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2012 | Buch

On the Mathematics of Modelling, Metamodelling, Ontologies and Modelling Languages

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Computing as a discipline is maturing rapidly. However, with maturity often comes a plethora of subdisciplines, which, as time progresses, can become isolationist. The subdisciplines of modelling, metamodelling, ontologies and modelling languages within software engineering e.g. have, to some degree, evolved separately and without any underpinning formalisms.

Introducing set theory as a consistent underlying formalism, Brian Henderson-Sellers shows how a coherent framework can be developed that clearly links these four, previously separate, areas of software engineering. In particular, he shows how the incorporation of a foundational ontology can be beneficial in resolving a number of controversial issues in conceptual modelling, especially with regard to the perceived differences between linguistic metamodelling and ontological metamodelling. An explicit consideration of domain-specific modelling languages is also included in his mathematical analysis of models, metamodels, ontologies and modelling languages.

This encompassing and detailed presentation of the state-of-the-art in modelling approaches mainly aims at researchers in academia and industry. They will find the principled discussion of the various subdisciplines extremely useful, and they may exploit the unifying approach as a starting point for future research.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
Modelling (and hence metamodelling as a special kind of modelling) focusses on linking symbolic representations to elements within a specific part of reality. For most software engineering models, we delimit this reality as the ‘system under study’ or SUS for short, while recognising that, while Hesse (2006) argues against its use, here we use the word system in a generic sense to mean the ‘original’ or ‘target’ of our modelling exercise. This target may be (a part of) reality or an artificial system (e.g. conceptual or design models)
Brian Henderson-Sellers
Chapter 2. Mathematics for Modelling
Abstract
In the following subsections, we introduce the mathematical formalisms that underpin modelling, metamodelling and ontologies. Basic to a mathematical formalism are set theory and morphisms. First we consider, in Sect. 2.1, how set theory can represent modelling concepts. In Sect. 2.2, we look at mappings (functions) between pairs of sets–whether such a mapping is one-to-one, onto or both can be critical in understanding many of the ‘problems’ identified in the software engineering modelling literature
Brian Henderson-Sellers
Chapter 3. Models
Abstract
A model is “an abstraction of reality according to a certain conceptualization” (Guizzardi 2005) and is therefore a set of statements about some ‘system under study’ (SUS) (Seidewitz 2003)—although note that Hesse disputes with Kühne the use of the word ‘system’. Statements can be shown to be either true or false (see also Gonzalez-Perez and Henderson-Sellers 2007).
Brian Henderson-Sellers
Chapter 4. Metamodels
Abstract
A metamodel has been defined as an explicit specification of an abstraction expressed in a specific language. This is similar to Seidewitz’s (2003) definition of a metamodel as a specification model for a class of SUSs, where each SUS is no longer a part of reality but is itself a valid model, i.e. “a metamodel makes statements about what can be expressed in the valid models of a certain modelling language”. It should be stressed that the abstraction needed here is the F-abs kind of abstraction since it creates type rather than token models.
Brian Henderson-Sellers
Chapter 5. Ontologies
Abstract
Aßmann et al. (2006, p. 253), quoting Gruber (1993), state that an ontology is a formal specification of a shared conceptualisation. Since it is both an abstraction and a simplification of some reality (domain of interest), an ontology is also a model (e.g. Nirenburg 2004; Guizzardi 2005, p. 6; Aßmann et al. 2006, p. 256) but one that is descriptive, domain-relevant and static as opposed to a system-focussed model, which does not require any shared understanding nor does it model the whole of the domain (Fig. 5.1). It is widely stated that, while an ontology uses an open-world assumption, a model uses a closed-world assumption (e.g. Aßmann et al. 2006; Atkinson et al. 2006) although some ontologies are based on a closed-world assumption (e.g. Wang et al. 2006).
Brian Henderson-Sellers
Chapter 6. Modelling Languages
Abstract
For a (conceptual) model to be communicated to others, it is necessary to represent it in some way. In much of software engineering and conceptual modelling, such a representation is through the application of a graphically based modelling language (ML)—a language appropriate for each of the three domains shown in Fig. 1.10. Such an artificial language has many of the attributes of a natural language in terms of possessing both syntax and semantics.
Brian Henderson-Sellers
Chapter 7. Linking Models, Metamodels, Ontologies and Modelling Languages
Abstract
In his investigation of the correspondences of terminology in the metamodelling and the ontology subdisciplines, Henderson-Sellers (2011b) concludes, as here, that there is a correspondence between a domain-specific ontology and the method domain in Fig. 1.​10 (roughly OMG layer M2 in the architecture of Fig. 1.​9). This ontology must be represented by a language, where this language may itself be defined by a metamodel (OMG M2 or ISO Metamodel Domain). This is named, variously, meta-ontology, ontology specification language, foundational ontology or higher level ontology (see Fig. 5.​7 and previous discussion). Such a language provides reasoning support not possible in software modelling languages such as UML or ER, which many researchers are tempted to use and/or extend for this purpose.
Brian Henderson-Sellers
Chapter 8. Other Related Work and Further Discussion
Abstract
Several authors discuss styles of SUSs, application domains and the strengths and weaknesses of models (e.g. Mellor et al. 2004; Hoppenbrouwers et al. 2005b; Jackson 2009; Hoppenbrouwers and Wilmont 2010). Selic (2003) proposes five characteristics for all engineering models: abstraction, understandability, accuracy, predictiveness and inexpensive. Seidewitz (2003) and Muller et al. (2009) examine how model truth can be ascertained from both forward-looking and backward-looking models. Henderson-Sellers (2011a), based on an overview of conceptual modelling, concludes with recommendations for enhancing research programmes in quality assessment of conceptual models (including metamodels). Since these are not strictly mathematically-related, we have deemed such discussions out of scope for this present discussion but encourage its adoption in a more mathematical format in future research. Furthermore, although model transformations are a key idea (e.g. Bézivin 2004) for model-driven engineering (MDE), including the OMGs MDA (OMG 2003), they are consequent on the mathematics presented here but are not discussed further in this book.
Brian Henderson-Sellers
Chapter 9. Conclusions and Further Work
Abstract
Although much has been analyzed regarding the relationships between models, metamodels, ontologies and modelling languages, without a mathematical underpinning, these analyses can be readily influenced by the imprecision of the natural language (usually English) used in their exposition. The introduction of simple set theory mathematical representations can aid researchers in understanding better the links between the various elements of the modelling scene and, especially in our context, how it all relates to modelling in software engineering.
Brian Henderson-Sellers
Backmatter
Metadaten
Titel
On the Mathematics of Modelling, Metamodelling, Ontologies and Modelling Languages
verfasst von
Brian Henderson-Sellers
Copyright-Jahr
2012
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-29825-7
Print ISBN
978-3-642-29824-0
DOI
https://doi.org/10.1007/978-3-642-29825-7