The Fantastic Combinations and Permutations of Coordinate Systems’ Characterising Options: The Game of Constructional Ontology Chris Partridge BORO Solutions Ltd University of Westminster 0000-0003-2631-1627 Hayden Atkinson CooperVision Ltd 0000-0002-5153-9116 Andrew Mitchell BORO Solutions Ltd 0000-0001-9131-722X Sergio de Cesare University of Westminster 0000-0002-2559-0567 Michael Loneragan QinetiQ Group PLC Portsmouth, Hampshire, UK mjloneragan@qinetiq.com Mesbah Khan OntoLedgy Ltd University of Westminster 0000-0002-1327-6263 … Conway's latest brainchild, a fantastic solitaire pastime he calls “life”. ... The basic idea is to start with a simple configuration of counters (organisms), one to a cell, then observe how it changes as you apply Conway's “genetic laws” for births, deaths, and survivals. ... Conway’s genetic laws are delightfully simple. ... It is important to understand that all births and deaths occur simultaneously. Together they constitute a single generation or, as we shall call it, a “move” in the complete “life history” of the initial configuration. ... Because births and deaths occur simultaneously, newborn counters play no role in causing other deaths and births. … You will find the population constantly undergoing unusual, sometimes beautiful and always unexpected change. ... ― Martin Gardner, The Fantastic Combinations of John Conway's New Solitaire Game “Life”, 1970 [1] Philosophy [nature] is written in that great book whichever is before our eyes -- I mean the universe - - but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth. ― Galileo Galilei, Il Saggiatore, 1623 No one shall expel us from the paradise that Cantor has created for us. ― David Hilbert – Mathematische Annalen 95, 1926, p. 170 -- 1 of 77 -- Abstract: The multi-level modelling community’s raison d'être is its vision of the ubiquity and importance of multi-level-types: the ascending levelled hierarchy of types in conceptual models; starting with types of things, then types of these types, then types of these types of types, and so on. The community both promotes this vision and investigates this hierarchy, looking at how it can be accommodated into existing frameworks. In this paper, we consider a specific domain, coordinate systems’ characterising options. While we recognise that, unsurprisingly, this domain contains a ubiquity of multi-level-types, our interest is in investigating a new and different approach to understanding them. For this we needed to develop a new framework. We devise one focussing on this case, based upon scaling down to simple compositional algorithms (called constructors) to form a new, radically simpler foundation. From the simple operations of these constructors emerges the scaled up multi-level structures of the domain. We show how the simple operations of simple constructors give rise to compositional connections that shape – and so explain – different complex hierarchies and levels, including the familiar multi-level-types and relatively unknown multi-level-tuples. The framework crystallises these connections as metaphysical grounding relations. We look at how simple differences in the shape and operation of constructors give rise to different varieties of these hierarchies and levels – and the impact this has. We also look at how the constructional approach reveals the differences between foundational constructors and derived constructors built from the foundational constructors – and show that conceptual modeling’s generalisation relations are secondary and dependent upon the more foundational instantiation relations. Based upon this, we assemble a constructional foundational ontology using the BORO Foundational Ontology as our starting point. We then use this to reveal and explain the formal levels and hierarchies that underlie the options for characterising coordinate systems. Keywords— Constructional Ontology; Ontological Space; Ontological Sandpits; BORO Foundational Ontology; Grounding Relations; Multi-Level Options; Geometric Coordinate System Ontology; Multi-Platform-Domain Sensor System. 1 Overview 1.1 Introduction The multi-level modelling community’s raison d'être is its vision of the ubiquity and importance of multi-level-types: the ascending levelled hierarchy of types in conceptual models; starting with types of things, then types of these types, then types of these types of types, and so on. The community both promotes this vision and investigates this hierarchy, looking at how it can be accommodated into existing frameworks. In this paper, we consider a specific domain, coordinate systems’ characterising options. While we recognise that, unsurprisingly, this domain contains a ubiquity of multi-level-types, -- 2 of 77 -- our main interest is in investigating a new and different approach to understanding them. For this we needed to develop a new framework. We devise one focussing on this case, based upon scaling down to simple compositional algorithms (called constructors) to form a new, radically simpler foundation. From the simple operations of these constructors emerges the scaled up multi-level structures of the domain. We show how the simple operations of simple constructors give rise to compositional connections that shape – and so explain – different complex hierarchies and levels, including the familiar multi-level-types and relatively unknown multi-level-tuples. The framework crystallises these connections as metaphysical grounding relations. We look at how simple differences in the shape and operation of constructors give rise to different varieties of these hierarchies and levels – and the impact this has. We also look at how the constructional approach reveals the differences between foundational constructors and derived constructors built from the foundational constructors – and show that conceptual modeling’s generalisation relations are secondary and dependent upon the more foundational instantiation relations. Based upon this, we assemble a constructional foundational ontology using the BORO Foundational Ontology [2, 3] as our starting point. We then use this to reveal and explain the formal levels and hierarchies that underlie the options for characterising coordinate systems. 1.2 The paper’s themes as two tropes This paper has a focus on examining the case of coordinate systems’ characterising options in the light of a BORO constructional ontology. The examination has two main underlying themes which we would like to make explicit. These themes are based upon two tropes; ubiquity revealed and radically simplify by scaling down to scale up – as we explain below. 