A Framework for Composition A Step Towards a Foundation for Assembly -- 1 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 2 Contents Preface 4 Summary 5 Introduction 6 Focus on the abstract general component breakdown structure 6 Structure of the paper 7 The composing operation 7 A simple constructional process 8 Associated part relation 11 A (very) simple example 13 The stricter (assembly) structure 16 The simplest atomic assembly structure 17 Formal constraints on the components being assembled 17 Material constraints on the components being assembled 19 Unique compositions and decompositions? 19 One atomic composition atop another 20 Atomic immediate parts 21 Dividing and combining atomic compositions 22 Hierarchies of strict atomic compositions 23 Assembling atomic compositions into a hierarchy 23 Criterion of identity 29 A new ‘mereology’ 29 Extending strictness to molecular compositions 29 Associated parts – extending dissection to molecular compositions. 29 Compositional immediate parts 30 Material constraints on the component hierarchy 30 -- 2 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 3 Overlap for molecular component hierarchies 30 Refining and collapsing constituents in a component hierarchy 31 Generating composition structures 31 Relaxing the strict (de-)composing structures 32 Relaxing disjointness 32 Accommodating partial cover 33 Accommodating families 34 Assessment framework 34 Further work 36 Summary 37 Appendices 38 Appendix A – Integral and replaceable parts 38 Appendix B – Some literature on modularity and components 44 Appendix C – A brief etymological analysis 48 Appendix D – A brief history of composition and whole-ness 50 Appendix E – Logical formulae 51 Appendix F – Foundational and conceptual ontology distinction 56 References 58 Acknowledgements 61 -- 3 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 4 Preface 1 https://constructioninnovationhub.org.uk/platform-design-programme-defines-the-need/ The Construction Innovation Hub’s Platform Design Programme1 is working in partnership with both government and industry to develop a platform construction system consisting of a standardised kit of parts to deliver social infrastructure buildings, such as schools, hospitals, and village halls. This document is another piece of work done by the National Digital Twin programme in collaboration with the Construction Innovation Hub towards developing the Information Management Framework that will form the foundation for sharing consistent data not only for the Platform Design Programme, but for use across government and industry. Industries involved in the creation and management of built assets require effective, resilient, and secure data sharing and aggregation. Much of this information is needed throughout the life of the asset and needs to be shared with a number of organisations. This is critical not only for asset management, but to support the services provided by the asset, as well as other considerations such as measuring the accumulating carbon emissions in order that a net zero footprint can be achieved. As a result, a formal mechanism to ensure that the right information can be made available at the right time, to the right people and that the quality of the information is known and understood, is required. The Information Management Framework is such a mechanism, the technical part of which comprises three main elements: • A Foundation Data Model • A Reference Data Library, and • An Integration Architecture. The Foundation Data Model (or ontology) and Reference Data Library define a common structure and meaning for information that is shared between organisations within and across sectors and domains. Together, therefore, they enable the consistent sharing and integration of information. The Integration Architecture comprises a combination of technologies that enables this sharing of data between data sources and the systems that use them. -- 4 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 5 Summary Component breakdowns are a vital multi-purpose tool and hence ubiquitous across a range of disciplines. Information systems need to be capable of storing reasonably accurate representations of these breakdowns. Most current information systems have been designed around specific breakdowns, without considering their general underlying formal structure. This is understandable, given the focus on devising the breakdown and that there is not a readily available formal structure to build upon. We make a step towards providing this structure here. At the core of the notion of a component breakdown is the component as an integral (dependent) part of the composite whole. This leads to a rich formal structure, one that requires careful consideration to capture well enough to support the range of specific breakdowns. If one is not sufficiently aware of this structure, it is difficult to determine what is required to produce a reasonably accurate representation – in particular, one that is sufficiently accurate to support interoperability. In this report, enabled by the Construction Innovation Hub, we describe this rich formal structure and develop a framework for assessing how well a data model (or ontology) has captured the main elements of the structure. This will enable people to both assess existing models as well as design new models. As a separate exercise, as an illustration, we develop a data model that captures these elements. Associated with the notion of component (as an integral, dependent part) is the notion of replaceable part (see Appendix A for more details). We do not characterise this here but will do so in a later report. -- 5 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 6 Introduction Engineers (for example) use many types of breakdown. They breakdown a product (a composite whole) into the components needed to assemble it and list these in a Bill of Materials (BOM). They breakdown the work on a project (another composite whole) into manageable chunks associated with deliverables and list these in a Work Breakdown Structure (WBS). There are a range of benefits of this approach. Managing the whole through its components is often easier than managing the whole on its own. The components may be ‘modules’ reusable in different products and projects – the products (or projects) may be ‘modules’ reusable across a range of bigger products and projects. It is not just engineers that use breakdowns, they are ubiquitous across a range of disciplines that deal with both the natural and artefactual (for more on the topic, see Appendix B which has a brief overview of literature – see also Appendix C which gives a sketch of the linguistic depth through the etymology). For engineers (and others) to collect and share information about breakdowns in computer information systems, they need to fit their representations accurately into a data structure. Here we explore how the general formal structure of component breakdown can be characterised in terms of a composing operation. We use this to develop a framework for assessing how data (or ontologies) have captured the structure. Focus on the abstract general component breakdown structure It turns out that a range of component breakdown structures – and their associated composing operations – emerge in many domains. In general, if a domain has a kind of (flat) part