Describing the correlations between metamodels and transformations aspects

Abstract

Metamodels are a key concept in Model-Driven Engineering. Any artifact in a modeling ecosystem has to be de ned in accordance to a metamodel prescribing its main qualities. One of the most important artifact is model transformation that are considered to be the heart and soul of MDE and as such advanced techniques and tools are needed for supporting the development, quality assurance, maintenance, and evolution of model transformations. Several works propose the adoption of metrics to measure quality attributes of transformation without considering any metamodel aspects. In this paper, we present an approach to understand structural characteristics of metamodels and how the model transformations depend on corresponding input and target metamodels.

Keywords

• Model Driven Engineering
• metamodeling
• metamodel metrics
• transformation metrics

1 Introduction

Metamodels are a key concept in Model-Driven Engineering [ 22 ]. Almost any artifact in a modeling ecosystem [ 13 ] has to be de ned in accordance to a metamodel, which represents an ontological description of application domains [ 10 ]. Metamodels are important because they formally de ne the modeling primitives used in modeling activities and represent the trait-d'union among all constituent components. One of this components are model transformations (MT), in fact MT play a key role since they permit to bridge di erent abstraction levels by automatically mapping source models to target ones. In [ 23 ] model transformations are considered to be the \heart" and \soul" of MDE and as such they require to be treated in a similar way as traditional software artifacts [ 2 ]. Understanding common characteristics of metamodels, how they evolve over time, and what is the impact of metamodel changes throughout the modeling ecosystem is key to success. Several approaches have been already proposed to analyse models [ 20 ] and transformations [ 3,28 ] with the aim of assessing quality attributes, such as understandability, reusability, and extendibility [ 7 ]. Similarly, there is the need for techniques to analyse metamodels as well in order to evalutate their structural characteristics and the impact they might have during the whole metamodel lifecycle especially in case of metamodel evolutions. To this end, some works propose the adoption of metrics for analysing metamodels [ 17,19 ] and transformation [ 28 ] as typically done in software development by means of object-oriented measurements [ 16 ]. Starting from our previous work [ 11 ], we are interested in better understanding metamodel characteristics and how metamodels and transformations are correlated by investigating the correlations of di erent metrics applied on a corpus of more than 450 metamodels and 90 transformations. On one hand we propose an approach for a) measuring certain metamodeling aspects (e.g., abstraction, inheritance, and composition) that modelers typically use; and b) for revealing what are the common characteristics in metamodeling that can increase the complexity of metamodels hampering their adoption and evolution in modeling ecosystems [ 13 ]. On other hand we propose an approach for identifying how the transformations are correlated to metamodels. The identi ed correlations permit to draw interesting considerations e.g. how a model transformation is typically structured depending on the considered metamodels, and how does the complexity of metamodels has an impact on the overall model transformations development. Such considerations can be preparatory to further analysis that are very common in software development [ 9 ], e.g., estimating the e ort required to develop model transformations by considering the structural characteristics of the source and target metamodels.

The paper is structured as follows: Section 2 describes the process we have conceived and applied to analyze metamodels. Interesting correlations are discussed in Section 3. Section 4 discusses related work and Section 5 concludes the paper and draws some research perspectives. 2

The correlation among metamodels and transformations

Software metrics have been proposed to assess and predict software e ort and quality [ 15 ] and recent research has proposed the adoption of metrics to measure transformations. In particular, metrics on transformations have been investigated [ 28,3 ] to support the measurements of model transformations with the aim of understanding transformations via quantitative evaluations. For instance, in [ 28 ] speci c metrics have been conceived to measure ATL transformations, and in [ 4 ] authors de ne the meaning of several quality attributes in the context of model transformations and align them to a set of metrics.

The adoption of metrics to measure metamodels has been recently proposed in [ 17,19,12 ]. In particular, in [ 17 ] authors apply object-oriented measurements to understand common structural characteristics of metamodels, whereas [ 19 ] proposes a measuring mechanism for assessing the quality of metamodels. To the best of our knowledge, none of the existing approaches calculate transformation metrics with the aim of correlating them.

Since it is reasonable to claim that the complexity of model transformations is somehow related to that of the source and target metamodels, in our opinion in order to have a complete measurement of model transformations, it is necessary to identify also possible correlations between transformation and metamodel metrics e.g., to gure out at what extent the number of matched rules of given ATL transformation depends on the number of metaclasses in the source and/or target metamodels.

