Developing an Ontological Sandbox: Investigating Multi-Level Modelling’s Possible Metaphysical Structures Chris Partridge Sergio de Cesare Andrew Mitchell BORO Solutions Ltd, Henley on Thames University of Westminster BORO Solutions Ltd University of Westminster, London, UK London, UK Henley on Thames, Oxfordshire, UK partridgec@borogroup.co.uk s.decesare@westminster.ac.uk mitchella@borogroup.co.uk Frederik Gailly Mesbah Khan Faculty of Economics and Business Tullow Oil plc. Administration London, UK Ghent University mesbah.khan@tullowoil.com Ghent, Belgium frederik.gailly@UGent.be Abstract— One of the central concerns of the multi-level modelled [1]. It has been recognized that these levels are modelling (MLM) community is the hierarchy of classifications sometimes ontological [2] and, where they are, that this that appear in conceptual models; what these are, how they are introduces constraints, such as anti-cyclicity. There have also linked and how they should be organised into levels and been attempts to characterise classification and how it differs modelled. Though there has been significant work done in this from generalisation [3-5] that include consideration of their area, we believe that it could be enhanced by introducing a different ontological natures. Though there has been significant systematic way to investigate the ontological nature and work done in this area, we believe that it could be enhanced by requirements that underlie the frameworks and tools proposed introducing a systematic way to investigate the ontological by the community to support MLM (such as Orthogonal nature and requirements that underlie the levels and so inform Classification Architecture and Melanee). In this paper, we introduce a key component for the investigation and the frameworks and tools proposed by the community to understanding of the ontological requirements, an ontological support MLM (such as Orthogonal Classification Architecture sandbox. This is a conceptual framework for investigating and and Melanee). comparing multiple variations of possible ontologies – without In the long term, we aim to provide support for the having to commit to any of them – isolated from a full investigation and understanding of the ontological commitment to any foundational ontology. We discuss the requirements and so guide the design of ontologies including sandbox framework as well as walking through an example of those used in MLM frameworks and tools. In this paper, we how it can be used to investigate a simple ontology. The example, introduce a key component of this, an ontological sandbox. despite its simplicity, illustrates how the constructional approach can help to expose and explain the metaphysical structures used This is a conceptual framework for investigating and in ontologies, and so reveal the underlying nature of MLM comparing multiple variations of possible ontologies – without levelling. having to commit to any of them – isolated from a full commitment to any foundational ontology. The sandbox helps Keywords— ontological sandbox; constructional ontology; to expose and explain the metaphysical structures of the ontological space; ontogenesis; multi-level modelling; candidate ontologies, the differences between them as well as generalisation; classification suggesting potential alternatives. This makes it useful for assessing different architectural choices. I. INTRODUCTION Here we aim to provide a sketch of the sandbox based upon One of the central concerns of the multi-level modelling a constructional approach outlined in [6, 7]. We discuss the (MLM) community is the hierarchy of classifications that sandbox framework as well as walking through an example of appear in conceptual models; what these are, how they are how it can be used to investigate a simple ontology. The linked and how they should be organised into levels and example, despite its simplicity, illustrates how the constructional approach can help to expose and explain the here will be constructional in the Finean sense). In these metaphysical structures found in ontologies, and so reveal the ontologies, objects are accepted into the ontology on the underlying nature of MLM levelling. It illustrates how grounds that they are constructed from elements already in the metaphysical choices [8] guide the construction of the ontology ontology; where a constructor is applied to the constructees to and how understanding the way metaphysical structures can produce constructs. An ontology can also contain given or vary helps to guide the investigation of possible ontologies. basic elements that are just accepted. We look at how to use these metaphysical structures to Hence an ontology can be divided into three domains (see derive, and so help to explain the kinds of levelled