structure then it also has the potential for a sibling (hierarchical) composition structure. There are two clear cases (illustrated in Figure 1) rooted in mereonomy (which deals with wholes and parts) and taxonomy. Firstly, the breakdown of a particular individual into a whole-part structure – where, for example, my car can be broken down into its body, chassis, engine, interior and wheels. Secondly, a super-set sub-set structure of classes in taxonomies – where, for example, engines are broken down by their energy sources into heat, non-thermal, electrical and physically powered engines. Much of the interest in component breakdowns has focused on specific breakdowns. There is some, but not much, work that looks at the component breakdown structure as such, abstracting away from particular examples (for example (Bittner et al., 2001)). One cannot help thinking that there is a common, mistaken, assumption that this general structure is in some sense obvious, or easily recoverable, so not worth examining closely. We start here by showing that the underlying formal structure, needed for information systems’ data structures is neither -- 6 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 7 obvious nor easily recoverable – though it is based upon a simple base structure, a composing relation. Accordingly, in order to develop a clearer general picture, in this initial section, we focus on the abstract general component breakdown structure, for the most part ignoring which specific breakdown it is and even which domain it might be in. Structure of the paper The body of the paper has two broad parts. In the first part, the general component breakdown structure is described in three sections. The first section describes the composing operation at the heart of the structure and how this is reified into a composition relation which we call an atomic composition. It notes the close connection between the atomic composition and whole- part relations into which it can be dissected. In the second section, it looks at constrained atomic compositions, where the composing objects are disjoint, what we call strict atomic compositions. It then looks at how these atomic compositions can be joined together to create hierarchies, which we call molecular compositions. In the third section, ways of relaxing the constraints are examined. In the second broad part, there are also three sections. In the first, the analysis work in the first three sections is translated into an assessment framework. Then, briefly, further work is discussed and finally the report is summarised. The composing operation The rich formal structure of component breakdowns has been much commented on. We describe this hierarchical structure, and its associated flat part structure using a simple composing operation. A key feature of breakdowns is a property of the composite wholes; completeness – in that they are composed of just their components, no more. This ‘covering’ composing property has been noticed and commented on since the Ancient Greeks (see Appendix D). It has an obvious practical side, if a component is missing from a BOM or a WBS, then the composite product or project is not whole, it is incomplete. In many engineering situations, it probably won’t work or won’t work properly. In a sense, studying breakdowns is studying wholes in relation to their parts and so involves a kind of ‘holology’ (or ‘wholology’) – see (Cotnoir et al., forthcoming, p. 17). This property of completeness is missing from the individual whole-part relations in the associated flat structure. By itself, the individual relation carries no implication that it is linked to any particular complete whole – and so does not involve wholology. Though, as we shall see, given a plurality of such relations (with the right background theory) one can construct a whole – and so the composition property for that whole. super-set sub-set structure of classes whole-part structure of individuals heat engines engines my car’s body my car my car’s chassis my car’s engine my car’s interior my car’s wheels Non- thermal engines electrical engines physically powered engines Figure 1: Two clear cases rooted in mereonomy and taxonomy -- 7 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 8 A simple constructional process One can devise a simple general constructional process to generate both the composition and its associated whole-part structures (Partridge et al., 2017), (Partridge et al., 2019), (Partridge et al., forthcoming). One starts with a plurality – technically, a simple collection with no order and no repetition – of objects in a domain. This is input as components into the constructional process which builds (‘assembles’) the composite. The left-hand side of Figure 2 shows this, using a funnel to symbolise the operation. The centre has a graph representing the result of the operation. The right-hand side shows the result of dissecting the composition into whole-part relations. In this structure, components are (logically) assembled/composed into a single composite whole – so, in a sense, the composing operation (which we reify as a relation – or state of affairs – and call an ‘atomic composition’ – Figure 2 shows the relation and composition views) is an n-to-1-ary operation between the composite and the components (where the elements on the n-ary side of the relation are unique and unordered, in effect, a plurality). Though there are multiple possible approaches to characterising composition, for expository simplicity, here we assume an underlying theory of parthood, from which composition emerges. Our favoured theory is classical mereology (to be found in many places, including (Cotnoir et al., forthcoming)). In this approach, the fusion operation is the engine for composition. The composition operation builds the composite as the fusion of the components. An alternative constructional approach might take the composition operation – and so composition – as primitive and derive the part relations. A transparent account of composition should make clear what approach it takes. n-to-1-ary composing operation part relations a b ab ab a b ab b a composite components composing relation (view) components composing operation composite part relations parts whole atomic composition (view) Figure 2: A simple constructional process -- 8 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 9 In practical terms, it makes little sense to physically assemble less than two components – to assemble a single, or (indeed) no, component. But, for overall symmetry and consistency one might want to allow these last two cases – visualised in Figure 3. If one is allowed an assembly with no components, then the result would be the null component (see (Fine, 2010)). If one allowed a single component, then as the composite would be the same single component, one would be allowing things to be improper components of themselves. A transparent account of composition should make clear where its structure stands on these two points. Going forward, our default assumption will be that we are dealing with ‘proper’ components, so each composition will have multiple components and there will not be a null component. Alternative approaches may decide to make different choices. Where it makes sense, we include the formalisation in the body of the text, highlighted in grey – as below. We collect all the formalisation into one place, for ease of reference, in Appendix E. We make the fundamental propositions a little more rigorous by formalising them (in first-order logic). Our aim is not to provide a complete theory, only to delineate for the sake of clarity. We introduce the vocabulary we need, or just find useful, in formulating these propositions as we go. We make no attempt to provide a complete or minimal formalisation and we leave open whether it could be replaced or condensed by an alternative axiomatisation. Let us assume that: AtomicComposition(x) means x is an atomic composition. isAtomicComponentOf(x,y) means x is an atomic component of y, where y is an atomic composition. isAtomicCompositeOf(x,y) means x is an atomic composite of y, where y is an atomic composition. We can clarify that only atomic compositions have components and composites: null single n null e a b c f … abc… a Figure 3: Formal overall symmetry and consistency cases -- 9 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 10 (P1) ∀x ∀y ((isAtomicComponentOf(x,y) ∨ isAtomicCompositeOf(x,y)) → AtomicComposition(y)) Furthermore, for a given composition, components and composites are distinct: (P2) ∀x ∀y (isAtomicComponentOf(x,y) → ¬ isAtomicCompositeOf(x,y)) We refer to a standard account of the classical mereological fusion operator that allows us to denote the sum of all objects that satisfy a given property, ϕ, assuming the property has instances. This is standardly defined in terms of the Parthood relation. Under these assumptions, given a property, the fusion of the objects satisfying this property exists and is unique. This implies a sophisticated formal apparatus that we will use here for the convenience of presentation but recognise that we may need to consider an alternative treatment in a fuller formalisation. σxϕ(x) denotes the fusion of the x’s such that ϕ(x). We can then use fusion to express the claim that the atomic composite of an atomic composition is the fusion of its atomic components: (P3) ∀x ∀y (AtomicComposition(x) → (isAtomicCompositeOf(y,x) ≡ (y = σz(isAtomicComponentOf(z,x))))) Notice that by construction of atomic compositions as (unique) fusions of atomic components, it is the case that every atomic composition has a unique atomic composite: (P4) ∀x ∀y ∀z ((isAtomicCompositeOf(x,y) ∧ isAtomicCompositeOf(z,y)) → x=z) (Unicity of Atomic Composite) To adopt the assumption that there are at least two components is to accept a proposition we will call the Multiplicity of Atomic Components principle. (P5) ∀x (AtomicComposition(x) → ∃y ∃z (isAtomicComponentOf(y,x) ∧ isAtomicComponentOf(z,x) ¬ (y=z))) (Multiplicity of Atomic Components) Furthermore, as there seems to be no practical requirement for infinite composition from, for example, engineering practice, we shall assume here that the plurality of components is finite, which simplifies the formal structure. Again, for transparency, one should make clear where one’s structure stands on this and the earlier overall symmetry and consistency points. Let us call the results of a single composing operation, atomic compositions (later on, we join atomic compositions together to construct a hierarchy, which we call a molecular composition). The criterion of identity for atomic compositions is the same (plurality of) components. Two atomic compositions with the same components are the same composition. We have already introduced the formalism for atomic compositions as fusions of components above. The criterion for identity of atomic compositions can now be formulated as follows: (P6) ∀x ∀y ((AtomicComposition(x) ∧ AtomicComposition(y) ∧ ∀z (isAtomicComponentOf(z,x) ≡ isAtomicComponentOf(z,y))) → x=y) (Atomic Composition Criterion of Identity) -- 10 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 11 Associated part relation Given an n-to-1-ary atomic composition relation, one can formally ‘dissect’ it into multiple binary dissected composing relations which correspond to part relations – as shown in Figure 4. In order to articulate this point, and others, we will use a formalisation in which we reify the particular binary relations between a part and a whole. PartRelation(x) means that x is a particular part relation between objects. We use the predicates partInRelation and wholeInRelation to write ‘partInRelation(a,b)’ (respectively, ‘wholeInRelation(c,b)’) for a (respectively, c) is the part (respectively, the whole) in the part relation b. Note that we can make the link to the parthood relation predicate as follows: wholePart(x,y) means that y is a part of x. (P7) ∀x ∀y ∀z ((PartRelation(x) ∧ partInRelation(y,x) ∧ wholeInRelation(z,x)) → wholePart(z,y)) dissectingPartInAtomicComposition(x,y) means x is an atomic composition for which y is one of the dissecting part relations, i.e., where y is a part relation instance and the whole in y is the composite of x and the part in y is an atomic component of x Assuming an extensional criterion of identity, one can characterise these part relations – one part relation is identical with another if their whole and part projections are the same. They never appear twice in the same composition, though the same part relation can appear in multiple composition relations. (P8) ∀x ∀y ∀u ∀v ((PartRelation(x) ∧ PartRelation(y) ∧ partInRelation(u,x) ∧ partInRelation(u,y) ∧ wholeInRelation(v,x) ∧ wholeInRelation(v,y)) → x=y) (Part Relation Criterion of Identity) The part relations corresponding to the dissected composing relations uniquely define the composition, there is no other composition with exactly the same relations – in this sense, the part relations are a signature for the composition. n-to-1-ary ab b a part relations dissected composing relations a ab b a ab b dissects Figure 4: Harmonised dissected composing and part relations -- 11 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 12 (P9) ∀x ∀y ((AtomicComposition(x) ∧ AtomicComposition(y) ∧ ∀z (dissectingPartInAtomicComposition(x,z) ≡ dissectingPartInAtomicComposition(y,z))) → x=y) There naturally results in a requirement for a kind of ‘mereological harmony’ (used in a similar, but different, sense in (Uzquiano, 2011)), where the part and composition relations point in the same direction. Given this harmony, the part relations corresponding to the dissected composing relations are the domain’s part relations – if the dissected composing relation were the ‘wrong-way around’ – were not in harmony – then this would not happen – see Figure 5. However, in the dissection one loses a clearly identified whole with integral parts. Just given a plurality of part relations corresponding to dissected composing relations for multiple compositions there is generally no way to recover the original composition or identify which elements are the original composites. Figure 6 illustrates this – the left-hand part relations can be used to reconstruct multiple compositions, without any indication of which one is intended. If one starts with a single atomic composition, dissect to its corresponding part relations, and then add another part relation – one can no longer recover exactly the original composition. n-to-1-ary ab a abc dissected composing relations abc ab a part relations abc ab a dissects Figure 5: Disharmonised dissected composing and part relations abc a b c ab ac abc b a c abc bc a abc b a c abc