To this end, in this section the measurement process shown in Fig. 1 is presented. In particular, the rst step of the process consists in applying a number of metrics on a representative corpus of transformations and corresponding metamodels. Afterwards the calculated metamodel and transformation metrics are correlated among them by using statistical tools. Finally, the collected data are analysed in order to cross/link structural characteristics of transformations and metamodels, e.g., how the di erent kinds of ATL rules (i.e., matched, lazy, and called) are typically used. It is important to remark that in the analysis step, metamodel metrics are also considered in order to identify possible correlations among transformation and metamodel metrics (e.g., how the number of metaclasses in the target metamodel impacts the structural characteristics of transformations in terms of number of matched rules, helpers, etc.). In [ 12 ], we describe the process, shown in 1, we have applied to identify linked structural characteristics and to understand how they might change depending on the nature of metamodels. In this work we have extended this process in order to calculate di erent set of metrics from di erent artifacts (metamodels and transformations) and to understand how the model transformations are dependent from corresponding input and target metamodels. 2.1

The proposed measurement process

The rst step of the proposed process consists of the application of metrics on a data set of metamodels and transformations. Concerning the applied metrics on metamodels we borrowed those in [ 17 ] and added new ones by leading to a set of 28 metrics. Due to space limitations, in the rest of the paper we consider only the metrics shown in Tab. 1 for metamodels and shown in Tab. 2 form transformation. The corpus of the analyzed metamodels and transformations has been obtained by retrieving artifacts from di erent repositories, i.e., EMFText Zoo [ 6 ], ATLZoo [ 5 ], Github, and GoogleCode. To perform such analysis we have automatize the process for metrics calculation using a eterogenous repository called MDEForge presented in [ 8 ]. The calculated data are exported in CSV les encoding the values of all the calculated metrics. Generating CSV les enables the adoption of statistical tools like IBM SPSS, Microsoft Excel, R and Libreo ce Calc for subsequent analysis of the generated data. 2.2

Calculation and selection of metrics correlations Correlation is probably the most widely used statistical method to detect crosslinks and assess potential relationships among observed data. There are di erent techniques and indexes to discover and measure correlations. In the following we overview the Pearson's and Spearman's coe cients that we have considered in this paper to measure the correlations among calculated metamamodel metrics.

The Pearson's correlation coe cient [18A] was deveBlopedC by Karl DPearson from a related idea introduced by Francis Galton in the 1880s. It is widely used in the sciences as a measure of the degree of linear dependence between two variables. In particular, the Pearson correlation coe cient is appropriate when it is possible to draw a regression line between the points of the available data (e.g., see the diagrams A and B in Fig. 2).

The Spearman's correlation coe cient [ 24 ] was used by Charles Spearman in the 1900s in the psychology domain. This coe cient is better than Pearson to manage situations when there is a monotonic relationship between the considered variables. For instance, in the cases shown in the diagrams C and D in Fig. 2, the Pearson coe cient would wrongly identify a very low correlations among the considered data. This is due to the fact that the assumption of linear relationships required by Pearson is not satis ed. Contrariwise, Spearman's correlation index would perform better in cases of monotonic relationships as in the diagrams C and D in Fig. 2

It is also important to note that A B C D the assumption of a monotonic relateiaornsrheliaptiiosnlsehsispre(astnriactsisvuemtphtaionnatlhinat- Fig. 2. Examples of scattered plots has to be met by the Pearson correlation). For this reason, we use Spearman only for highlighting curvilinear correlations. Finally, both Pearson's and Spearman's correlation indexes assume values in the range of -1.00 (perfect negative correlation) and +1.00 (perfect positive correlation). A correlation with value 0 indicates that there is no correlation between two variables. In order to assess the strength of correlations it is possible to consider the guide that Evans [ 14 ] suggests for the absolute value of the correlation indexes, i.e., [0.0,0.19 ] very weak, [0.20,0.39 ] weak, [0.40,0.59 ] moderate, [0.60,0.79 ] strong, and [0.80,1.0 ] very strong.