structures frequently found in conceptual modelling; including taxonomic structures such as the Linnaean hierarchy and component TABLE I. ONTOLOGY DOMAINS (SEE [6]) breakdowns common in engineering. This reveals a common Acronym Domain Names for Members underlying foundation for the levelled structures that suggests B basis domain bases, givens, basic elements or there may be a wider application for MLM techniques. given elements C constructor constructors We look at the MLM structures themselves. We use the domain example as a foundation for characterising the family of formal E constructed constructs or constructed structures associated with MLM classification and domain elements generalisation. We show how understanding the way they are derived from fundamental structures helps to explain what they Table I). are and how different derivations give rise to the variety of members of the family. II. THE SANDBOX’S UNDERLYING FRAMEWORK The basis and constructed domains combine to form a domain whose members are called elements. All three domains The sandbox adopts the algebraic constructional combine to form the ontology’s universe, whose members are ontological framework outlined by Fine in [6] and further called items. developed, with a focus on wholes and parts, in [7]. Fine sees the advantage of his framework is that it naturally reveals the The core theory uses ontological principles to show that underlying metaphysical structure of reality. As an example, he one can generate all the constructed elements (the constructed comments on levels: “there is an intuitive distinction between domain) from the basis and the constructor domains. As Fine wholes which are like sets in being hierarchically organised notes, this means we do not need to use the constructed and those which are like sums in being ‘flat’, or without an domain, E, to characterise an ontology, we can just use the internal division into levels. The distinction, under the couple < B, C>; E can then be generated from B and C. operational approach, can be seen to turn on whether repeated Though the order of analysis may well be the opposite; where applications of the operation are capable of yielding something one starts with the elements and works out what the new” [7]. (One can see the same flat/hierarchy distinction constructors are and so the bases. made in the literature between generalisation and classification This generation of E relies upon an exhaustive application – a topic we return to below). In the same paper, he talks about of the constructors; where anything that can be generated is its power and beauty; its ability to provide a single and elegant generated. In our approach, we find it useful to ‘construct’ this account of a variety of structures. One cannot see this in the process. We call this the ontology’s ONTOGENESIS and logical characterisation to these hierarchies, such as [9]. This characterise the ontology as the couple < B, C > plus makes it a better tool for the task of investigating the ONTOGENESIS; so, in a sense, ONTOGENESIS stands in for metaphysical structure of possible ontologies as well as the E. ontological content of MLM’s levels. Fine’s extended theory is about how ontologies fit into an Here we outline, with some minor clarifications, the ontological space; where this is a nonempty collection of relevant portion of Fine’s framework. This is not intended to be ontologies that conforms to certain principles. It shows how, a detailed exposition; this can be found in Fine’s papers. We given ontologies in a space, similar ontologies with broadly follow Fine’s notation, with some amendments to permutations of basis and constructor domains also exist in the make this short exposition clearer. We divide the outline into space. We extend this to permutations of the individual two sections; the first dealing with the general ontological constructors in the domain. framework and the second with the general composition framework. This provides a framework for an incremental sandbox approach to explaining and understanding an ontology. If one A. Finean General Ontological Framework wishes to understand an ontology (the target), these principles Fine’s general framework has two theories; a core theory allow one to initially pick out from the ontological space about ontologies and an extended theory for ontological spaces ontologies that contain just the basis domain or just a single containing ontologies. constructor from the target ontology and examine these. One can then pick and examine richer combinations, seeing how Fine’s core theory deals with constructional ontologies. these lead to richer structures, until one arrives at the target (From now on we will feel free to drop the qualification ontology. ‘constructional’ from ontology – as all ontologies discussed B. Finean General Composition Framework structure; how wholes are related to parts and by which kind of Using this general ontological framework, in [7] Fine whole-part. However, not every application of a constructor sketches a general unified framework for composition; the constructs new elements; some of these non-generational ways in which one object can be a part of another. constructions map new composition structures (others neither construct new elements or new mappings). Sum provides us It is distinctive in several ways (all of which suit our with a good example of this. If we start with a molecule abc of current purpose). It takes a very liberal notion of part that three atoms as the only given, then a single sum-decomposition encompasses both traditional mereological relations as well as into all its parts will construct new elements; a, b, c as well as others, such as set-membership, that are not usually thought of the molecules ab, ac, and bc. It also give us the composition as whole-part relations; in other words, for Fine set- mapping for abc; the relations between abc and all its parts. If membership in another kind of whole-part. (The term we apply sum-decomposition to ab we get a and b – but these mereological whole-part will be reserved for the traditional are not new, they are in a sense re-constructed as they were mereological relations). The formulation of the framework newly constructed in the first decomposition. However, it gives takes the operation of composition (and decomposition) as us a new composition mapping for ab, something that was not primitive rather than the more familiar relation of whole-part. given in the first decomposition. It also shows, in some sense, These primitive operations are then treated as constructors ab is prior to a and b. To capture these characteristics, we call within the general ontological framework; where each the initial types of construction generative. The second fundamental constructor generates a different kind of whole- decomposition does some work, as it shows that ab is part. As noted in Bennett [10], one key choice is the direction composed of a and b, so we call this type of construction of generation for these composition constructors; whether parts compositional. Earlier we introduced the process generate wholes or the other way around. Standard mereology ONTOGENESIS, which exhausts the ontology’s generative and cumulative set theory generate wholes from parts. Bennett power. However, as the example shows, this may not exhaust offers Schaffer [11] as an example of choosing the other the compositional power. To do this we extend direction. ONTOGENESIS to cover all the possible compositional Fine formulates his framework in terms of compositional relations, and call this ONTOGENESIS+. This is needed to principles; which act like axioms. His first broad division is extract the full power of the constructors. formal and material. Formal principles can be further divided Generative Hierarchies. The generative power of into those that deal with conditions of application and those constructors provides a way of organising elements into levels that provide identity conditions. Material principles provide, within hierarchies. Every constructed element is constructed by for example, conditions for the presence of a whole or part in a sequence of generative constructions. This provides a simple space and time or at a world. way to organise them into a hierarchy: each level is Fine develops a simple way of characterising the formal characterised by the number of times the constructor has been identity principles for summative identity (identity for the applied. The additional compositional constructions are not mereological sum operation) based upon a notion of regular considered here. identity conditions (the reader can find the details in [7]). The One can also see this as a form of ONTOGENESIS, where result is the four CLAP principles in Table II, so-called one is given a START collection and a constructor, and because of their initials. elements are constructed by repeatedly applying the A constructor’s formal identity can be characterised by constructor. One needs to be clear what the constructor is being whether these principles or the counter-principles hold for it – repeatedly applied to. For this it is useful to introduce the one can summarise this into a CLAP profile, with a mnemonic notion of stage and to help define it the notion of generation. A where the appropriate letter is struck through when the counter- generation is all the (new) elements generated at a level–this principle holds. Its application is also characterised by its excludes cases where the element is re-constructed. And a direction of generation. Table