a b c ab bc ac bc added Figure 6: Example of a loss of composition information in part relations -- 12 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 13 Though, as the dissected parts are a signature for the composition, given the correct set of part relations one can recover the original composition. The set contains enough information to reconstruct the whole. More generally, there are sets of part relations that correspond one-to- one with the compositions. For atomic compositions, this one-to-one correspondence can be formalised as the equivalence between a component decomposition and the existence of a part relation for each atomic component of the atomic composition such that the atomic component is the part and the whole is the atomic composite. (P10) ∀x ∀y ∀z ((AtomicComposition(x) ∧ isAtomicComponentOf(y,x) ∧ isAtomicCompositeOf(z,x)) ≡ ∃r (PartRelation(r) ∧ dissectingPartInAtomicComposition(x,r) ∧ partInRelation(y,r) ∧ wholeInRelation(z,r))) Hence, every component of an atomic composition is a part of the composite of that composition (while, in general, the converse is not the case in the absence of a non- dissecting part relation). (P11) ∀x ∀y ∀z ((isAtomicComponentOf(y,x) ∧ isAtomicCompositeOf(z,x)) → wholePart(z,y)) A (very) simple example It is perhaps easier to illustrate this with a simple example. Consider a case where there are two mereological atoms (objects with no parts). Assume we have a simple composing operation with no restrictions on what can be composed. As Figure 7 shows, we can fuse a and b to construct ab. This gives us a single composing relation which we can dissect into two (associated) binary whole-part relations. This is too simple. Things get more interesting with three mereological atoms. We use fusion to construct all the possible wholes. After we have exhausted it (completed all the possible fusions) we end up with the part relations shown in Figure 8. As noted earlier, compositions are atomic where there is only one composing operation and molecular when there is more than one – examples of these two kinds are shown in Figure 9. We consider atomic compositions initially. ab ab ab b a b a a b Figure 7: Simple case of two mereological atoms -- 13 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 14 abc a b c ab bc ac Figure 8: Simple case of three mereological atoms – all part relations abc b a c b a c abc abc bc a c b b c bc a abc atomic composition molecular composition Figure 9: Examples of atomic and molecular composition -- 14 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 15 Let us call atomic compositions strict when the components of the composition are disjoint. As, in our example, the mereological atoms have no parts and so are, by definition, disjoint, we have a simple way, in this example, of determining disjointness. We look at their ultimate parts, the atoms from which they have been composed. If they share no atoms, then they are disjoint (this of course, only works where things are composed of mereological atoms). StrictAtomicComposition(x) means that x is a strict atomic composition. disjointFrom(x,y) means that x and y are mereologically disjoint, that is to say, according to standard mereological definitions, they do not share a part. (P12) ∀x (StrictAtomicComposition(x) ≡ (AtomicComposition(x) ∧ ∀y ∀z ((isAtomicComponentOf(y,x) ∧ isAtomicCompositeOf(z,x) ∧ ¬ (y=z)) → disjointFrom(z,y)))) In our example, there are four strict compositions with a as one of the components. The algorithm for finding this is: start with a in the part hierarchy on the left of Figure 10, find each whole of which it is a part then find all the disjoint parts of that whole. This clearly illustrates how a component can be composed in multiple ways. Furthermore, there are four strict compositions with abc as their composite – as illustrated in Figure 11. This clearly illustrates how a composite can be strictly decomposed in multiple ways (there are multiple other weaker decompositions). One can view the last three atomic compositions in Figure 11 as the final stage in the construction history – where there are prior compositions. Figure 12 shows this for one of the cases. This shows the composition recapitulating the final stage of one of the composing histories of the composite. abc a b c ab bc ac abc b a c ab b a ac c a abc bc a Figure 10: Example of four strict compositions with ‘a’ as one of the components -- 15 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 16 The stricter (assembly) structure Engineers building a BOM or WBS will want their breakdown to be both complete (to cover everything relevant) and for the components to be separate, disjoint. So, together these make components a kind of inventory (or simple single level bill of materials) of the composite. Where a good inventory must be complete: everything must show up somewhere. But it must also be judicious: nothing should show up more than once. Thus, the inventory should cover the composite short of overlap: every component should be disjoint from every other component – and the components should cover the composite completely. The strict atomic compositions have exactly these properties. The components cover the whole composite, and the components are separate, disjoint. We start by looking at their structure more carefully. Breakdowns are often multi-level, so we then look at how these atomic compositions can be assembled into a hierarchy, which we have called a molecular composition, with the same properties. abc b a c abc ac b abc ab c abc bc a Figure 11: Example of four strict compositions with ‘abc’ as their composite b c a abc bc abc bc a c b Figure 12: Example of the final stage in the construction history -- 16 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 17 The simplest atomic assembly structure To recap, in this structure, disjoint components are (logically) assembled/composed into a single composite whole. One can reify the composing operation as a composing relation – an n-to- 1-ary relation that partitions the composite into components (where the elements of the n-ary side of the relation are unique and unordered, in effect, a plurality). This composing relation is called an atomic composition. As with all relations, this can be viewed as the relata linkage between the composite and its components – or as the linked relata including the composite and its components as visualised in Figure 13 – however, these cash out to the same as the linkage includes the relata. The composite is the fusion of the components – furthermore, the composition (relation) can be dissected into corresponding part relations; one for each component-composite pair. This is also visualised in Figure 13. Formal constraints on the components being assembled We have made a link between fusion and composition. In classical mereology a principle of unrestricted fusion is adopted, where any parts can be fused into a whole. But one does not have to adopt this, one can place restrictions on what can be fused (either in the underlying structure or in this structure). Firstly, the association with fusion brings a constraint. Fusion comes with an in-built constraint, it ensures that the components cover the composite; in other words, there is no part of the composite that does not overlap one or more of the components. So the partial cover, partial separation