Multilevel modeling: What's in a level? A position paper

Abstract

Multi-Level Modeling (MLM) conceptualizes software models as layered architectures of sub-models whose elements are inter-related by the instance-of relation, which is radically di erent from the traditional two level representations that consist of a class model and an instance model, where the former represents the schema and the latter the population. Key to the MLM representation is the notion of a level, which is the subject of this position paper. We argue that a theory of MLM must have a distinguished notion of level, which should have both a syntactic and a semantic status. Moreover, the notion of a level must represent a class model, and the overall MLM theory must be compositional, being composed from the theory of class models.

Introduction

Multilevel system modeling (MLM) views the enterprise as a layered collection of models that are inter-related by the instance-of (or membership) relation among objects and classes. The grounds for this approach are both philosophical and pragmatic. On philosophical grounds, researchers have been arguing that faithful modeling of real world domains cannot be restricted by the standard two-layer architecture of the OMG meta-modeling approach. They claim that natural domains have multiple levels of classi cation, and an arti cial restriction to two layers yields models that are too weak [ 3, 14 ]. On pragmatic grounds, researchers have argued that a multilevel architecture of models simpli es the management and evolution of complex software [ 4, 10, 12 ].

Classes and associations in adjacent layers in the multilevel architecture can be related by the standard instance-of relationship, while within each layer, classes are organized via subclass and general association relationships. A multilevel model consists of multiple, possibly overlapping, ontological dimensions, each being a multilevel organization of classes and associations, as discussed above. A potency mechanism, introduced in [ 2,11,19 ], is a kind of an inheritance mechanism between levels.

E orts to formulate multilevel modeling has led to a variety of approaches, including a category-theory-based framework [ 19 ], a UML/OCL formulation [ 13 ], multilevel objects and relationships [ 17, 18 ], an axiomatic speci cation of multilevel types [ 9 ], a set-theoretic formulation [ 5 ], and more. Most approaches have some form of an instance-of relation that stands for set membership, and of subtyping relationship that stands for set inclusion.3

In [ 1 ] the authors claim that a multilevel framework can be either level blind, i.e., enable multilevel arbitrary classi cation structures, as in Telos [ 16 ], F-Logic [ 15 ] and FOML [ 6 ], or level adjuvant, i.e., recognize the utility of levels, as in the aforementioned frameworks. In these approaches, a level is a numeric attribute de ned for all elements in a multilevel model, which enables layering elements based on their level value. Yet, in most approaches the concept of a level has no independent syntactic or semantic status.

The main theme of this position paper is the claim that a multilevel theory must have a distinguished notion of level with both a syntactic and a semantic status. Moreover, levels should re ect the traditional notion of a class model, and the overall theory should be compositional with respect to class models, i.e., be constructed|syntactically and semantically|from the theory of class models. 2

Multilevel Formalisms

In this section we analyze several published approaches to MLM with respect to the following three questions: (1) Does the formalism include a clear and well-de ned syntax and semantics for multilevel models (the latter meaning wellde ned notion of legal instances)? (2) Does the formalism include a distinguished syntactic and semantic notion of a level ? (3) Is the formalism compositional with respect to the class model language? The category-theory-based formulation of [ 19 ]. This approach formalizes the MLM approach of [ 2, 11 ] using category theory. It supports a syntactic de nition of layered class models, extended with a rich potency mechanism that enables cross-level constraints and cross-level instantiation (termed here single potency ). The paper does not show how subtyping and standard class model constraints, like association classes, GS-constraints, aggregation/composition and inter-association constraints are handled.

With respect to the above three questions: (1) this formalism provides a formal de nition of multilevel models, with deep constraints and cross-level instantiation. It is unclear, however, what is the semantics of a multilevel model, i.e., what is the set of legal instances. For example, what is a legal population of a model at some level, say, L3? How it is related, if at all, to the syntactic instances in level L2? (2) This formalism includes a distinguished syntactic notion of level : The syntactic elements of a level form a valid instance of the next higher level, that functions as its class model. Yet, the semantic status of a level is left unclear. (3) The formalism is syntactically compositional. The UML/OCL constraint-based approach of [ 13 ]. This paper suggests formulation of MLM using UML/OCL. The idea is to join metamodels of several levels 3 Some languages that support these relationships are Telos [ 16 ] and the executable logic language FOML [ 6 ], which is derived from Flogic [ 15 ]. into a single model, using a concept of superstructure, which is modeled using generalizations, associations and UML/OCL constraints.