III gives examples of stage is all the elements in a level’s generation and all the prior constructors for each CLAP profile and direction. levels’ generations. Then a stage level hierarchy is generated by repeatedly applying the constructor to stages. The composition constructors construct elements – the constructed elements. They also map out the composition TABLE II. CLAP (FORMAL IDENTITY) PRINCIPLES (SEE [7]) C Collapse ∑(x) = x If Collapse holds then any whole composed of a single part is identical to it. L Levelling ∑(… ,∑(x, y, z,...),… ,∑(u, v, w,...),...) = ∑(… , x, y, z,… If Levelling holds then when the parts of whole have parts, these parts’ parts ,… , u, v, w,… , ...) are also parts of the whole. A Absorption ∑( … , x, x, … , … , y, y, … , … ,) = ∑( … , x, … , y, ...) If Absorption holds then the repetition of parts is irrelevant to the identity of the whole. P Permutation ∑(x, y, z, ...) = ∑(y, z, x, ...) (and similarly for all other If Permutation hold then the order of the parts is irrelevant to the identity of permutations) the whole. TABLE III. SOME POSSIBLE FORMS OF COMPOSITION (SEE [7]) Profile Whole Example Parts-to-Wholes Example Wholes-to-Parts Constructor Constructor CLAP Sums SUM-BUILDER SUM-DECOMPOSER CLAP Sets SET-BUILDER SET-DECOMPOSER CLAP Strings STRING-BUILDER STRING-DECOMPOSER CLAP Sequences SEQUENCE-BUILDER SEQUENCE-DECOMPOSER We can make the process a little more formal; we need to BUILDER is a good example. An element is of the kind set if account for the cases where the START collection has multiple and only if it is constructed by SET-BUILDER. One can also elements and there are multiple ways to choose the collections base kinds of whole-parts upon the constructor from which to which the constructor is applied. For this we define an they emerge – so set membership is the kind of whole-part that operation POWER that takes a collection and selects all the emerges from SET-BUILDER; relating the (whole–set) possible sub-collections from this (in our example, we do not constructs with the (parts–members) constructees. Kinds are need to take account of sensitivity to duplications and useful in defining the application scope of the constructor permutations). In the case of sets, there is a connection with the operations; for example, SUM-BUILDER does not apply to set-theoretic powerset axiom. Then, given a constructor, one elements of the kind set – nor does it construct elements of this can create a stage level hierarchy in steps (using constructor kind. and start variables) – where any element that can possibly be In the case of sets, strings and sequences, there is a unique generated from the constructor will be generated at some level - as follows: collection of parts that compose the wholes – the principles merely regulate the permutations and duplicates of the parts Generation 0: start and the levelling of the constructors. In standard mereology, Generation 1: constructor (POWER (Stage 0)) which is based upon the mereological whole-part relation, … sums work in a different way. There are multiple ways in Generation N+1: constructor (POWER (Stage N)) which a whole can be a sum of its parts. Our framework starts with composition (and de-composition) rather than whole-part where N is unbounded and the constructor variable ranges over and this provides us with a way to ensure a unique all the constructors (this is needed where there is more than one decomposition. One can decompose a sum uniquely into a constructor). For this example, we do not need to consider collection of parts. A good illustration of how this would work actual infinite or transfinite recursion. Note that this hierarchy is a universe of mereological atoms. All the wholes that are is mixed in the sense that its component parts can be from sums of mereological atoms would be the sum of a unique multiple levels. collection of these atoms. If one did not want to assume There is a variant hierarchy based upon generations rather mereological atoms, one can take the decomposition to be all than stages - a generation level hierarchy. This is of interest the parts. We adopt this latter approach for the example. because the multi-level hierarchies considered in some multi- III. ANALYSING A SIMPLE SANDBOX EXAMPLE level modelling seem to be of this type – see, for example, the ‘level respecting’ principle in [12] and the discussion of strict We now describe a simple example ontology, called metamodelling in [13]. We look at this in Section 4. (unsurprisingly) SIMPLE. The SIMPLE ontology is intended to illustrate how our approach is useful in explaining the A generation level hierarchy is one where the next level is formal requirements for metaphysical structures in models. generated by applying the constructor to the previous level’s