1.2.1 Ubiquity revealed There is a common trope for scientific and engineering advances, which we call ubiquity revealed. In this narrative, at the start people work with data but see no pattern, they then learn to recognise the pattern and as a result see it everywhere in the data. The community goes from a state of not being able to see the pattern anywhere, to seeing it everywhere. The surprise here is both that knowledge is hiding in plain sight and that once one learns how to look, it is not only really obvious but everywhere. A reasonably well-known example is complex systems and its many features. Let’s pick one; say, scaling/power laws. Once scientists developed an understanding of complex systems and scaling laws, these laws were revealed as ubiquitous in complex systems, visible everywhere. The multi-level modelling (MLM) community’s vision has this ubiquity revealed narrative; once one learns to recognise multi-level-types in information systems, it becomes apparent that they are ubiquitous. Part of the work of the community is helping the wider information systems community learn to see these patterns. Another part of the work is examining the phenomena; aiming to explain what these multi-levels are (often the answer is ontological) and the implications of accommodating this pattern with other structures in the scheme of things – for example, how attributes have a potency -- 3 of 77 -- reflecting their mode of association with their multi-level-types. Our paper will illustrate the MLM narrative for a specific case, coordinate systems’ characterising options for sensor data, showing how a close look at these exposes the ubiquitous multi-level-type pattern. 1.2.2 Radically simplify by scaling down to scale up There is another common trope associated with scientific and engineering advances, which we call radically simplify by scaling down to scale up – abbreviated to scaling down. In this reductionist narrative, at the start a community is familiar with a complex, often ubiquitous, pattern, but cannot explain it. They then find a way of explaining this in two steps. They firstly decompose the elements of the pattern into simple components – scaling down. Then they consider the patterns that emerge from the simple interactions of these components – scaling up. And they see how their complex pattern, and often other new patterns emerge. In this way, the components and their interactions provide a new, radically simpler (down-scaled) foundation. The community goes from a state of seeing but not being able to explain ubiquitous complex patterns, to having a simple explanation based upon patterns emerging from simple interactions of a scaled-down foundation. The novelty here is the way the scaling down to a simple foundation dissolves and explains the scaled-up complexity. There are many examples of this in physics’ search for fundamental particles – starting as far back as Ancient Greece, with Democritus’ atomism. A modernish scientific example starts with the ubiquitous patterns of periodic trends in Mendeleev’s periodic table of elements. Empirically, there were clearly patterns, but no real explanation of how they arose. Also, there were oddities; for example, Mendeleev ordered cobalt and nickel based upon their properties rather than atomic mass to make the trend patterns work. Scientists scaling down, looking inside the atomic elements, developed simple structures that explained how these patterns, including the oddities, emerged. Rutherford and Bohr’s work [4] on the model of the atom led to electron configurations. While Henry Moseley’s work on atomic number – the positive charges on the nucleus – helped to explain the ordering, including that of cobalt and nickel. Breaking down the atomic elements into sub-atomic components provided a foundation which simply explained the patterns. It also created the possibility for new patterns (new periodic trends) based on the components, such as electronegativity. A more recent example from complex systems is Conway’s Game of Life [1]. This very neatly illustrates how very simple algorithms give rise to – and so help to explain – the complex ‘lives’ of cell automata and so perhaps complex natural life. In this paper, we tell a radically simplify by scaling down to scale up story. Given MLM’s revealed ubiquitous multi-level-types pattern, we outline a new, radically simpler (down-scaled) foundation that explains not just how the multi-level-types pattern emerges, but also how the whole underlying formal grounding structure, including other patterns of levels and hierarchies, emerges. Much like in Conway’s Game of Life, we start with a judicious choice of simple construction processes. These then give rise to a fully-fledged formal domain containing hierarchies and levels, including multi-level- -- 4 of 77 -- types. This draws upon research done in philosophy, particularly Kit Fine’s work on constructional ontology, composition (part-whole), procedural postulation and formal grounding. (This is quite technical, so we found we needed to devote a significant portion of the paper to giving an exposition sufficiently rich to enable a reasonable understanding.) With this understanding in place, we examine the case of coordinate systems’ characterising options to show how this approach can be deployed. 1.3 The structure of the paper The rest of the paper is divided into six broad parts. The first part aims to give a general context by briefly outlining the overall project that framed this work, describing the specific coordinate system characteristics challenge and how we aim to address it. It then describes the initial project that focuses on this specific challenge. The next two parts of the paper contain the technical analysis. The second part is devoted to outlining the general constructional framework (based upon work by Kit Fine, particularly [5], [6].) The third part uses this framework to assemble a constructional BORO foundational ontology. The next two parts of the paper focus on using the technical analysis to develop a constructional ontology for coordinate systems’ characterising options. The fourth part gives a brief overview of the approach to the building of the ontology. The fifth part looks in more detail at the ontology’s examples of the multi-level options – both type and tuples. A final summary concludes the paper. 2 The requirement for our case The need to understand the characterising options for coordinate systems arose from a specific requirement which we outline in this section. It is well-recognised that a major challenge currently facing the deployment of collaborating unmanned vehicles is semantic interoperability [7, 8], and that as this technology develops, the requirements for interoperability are likely to become both more stringent and complex. A common (preferably open) data architecture is seen as key to resolving this [7, 8]. Many of these current vehicles use systems and data structures that are proprietary and have a single platform – single domain heritage. These typically made no use of conceptual models in their development, and so have a lightweight (sometimes, non-existent) conceptual framework. It is a situation with substantial opportunities for improvement [9]. In most vehicles, sensors are the major producers of data, with a significant proportion of this being sensed position data. Sensed positions (typically structured as a triple of coordinates) are relative to a coordinate system. Where there are multiple platform/domains, their sensors will use different local coordinate systems. To be able to integrate this sensor data into a single common picture, the integrating system needs to know the various sensed positions’ coordinate systems. For example, the integrating system might receive two sensed position triples from different platforms’ sensors with the same coordinate numerals (such as <10, 20, 30>). These would typically use different local coordinate systems and if the integrating system does not know which coordinate system each is relative to, it cannot interpret and integrate them. The ‘20’ coordinate in the first triple might be relative to a -- 5 of 77 -- Cartesian coordinate system and so refer to a distance and the same ‘value’ in the second triple be relative to a Spherical coordinate system and so refer to an azimuthal angle – or maybe vice versa. In general, if the integrating system does not know enough about the ‘owning’ coordinate systems, it cannot interpret the position triples and so integrate them. More generally, what is required is an understanding of which characteristics of the coordinate system need to be known so that the position triple can be interpreted. Unfortunately, little work has been done on determining what a full characterisation would look like. In practice, coordinate systems are often not explicitly defined at all: it is assumed that users of the sensors know enough about what their coordinate system is. Where coordinate systems are defined, the characterisation is partial and pragmatically ad hoc. 2.1 The project We are working on a project that is assessing a radical approach to developing a suitable conceptual architecture for articulating the requirements for a full characterisation. The aim is to uncover the underlying conceptual foundations and reveal a clear fundamental picture. And, in so doing, to strip away any pre-conceptions remaining from the single platform/domain heritage. To uncover its conceptual foundation, we are taking a close technical look at the geometric foundations of the world described by the sensor position data. The project showcases an approach to building a foundational conceptual model that should be capable of resolving the semantic integration problems that multi-platform/domain sensor systems are currently facing. The project’s prime analytic tool is a constructional ontology based upon the BORO Foundational Ontology. Early work has focused on three simple coordinate systems for sensed positions and is clearly exposing an underlying compositional geometric structure; one where systems are built from a common set of ontological components whose construction processes follow broadly similar stages. In this paper, we focus on one aspect that is interesting from a multi-level modelling perspective. We investigate how to characterise the variety of coordinate systems as a series of sets of options. We focus on how these sets of options are embodied in the ontology by generating option objects at a higher level. These option objects are of two kinds. One is the well-known ‘type’ multi-levels (associated with Powertypes [10]). The other is a second kind of level-ascending based upon ‘tuples’, also known as ‘relations’ – which in turn gives rise to another levelled hierarchy. This second pattern is as far as we can determine new to the MLM community; so, from the perspective of the scaling down narrative, they are new patterns. We use the types and tuples hierarchies to characterise two kinds of options: combinations (types) and permutations (tuples/relations). These provide a basis for the fundamental characteristics needed to interpret the sensor position. -- 6 of 77 -- In the following sections, we look at the context in more detail and then give an overview of the requirements of the project. We then note our insights and show how this motivates our approach. 2.2 Context Unmanned vehicle collaboration across multiple domains/environments (air, surface, land, underwater and space) is recognised as a difficult engineering problem – Figure 1 shows examples of both single and multiple platform/domain manned and unmanned collaborating vehicles. Figure 1 – Single and multi-platform with deictic axes One challenge is the semantic integration of the sensing data into a single common picture – sometimes known as ‘ground truth’. A common data architecture with agreed data structures and APIs would simplify the challenge at the syntactic level. But this needs to be supported by a common semantic model to ensure shared semantics. The scope of such a model extends beyond the APIs, as their semantic integrity depends upon the systems behind them respecting (and so understanding) it. The unmanned vehicle sensors process the raw data and pass this on to other, typically centralised, systems for further processing. The level of local onboard processing varies depending upon various factors; for example, low bandwidth restrictions might lead to a preference for onboard over centralised remote processing. The sensors work on a local basis of own position and measure other positions relative to themselves – they may further process these measurements before reporting or directly report a sensed position relative to themselves. Directly reporting the local positions may be preferred as this allows a centralised, consistent calculation of errors. At the core of these reports is a sensed position recorded using coordinates. The data format of these coordinates is apparently simple and easy to specify – a triple of numbers, with a timestamp. However, it has emerged that it is more of a challenge to find a common data (and semantic) format to characterise all the coordinate systems to which these coordinates can (or could) belong. In large part, this is because the common format will need to be able to accommodate significantly more variety and complexity than the current single platform/domain systems – and include enough detail to make -- 7 of 77 -- coordinate conversions between the systems, or to a common system. The ways in which the systems vary include: • Coordinate system. Unmanned vehicles should be easy to add to (and remove from) the collaborative sensor systems – whatever coordinate system they use. These vehicles are likely to use new types of coordinate systems which will need to be supported. So, some general structure for coordinate systems needs to be developed. • Position and orientation. The unmanned platforms are moving (with both linear and angular velocity) relative to the main platform in all three dimensions – so the position and orientation of their coordinate systems will be both different and varying. So, some general structure for position and orientation needs to be developed. • Angle and unit. There will typically be limited governance over suppliers of the unmanned vehicles, who are likely to use their own configuration for the coordinate systems. For example, they may use different distance units; one using kilometres, the other miles. So, some general structure for distance and angle units as deployed in coordinate systems needs to be developed. • Domain-specific simplifications. Platforms in the sea domain have, in the past, often had more basic requirements than other domains. For example, they have typically used small angle correction, and some even only considering yaw angular movements, ignoring roll and pitch – as these are not so relevant for single platforms in the sea domain. (The papers [11, 12] describe another simplification for position calculation.) More generally, this raises the requirement, in multi-domain systems, for these domain-specific simplifications to be harmonised to avoid error-generating inconsistencies. • Direction. Different platforms and sensors will use different directions within the orientations. For example, the Cartesian z-axis often has a down direction for underwater and aerial platforms and an up direction for surface platforms (in the maritime domain, this can vary from ship to ship). So, some general structure for directions as deployed in coordinate systems needs to be developed. 