=Paper=
{{Paper
|id=Vol-2019/multi_3
|storemode=property
|title=Developing an Ontological Sandbox: Investigating Multi-level Modelling’s Possible Metaphysical Structures
|pdfUrl=https://ceur-ws.org/Vol-2019/multi_3.pdf
|volume=Vol-2019
|authors=Chris Partridge,Sergio de Cesare,Andrew Mitchell,Frederik Gailly,Mesbah Khan
|dblpUrl=https://dblp.org/rec/conf/models/PartridgeCMGK17
}}
==Developing an Ontological Sandbox: Investigating Multi-level Modelling’s Possible Metaphysical Structures==
https://ceur-ws.org/Vol-2019/multi_3.pdf
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. This is also common in conceptual modelling for the frameworks but also identify possible ontologically
contexts, where typically only the transitive reduction is driven improvements.
modelled, and the transitive closure is rarely if ever mandatory.
In some contexts, the hierarchy is restricted to a tree-structure,
whereas in others it is more usual to allow lattice (multiple
inheritance) hierarchies. Submitting these different structures
to a sandbox analysis would capture their different
commitments in derived constructors (typically using filters)
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