Given this goal, we aim to make the example as simple as generation. Each step of the generation level hierarchy only possible while still being able to illustrate how the structural considers the preceding level’s generation and so ignores any whole-part patterns emerge, with a focus on the hierarchies and earlier levels. This means it does not use the full generative levels (such as generation and stage level hierarchies) within power of the constructor. We can formalise this by replacing the patterns. stage with generation in the earlier process as follows: A. Building Up the SIMPLE Ontology Generation N+1: constructor (POWER (Generation N)) As noted earlier, in the Finean ontology framework, we can Note that this hierarchy is pure (that is, unmixed) in the characterise an ontology in terms of its basis and constructor sense that its component parts at each level are from a single domains, which we now do. Our full example ontology, called level. SIMPLE, has a non-empty basis domain and a constructor domain containing two constructors (SET-BUILDER and C. Framework Clarifications SUM-DECOMPOSER). There are a few points that we clarify as scene-setting for the sandbox example; kinds and unique decomposition. Under our approach, based upon the Finean Extended Theory, we construct an ontological space with ontologies that Constructed elements are linked to the constructor that take us in small incremental steps from the NULL ontology to constructs them; this can form the basis for kinds. SET- the final SIMPLE ontology. This is essentially all the TABLE IV. SIMPLE’S ANALYSIS ONTOLOGICAL SPACE Ontology Acronym Basis Domain BUILDER SET- R DECOMPOSE SUM- Includes Analysed NULL NULL NO NO NO None NO Simple Basis only SB YES NO NO NULL YES SET-BUILDER only SET NO YES NO NULL YES SUM-DECOMPOSER only SUM NO NO YES NULL NO SET-BUILDER and SUM- SET+SUM NO YES YES SET, SUM NO DECOMPOSER Simple Basis plus SET-BUILDER SB+SET YES YES NO SB, SET NO Simple Basis plus SUM- SB+SUM YES NO YES SB, SUM YES DECOMPOSER SIMPLE SIMPLE YES YES YES SB+SET, YES SB+SUM, SET+SUM permutations of basis domain and individual constructors (as SET-BUILDER Only (SET) Ontology. This ontology has Table IV shows). Some of the permutations are not an empty basis domain and a single constructor SET- illuminating, so are not visited in the analysis – these are BUILDER, mentioned earlier, so SET = < (), (SET-BUILDER) marked in Table IV. >. This constructor is introduced in [6] and described in detail in [7]. We mentioned SET-BUILDER earlier in Table IV, One could regard the other ontologies as partial versions of noting it is formally a SET constructor where the direction of the SIMPLE ontology, or subontologies of it. We start the generation is part to whole and its form of identity is CLAP; it analysis with the Simple Basis Only (Sub-)Ontology. works as follows: Simple Basis Only (SB) Ontology. From a metaphysical • It has the associated kind, sets. An element is a set if viewpoint, the choice of bases typically involves important and only if it is constructed by SET-BUILDER. architectural commitments [8]. There are a variety of options. One could start with a basis domain of mereological atoms [14] • It is presented with a collection (possibly empty) of and build up the ontology from them. Or one could adopt elements (parts) and it constructs a set (the whole). priority monism [11], then the given will be everything (which may be a single actual or an infinity of possible worlds) and the • If presented with a collection of zero elements it ontology is built by decomposing this. Choosing one of these generates the empty set, {}. would then dictate the direction of the intended SUM • Its CLAP profile means that if presented with a non- constructor; whether to start with mereological parts and zero collection of elements it generates the set of those construct wholes – or vice versa, start with a mereological elements, ignoring duplicates and order. whole and construct the parts. This provides us with sufficient resources to develop a To keep things simple, we follow the priority monism route good example of generation and stage level hierarchies. Using and have a basis domain consisting of a single object – a the process schema defined earlier, we can create SET’s pluriverse of possible worlds [15], which we will abbreviate as generation level hierarchy in steps as follows: PV. Again, to keep things simple, we adopt super- substantivalism [16], which [17] calls monistic Generation 0: () substantivalism; this considers matter to be identical to the Generation 1: SET-BUILDER (POWER spacetime region is occupies. (Generation 0)) … We can now define the ontology as SB = < (PV), () >. As Generation N+1: SET-BUILDER (POWER there are no constructors in this ontology then (Generation N)) ONTOGENESIS is the NULL process – and PV is the only element (and item) in the ontology. From a hierarchy perspective, this is a limit case. There is a single object which can be regarded (pathologically) as a stage level hierarchy. TABLE V. SET-BUILDER GENERATION LEVEL HIERARCHY Level 0 1 2 3 4 Generation {} {{}} {{{}}} {{{{}}}} Stage {} + {{}} + {} + {{{}}} + {{}} + {} + {{{{}}}} + {{{}}} + {{}} + {} + TABLE VI. SET-BUILDER STAGE LEVEL HIERARCHY Level 0 1 2 3 Generation {} {{}} {{{}}}, {{}, {{}}} Stage {} + {{}} + {} + {{{}}}, {{}, {{}}} + {{}} + {} + The first four levels’ generations and stages are shown in The choice of this constructor naturally complements the Table V. choice of PV as a basis. Given the super-substantival choice for PV, SUM-DECOMPOSER takes a material element and This provides us with an example of the point made earlier, uniquely decomposes it into all its material, spatiotemporal that the generation level hierarchy is not necessarily the whole parts. One can see the constructor as establishing the parts of constructed domain as, at each level, the constructor is only which the whole is composed. applied to the previous generation. For example, at stage 2, the universe contains both generation 1 and 2 elements, but SET- This ontology has a simple two level hierarchy, with PV as BUILDER is only applied to the generation 2 elements. generation 0 and all its parts as generation 1. And only one application of the constructor gives all the constructed Of course, we could take a related constructor elements; the parts of the whole PV. So: ONTOGENESIS GENERATION-SET-BUILDER whose application is (SB+SUM): SUM-DECOMPOSER (PV). ONTOGENESIS+ restricted to generation collections of elements, then the (SB+SUM) needs to consider these parts as wholes and hierarchy would cover the domain. However, we would then establish their parts, and this is achieved by applying sum- have to metaphysically justify the choice of this constructor. decomposer to each of them. We specify this process using an We can reinforce this incompleteness point by generating iterative FOR EACH component process – as follows: the corresponding stage level hierarchy by replacing generation Generation 0: (PV) with stage as noted earlier, giving: Generation 1: SUM-DECOMPOSER (PV) Generation N+1: SET-BUILDER (POWER (Stage N)) Generation 2: FOR EACH X in Generation 1 (SUM- DECOMPOSER (X)) The first three levels are shown in Table VI. The full process exhausts the generative power of the basis domain and The two generations exhaust the compositional power of constructors and so; ONTOGENESIS (SET): SET-BUILDER the constructor. Stage Level Hierarchy. The SIMPLE Ontology. This ontology is Simple Basis Every application of the SET-BUILDER is generative, so plus SET-BUILDER and SUM-DECOMPOSER, so SIMPLE there is no difference between ONTOGENESIS (SET) and = < (PV), (SET-BUILDER, SUM-DECOMPOSER) >. It is a ONTOGENESIS+ (SET). combination of all of the earlier ontologies, where the basis domain and constructors were defined and much of the analysis Simple Basis plus SUM-DECOMPOSER (SB+SUM) done. Ontology. This ontology is an extension of the Simple Basis Only Ontology with the SUM-DECOMPOSER constructor, so; This is the first ontology with multiple constructors. For the SB+SUM = < (PV), (SUM-DECOMPOSER) >. We see from single constructor situations, we had two simple levelling Table IV, that SUM-DECOMPOSER is a SUM constructor schemes based upon the number of applications of the where the direction of generation is whole to part and its form constructor; we combine these for multiple constructors to give of identity is CLAP. It works as follows: us ONTOGENESIS, as follows: • It has the associated kind, material elements. PV is a Generation 0: (PV) material element, all the elements constructed by SUM- Generation 1: SUM-DECOMPOSER (PV) + SET- DECOMPOSER are also material elements and these are BUILDER (POWER (PV)) the only material elements. Generation 2: SET-BUILDER (POWER (Stage 1)) … • It is presented with a single material element (the whole) Generation N+1: SET-BUILDER (POWER (Stage N)) and it constructs a collection of material elements (parts). And we build ONTOGENESIS+ (the full composition • Its CLAP profile means that it generates a collection of map) in the same way as in SB+SUM adding this to generation elements with no duplicates or order. 