular way. Figure 1 presents a PCM which describes four products against two features, taken from Wikipedia.
We notice that the cells of this PCM lack of formalisation: in this, different values have the same meaning (Object-Oriented and OO ) and it seems that there are no rules on the use of value separator (’,’ or ’&’ or ’and’).
If we want to automatically build concept lattices from this kind of descriptions,
we need to make the cell values respect a format in order to extract and process
them. In [
We clean each type of cells as follows. Empty cells are not a problem, even though they indicate a lack of information. Inconsistent values should be detected, then clarified or erased. Unknown values and extra informations will be simply erased. Other types of cells could have either one single value or a list of values. We will always use a coma as value separator. Values with the same meanings will be written in the same way. Once we have applied these rules on a PCM, we consider that this PCM is cleaned. A cleaned PCM can own three types of features: – simple boolean, – constrained boolean, – non-boolean.
Since automatic process is difficult on original PCMs, we clean them manually. When we clean the PCM in Figure 1, we obtain the PCM in Figure 2.
We can now automatically extract values from cleaned PCMs to generate formal contexts. Given a set of objects and a set of binary attributes, a formal context is a binary relation that states which attributes are possessed by each object. In summary, we want to convert a set of multivalued features (PCM) into a set of binary attributes (formal context).
Scaling technique [
because of empty cells, the fact that an object does not possess an attribute could also mean that the cell from the PCM was left blank. To be more precise, we can choose to generate two attributes: one to indicate that the object possesses the feature, and one to indicate that the object does not possess the feature. Constrained boolean features can be processed in the same way than simple boolean features by producing one or two attributes, with the difference that we can keep constraints in the form of attributes. Non-boolean features will simply produce an attribute per value or group of values. We applied scaling technique on the cleaned PCM of Figure 2 and got the formal context in Figure 3. e g a u g n a
L
Php C# x :sedY :odN teedn irzead irzead iloan lrau itev i-rO tandS tandS tcnuF rceodP Ireapm jtcebO
x x x x x x x x x x x x
x x x x The structure of concept lattices and AOC-posets permits to highlight interesting properties from variability point of view: they classify objects depending on their attributes and emphasise relations between these attributes (e.g. require, exclude). For instance: attributes introduced in the top concept are owned by all objects; attributes which are introduced in the same concept always appear together; if an object o1 is introduced in a sub-concept of a concept introducing an object o2, o1 possesses all the attributes of o2 and other attributes; two objects introduced in the same concept possess the same attributes. Feature models show part of this information, mainly reduced to relations between attributes, as they do not include the products in the representation.
Figure 7 represents the concept lattice from the formal context of Figure 3. Figure 4 represents the AOC-poset from the formal context of Figure 3. In these two structures, we can see that: all languages permit to write programs according to functional and imperative paradigms; the product Php has all the attributes of Perl in addition to the attribute Object Oriented ; all standardised languages are object-oriented.
This section proposes an approach to model the case where a PCM possesses a feature whose value domain corresponds to a set of products described by another PCM. 3.1
We illustrate the modeling of interconnected families of products with an extension of our example. We can find on Wikipedia a PCM on Wikisoftwares that refers to Programming Languages: we want to structure wikisoftwares according to programming languages in which they are written. We assume a PCM about wikisoftwares that owns a boolean feature Open Source and a constrained boolean feature Spam Prevention. We applied Section 2 approach and obtained the formal (objects-attributes) context in Figure 5. Figure 8 presents the concept lattice associated with the context in Figure 5.
Relational Concept Analysis (RCA) [
We want to express the relation isWritten between objects of Wikisoftwares and objects of Programming languages. TeamPage and EditMe are written in Java, SocialText in Perl, DokuWiki in Php and Mindtouch in both Php and C#. We link each wikisoftware with corresponding programming languages in an objects-objects context, and we present it in Figure 6.
isWritten Java Perl Php C#
Given an objects-objects context Rj = (Ok, Ol, Ij ), there are different ways for an object from Ok domain to be in relation with a concept from Ol. For instance: an object is linked (by Ij ) to at least one object of a concept’s extent (existential scaling); an object is linked (by Ij ) only to objects of a concept’s extent (universal scaling). For each relation of R, we specify which scaling operator is used.