and full separation pairs in Figure 14 cannot be composed into x, cannot have x as their composite. This is a logical consequence of the link to fusion (from the underlying part structure). Secondly, we have a strictness constraint on the components. They need to be pair-wise disjoint – in other words, no pair can share a common part. Hence the overlapping and inside pairs in Figure 15 cannot be strictly atomically composed – though they could be non-strictly composed. Tiling is a geometric procedure with the slogan ‘no gaps, no overlaps’ – which clearly applies here. This suggests another formal way of looking at this. As a kind of mereological (that is, sub- metrical) tiling constraint. Where, formally, a tiling is a collection of disjoint open regions, the closures of which cover the selected region, typically the whole plane (for more details see e.g. (Stein et al., 1994)). n-to-1-ary part relations composing operation components composing operation composite part relations parts whole composite components composing relation (view) atomic composition (view) Figure 13: Three ways to visualise a simple atomic assembly structure -- 17 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 18 The requirement that the atomic components cover the atomic composite can be expressed as the requirement that the fusion of all the atomic components is the atomic composite whole. We do not formally express this requirement here. The requirement that the components do not overlap (are disjoint) is the requirement that no two components’ entities have a common part or equivalently that any two components of a composition are mereologically disjoint: (P13) ∀x ∀y ∀z ((isAtomicComponentOf(x,y) ∧ isAtomicComponentOf(z,y) ∧ ¬ (x=y)) → disjointFrom(x,z)) cover partial cover partial separation separation x a ab b x a b a b x a b Figure 14: Cover and ways to not cover disjoint overlapping inside b a b a b a common part Figure 15: Disjointness and ways to not be disjoint -- 18 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 19 Material constraints on the components being assembled The constraints to fusion may be material as well as formal; one that only allows some types of object to be so composed. (Fine, 1999, p. 72) describes one kind of system that emerges from imposing a material constraint (his system of embodiment): “The majority of material objects […] will submit to a hierarchical division into parts. Just as a car will have an engine, a chassis, and a body as immediate parts (these being the components of the rigid embodiment that is the current manifestation of the car), these immediate parts will themselves have further immediate parts, and so on all the way down until we reach the most basic forms of matter. Thus a material object will be like a set, with its hierarchical division into members, members of members, and so on. In this Finean world, only some of the objects participate in the composing hierarchy – so only some objects can be composed into whole composites. If one pursues this route, then one needs to make the material constraints explicit – as Fine does. Unique compositions and decompositions? Once one has settled on the constraints, formal and material, one knows which pluralities of objects qualify for use as components in a composing relation. Given one of these qualifying pluralities then a natural assumption is that its members’ fusion uniquely composes a composite whole. In inventory-speak, each inventory is for a single unique object. One can then ask questions about, in terms of the TLO Survey (Partridge et al., 2020, sec. 4.2.1.1), whether the child-arity and parent-arity are single or multiple. Here we just consider the atomic compositions. Later, when we have introduced hierarchies, we ask the same questions relative to a hierarchy. The first relation to consider is the (atomic) composing relation. Within the scope of a composition, this is defined as having single parent-arity and of multiple child-arity. The next relation to consider is the participation (as composite or component) relation. In terms of child-arity and parent-arity, this is whether an object can participate as a component in multiple atomic compositions (parent-arity) and whether an object can participate as a composite in multiple atomic compositions (child-arity) – see Figure 16. abc b a c abc bc a abc multiple-(participation) child-arity a multiple-(participation) parent-arity Figure 16: Participation (as composite or component) relation -- 19 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 20 Earlier (see Figure 10) we established that with unrestricted fusion an object can participate in multiple compositions. In the survey’s terms, this would be a case of multiple child-arity. This would seem to be an essential feature of composition. This raises the question whether (symmetrically) there could also be single, unique, decompositions, where each composite whole uniquely decomposes into a plurality of components. Whether, in the survey’s terms, the parent-arity is single or multiple. Fine, in the next paragraph after the one quoted in the previous section, writes: “… this division into parts will be largely unique”. So he envisages a structure where the participation (as composite or component) relation is largely unique – where the child-arity and parent-arity is largely single. However, this is an unusual position. In practice, as (Wimsatt et al., 2007, p. 182ff) notes scientists break down natural objects in many ways – in fact, he suggests this multiple decomposition is the mark of complex natural structures. Similarly, engineers typically provide multiple breakdowns of complex systems. In this case, the decomposition is not unique. When one proposes a system of compositions, one should be clear about these aspects of its formal structure. In the same spirit of plenitude (and simplicity) that infuses classical mereology and set theory – noted in the TLO Survey (Partridge et al., 2020, sec. 3.2.1), one could say that, given a part relation in a domain, then every possible partition based upon it is also a composition – in other words, that every disjoint plurality of objects in the domain can form a composite. This would allow for every possible multiple composition. It would only remain to highlight the more interesting ones. One atomic composition atop another Atomic compositions can be atop one another – where the composite of one composition is a component of another. We call this the atop relation. It is a relation between the compositions though it reflects a relation between their contents – as shown in Figure 17. This relation induces a partial order over the compositions. transparent opaque abc a bc b c bc atop relation Figure 17: Atop relation -- 20 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 21 These atop relations can be seen as a composing sequencing relation, relating one composing operation to another to tie together the compositional history – as shown in Figure 18. This also shows a composition with multiple atop relation children – one corresponding to each of the possible compositions the element can appear as a composite – so it can have multiple child-arity. And, though not shown, a composition can have multiple atop relation parents – one corresponding to each of the possible compositions the element can appear as a component. In