Despite being one of the fundamental concepts of multi-level modeling - to the extent of occurring in the name of the discipline - the concept of “level” has no universally agreed upon meaning among multi-level modeling researchers. There is no consensus on what the nature of a level is nor on how levels should be used to organize modeling elements. In this paper, I aim to initiate a discussion on what the options for defining levels in multi-level modeling are and how they could be systematically characterized.
Multi-level modeling has demonstrably matured from a research idea to an approach
of practical significance. The complexity-reducing properties of multi-level modeling
[
However, there is no discipline-wide consensus on what the term “Level” in
“MultiLevel Modeling” represents. The majority of approaches align their “level” concept
with the notion of a classification stratum [
An investigation into the differences of these various interpretations of “level” and what their impact on multi-level modelers is indicated for a number of reasons: i) level underpinnings should be made explicit in order to support an informed growth of approaches and to avoid misunderstandings and unwarranted debates between proponents of different approaches. ii) a potential consolidation of ideas would lend more strength to particular schools of thought. iii) trade-off analyses regarding the impact on multi-level modelers should give the latter a way to choose the optimal approach for their application. iv) leveraging the sanity-checking ability of some level-based well-formedness constraints has significant potential to reduce errors in modeling.
Regarding the last aspect, Brasileiro et al. have shown that taxonomic hierarchies in
Wikidata contain an alarming amount of statements that are inconsistent with each
other [
Considering the aforementioned reasons, I therefore could not agree more with
Almeida who stated “it is paramount for multi-level modeling as a discipline to
investigate the guiding notion of ‘level’” and with his proposed five point investigation
into the nature of levels [1, Sec. 3.2]. Due to space constraints, I will only consider
simple linear levels schemes, in particular those used for the ontological dimension of
multi-level modeling approaches. The treatment of !-levels [25] and/or spanning levels
[
The classic OMG four-layer architecture (see Fig. 1(a)) has been a popular subject of study regarding metamodeling principles and also serves an example for how drastically an initially nondescript layer or level hierarchy may change its appearance upon closer examination. Figure 1(a) shows the original depiction of the four-layer architecture that M3 M2 M1 M0
(a) M2 M2 M1 W
(b)
used three instance of arrows of the same kind. A closer examination, however, revealed
that three different relationships are involved, of which one (connecting the system
under study with the model stack proper) is a representation relationship that often is
not related to classification at all [
The first metamodeling principle that aimed at introducing sanity-ensuring rules
was the so-called strict metamodeling doctrine [
Note that the so-called orthogonal classification architecture ([
In general, the rationale for employing levels is to achieve a grouping of elements that share some commonality with each other. For instance, in the case of the OMG’s fourlayer architecture, all elements within a particular layer belong to one language definition or represent usages of one language. For an explicit description of a level-based organization scheme it is therefore helpful to explicitly state a level cohesion principle which characterizes why elements are grouped with other elements in the same level. 3.1
Level Cohesion Principle
Cohesion between elements in a level reflects a notion of semantic proximity. For
instance, as mentioned before, Henderson-Sellers and Gonzalez-Perez proposed a level
scheme whose level cohesion principle is stakerholdership. For schemes that are
ordersynchronized (e.g., MLT [
A respective grouping of elements implies a segregation of elements (into their disjunct levels) which is considered helpful for – maintaining an overview by systematically organizing elements, and – detecting potentially problematic relationships by recognizing them as “level-crossing”.
If a level scheme is constructed such that one would only expect one kind of
relationship – e.g., instance of for order-based schemes – or only few kinds of relationships
that are related in nature ([
From the above it follows that an alternative way to describe a level-based organization scheme is to explicitly state its level segregation principle which characterizes how elements in adjacent levels are related to each other. Note that while it may initially appear as if a segregation principle and a cohesion principle for the same level hierarchy were just dual formulations of each other, I will later show that this is not the case. 3.2
Level Segregation Principle
In a survey conducted ahead of Dagstuhl seminar 17492, 16 participants (out of 18
respondents) indicated in their response to the question “What is multi-level modeling?”
that the segregation principle used in multi-level modeling is abstraction [
Within order-based level schemes there is still room for variability regarding the
placement of elements in such level hierarchies. Even when a so-called level-respecting
scheme ([
LS1: element.order = element.level, or
LS2: element.order element.level [
I refer to LS1 as order-synchronized and LS2 as order-aligned.
It is important to observe that all aforementioned level schemes imply the same relative level differences between elements. The relative level distance ( level) between related elements – as measured by the difference in level values between elements in the same classification branch – is always the same, independently of the choice of LS1 or LS2. The latter, i.e., order-aligned schemes, simply enable elements to be optionally shifted up the level hierarchy to any desired height.
As a result, it is adequate to regard order-synchronized schemes as reflecting logical classification strata in an absolute sense, whereas the contents of order-aligned schemes are always locally equivalent to that of order-synchronized schemes but additionally support varying absolute localization.
Due to the fact that order-synchronization is simple, easy to formally capture, and references a tried and tested principle of organization, to which order-alignment represents a variation that has hitherto not been explicitly motivated, I will focus the following discussion on a comparison of these two order-based level schemes. 4
Since both order-synchronization and order-alignment use the same level segregation
principle (classification), they both support the same level of sanity-checking, i.e., are
equivalent with respect to the kinds of anti-pattern (as employed by Brasileiro et al. [
However, order-synchronization and order-alignment differ in the level cohesion principle they use. In order-synchronization the cohesion and segregation principles are duals of each other. Any element that is segregated from a reference element is treated as being in cohesion with all other elements that are segregated from the reference element in the same way (here, that have the same level distance to the reference element). Likewise, any two elements that are viewed as being cohesive, are segregated to all other elements in the same manner respectively (here, by the same level distance).