2.3 The project’s aims The project aims to build a conceptual model that will support the semantic unification requirements of multiple platform/domain systems. More generally, it aims to showcase a general methodology for designing the data architecture of this domain; one that involves a principled, repeatable, auditable, extendable process. Such a process should be able to identify the range of possible coordinate systems characteristics (possibly exposing their foundations) and design a parsimonious and elegant conceptual model for representing them. -- 8 of 77 -- This should provide a degree of comfort that the data architecture built from the conceptual model not only accurately covers current requirements but is also relatively future-proofed: • that the process will identify a reasonably complete range of possible configurations and • that it will be easily extendable to new coordinate systems. It should also provide a benchmark for identifying gaps in the existing data architectures. 2.4 Our insights The following three insights motivated the approach for developing the conceptual framework outlined in this paper: 1. Each characteristic of the coordinate systems can be thought of as an exhaustive set of independent options. For example, the coordinate system’s surface configuration type may be Cartesian, cylindrical, or spherical – one of these options needs to be selected. Generally, the sets of independent options seem to come in two varieties (kinds), combinations and permutations. These correspond, respectively, to ways the system can be and to ways of organising the system. 2. Currently, there is no obvious parsimonious and elegant framework for organising these characteristics waiting to be plucked off the shelf. Standards, such as OMG’s [13] and ISO’s [14], do not (upon inspection) provide the right kind of help. Neither does theoretical work such as [15]. Though, of course, all of these provide useful input. In some ways, this is a surprising situation, as Euclidian coordinate geometry has been researched extensively for millennia. In other ways, it is not so surprising, as the motivation for this research has not been to unearth the characteristics that should drive a conceptual model to support a data architecture. 3. The coordinate system characteristics that drive the conceptual model are grounded in the system’s geometry and that an understanding of these characteristics will emerge from a clear picture of its foundational geometrical features. (As a side note, there is a revived interest in geometry as a mathematical foundation for space and time – see, for example [16] – as well as one in the foundations of Euclid’s original geometric work – see, for example, [17]. 2.5 Our approach We decided to start with an ontological conceptual model which would give us a technology agnostic picture. Given the importance of exposing the geometric foundations, we recognised the need to be geometry friendly. We chose a foundational ontology that is extensional and four-dimensional, the BORO Foundational Ontology [3], and are deploying it using a constructional approach [2]. We expected this to not only expose the foundations of the range of possible coordinate systems characteristics but also provide a workspace for exploring the relative parsimony and elegance of -- 9 of 77 -- different conceptual structures. We also adopted the goal of understanding what the ontology of the coordinate system options is; to enable us to use this to design the data architecture. As a first stage, we started an initial project for the limited set of the three simplest local coordinate systems; Cartesian (sometimes called rectangular – though from our perspective planar would be more accurate), spherical and cylindrical. We also assumed that we could simplify the geometry to Euclidian affine space-time. We build upon earlier work we have done with coordinate systems [18, 19]. This initial project is under way, and this paper is based upon early results. 3 The general constructional framework This section is a technical overview devoted to explaining the general constructional framework we have devised for this case. It is divided into six subsections. The first provides a general background. The next two provide the general background framework; the general ontological framework and the general composition framework. The final three sections build an application framework upon this. 3.1 Background This framework is motivated by a Neo-Aristotelian view of ontology and based upon a specific constructional ontology developed by Kit Fine. 3.1.1 Philosophical motivation There have been developments in ontology in the last few decades often grouped together under the label ‘Neo-Aristotelian’. This typically takes issue with the flat Quinean view of ontology [20] reflected in Jonathon Lowe’s description in The Oxford Companion to Philosophy of ontology as “the set of things whose existence is acknowledged by a particular theory or system of thought.” Instead, it proposes that reality has an underlying metaphysical structure [21, p. 354] – and that this structure is more than a set, more than a sorted list of categories, rather that it is ordered by some sort of metaphysical grounding – these different structures are illustrated in Figure 2. In this paper, we build an ordered ontology – the constructional BORO ontology – and show how it explains the structure of our coordinate systems example. Figure 2 – Types of structure (from [21, p. 355]) -- 10 of 77 -- 3.1.2 Fine’s constructional ontology Constructional ontology has a history that can be traced back through Goodman to Carnap [22]. The particular algebraic constructional ontological framework used here is based upon that outlined in Fine’s [5] and further developed in [6] as a general compositional framework. Fine makes plain his focus [6, p. 560] is on “the ‘pure’ theory of part whole rather than its application to our actual ontology.” Our focus here is on understanding an actual case – coordinate systems – through the application of an actual ontology. Fine’s pure theory provides us with the starting point for the framework for our endeavour. Fine sees the specific advantage of what he calls the ‘operational’ nature of his framework is that it naturally leads one to consider the nature of the metaphysical structures. As an example, he comments on levels (a central topic for MLM): “there is an intuitive distinction between wholes which are like sets in being hierarchically organized and those which are like sums in being 'flat', or without an internal division into levels. The distinction, under the operational approach, can be seen to turn on whether repeated applications of the operation are capable of yielding something new.” [6, p. 566]. Elsewhere [6], he talks about its power and beauty; its ability to provide a single and elegant account of a variety of structures. This makes it an ideal tool for the task at hand. For investigating the metaphysical structure of possible ontologies and, in so doing, investigating the ontological content of multi-level-types while developing a new foundation that explains their ontological structure. In this paper, we build upon Fine’s work to develop a framework for building constructional ontologies – and examine and investigate their constituent constructional components in sandpits. We use this to develop the constructional BORO ontology to help us analyse the example of coordinate systems. 