other words, the atop relation can have multiple parent-arity. The atop relation can be seen (from a graph-theoretic perspective) as a branch re-composition. Where the branches are the histories, paths that trace out a tree. And the atop structure is the directed graph that is walked to produce these paths – hence a branch re-composition. Atomic immediate parts If one reads the atop relation as relating the composites of its related atomic compositions, then this is a kind of part relation – one that tracks only the underlying part relations between the composites. Read this way, the atop relation mereology has several distinctive features. For example, it is a mereology of ‘immediate parts’. Where an immediate part has no intervening mediate parts – as shown in Figure 20. This distinction of immediate and mediate parts goes back to (Husserl, 1970). The structure is (unlike normal part relations) antitransitive (if a is part of b and b is part of c, then a is not part of c). So in Figure 18, the lowest level atomic composition cannot be atop related to the top level composition. a abc bc c b ef f e abc def d ef d abc abcdef e f a bc b c Figure 18: Example compositional history an ‘atop’ directed graph Figure 19: The branch recomposed atop directed graph -- 21 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 22 Dividing and combining atomic compositions Atomic compositions can be divided and combined. The division involves taking a subset of the composition components and replacing it with another atomic composition – which gives two compositions one atop the other. Combining is effectively the reverse – as shown in Figure 21. One takes two compositions, one atop the other, and combines them, removing the overlapping element. abc a b c ab bc ac a mediate part an immediate part Figure 20: immediate and mediate parts Figure 21: Dividing and combining atomic compositions -- 22 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 23 One can see these operations as shifting the path chosen through the construction history between the component leaves and the composite root – as shown in Figure 22. Another way to look at this is as navigating the inherent transitivity in the underlying sibling part hierarchy. Under transitivity if b and c are parts of bc and bc is part of abc, then b and c are parts of abc. Hierarchies of strict atomic compositions We now have an informal sketch of the structure and options for atomic compositions. With these in hand, we can build hierarchies of them, what we call molecular compositions. Assembling atomic compositions into a hierarchy The process of joining together some atomic compositions into a ‘molecular’ hierarchy composition is simple. Firstly, we use the notion of ‘atop’ to characterise molecular compositions built from two atomic compositions where one is atop the other (in other words, where the atomic composite of one is also an atomic component of the other). Consider the molecular composition that results from joining these two. It has the two atomic compositions as atomic constituents. One can characterise this through the single atop relation that binds it together. More generally, subject to other constraints, associated with each atop relation, there is a doubleton molecular composition with two atomic constituents. If the atomic compositions being joined are disjoint, then the resulting doubleton molecular composition is too – through the shared element that ‘supports’ the atop relation. multiple composition history single composition history a abc bc c b a abc b c abc bc b c a abc bc b c bc a abc b a c Figure 22: Example shifting the path chosen through the construction history -- 23 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 24 atop(x,y) means that atomic composition x is atop atomic composition y. atopAt(x,y,z) means that atomic composition x is atop atomic composition y and z is a component of x and the composite of y. (P14) ∀x ∀y (atop(x,y) ≡ (AtomicComposition(x) ∧ AtomicComposition(y) ∧ ∃z (isAtomicComponentOf(z,x) ∧ isAtomicCompositeOf(z,y)))) MolecularComposition(x) means x is a molecular composition. atomicJoinedInto(x,y,z) means that (the composite of) atomic composition x and (a component of) atomic composition y are joined into (doubleton) composition z. isAtomicConstituentOf(x,y) means that x is an atomic constituent of y. When two atomic compositions are in the atop relation, there is a molecular composition (though only the doubletons): (P15) ∀x ∀y (atop(x,y) → ∃z (MolecularComposition(z))) The above is weak and we use joinedInto to express the stronger joining of the atomic compositions in a molecular one. (P16) ∀x ∀y (atop(x,y) → ∃z (MolecularComposition(z) ∧ atomicJoinedInto(x,y,z))) That the joined atomic compositions are atomic constituents of the molecular composition resulting from the joining operation can be expressed as: (P17) ∀x ∀y ∀z (atomicJoinedInto(x,y,z) → (isAtomicConstituentOf(x,z) ∧ isAtomicConstituentOf(y,z))) In fact, we can define isAtomicConstituent in terms of joinedInto: (P18) ∀x ∀y (isAtomicConstituentOf(x,y) ≡ ∃z (atomicJoinedInto(x,z,y) ∨ atomicJoinedInto(z,x,y))) Then we need to extend some of the atomic vocabulary to cover molecular compositions. The elements of this molecular hierarchy can be classified in three ways, based upon their ‘roles’ in the constituent atomic compositions. A single top ‘composite’ element – which belongs to only one atomic constituent and is a composite in that atomic composition. This constituent we call a top atomic constituent. Joined elements, which each belong to two atomic constituents, where it is a composite in one of these atomic compositions and a component in the other. All component elements that are not joined are free elements. This classification enables us to say that one can join a molecular composition to an atomic composition at its free elements to build a new molecular composition. Atomic constituents that have one or more free component elements are called open atomic constituents. Atomic constituents that are not open are called closed atomic constituents. Atomic compositions are their own atomic constituents. These new classifications are visualised in Figure 23. isCompositeOf(x,y) means that x is the composite element of (the composition) y. isFreeComponentOf(x,y) means that x is a free component of y. isJoinedElementOf(x,y) means that x is a joined element of y. isTopAtomicConstituentOf(x,y) means that atomic composition x is a top atomic constituent of y. -- 24 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 25 isOpenAtomicConstituentOf(x,y) means that atomic composition x is an open atomic constituent of y. isClosedAtomicConstituentOf(x,y) means that atomic composition x is a closed atomic constituent of y. Every molecular composition has a single top atomic constituent: (P19) ∀x ∀y ∀z ((MolecularComposition(x) ∧ isTopAtomicConstituentOf(y,x) ∧ isTopAtomicConstituentOf(z,x)) → y = z) It follows that every molecular composition has a unique composite: (P20) ∀x ∀y ∀z ((MolecularComposition(x) ∧ isCompositeOf(y,x) ∧ isCompositeOf(z,x)) → y = z) The composite of a molecular composition is the atomic composite of its top atomic constituent: (P21) ∀x ∀y ∀z ((isCompositeOf(x,y) ∧ isAtomicConstituentOf(x,z)) → isTopAtomicConstituentOf(z,y)) It follows that the composite belongs to a single constituent: (P22) ∀x ∀y ∀z ((isCompositeOf(x,y) ∧ isAtomicConstituentOf(y,z) ∧ isAtomicConstituentOf(u,z)) → u=y) We can break down the definition of free component above into 3 propositions possibly overlapping. Firstly, free components in a molecular composition are atomic components in one of its atomic constituents: (P23) ∀x ∀y (isFreeComponentOf(x,y) → ∃z (isAtomicConstituentOf(z,y) ∧ isAtomicComponentOf(x,z))) Furthermore, a free component belongs to only one atomic constituent: (P24) ∀x ∀y ∀u ∀v ((isFreeComponentOf(x,y) ∧ isAtomicConstituentOf(u,y) ∧ isAtomicConstituentOf(v,y) ∧ isAtomicComponentOf(x,u) ∧ isAtomicComponentOf(x,v)) → u = v) The atomic constituent to which a free component belongs is in fact an open atomic constituent: (P25) ∀x ∀y ∀z ((isFreeComponentOf(x,y) ∧ isAtomicConstituentOf(z,y) ∧ isAtomicComponentOf(x,z)) → isOpenAtomicConstituentOf(z,y)) We can define joined elements as above: (P26) ∀x ∀y (isJoinedElementOf(x,y) ≡ ∃u ∃v (isAtomicConstituentOf(u,y) ∧ isAtomicConstituentOf(v,y) ∧ ¬ (u = v) ∧ isAtomicCompositeOf(x,u) ∧ isAtomicComponentOf(x,v))) Open atomic constituents are atomic constituents with free components: (P27) ∀x ∀y (isOpenAtomicConstituentOf(x,y) ≡ (isAtomicConstituentOf(x,y) ∧ ∃z isFreeComponentOf(z,x))) Closed atomic constituents can also be defined as above: (P28) ∀x ∀y (isClosedAtomicConstituentOf(x,y) ≡ (isAtomicConstituentOf(x,y) ∧ ¬ isOpenAtomicConstituentOf(x,y))) -- 25 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 26 With these definitions, we can define a general joining relation using atop. Given any composition, we can join another atomic composition to open atomic constituents if they are in the right kind of atop relation. The joined atomic composition becomes a constituent of the new molecular composition. In this way, step by step, atomic constituent by atomic constituent, we build up all possible molecular hierarchies. A simple example is shown in Figure 24, where atomic compositions are added one by one (we use a ‘+’ sign in the name to mark this addition). This process makes clear that compositions can be characterised by their atomic constituents (where atomic compositions are deemed to have just themselves as atomic constituents). atomic constituent view element view top ‘composite’ element joined element free element open atomic constituents closed atomic constituent molecular composition top atomic constituent Figure 23: Molecular composition views and classifications -- 26 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 27 Composition(x) means x is a composition. topJoinedInto(x,z,w) means that the composition x is joined at the top to atomic composition z to build composition w. Compositions can be joined with atomic compositions at the top into new compositions: (P29) ∀x ∀y ∀z ((Composition(x) ∧ isTopAtomicConstituentOf(y,x) ∧ AtomicComposition(z) ∧ atop(z,y)) → ∃w (topJoinedInto(x,z,w) ∧ ∀u (isAtomicConstituentOf(u,x) → isAtomicConstituentOf (u,w)) ∧ isAtomicConstituentOf(z,w))) bottom joined into top joined into (atomic) joined into abc +bc abcdef abc +bc ef abc def d ef d abc abcdef abc bc a abc bc c b abc a bc b c abc+bc abc +bc abc +bc abc +bc abcdef +abc+bc abc abcdef +abc+bc abcdef +abc+bc ef ef f e ef e f abcdef +abc+bc +ef abcdef +abc+bc +ef abcdef +abc+bc+ef abcdef +abc+bc abcdef +abc+bc abcdef +abc+bc Figure 24: Step-by-step construction of molecular compositions -- 27 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 28 In these cases, the ‘new’ top constituent, the top constituent of the result of joining is the atomic composition used in the joining. (P30) ∀x ∀y ∀z ∀w ((Composition(x) ∧ isTopAtomicConstituentOf(y,x) ∧ AtomicComposition(z) ∧ atop(z,y) ∧ topJoinedInto(x,z,w)) → isTopAtomicConstituentOf(z,w)) bottomJoinedInto(z,x,w) means that atomic composition z is joined to the bottom of composition x to build composition w. Compositions can be joined with atomic compositions at the bottom into new compositions: (P31) ∀x ∀y ∀z ((Composition(x) ∧ isOpenAtomicConstituentOf(y,x) ∧ AtomicComposition(z) ∧ atop(y,z)) → ∃w (bottomJoinedInto(z,x,w) ∧ ∀u (isAtomicConstituentOf(u,x) → isAtomicConstituentOf (u,w)) ∧ isAtomicConstituentOf(z,w))) In these cases, the ‘new’ top constituent is the ‘old’ top, that is the top atomic constitutent of the composition used in the joining. (P32) ∀x ∀y ∀z ∀w ∀u ((Composition(x) ∧ isOpenAtomicConstituentOf(y,x) ∧ AtomicComposition(z) ∧ atop(y,z) ∧ bottomJoinedInto(z,x,w) ∧ isTopAtomicConstituentOf(u,x)) → isTopAtomicConstituentOf(u,w)) Compositions resulting from joining are made of the constituents of the compositions joined: (P33) ∀x ∀y ∀z ∀u (joinedInto(x,y,z) → (isAtomicConstituentOf(u,z) ≡ (isAtomicConstituentOf (u,x)) ∧ isAtomicConstituentOf(y,z)) Compositions are atomic compositions or molecular compositions resulting from the general joining operation: (P34) Composition(x) ≡ (AtomicComposition(x) ∨ MolecularComposition(x)) Another way to visualise this is as taking a connected tree sub-graph of the atop relation graph, where every branch relates to different components – see Figure 25. As this last way of visualising makes particularly clear, molecular compositions can be characterised (in part) by the atop relations they involve. Figure 25: visualising connected tree sub-graph histories -- 28 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 29 Criterion of identity The earlier comments translate into a criterion of identity for compositions – that of being constituted by the same atomic constituents. Two molecular compositions with the same atomic constituents are the same composition. Two atomic compositions with the same atomic constituents (that is, themselves) are plainly the same composition. Based on the formalism introduced already, the criterion for identity of compositions can now be formulated as follows: (P35) ∀x ∀y ∀z ((Composition(x) ∧ Composition(y) ∧ ∀z (isAtomicConstituentOf(z,x) ≡ isAtomicConstituentOf (z,y))) → x=y) (Composition Criterion of Identity) A new ‘mereology’ For those familiar with the history of mereology, the connected graph visualisation suggests the possibility of a mereology of connected regions. One similar to Whitehead’s region mereology in variously (Whitehead, 1916), (Whitehead, 1919), (Whitehead, 1920), (Whitehead, 1929) and the associated (De Laguna, 1922) and (Clarke, 1981). This should not be surprising as, for example, (Cotnoir et al., forthcoming) notes how Whiteheadian mereologies can arise through relativisation. This is interesting as Whitehead was aiming for a structure of well-behaved regions, and this suggests that compositions may behave in a similar fashion. Whitehead originally used ‘extends over’ as his primitive but revised this after Laguna’s paper to use ‘connection’ and Clarke followed suit. The atop relation could be regarded as a similar kind of connection relation. There is a difference, but it is not relevant – Whitehead’s theory was atomless and this is atomic – being built from atomic compositions. Another approach would be to define a part relation between compositions based upon all the atomic constituents of the part also being parts of the whole. This structure allows for mediate parts (unlike the earlier atomic atop structure). Its fusion operation – joins – is not unrestricted. It only joins atop-related compositions. This ensures that the result is atop-connected (in a similar