In order-alignment, two cohesive elements need not have coincidental segregation
properties. Figure 3 illustrates an advantage of the relaxed rules of the order-alignment
scheme. In this scenario, which features two classification hierarchies – one for
activities and one for products (cf. [
Note that an order-synchronized allocation of the elements in Fig. 3 would have required BobDesigns to reside at level zero, thus creating an association that would have crossed a level boundary. As discussed before, a potential advantage of level-based schemes is that they allow potentially problematic relationships to be easily recognized by the fact that they cross level boundaries even though they are not of the kind that gives rise to the level segregation principle. With respect to cases like BobDesigns and BobsObject – in which an alignment of separate classification hierarchies can be easily achieved by shifting them against each other – order-aligned schemes thus are able to reduce the number of false positives.
This desirable property of order-aligned schemes can be justified by elaborating their associated level cohesion principle as grouping elements that share semantic proxl e v e l ≤ r e d r o l e v e l ≤ r e d r o level = 2 order = 1 level = 1 order = 0
duration1: Integer
duration = 50
validated = True
BobsObject validated = True o r d e level = 1 r
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order = 1 l
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level = 0 re=
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imity – here in the sense of participating in relations as implied by associations –
without causing a violation of a strict (cf. [
In summary, order-aligned schemes allow the same sanity-checking capabilities as order-synchronized schemes but support a more accommodating level cohesion principle. On the one hand, the implicit level allocation of elements in an order-synchronized scheme relieves modelers from making placement decisions. On the other hand, the level hierarchy only reflects the segregation principle at the expense of not accommodating a richer cohesion principle. While order-alignment may require manual vertical adjustments to local classification ensembles, the placement of elements could be largely automated by choosing placements that minimize the overall sum of levelcrossing relationships.
The optimization principle has to be one of minimization as opposed to complete removal, since it is unfortunately not the case that order-aligned schemes can avoid all non-problematic relationships from crossing level boundaries. In Fig. 3, BobDesigns also entertains a relationship with BobsObject which has to reside on a different level in order to respect its local order-level-locking ( order = level) with respect to BobsClass. This means that the association between BobDesigns and BobsObject has to cross a level boundary, making it a “false positive” in the aforementioned sense.
Atkinson and Ku¨hne have argued that diagrams exhibiting such “non-strict”
scenarios can be regarded as the superimposition of two locally strict diagrams [
It seems plausible that the perimeters of modeling spaces can be straightforwardly defined based on connected classification ensembles, but the exact details and the empirical validation of the assumed advantages of order-aligned schemes over ordersynchronized schemes remains future work.
A2 A1 A0
P2 P1 P0
Dagstuhl seminar 17492 working group “Formal Foundations and Ontology Integration” concluded that fundamental notions such as the organization of elements into levels, and the question as to which sanity-checking approaches are expedient, still require further investigation in order to support an explicit understanding and a solid platform for coherent multi-level modeling frameworks [1, p. 23, p. 33].
This paper aimed at taking steps towards addressing these important questions. In particular, Almeida’s question: “What does it mean for an entity to be in a ‘level’?” [1, Sec. 3.2] can now be answered in two ways based on the framework developed in this paper. One way to answer the question is to state how an entity is segregated from other entities in adjacent levels. The second way to answer the question is to state the level cohesion principle which groups the entity with other entities in the same level.
I maintain that multi-level modeling research would benefit from each approach having both its level segregation principle and its level cohesion principle explicitly formulated. For instance, such an explication would avoid criticizing sanity-checking strictness schemes for their apparent inflexibility when their rules are being applied to level schemes whose level segregation principle is simply not amenable to strictness enforcing rules. Any multi-level modeling approach with a level segregation principle that is akin to generalization will certainly not benefit from enforced strictness. However, it is undoubtedly the case that order-based level schemes can significantly benefit from strictness schemes by effectively trivializing the detection of a whole class of antipatterns.
This paper showed that there is merit in separating a level segregation principle from a level cohesion principle because there are useful examples where these principles are not simply duals of each other. Instead, a level cohesion principle may enjoy a constructive cohabitation with a level segregation principle, enriching the latter. I discussed the particular case of order-alignment, which has some interesting advantages over order-synchronization that deserve further investigation.
I maintain there is promise in identifying more anti-patterns whose detection would reveal serious inconsistencies and illogical constellations in multi-level models. Such anti-patterns could then inform further constraints to be used for tightening level membership principles. However, until such anti-patterns are identified within scenarios enabled by order-alignment schemes (compared to order-synchronized schemes) it appears to be justified to not prematurely tighten level membership rules. Acknowledgments This paper draws considerably from numerous insightful debates with Colin Atkinson. I would furthermore like to thank all members of the “Formal Foundations and Ontology Integration’ group of Dagstuhl Seminar 17492 for the stimulating discussions. 25. Mylopoulos, J., Borgida, A., Jarke, M., Koubarakis, M.: Telos: Representing knowledge about information systems. Information Systems 8(4), 325–362 (1990) 26. Neumayr, B., Schuetz, C.G., Jeusfeld, M.A., Schrefl, M.: Dual deep modeling: multi-level modeling with dual potencies and its formalization in f-logic. Software & Systems Modeling pp. 1–36 (2016) 27. Rossini, A., de Lara, J., Guerra, E., Nikolov, N.: A comparison of two-level and multi-level modelling for cloud-based applications. In: Proceedings of ECMFA 2015. pp. 18–32. LNCS 9153 (2015)