3.2 Finean general ontological framework Fine’s general framework has what he calls two theories; a core theory about what individual constructional ontologies are and an extended theory for multiple ontologies in an ontological space. 3.2.1 Finean core theory For Fine an ontology is constructional if some of the objects of the ontology are accepted (that is, included within the ontology) on the grounds that they are constructed from other objects within the ontology. It is their status as constructs which earns their admission into the ontology. For him the paradigm of a constructional ontology is the cumulative hierarchy of sets [23]. Sets are admitted into the ontology on the grounds that they are constructed from their members. (From now on we will feel free to drop the qualification ‘constructional’ from ontology – as all ontologies discussed here will be constructional in this Finean sense. We will also feel free to deviate slightly from Fine’s terminology and structure where this suits our exposition here.) -- 11 of 77 -- Fine calls this constructional approach ‘operationalism’ [6]. He sees it as an example of a specific kind of operationalism which he calls ‘proceduralism’ or ‘procedural postulationism’ [24], [25, pp. 36, 56, 100], where the existence of an object is postulated according to construction rules. Fine explicitly compares these rules to a computer program [24, pp. 90–1] for going from one state of a domain to another and suggests links to dynamic programming logic [26]. He describes the construction process as creative or expansive – as expanding the ontology [24, p. 103] – a point we return to later. He also introduces the metaphor of a genie that automatically executes the procedures. This metaphor of a genie helps us to visualise construction operations, but we need to be careful to visualise the genie’s work in a platonic, ideal way. We should not think of it happening in space or time. If we do, it would (for example) seem possible for the genie to execute the same operation, creating exactly the same object, at different places or times. This seems a clear case of two executions, where presumably the first execution creates the object and the second merely reconstructs it. To avoid thinking like this, it is better to idealise the metaphorical genie’s work. One can assume the genie surveys everything that exists and determines all possible operations and executes them simultaneously. This may generate new objects, which may open up the possibility for new executions. The genie will again survey everything that exists and determines all possible operations and execute them simultaneously. This sequence of simultaneously operations is a feature of constructional approaches. Conway’s Game of “Life” which is constructional (though not an ontology) works in a similar way – (Gardner [1]) – quoted at the start of this paper – says “It is important to understand that all births and deaths occur simultaneously” and “each simultaneous execution is called a generation or “move””. In Boolos’ constructional cumulative set theory [23], they are called ‘stages’. Fine describes how using construction operations shape the architecture of the ontology. They make objects – the constructs. The acceptance of a construct into an ontology requires firstly that, if required, there are constructees (the objects, if any, from which the construct is constructed) and secondly the constructor (the means by which the construct is constructed from the constructees). Of course, constructs, once constructed, can be constructees in later construction operations. Not all the objects in an ontology need to be constructs, some can be just accepted into the ontology. These are given objects (givens), which (along with the constructors) seed the ontology. From these all the constructs are constructed. An ontology does not need to have givens. For example, in pure cumulative set theory, there are none; the empty set is built from zero input. Similarly, an ontology does not need to have any constructors, all its objects could be givens – though for us this would be an uninteresting limit case – and not constructional under Fine’s definition. -- 12 of 77 -- This gives us a framework where objects are accepted into the ontology on one of two grounds, that they are 1. given objects that are just accepted 2. constructs or constructed objects which have been constructed from objects already in the ontology; where a constructor is applied to the constructees to produce constructs. This leads to a four domain architecture for constructional ontology universes (see Table 1 and Figure 3). Table 1 – Ontology domains in the ontology universe (see [5]) Acronym Domain Names for Members Descriptions U ontology universe items contains the ontology domains O object domain objects all objects in the ontology CR constructor domain constructors G given domain givens or given objects a sub-domain of the object domain CD constructed domain constructs or constructed objects a sub-domain of the object domain Figure 3 shows the domain composition of the universe more directly. The tree view shows how the given and constructed domains combine to form the object domain. It also shows how the object and constructor domains combine to form the ontology universe. The iconic (Euler) view shows more concretely how the universe is composed. Figure 3 – Two views of the ontology universe’s domains The core theory uses ontological principles to show that one can generate all the constructed objects (the constructed domain CD) from the given (G) and the constructor domains (CR). As Fine notes, this means we do not need the object domain, O, to characterise an ontology, we can use just the -- 13 of 77 -- domain couple <G, CR> as seeding. We call this the ontology signature – as shown by the hexagonal icon in Figure 3. We will use this signature to characterise ontologies in the rest of the paper. Given the signature, CD can then be generated from G and CR – and combined with G to give O. Though the order of analysis may well be the opposite; where one starts with the objects and works out what the constructors are and so the given objects. Furthermore, the signature couple <G, CR> has more structure than just plain O, as it picks out the given objects and the constructors – which gives the rules for how the other (constructed) objects are constructed. This generation of CD relies upon an exhaustive application of the constructors; where anything that can be generated is generated. In our approach, we find it useful to make this process explicit as it helps us to see the structure of CD. We call this the ontology’s ONTOGENESIS and adopt the triple <G, CR, ONTOGENESIS> as the extended ontology signature, where required. We will be looking at examples of ONTOGENESIS later. 