fashion to Whitehead ensuring all his regions are connected). It also has simple boundary – connection conditions. Under the hood, the atop- connected compositions share an object. Extending strictness to molecular compositions The extension from atomic to molecular is simple. Strict molecular compositions are those whose atomic constituent compositions are all strict. One can generalise the definition. A composition is strict if all its atomic constituents are strict. Associated parts – extending dissection to molecular compositions One can dissect the associated parts of a molecular composition by collecting the dissected parts of all its atomic constituents. Molecular compositions share many of the properties of atomic compositions, making these general composition properties. For example, dissected part relations do not appear twice in the same molecular composition, though they can appear many times in different compositions, molecular or atomic. The dissected part relations uniquely define the molecular composition, there is no other composition with exactly the same relations – in this sense, the relations are a signature for the molecular composition – and more generally for compositions. -- 29 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 30 The mode of construction of molecular compositions preserves the ‘mereological harmony’ of their atomic constituents. So there is no dissected relation that is the ‘wrong-way around’. Compositional immediate parts We mentioned earlier that the atop relation induced an immediate part structure over the atomic compositions. The molecular compositions (due to their mode of construction) have a parallel immediate part structure. One way of visualising this is starting with all the part relations (which, if dense, might not have any immediate parts) and pick out sub-sets of these that correspond to molecular compositions. These will only contain connected immediate parts. Material constraints on the component hierarchy The material constraints on the atomic compositions are also inherited by the hierarchies – as only the compositions allowed by those material constraints can participate in the hierarchy. Additional material constraints can be added if required. Though, in practice these seem to be local to specific hierarchies rather than global. Where, for example, a location breakdown structure may restrict its nodes to locations, whereas a systems breakdown structure might restrict its nodes to systems. Overlap for molecular component hierarchies There are various ways two molecular compositions can overlap. The most liberal is where two compositions share elements: this does not imply they share any atomic constituents. Atop is a specialised case of this. A less liberal, coarser grained, way is where two compositions share atomic constituents: this does not imply that the full overlap is a composition. An even less liberal, coarser grained, way is where two compositions’ overlap is another composition. These are illustrated in Figure 26. constituent overlap ef ab cd composition overlap ab cd ef gh element overlap ab bc a ab c bc b atomic compositions ab a b ab cd c d cd ef e f ef gh g h gh Figure 26: Overlapping molecular component hierarchies -- 30 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 31 Refining and collapsing constituents in a component hierarchy The dividing and collapsing of atomic compositions form the basis for the refining and collapsing of molecular hierarchies. When an atomic constituent of a molecular hierarchy (an atomic composition) is divided this results in a refined molecular hierarchy. When the reverse happens and two atomic constituents are combined, the result is a collapsed hierarchy. In the first case, there is an insertion of an element (refinement) and the second a removal of an element (collapse) – see this visualised in Figure 27. One can concatenate a series of refinements or collapses – these then zoom us in and out of the underlying part structure. Interestingly, it is feasible to zoom in and then zoom back out to a different hierarchy. More specifically, the process of reduction involves collapsing (a process described above) a joined atomic constituent and into two joined atomic constituent in the composition hierarchy. The process of extension involves dividing (also described above) an atomic constituent. Obviously, there can be a series of these processes taking one from one hierarchy to another. Generating composition structures One can approach the question of formal generation (see the TLO Survey, Section 4.2.1.5 (Partridge et al., 2020) in a number of ways. The way we favour would be, for each domain, to bind the part and composition relations, including their formal generations. Where every disjoint plurality of parts is input for an atomic composition and every atomic composition breaks down into part relations. If one adopts a plenitudinous approach to part relation generation, then this is inherited by composition. Given this one then (plenitudinously) formally generates every possible composition hierarchy. There are other possible strategies. One could avoid formal generation and only recognise the compositions that are ‘valid’. One could generate the part relations from the valid compositions. Whichever decision is made, it is good to make this explicit. Figure 27: Refining and collapsing constituents example -- 31 of 62 -- A Framework for Composition: A Step Towards a Foundation for Assembly 32 Relaxing the strict (de-)composing structures In many cases, the strict structures make sense. In the case of physical assembly, the process of physical assembly involves putting disjoint parts together to make a complete whole. Similarly with physical tiling or tessellation, the individual tiles are disjoint and are combined to cover the whole floor. In fire safety engineering, the physical asset is similarly divided into fire zones. However, there are cases where the strictness may seem too constraining. We look at two approaches, firstly at one of relaxing then at one of adaption. Relaxing disjointness There are cases where disjointness seems too strict. For example, consider a car whose systems breakdown includes a fuel system and an electrical system. The systems are largely disjoint. But at the lower levels they overlap. For example, the fuel pump is part of both the fuel system and the electrical system – as shown in Figure 28. There is some leeway for increasing the disjointness; one could argue that the pump is not really part of both the fuel system and the electrical system, and that it is only the pump head that is part of the fuel system and the electric motor that is part of the electrical system. But this only works so far, eventually there is some irreducible overlap; the shaft that links the motor to the pump head must be a part of both or nothing happens, and it is the load on the fuel pump that determines the draw on the battery. One cannot argue that compositions are not necessarily disjoint. In the simple examples at the beginning of the report, we chose to filter out the non-disjoint compositions. So we have options to relax the disjointness constraint. A simple relaxation that still places a degree of constraint is one that, within an atomic composition, while the components can overlap, no components are parts of one another. (Schaffer, 20