With respect to the above three questions: (1) Syntax and semantics are given by UML/OCL; (2) elements are assigned a level, but there is no syntactic unit that combines all elements with a common level value, and assigns it a semantic status; (3) the formalism is not compositional.

The multilevel objects and relationships approach of [ 17, 18 ]. This approach suggests a representation that is based on constructs that integrate views at di erent abstraction levels. An overall model consists of interrelated clabject hierarchies, in which clabjects are classi ed into di erent abstraction levels. The representation is formulated and implemented in F-logic [ 15 ].

With respect to the above three questions, the approach is similar to the previous one: (1) syntax and semantics are given by the de nition of abstraction hierarchies and the F-Logic translation; (2) elements are assigned to an abstraction level, but there is no syntactic/semantic unit of level; (3) the formalism is not compositional.

The FOL axiomatization of multilevel domains in [ 9 ]. This paper presents an axiomatic FOL speci cation of a domain of types. The domain is structured in linearly ordered levels, starting from an initial level of individuals. A level is a set of all types whose elements belong to a lower level.

Concerning our three questions: (1) the paper deals only with axiomatization of sets of instances at each level and their types; (2) there is a distinguished semantic notion of level; (3) the compositionality issue does not apply, since this approach lacks a concrete language for MLM.

The set-theoretic MLM theory in [ 5 ]. The formal executable theory for MLM is based on set-theoretic semantics of class models [ 7,8 ]. Class models are combined to form ontological dimensions, which are the backbone of a multilevel model. The formal theory has a direct semantics for multilevel models, which accounts for complex model interactions, including deep constraints and inference. Legal instances are de ned using Herbrand instances in which the syntactic symbols comprise the semantic domain. The MLM theory is seamlessly embedded in a provably correct way into the executable logic language FOML [ 6 ].

Regarding our three questions, the answers are all positive, owed in part to the distinguished syntactic and semantic status of the concept of a level. A level is represented by a schema, which is a pair of an instance model and a class model. The instance model is syntactically included in the class model, using a mapping, called a Herbrand mediator, of objects to classes, links to associations, and property renaming. This enables renaming of properties and classes between levels, which is absolute necessity for practical applications of MLM. In addition, the instance model of a level is required to be a partial instance of the class model in the next higher level. This partiality is important for proper semantics of potency, but we have no room to expand on this issue here.

The semantics of a multilevel ontological dimension de nes a legal instance of a syntactically correct dimension as a sequence of legal instances of the class models in the schemas, where (1) an instance of the class model in level i must include the instance model in level i 1 (which is a partial instance of this class model); and (2) satisfy all cross-level constraints in higher level schemas. Therefore, the formalism has a well-de ned syntax and semantics, a distinguished notion of level, and is compositional with respect to class models. 3

On the Necessity of Level-based Compositional Theory for Multilevel Modeling Much of the literature on MLM is plagued by a confusion between true multilevel modeling in which levels correspond to class models with each level being a partial instance of the next, and modeling that is simply based on formalisms with higher-order features where classes are uniformly treated as objects that can be members of other (meta)classes (like in F-logic [ 15 ], RDF, Telos [ 16 ], etc.). With respect to the approaches discussed in Section 2, an example of the former kind of true MLM is [ 5 ], while [ 13 ] is an example of the latter kind of approaches (which, in our view, are not truly MLM). Approaches in the middle of the spectrum, which attempt to include the concept of a level (with partial or no semantics), are [ 9, 19 ]. [ 17, 18 ] has a level notion that prevents compositionality with respect to class models.

We claim that MLM is much more than just a class model with higher order features and constraints. It is a collection of levels bound together by the instance-of relation, where each level is a class model, not just a numeric attribute. Besides the instance-of relations, inter-level mediation includes terminology changing mappings that permit renaming of concepts. Thus, levels should be expressed both syntactically and semantically as an integral part of the structure. A compositional4 MLM theory on top of class models turns MLM into a natural extension of the class model in two ways: (1) a class model is a degenerate case of a simple MLM; (2) a composite multilevel model consists|syntactically and semantically|of class models and the theory of MLM should reuse the theory of class models. Thus, in [ 5 ] we de ne consistency, nite satis ability and optimization of MLM as natural extensions of their class model variants, and show that intra-layer correctness and optimization can be checked using class model algorithms. 4 By compositionality we mean that the semantics of MLM consists of the semantics of the class models of the layers, properly stitched together by inter-level relations.

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