Metamodel metrics correlations Once the metamodel metrics have been calculated, the most correlated ones are identi ed and selected. In particular, we have calculated the Pearson's correlation indexes for all the values of the metamodel metrics. The outcome of this operation is a correlation matrix as the one shown in Fig 3. The discussion is based on the correlation matrix shown in Fig 3 and by considering the most interesting correlations having value greater than 0.60 (thus strong or even very strong). Because of lack of space it is not possible to discuss all the identi ed correlations that include the metrics shown in Table 1 and 2. However, interested readers can refer to the spreadsheet available online1 containing all the obtained results. For instance, the number of MC2 (number of metaclasses) is strongly correlated with the number of CMC 1 http://www.di.univaq.it/ludovico.iovino/data-mise2015.html 2 For the complete list of acronyms in the table we refer to [ 11 ] (number of concrete metaclasses) as testi ed by their Pearson's correlation index having value 0.997.

#MC #AMC #CMC #IFLMC #SF #ASF #TCWS #MGHL #MHS

LNS #CMC #SF #ASF #MGHL #MHS LNS Model transformation and metamodel metric correlations The interesting part of our analysis relies on correlating model transformation and metamodel metrics. To this end a correlation matrix based on the Spearman's index has been calculated and a fragment is shown in Fig 4. The matrix relates model transformation metrics with metrics calculated on the corresponding source and target metamodels. For instance, according to the calculated matrix, the number of output patterns (OP) of a model transformation is strongly related with the number of metaclasses (MC) contained in the output metamodel. MC AMC CMC SF MC AMC CMC SF

B

IP

OP

TR

MR

LR

CR

RWF

RWD

H

HWC

HNC

CRT about how the constructs of the ATL language are typically used by developers. Moreover, by considering the correlations of both transformation and metamodel metrics (see Section 3.2), further considerations can be drawn about how structural characteristics of metamodels a ect the structure of the corresponding model transformations. 3.1

Metamodels correlation analysis

In this section we brie y present the most representative metrics and correlations we have discovered in this process. We present the metrics correlation discussing the meaning and highlighting the results in the graphical representation. How the number of metaclasses is related to the adoption of abstraction constructs In this section we discuss how the size of metamodels expressed in terms of number of metaclasses is related to the adoption of abstraction constructs, i.e., abstract metaclasses, and supertypes.

In particular, as shown in Fig. 5 the number of metaclasses (MC) and the number of those with su- 100 per types (MCWS) are strongly correlated (with Pearson index 0.99). 80 More speci cally, when the number of metaclasses grows, typically also the MGHL 60 number of classes with supertypes in- MHS creases. In other words, as expected, the adoption of inheritance is propor- MCWS 40 tional to the size of metamodels expressed in terms of number of meta- 20 classes. Interestingly, metamodel delsceihgvineelsesr.isnTpsthreeiasfedirsottofeaasdtdiddiensdigblbniynegwFsihigni.ehr5aiertrchaharyt- lFevige.l 50.0 Analyz5Ni0unmgber omf meett1a0ac0lamsseos (dMCe)l 15a0bstrac2t0i0on shows the values of the MHS (Max Hierarchy Sibling) and MGHL (Max generalization hierarchical level) metrics. Such conclusions are con rmed by the Pearson correlation indexes between MC and MHS (0.70) and the one between MC and MGHL (0.66). Finally, Fig. 5 reveals that in metamodels with at most 50 metaclasses, i) the number of supertypes in hierarchy is in between 0 and 20, ii) the number of siblings in a hierarchy is in between 0 and 10, and iii) the maximum height of a hierarchy is in between 0 and 5. These data represent a pattern charactering the typical typical metamodel de nition.

How structural features are used with hierarchies This section aims at comprehend how structural features are used in presence of class hierarchies. To this end, we can consider the average number of features (ASF) and the total number of metaclasses with supertypes (MCWS) metrics. Even though the correlation index of these two metrics is low, according to the matrix in Fig 3, the Spearman approach permits to identify a greater correlation index. As shown in Fig. 6 it is evident that ) 12 increasing the number of metaclasses FSA cbwleaitrshsodfseusctprreeuarctstyeupsr.eaMsl, ofetrahetoeuvreaervs,earinnagiaentmenrueetmsat---    li(trsscaaaeenu 1860 ing statistical result obtained by con-  lfra sidering the correlation between the ttrcuu 4 MC and ASF metrics is that by con-  rsgae 2 sidering metamodels having the num- veA 0 ber of metaclasses in the range be- 0 50Number 1o0f0 class wit1h5 0a super t2y0p0e (MCWS2)50 300 tween 1 and 50 , the average num- Fig. 6. Analyzing structural features introber of features (excluding the inher- duction in hierarchies ited ones) of a metaclass ranges between 1 and 5.