In RCA, objects-attributes contexts are extended according to objects-objects contexts to take into account relations between objects of different sets. For each objects-objects context Rj = (Ok, Ol, Ij ), RCA extends the objects-attributes context of the set of objects Ok by adding relational attributes according to concepts of the lattice associated with the objects-attributes Ol. Each concept c from Ol gives a relational attribute q r :c where q is a scaling operator and r is the relation between Ok and Ol. A relational attribute appears in a lattice just as the other attributes, with the difference that it can be considered like a reference to a concept from another lattice.
As shown on the example, data are represented in a Relational Context Family (RCF), which is a tuple (K, R) such that K is a set of objects-attributes contexts Ki = (Oi, Ai, Ii), 1 ≤ i ≤ n and R is a set of objects-objects contexts Rj = (Ok, Ol, Ij), 1 ≤ j ≤ m, with Ij ⊆ Ok × Ol. Given an objects-attributes context K = (O, A, I), we define rel(K) the set of relations (objects-objects contexts) of R which have O for domain, and ρ a function which associates a scaling operator to each objects-objects context of R. For each step, we extend the context K by adding relational attributes from each context of rel(K): we obtain the complete relational extension of K. When we compute the complete relational extension of each context of K, we obtain the complete relational extension of the RCF.
RCA generates a succession of contexts and lattices associated with the RCF (K, R) and ρ. In step 0, RCA generates lattices associated with contexts of K. K0 = K. In step e + 1, RCA computes complete relational extension of the Ke contexts. The obtained extended contexts (Ke+1) possess relational attributes which refer to concepts of lattices obtained in the previous step.
In our example, the RCF is composed of Wikisoftware, Programming Language and of the objects-objects context isWritten. When we compute the complete relational extension of this RCF, we extend the objects-attributes context of Wikisoftware with relational attributes which refer to each concept of Programming language. Figure 7 and Figure 8 represent lattices of Wikisoftware and Programming language at step 0. Figure 9 presents the concept lattice from the extended objects-attributes context of Wikisoftwares, at step 1. In this example, we cannot go further than step 1.
In Figure 9, relational attributes are read like references to concepts in the lattice of Programming languages at step 0 (Figure 7). An extended concept lattice gives us the same kind of informations that are emphasised in a basic concept lattice, but it takes into account attributes from other product families. This brings a new dimension to research and classification of products from a same family. In our example, it permits us to select a wikisoftware depending on the paradigm of its programming language. In Figure 9, we can read for instance: – DokuWiki (concept 1) is written in a programming language characterised by concepts 6, 7, 8, 3, 5 and 0 of Programming languages, corresponding to attributes Standardized:No, Procedural, Functional, Object Oriented and Imperative; – Team Page and EditMe are equivalent products because they are introduced in the same concept (same attributes and same relational attributes); – a wikisoftware not Open Source is written in a standardised language; – an Open Source wikisoftware is written in an unstandardised language. 4
Until now, we proposed a first method to obtain a concept lattice from a PCM’s description and a second method to depict the case where features possess their own PCM with interconnected lattices. In this section, we evaluate these two methods on existing PCMs from Wikipedia to get an order of magnitude of the obtained structures size.
The number of generated concepts from a formal context depends on the number of objects, the number of attributes and the form of the context: this number can reach 2min(|O|,|A|) with a lattice, and |O| + |A| for an AOC-poset.
In the following tests, we generate both concept lattices and AOC-posets to emphasise the impact of deleting concepts which introduce neither attributes nor objects on the size of the structure. Each test was made twice, a first time with full formal contexts (scaling technique giving two attributes for each boolean feature, and keeping constraints in the form of attributes for constrained boolean features) and a second time with reduced fomal contexts (scaling technique giving one attribute for each boolean feature). 4.1
Firstly, we analysed 40 PCMs from Wikipedia without taking into account relations between products. These 40 PCMs were converted into formal contexts with the method of Section 2. Results are presented in Figure 10. We have analysed 1438 products, generated 4639 binary attributes and 26002 formal concepts.