2: Generation 2+: FOR EACH X in Generation 1 (SUM- DECOMPOSER (X)) TABLE VII. EXAMPLE MULTIPLE CONSTRUCTOR HIERARCHY Level 0 1 2 Generation PV {}, {PV}, p1, p2, … {{}}, {{PV}}, {p1}, {p2}, … The ONTOGENESIS hierarchies do not have the same Generation 0: (set) symmetry as the cumulative set hierarchy. For example, the Generation 1: SUBSET-DECOMPOSER (set) standard cumulative set hierarchy all the singleton sets are at Generation 2: FOR EACH X in Generation 1 (SUBSET- the same level (see SET above). However, as shown in Table DECOMPOSER (X)) VII (where p1, p2, … are the parts of PV), the combination of the two constructors produce, at level 1, not only all the parts One can begin to see the structural similarities between SUBSET-DECOMPOSER and ordinary SUM- of PV but also the empty set and singleton PV – and produces at level 2 singletons of the empty set and the parts of PV as DECOMPOSER noted in [18]. For example, like the SUM- DECOMPOSER constructor, the SUBSET-DECOMPOSER well as singleton-singleton PV. constructor is not levelled (in CLAP terms) and so has a flat This suggests another way to construct hierarchies; by structure. However, one can also see, as [7] notes, that in this calculating the levels using a single constructor. In this case, by space the first is fundamental and the other is derived. only considering the generative applications of SET- BUILDER– ignoring SUM-DECOMPOSER. This recaptures B. Deriving SIMPLE Taxonomies and Component the symmetrical hierarchy. Breakdowns One can derive a new hierarchy by subsumption from a IV. DERIVING STRUCTURES FROM SIMPLE whole-part hierarchy – and this can be levelled in ways that the The example shows how fundamental multi-level original hierarchy is not. Taxonomies, including ones such as hierarchies emerge from the pattern of construction. However, the Linnaean classification, and component breakdowns, in so this is not the only way that these hierarchies, multi-level or far as they are ontological, are good examples. The procedure otherwise, can emerge. The fundamental structures can be used is simple. One chooses a subset of the elements and then a to derive other types of compositions and their associated subset of their whole-part compositions so that the result hierarchies. conforms to the desired whole-part structure; if one wants a levelled structure, then one ensures the structure conforms to A. Deriving SIMPLE’s Subset Constructor the right CLAP principles – typically that it does not adhere to One of these derived types of composition is subset or set- levelling. inclusion. For our purposes, it makes sense to do this using a Consider (one version of) the Linnaean classification derived constructor SUBSET-DECOMPOSER (set) that works hierarchy that has ‘Natural Things’ at the top and is divided as follows: and sub-divided through the levels until reaching ‘Felis leo’ • Its direction of generation is whole to parts. and ‘Felis tigris’ as the bottom. Within the SIMPLE example, ‘Natural Things’ is a set of (spatiotemporally extended) • Its form of identity (like SUM-DECOMPOSER) is CLAP, material elements and the other classifications are subsets of it which means that it generates a collection of elements with [5]. This collection of classifications are the elements used in no duplicates or order. the hierarchy. These can be derived using SUBSET- • It is presented with a single set element (the whole) and it DECOMPOSER and a filter, FILTER-LC, that given a constructs a collection of set elements (the subset parts) – collection selects those elements that are Linnaean hence it can only be applied to elements of kind set and it classifications; LINNAEAN-C: FILTER-LC (SUBSET- constructs elements of the same kind. DECOMPOSER (Natural Things)). Applying this constructor to a set will produce a collection Traditionally, the classification structure is levelled by of all its subsets. This can be regarded as the initial stage in the taking a transitive reduction of the underlying whole-part set-theoretic power set axiom. To formalise this, we specify the hierarchy. We can derive the LINNAEAN-SUBSET- inverse of SET-BUILDER; SET-DECOMPOSER (set) which DECOMPOSER constructor for this by taking a constrained takes a set and produces a collection of its members. Using this version of SUBSET-DECOMPOSER that can only be applied we can specify; SUBSET-DECOMPOSER (set) = SET- to Linnaean classifications and when applied only reruns the BUILDER (POWER (SET-DECOMPOSER (set))). This, next level – other levels are filtered out. This gives us a natural when given a set takes its members and constructs sets from generation level hierarchy LC, which is defined as having them, constructing all its subsets. As with SUM- ‘Natural Things’ as its base and LINNAEAN-SUBSET- DECOMPOSER, one then needs to apply the same operation DECOMPOSER as its sole constructor – and; to each of the parts to establish the full subset composition mapping, as shown below. with their appropriate CLAP profile – as well as their common underpinnings. Similarly, there are varieties of classification. As noted Generation 0: (Natural Things) earlier, [12] introduces the property of ‘level respecting’ in a