How the number of featureless metaclasses is related to hierarchies height The correlation between the number of metaclasses with supertypes (MCWS) and the number of concrete metaclasses without features (IFLMC) is interesting for understanding how specializations of metaclasses can introduce or reduce structural features in metamodels. To this end, MCWS and 10 IFLMC are strongly correlated as supported by the 8 r0Pe.e8laa9rt0si.oonnT'hsiseinsehdoeexwctnhoaifvnisnuFgcihgv.acl7our3e-.  I)(LLFgoCCM64 In particular, by increasing the number of metaclasses 2 bweitrh osfupmeretatyclpaesss,esthweitnhuomut- 0 0 2 4 Log (MCWS) 6 8 10 attributes or references in- Fig. 7. Analyzing hierarchical height and featureless creases too. This means that metaclasses when hierarchies are introduced, usually existing features are subject to refactoring operations. Usually, what is done is to move them to super classes and to create leaves in the hierarchies inheriting features from the super types. This is in line with the typical usage of hierarchies for factorizing common aspects in superclasses. 3.2

How metamodel characteristics a ect model transformations By exploiting the matrix obtained by correlating transformation and metamodel metrics, in this section we discuss how metamodels a ect the development of 3 This scattered plot diagram use date logarithmic scale for empathize the correlation model transformations. The discussion is based on the correlation matrix shown in Fig. 4 and by considering the most interesting correlations having value greater than 0.65.

How transformation rules are in uenced by target metamodels This aspect can be investigated by considering the correlation between the number of metaclasses in the target metamodel (OUT MC) and the number of TR (Transformation Rules). Such two values are correlated because of the Spearman's index having value 0.746. The graph in Fig 84 represents how these two values are in uenced by each other in our corpus. According to the graph it is evident that increasing the num- Fig. 8. How TR are in uence by number of MC in ber of the MC in the target target metamodel metamodel the number of TR increases too. This is generally true, since the transformation writing is outputdriven when the developer tries to cover all the metaclasses of the target metamodel. We can also state that the common concentration in the corpus is in the range between 1 and 20 metaclasses and 1 and 15 transformation rules, again con rming the declarative style of transformation as common choice of the developers.

How the structural features in the target metamodel in uence the number of bindings According to the calculated Spearman correlation, the structural features (SF) of the target metamodel can in uence the number of bindings (B) written in the rules of the transformations. The plot in Fig 94 shows that increasing the value of SF in the output metamodel (OUT SF), the number of binding grows too. The distribution is common for the number of SF between 0 and 20 distributed for Fig. 9. How the SF in the target metamodel in uthe value of B that goes from ence the number of B 1 to about 75. 4 The scattered plot diagram use date logarithmic scale for empathize the correlation How the total number of output patterns are in uenced by the target metamodels According to the calculated matrix the Spearman's correlationindex between the value of OP (Output Patterns) in the rules and the number of metaclasses in the target metamodels has value 0.783. This correlation occurring in our corpus is depicted in Fig 104 where the value of OP in the rules of our transformations increases at the raising of the value of MC in the target metamodels. The most dense concentration is in the range Fig. 10. How OP are in uenced by the target metaof 1-10 output patterns and 1- models 10 metaclasses in output.

How the total number of input pattern are in uenced by the source metamodels As anticipated in the previous sections the IP (Input Pattern) of the transformations are related to the value of MC in the source metamodel (IN MC). This is con rmed by the Spearman's correlation that results 0.692. In the graph in Fig 114 the distribution is less clear than the previous case but the trend is similar: increasing the value of MC in input, the value of IP increases too. This again con- Fig. 11. How IP are in uenced by the source metarms the use of declarative models style as the preferred one in our corpus.