Most of the concept lattices possess between 50 and 300 concepts, but some of them can reach about 5000 concepts: this number is very high, and it would be difficult to quickly process these data. Reduced contexts (blue plots) give smaller structures, but some of them remain considerable. Thus, results of AOC-posets are encouraging: the highest number of concepts obtained is 161. Most of them possess between 30 and 60 concepts. 4.2
Secondly, we made the same type of tests on structures which have been extended with a relation, using method of Section 3. We wanted to realise these tests on a quite important number of relationships. Yet, it is simple to find PCMs on Wikipedia but it is more difficult to automatically find relationships between these PCMs. To simulate relations between PCMs we used in the first test,
we choose to automatically generate random objects-objects contexts based on two existing relations we found on Wikipedia. We analysed these relations and found out that about 75% of objects are linked to at least another object and that 95% of linked objects are linked to a single other object. We formed 20 pairs of objects-attributes contexts and generated object-objects contexts for each pair according to our analysis.
Results are presented in Figure 11. Concept lattices are huge (some can reach 14000 concepts) whereas AOC-poset remain relatively affordable (about 200 concepts).
These results match with the products used in a very simplified form for illustrating the approach. In real data, the first existing relation is between LinuxDistribution (77 objects, 25 features) and FileSystem (103 objects, 60 features). With a lattice, we obtain 1174 concepts (full context) and 1054 concepts (reduced context). With an AOC-poset, we obtain 188 then 179 concepts. The second relation is between Wikisoftware (43 objects, 12 features) and Programming Language (90 objects, 5 features). With a lattice, we obtain 282 concepts (full context) and 273 concepts (reduced context). With an AOCposet, we obtain 87 and then 86 concepts. All the datasets (extracted from
Lattices (F) Lattices (R) AOC-posets (F) AOC-posets (R) 2263.5 878.2 89.95 72.55 793 194 82 61.5 61 24 33 16 14083 7478 201 189 359.25 108.25 67.5 47.25 2719.5 597.75 101.25 87.25 wikipedia and generated) are available for reproducibility purpose at: http: //www.lirmm.fr/recherche/equipes/marel/datasets/fca-and-pcm.
According to these results, we can deduce that the concept lattices obtained possess a majority of empty concepts (which introduce neither attributes nor objects): AOC-poset appears like a good alternative to generate less complex structures, and keeps variability informations in the same way that concept lattices. These results on product description datasets, that are either real datasets, or datasets generated with respect to existing real one profile, allow us to think that is is realistic to envision using FCA and RCA for variability representation within a canonical form integrating features and products. 5
To our knowledge, the first research work using Formal Concept Analysis to
analyse relationships between product configurations and variable features to
assist construction of a product line has been realised in Loesh and Ploederer
[
In the domain of product lines, another category of works is interested in
identification of features in source code using FCA [
Finally, a last category of works study feature organisation in FM with FCA.
Some approaches [
In this paper, we analyse the feasibility of using Formal Concept Analysis and Concept Lattices as a complement to Product Comparison Matrices to represent variability in product lines. We propose a method to convert a description from a product comparison matrix to a concept lattice using the scaling technique. We also propose a way to model the special case where a product family is described by another product family with Relational Concept Analysis. We obtain interconnected lattices that bring a new dimension to research and classification of products when they are in relation with other product families.
Subsequently, we realise series of tests to determine an order of magnitude of the number of concepts composing the structures obtained firstly with FCA by converting PCMs into formal contexts, and secondly with RCA by introducing relations between these contexts. In these tests, we compare two structures: concept lattices which establish a sub-order among all the concepts and AOCposets which establish a sub-order among the concepts which introduce at least an attribute or an object. It seems that most of the concepts do not introduce any information and AOC-poset appears like a more advantageous alternative, in particular for presenting information to an end-user. Concept lattices are useful too, when they are medium-size, and for automatic data manipulations. We also show the effect of two different encoding of boolean values.
In the future, we will use the work of [