Generation 1: LINNAEAN-SUBSET-DECOMPOSER way that leads to a similar structure to generation hierarchies. (Natural Things) The appropriate constructor for this shape of structure, Generation 2: FOR EACH X in Generation 1 GENERATION-SET-BUILDER, is similar to SET-BUILDER, (LINNAEAN-SUBSET-DECOMPOSER but with a filter on what it applies to. This analysis of the (X)) structures in terms of how they are derived from more … fundamental constructors helps to both formalise their Generation N+1: FOR EACH X in Generation N differences as well highlighting them. (LINNAEAN-SUBSET-DECOMPOSER (X)), It also gives rise to natural enquires as the motivations for the choices; for example, are they adopted for Of course, the classification structure is richer; which metaphysical/ontological or pragmatic reasons, and if so, what means more constructors are needed. For example, the levels are these reasons? So, for example, it makes clear that adopting (e.g. Kingdoms) are explicit as ranks (this is described in [5]), a GENERATED-SET-BUILDER (as strict metamodelling so a RANKER constructor is required. However, this skeleton does) filters out mixed sets. One can ask what the motivation outline should be sufficient to indicate how this could be done. for this is and whether the cost of excluding them is The same derivation process can be used on other kinds of worthwhile. whole-parts. Component breakdown structures (such as car X breaks down into body and engine components and engine into V. CONCLUSIONS so on) would be based upon mereological whole-part and We have provided an outline of the underlying framework SUM-DECOMPOSER [19]. upon which the ontological sandbox is built and indicated where further details can be found [6, 7]. We have used the This ability to derive levelled hierarchies from the example SIMPLE ontology to show how one can use this fundamental structures not only helps to explain (ontologically) sandbox to build up an understanding of the target ontology in what these hierarchies are but also suggests there may be new steps through ontological space. We have shown how the areas for MLM tools to be deployed. constructional approach exposes the underlying compositional C. Deriving MLM Generalisation and Classification metaphysical structures of ontologies; for example, how, in the Hierarchies SIMPLE ontology at least, the set-inclusion hierarchy is Analyses of classification and generalisation have noted the derived from the fundamental set-membership hierarchy’s similarities with, respectively, set-membership and subset [3]. constructor. We have shown how understanding the CLAP A common point made is that generalisation (like subset) is principles of composition can expose the architectural choices transitive and does not have levels whereas classification (like made and suggest alternatives. We have shown how familiar set-membership) is anti-transitive and has levels [12]. As noted hierarchical structures, such as taxonomies and component earlier, the sandbox analysis both captures these formal breakdowns, can be derived from more fundamental structures. structures and exposes the close relationship between the two We have shown how the same techniques can be applied to (when seen as compositions based upon constructors); that MLM classification and generalisation structures. We have SUBSET-DECOMPOSER is derived from the fundamental provided a clear picture of the trade-off between order and SET-BUILDER. expressiveness that drives the choice of strict metamodelling (effectively a generation level hierarchy) – as well as an However, classification is not plain set-membership nor alternative – stage level hierarchy. Hopefully, this is sufficient generalisation plain subset – and the differences can guide us to provide a good idea of the potential for this sandbox to help on how to build the appropriate derived constructors. One key us understand and improve the ontological underpinnings of difference is that the conceptual models typically restrict their conceptual models. relations to a sub-domain of interest. As with the taxonomies above, this can be captured by a filter on the constructor – This paper provides an outline of the ontology sandbox. A where the filter clearly delineates the scope of the domain. lot more work needs to be done showing how it can be used. One area where we think it will be fruitful to develop the In UML and MLM there are many varieties of approach is investigating the range of possible ontological generalisation. In some MLM contexts, the generalisation commitments of existing MLM frameworks and tools. This hierarchy is, like in taxonomies above, restricted to a transitive could not only suggest possible ontologies (and so semantics) reduction. 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