3.2.2 Finean extended theory Fine’s extended theory deals with how ontologies fit into an ontological space; where this is a nonempty collection of ontologies that conforms to certain principles. It describes how, given ontologies in a space, similar ontologies with permutations of given and constructor domains also exist in the space. We extend this to permutations of the individual constructors in the domain. For us, this provides a framework for an incremental sandbox approach for explaining and understanding an ontology. Under this approach, one might start with an idea for a target ontology; characterised by its signature with the given and constructor domains. One can then build an ontological space that has ontology universes for each possible permutation of givens and constructors – as one knows their signatures. One would typically start with the simplest permutations, ontology universes that contain just the given domain or just a single constructor and examine these – looking at the results of applying ONTOGENESIS. One can then pick and examine richer and richer combinations, seeing how these lead to richer structures, until one arrives at the target ontology. Let’s make this more concrete with a very simple example. Consider a target ontology with the signature <(g), (cr)>; where the given domain has a single object g and the constructor domain a single constructor, cr. The possible permutations are: NULL = <Ø, Ø>, <(g), Ø>, <Ø, (cr)>, TARGET = <(g), (cr)> We can construct an ontological space, where each of these permutations is the signature for an ontological universe. The space is shown in in Figure 4, along with arrows showing how the permutations build up to the target. Working through the ontologies, starting from the simplest combinations allows one to see what an individual contributes to an ontology and how it interacts with other items usually giving an insight as to how it contributes structure to the target ontology. -- 14 of 77 -- Below, when we use an ontological space to build the BORO Constructional Ontology, there will be specific examples of this ability to provide insight. Figure 4 – Example ontological space 3.3 Finean general composition framework Fine [6] sketches a general unified framework for composition; the ways in which one object can be a ‘component’ of another, within his general ontological framework. 3.3.1 A liberal notion of composition Fine’s framework is distinctive in several ways (all of which suit our current purpose). The composing or part relation is usually restricted to mereological parts – where, for example, my hand is a part (component) of my arm. Fine proposes a very liberal notion of composition that includes many types of component, where the traditional mereological relations are just one type. So, for example, it would include a member of a set being part of the set in a similar way to my hand being part of my arm – though these would be different kinds of part. Fine [6] makes a strong case for this position. We take it as a starting point in this paper. (Terminologically, Fine chose to use the terms ‘part-whole’ and ‘composition’ to characterise the family of relations, to reinforce their unity. For our purposes, it is more convenient to use ‘composition’ as the term for the family and (following standard practice) reserve the terms ‘whole- part’, and its equivalent alternative ‘part-whole’, for the mereological relation. This is just a terminological matter and in no way undermines Fine’s general unification thesis.) The formulation of the framework follows the general framework, described above, of givens and constructors. Central to this is identifying the variety of constructors that account for the different ways things can be composed. -- 15 of 77 -- 3.3.2 Compositional principles Fine formulates his framework in terms of compositional principles. He first divides these broadly into formal and material principles. Our interest here is in the formal principles. He further divides formal principles into those that deal with conditions of application and those that provide identity conditions. Fine develops a very simple way of characterising the formal identity principles for compositional identity based upon a notion of regular identity conditions (the reader can find the details in [6]). The result is the four CLAP principles – so-called because of their initials –described in Table 2. Table 2 – CLAP (formal identity) principles (see [6]) C Collapse ∑(x) = x If Collapse holds then any composite composed of a single component is identical to it. L Levelling ∑(… ,∑(x, y, z,...),… ,∑(u, v, w,...),...) = ∑(… , x, y, z,… ,… , u, v, w,… , ...) If Levelling holds then when the components of composite have components, these components’ components are also components of the whole. A Absorption ∑( … , x, x, … , … , y, y, … , … ,) = ∑( … , x, … , y, ...) If Absorption holds then the repetition of components is irrelevant to the identity of the composite. P Permutation ∑(x, y, z, ...) = ∑(y, z, x, ...) (and similarly for all other permutations) If Permutation holds then the order of the components is irrelevant to the identity of the composite. A constructor’s formal identity can be characterised by whether these principles are adopted or rejected – one can summarise this into a CLAP profile, with a mnemonic where the appropriate letter is struck through when the principle is rejected. There are two conflicting senses of the term ‘level’ that make Fine’s choice of the name ‘Levelling’ less than ideal for the purposes of this paper. There is the sense of making flat, (into a single) level that motivates Fine’s use and then there is the almost opposite sense MLM uses of not being flat and having multiple different levels (also used by Fine [6, p. 566] – “in being 'flat', or without an internal division into levels”). We have stuck with Fine’s choice – partly to preserve the easy to remember CLAP acronym (CFAP does not have the same ring). And we have tried to make clear in the paper which sense is being used – restricting the present participle ‘Levelling’ to the Finean sense of making one thing level and the noun ‘level’ for MLM’s sense of arranged in levels. The other formal element in the framework is the conditions of application – typically what objects can be applied to the constructor. One example, often mentioned in Fine, is whether an empty application is allowed – in other words, no constructees are supplied to the constructor. In this case, it constructs a null object for that type of constructor. There are cases of this in mathematics; for example, the null set for sets and the null thing for things. Ontologies are simpler without null objects. So, for simplicity, in the example ontologies (and later the BORO constructional ontology) in the rest of the paper a null application will not be allowed unless explicitly specified. -- 16 of 77 -- The formal identity principles may appear arcane, but they have simple, intuitive, constructive tests. For Levelling, one can consider what happens when there are two applications of the constructor to a simple object. And whether the initial object is a component of the object created in the second application. For example, if one starts with Socrates and creates a set from it then one gets {Socrates}. If one then creates a set from this, one gets {{Socrates}}. And Socrates is not a part/component/member of {{Socrates}} – so the Levelling principle has not been adopted. Permutation is about whether order is considered. We will be talking about order quite a bit going forward: to make it clear when we are representing order, we use the less-than sign symbol ‘<’; where (a<b) means (a then b) in that order. For Permutation, a simple test is to consider whether changing the order of a couple of objects gives rise to a different object. So, whether the constructor when applied with (a<b) –– constructs a different object than when applied with (b<a). It does for STRING-BUILDER but not for SET- BUILDER – so the first adopts Permutation and the latter does not. Similarly, for the other two principles. The formal conditions of application also mostly have simple intuitive constructive tests. In the null object example above, one just needs to ask whether one can make an empty application to the constructor. 