4 Related works

In [ 28 ] the authors introduces metrics to measure ATL transformations and the adoption of metrics to measure quality attributes of transformation without considering any metamodel aspects. In other approaches the main topic is the quality attribute driven by the metric [ 4 ], for example making the quality of model transformations measurable. In [ 25 ] the authors have focused on transformation model measurements in order to better understand transformations via a quantitative evaluation, like the declarative factor of modules and rules. In [ 27 ] an analogous approach for measuring model repositories is shown, simply considering models in the evaluation. The authors in [ 26 ] investigate factors having impact on the execution performance of model transformations and they extracted metrics for the analysis. Van Amstel et al. propose a set of six quality attributes to evaluate the quality of model transformations [ 1 ]. All cited works propose the adoption of metrics to measure quality attributes of transformation without considering any metamodel aspects. The authors of [ 21 ] worked on how model transformations can improve the quality of models using metrics. A similar approach for understanding structural characteristics of metamodels and their relationships has been presented in [ 11 ]. Williams et al. in [ 17 ] is the rst one to discuss metrics related to a large metamodel collection exposing how metamodels are commonly structured, and how they evolve over time.

5 Conclusions and future work

In this paper, we proposed a number of metrics which can be used to acquire objective, transparent, and reproducible measurements of metamodels and transformations. The rst goal is to better understand the main characteristic of metamodels, how they are coupled, and how they change depending on the metamodel structure. We have also proposed an approach to analyze model transformations by considering also the corresponding metamodels. The approach relies on the correlation of di erent metrics and has been applied on a corpus of 450 metamodels and 90 transformations and permitted to draw interesting considerations that we intend to extend in the future. ASF CMC

IFLMC LNS MC

MCWS

MGHL MHS SF

Appendix

Name Description

Number of abstract MetaClass Number of metaclasses that cannot be instantiated in models

Average Structural Features Average number of attributes and references in a metaclass

Number of concrete MetaClass Number of metaclasses that can be directly instantiated

Number of concrete Immediately The number of concrete metaclasses that have no attributes

Featureless MetaClass or references, but may inherit features from a superclass

Isolated metaclasses It is the percentage of metaclasses that are not connected with any other one Number of total MetaClass Number of metaclasses in the metamodel (MC = AMC + CMC)

Number of class with a super Number of metaclasses having at least one super type type

Maximum generalization hierar- Maximum hierarchical depth in the metamodel chical level

Max Hierarchy Sibling

Maximun number of classes inheriting from a generic superclass

Number of structural features Number of attributes and references in the metamodel

Table 1. Some of the used metrics for measuring metamodels Name Description

Number of bindings Number of bindings in all output pattern

Number of Input Pattern The metric number of input pattern elements measure the size of the input pattern of rules. Note that since called rules do not have an input pattern, the metric number of input model elements does not include called rules.

Number of Output Pattern The metric number of output pattern elements measure the size of the output pattern of rules.

Number of Transformation Rules A measure for the size of a model transformation is the number of transformation rules it encompasses. In ATL, there are di erent types of rules, viz., matched rules, lazy matched rules, unique lazy matched rules, and called rules.

Number of Matched Rules (Ex- Number of matchad rule exzcluding lazy matched rule. If cluding Lazy Matched Rules) this matrics are equals to number of transformation rule the transformation are de ned completely declarative

Number of Lazy Matched Rules Number of lazy rule including unique (Including Unique)

Number of Called Rules Number of Called Rules

Number of Rules with a Filter Number of rules with a lter condition on the input pattern.

Condition on the Input Pattern The input pattern has a condition. This implies that not all model elements in the source model may be transformed.

Number of Rules with a do Sec- ATL allows the de nition of imperative code in rules in a

tion do block. This can be used to perform calculations that do not t the preferred declarative style of programming. To measure the use of imperative code in a transformation, we de ned number of rules with a do section

Number of Rules with a using ATL allows the de nition of local variable in a rule. This

clause can be used to perform calculations that do not t the preferred declarative style of programming. To measure the use of imperative code in a transformation, we de ned number of rules with a using clause

Number of Helper Number of total helper in the transformation

Number of Helpers with Context Number of helper with context in the transformation

Number of Helpers without Con- Number of helper without context in the transformation text

Number of Calls to resol- The resolveTemp() function is used to look-up references

veTemp() to non-default output elements of other rules. Therefore, it is to be expected that model transformations with a large number of calls to the resolveTemp() function are harder to understand.

Table 2. Some of the used metrics for measuring transformations

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