3.4 Developing the CLAP formal identity principles 3.4.1 Four familiar kinds To see how the CLAP principles work and how constructors fit a CLAP profile, consider the four familiar (to mathematicians and computer scientists) cases of composition – described in Table 3. Table 3 – Four familiar cases of composition (see [6]) Kind Form of composition Things The form of composition is fusion. A typical example is mereological fusion – where the fusion of the parts is the whole. Note: the mereological fusion of a single part is identical to the part itself. Sets The form of composition is collection. A plurality of objects is collected into a single new object. Note: the set with a single member – so, for example, the set composed of Socrates – {Socrates} – is not identical to the member. Strings The form of composition is simple concatenation. Concatenating two strings, say xy and uv, is the same as concatenating some combination of their components; so x, y, u, and v or xyu and v or … and so on. Similar to list data structures. Sequences The form of composition is sequence-building. Sequencing two sequences (xy) and (uv) to obtain ((xy)(uv)) is to be distinguished from sequencing x, y, u, and v to obtain (xyuv) (in contrast to the case of strings). Similar to list of lists data structures. -- 17 of 77 -- 3.4.2 Four CLAP Profiles for the familiar cases These familiar cases can be characterised by CLAP profiles, which we call orthodox – as shown in Figure 5. This captures their formal identity principles, but not the formal conditions of application. So, not, for example, whether they accept null applications. Figure 5 – CLAP profiles of orthodox versions of the familiar cases 3.4.3 Levelling and Permutation divide up the cases The four CLAP principles have fourteen potential valid combinations (of the sixteen possible combinations, CLAP and CLAP should not be allowed as they are invalid – they lead to cycles [6]). The four profiles in Figure 5 give four formally orthodox cases – which seem to construct different kinds of objects. How do the remaining ten combinations work? Do they, for example, construct the same or different kinds of object? Firstly, let’s establish that two constructors with different CLAP profiles can construct identical objects. Intuitively the answer seems to be that it is possible. The following example confirms this intuition. Consider these examples of different CLAP profiles listed in Table 4 that appear to be set variants. There is a partial taxonomy for them shown in Figure 6. -- 18 of 77 -- Table 4 – Example variants of set Kind Profile Name Description Set CLAP formally orthodox sets defined by the CLAP profile Set CLAP Quinean non-multi- sets like orthodox sets but with the singleton identical to its sole component Set CLAP non-Quinean multi- sets like orthodox sets but allowing multiple occurrences of the same component Set CLAP Quinean multi-sets like both Quinean sets and multi-sets Figure 6 – Partial taxonomy of set variants Table 5 shows that for a standard input of (a, b, c) there is a standard output of {a, b, c}. It appears to make sense that the different constructors are creating the same set {a, b, c}. The edge cases, where the input to Quinean sets is singular or the input to multi-sets includes repetition, end up having objects generated by one constructor but not the other. But they do not appear to generate counterexamples or inconsistency. Table 5 – Example inputs and outputs Profile Name Input Output CLAP formally orthodox sets (a, b, c) {a, b, c} CLAP Quinean non-multi-sets (a, b, c) {a, b, c} CLAP Quinean multi-sets (a, b, c) {a, b, c} CLAP Non-Quinean multi-sets (a, b, c) {a, b, c} Alternatively, one could argue that the constructor was part of the identity of the constructed object. In this case, CLAP: {a, b, c} and CLAP: {a, b, c} (where the CLAP prefix is a label for the generating constructor) would be different objects – despite them being effectively indiscernable in terms of kind and members. This would greatly inflate the framework with redundant objects, so we discount it here. -- 19 of 77 -- It turns out that some principles influence what kind of object is constructed – these are kind- characterising. Furthermore, it also turns out that the group of kind-characterising principles are sufficient to uniquely characterise the kinds. So, where two constructors similarly adopt or reject this group of principles, they build the same kind of object – where they are not similar, they generate different kinds of objects. Two of the four principles, Levelling and Permutation, form the kind- characterising group. Collapse and Absorption (and the conditions of application) are not kind- characterising. We can explain this by looking at when two constructors with different CLAP profiles construct identical objects. Assume that they do. Then one would expect that applying identical components to the two constructors would generate identical objects (as in the set example above). This creates a one-to-one mapping between the identical objects. As the example above shows, there can be edge cases where there is no identity and so no mapping. If two constructors have different Levelling or Permutation principles, then it is not possible to have an identity mapping. Consider Permutation first. We start with two components a and b as givens and two constructors one adopting, one rejecting Permutation. Assume that there is an identity mapping. Take the Permutation-adopting constructor; at the first stage, the genie applies (a<b) and (b<a) constructing two different objects – ab and ba. If a constructor rejects Permutation, then the genie can only apply (a, b) – as it does not recognise order – constructing a single object – {a, b}. Either this single object is identical to both the Permutation-constructed objects or neither. It cannot be identical to both, as this would imply the two Permutation-constructed objects are identical – given identity is transitive. So, it is identical to neither, in other words, there is no identity mapping. Similar arguments apply to Levelling. Thus, the four combinations of Levelling and Permutation – shown in Figure 7 – characterise four different kinds of objects. Fine sees these four kinds as just a small sample of the possible kinds – for our purposes here we restrict ourselves to them. Figure 7 – Kind characterising principles (based upon [6, p. 574]) Furthermore, the degree or similarity is dictated by whether they share principles. Here the rows in the table in Figure 7 (things and sets or strings and sequences) and columns (things and strings or sets and -- 20 of 77 -- sequences) indicate similarity (as they both either adopt or reject Levelling or Permutation). Whereas the two diagonals (things and sequences or strings and sets) take opposing positions on these principles. With this in place, there is a framework to let the constructors determine the (sort) kind of the objects they construct. There is not quite enough structure yet, to let them determine the kinds for the whole ontology. The kinds of the given objects cannot, in principle, be given by the constructors – as they are not generated by constructors – though they might be re-constructed. In this case, the simplest solution is to mandate that the kind of each given must be stipulated. 3.4.4 Stipulated conditions of application Given these four kinds, one can ask whether there are conditions of application based upon them. For three of the kinds, it is customary for there to be no constraints; to allow any kind to be included in the application. However, for one kind, thing, there is a constraint. Only objects of the kind thing can be applied to thing constructors. These conditions are listed in Table 6. Table 6 – Customary stipulated conditions of application Kind Component Kind Composite Kind Thing Thing Thing Set Any Set String Any String Sequence Any Sequence 3.4.5 Syntactic expansion and exhaustion As discussed earlier, these applications are executed as constructions in simultaneous stages. The metaphorical genie executes all possible constructions and no more, at each stage as part of the ONTOGENESIS process. As is traditional (see [23, p. 221]), we allow a finite, infinite or even transfinite number of stages. The genie is powerful enough to execute these stages if required. What counts as a possible construction? At any stage, the genie could take any combination of objects in the ontology and apply them. However, this leads to a situation where if there is a possible combination at any stage, then it could be a possible combination at all later stages. So, the process is inexhaustible, even when the process arrives at a point where the same combinations are being executed every time. For deterministic constructions, a more economical strategy is to only consider combinations possible in the stage they first appear (see [23, pp. 221–2] “… sets are formed over and over again … We could continue to say this if we liked; instead we shall say that a set is formed only once, namely, at the earliest stage at which, on our old way of speaking, it would have been said to be formed.”). So, the genie only executes each combination of input and constructor once – as this combination always gives the same output, so executing again is pointless. -- 21 of 77 -- For example, assume that we have an ontology STRING-EXAMPLE-1: <(a, b, c), (STRING- BUILDER)>. The orthodox string constructor (see Figure 5) rejects absorption so allows repetition. Hence, any given can be repeated any number of times and applied to the constructor. This would lead to very crowded figures and tables. To keep things simple, this string ontology adopts absorption – ignoring repetition – with this profile CLAP. At the first stage, one of the operations the genie applies is the ordered givens (a<b<c) generating the new string abc. At the second stage, no new objects are generated. For example, a, b and c are available for application, but the genie no longer applies (a<b<c) to STRING-BUILDER as it has already executed this operation before. Some objects are reconstructed though, and this is visible in Figure 8. Figure 8 – STRING-EXAMPLE-1 – visualisation of stages 0 to 2 For non-deterministic constructions, the economical combination is expanded to consider outputs as well as inputs. A possible construction is one where the combination of both the input and the output have not yet been constructed. A constructor is exhausted where, for each of all possible input combinations, all the possible output combinations for a particular input combination have been constructed. For example, assume that we have an ontology NON-DETERMINISTIC-EXAMPLE: <(abc), (RANDOM-STRING-SELECTOR))>, where the constructor returns a random atomic part of the input. At the first stage, the genie applies (abc) to RANDOM-STRING-SELECTOR randomly generating the string b. At the second stage, the genie again applies (abc) to RANDOM-STRING- SELECTOR generating the string c. At the third stage, yet again the genie applies (abc) to RANDOM-STRING-SELECTOR randomly generating the string a. Then the possible combinations of input and output are exhausted – so there are no possible constructions. With this ‘economical’ sense of possible in place, the process is complete when all the possible combinations have been executed once. We call this syntactic exhaustion as it only considers the combinations. -- 22 of 77 -- 3.4.6 Levelling: single or multiple possible build constructions One result of adopting the Levelling principle is that it allows multiple possible constructions for the same object; and rejecting Levelling leads to a single possible construction. Consider these two examples. Firstly, take the Levelling-adopting STRING-EXAMPLE-1 ontology again. The string abc is built (constructed) in multiple ways; by applying (a<b<c) or (ab<c) or (a<bc). In the ontology STRING- EXAMPLE-1, (a<b<c) is constructed at both the first and second stage. Clearly, the operation at the first stage will generate the object. The subsequent, multiple second stage constructions merely re- construct the (already generated) object. This allows us to distinguish between two types of construction; generative, where new objects are created and merely compositional, where no new objects are created. The compositional constructions have a role to play, as they reveal the composition structure. In this case, two composition relations are revealed; that ab and bc are components of abc – as shown in Figure 8. However, note that the overall operations are only partially compositional, as a and c composing abc has already been executed. It is not always possible to make this distinction. In cases where the multiple constructions are effectively simultaneous, there is no way to order them and say which is first. One could say they are jointly generative. Now assume that we have an ontology SET-EXAMPLE-1: <(a, b, c), (SET-BUILDER)> with the Levelling-rejecting constructor SET-BUILDER – see Figure 9. At the first stage, one of the operations the genie applies the (unordered) givens (a, b, c) to SET-BUILDER generating the new set {a, b, c}. This is the only way this set can be constructed – and with the ‘economical’ sense of possible, the only time it is constructed. Figure 9 – SET-EXAMPLE-1 – visualisation of stages 0 and 1 3.4.7 Direction of operation Bennett [27] identifies a key choice not captured by the CLAP principles: the metaphysical direction of operation. The operation of a constructor can either start with components and end with their composite (composing) or start with a composite and end with its components (decomposing). In the -- 23 of 77 -- literature [27], the direction is typically composing. Bennett offers Schaffer [28] as an example of decomposing. For three of the kinds we are looking at (sets, strings and sequences), the construction of the composite from its components is a formal affair with a natural composing direction of operation. Cantor [29] famously wrote about a set being “a gathering together into a whole of definite, distinct objects” – suggesting a natural direction of gathering together rather than splitting apart. In the constructional approach, this gathering is a construction that postulates the existence of the set built from its members using formal rules: the postulation of strings and sequences similarly involve formal building rules. Also, at least in the simple cases, the constructor is complete – it constructs all objects of these kinds in the same way – no objects of these kin