A 4-Dimensionalist Top Level Ontology (TLO): Mereotopology and Space-Time 4-Dimensionalism in Large Scale Data Sharing and Integration Newton Gateway to Mathematics Chris Partridge – BORO Solutions -- 1 of 57 -- Team effort: the BORO Team http://www.borosolutions.net/who-we-are 2 -- 2 of 57 -- Top Level Ontology Foundation Data Model Industry Data Models – Reference Data Integration Architecture Process Model based Information Requirements Information Quality Management Core Constructional Ontology Where in the Seven Circles of Information Management 3 -- 3 of 57 -- Preliminaries - overall approach How, broadly speaking, do we develop the ontology? 4 -- 4 of 57 -- Target structure A modular, component-based architecture – motivated by the usual reasons: • complexity management • understandability • encapsulation • simpler substitution/replacement • recombinability • expandability • resilience 5 -- 5 of 57 -- coupling cohesion Modular: loosely coupled, highly cohesive modules coupling • about the relations between the modules cohesion • about the relations between the elements within the module loose coupling, high cohesion aims for a structure where relations cluster inside the modules looser higher tighter lower 6 -- 6 of 57 -- ‘Agile’ development: an iterative, adaptive approach At each stage, the ‘product’ needs to be: • something useful and usable • a step towards the target structure 7 mereotopology and space-time -- 7 of 57 -- Situating 4D in ontological space A requirement for space-time is central 8 -- 8 of 57 -- Reminder: 4D ontologies’ choices category type choice 4D Ontologies horizontal aspect spacetime unifying or separating unifying locations unifying or separating unifying properties unifying or separating unifying endurants unifying or separating unifying immaterial unifying or separating unifying 4D ontologies are maximally unifying Motivation: • the perceived benefits of parsimony and cost of separation • the easy fit with plenitude 9 https://digitaltwinhub.co.uk/a-survey-of-top-level-ontologies/#a_survey_of_TLOs_contents -- 9 of 57 -- Two (of many) levels of unifying temporal locations spatial locations spatio-temporal locations more stratified more unified happens at locations in space and locations in time are unified as locations in space-time supersubstantival objects spatio-temporal locations material entities supersubstantival objects located at matter and the space-time it is located in are unified as supersubstantival objects – matter is then a way space- time can be 10 -- 10 of 57 -- Broad modularisation context 11 -- 11 of 57 -- Data perspective: four coarse grained layers Reference Data Foundation Data Model Top Level Ontology Core Constructional Ontology scope 12 -- 12 of 57 -- Basis: Core Constructional Ontology Provides the ‘base’ for the whole ontology: • object completeness – builds all the objects • categorical completeness – establishes all the (ontological) categories of objects • extensional identity criteria – establishes an extensional criteria of identity for the objects 13 -- 13 of 57 -- First iteration: scope What should the scope of the first ‘MVP’ be? 14 -- 14 of 57 -- Industry Data Model survey provides a context ISO 10303 Part 42 INSPIRE OS Open Names A core set of standards and associated data 15 These cover ‘geometry’ from CAD and geographical perspective https://digitaltwinhub.co.uk/a -survey -of-idms -and -rdls -intro/ -- 15 of 57 -- Geometry - CAD – examples from BuildingSMART https://standards.buildingsmart.org/IFC/RELEASE/IFC2x/FINAL/HTML/ifcproductextension/lexical/ifcspace.html multiple coordinate systems multiple spatial objects 16 -- 16 of 57 -- Geometry - CAD – examples from STEP (ISO 10303 Part 42) coordinate systems multiple spatial objects ISO 10303 Part 42 17 -- 17 of 57 -- Geometry - Geography – examples from TC211/INSPIRE coordinate systems multiple spatial objects INSPIRE OS Open Names 18 -- 18 of 57 -- Two (three) clear candidates emerge spatial objects one, two or three (spatial) dimensional objects either eternal or recurring (at each snapshot) technically rigid, in the sense of no (or practically no) deformation over time spatial locations where the spatial objects are located • typically expressed as a (2/3D) coordinate system note: these have the same characteristics as spatial objects (eternal/recurring and rigid) (names things, including the objects and locations have names) Current situation: • space appears to be Euclidean • no clear way of dealing with time (mostly, not even mentioned) • no deep formalization 19 -- 19 of 57 -- Some examples ISO 10303 Part 42 INSPIRE OS Open Names places spatial objects geography geometry roads spatial objects geography geometry solids spatial objects CAD geometry surfaces spatial objects CAD geometry points spatial objects CAD geometry Eastings and Northings spatial locations geography geometry coordinates (2D and 3D) spatial locations CAD geometry examples 20 -- 20 of 57 -- Proposed scope: ‘space-time’ Proposed scope: • a formalized spatio-temporal account of spatial objects and locations • hence named ‘space-time’ Spatio-temporal challenge: 1. provide a clear way of dealing with space-time, space and time 2. provide a deep formalization of this Spatial challenge: 1. provide, an account of spatial objects and locations in space-time • there is a nexus of things needed for this, including: • rigid objects and being relatively at rest 21 -- 21 of 57 -- space-time The space-time ‘module’ in context relations (foundation extension) names 22 ST: ISO 10303:42 ST: IFCs ST: INSPIRE core foundation focus of this presentation for another time next presentation needed for some of the space-time data -- 22 of 57 -- Top-down and bottom-up approach 23 -- 23 of 57 -- Approach – two workstreams 1. Top-down workstream: • (where top is the more general or abstract) • builds the formal ontological skeleton • discuss in following sections 2. Bottom-up workstream: • (where bottom is the more concrete) • data driven • mine an ontology from current industry standard schemas and data • Evidence-based Ontological Requirements Elicitation • also called: bCLEARer 24 -- 24 of 57 -- Two mutually supporting workstreams The two workstreams validate and inform each other • top-down workstream guides the bottom-up workstream • bottom-up workstream validates the top-down workstream; • its completeness and fit • early de-risking top-down workstream bottom-up workstream 25 -- 25 of 57 -- Bottom up workstream – breakdown - visualisation CAD Geometry ISO 10303 Part 42 buildingSMART IFC geographic information INSPIRE – OS Open Names geoSPARQL Bottom up Workstream Spatial Objects Coordinates Names 26 NB: As noted earlier, coordinates are a form of spatial location -- 26 of 57 -- Bottom-up bCLEARer process: visualisation data load evolve assimilate collect load evolve assimilate collect load evolve assimilate collect increasing semantic maturity reuse reuse Foundational ontology A repeated sequence of processes: increasing semantic maturity 27 -- 27 of 57 -- Bottom-up workstream – stages b(e) Collect collect the datasets in scope in order to establish the broad scope of the process – establishing a bCLEARer master dataset Load define the detailed scope by selecting from the Collect dataset the data in scope Translate the dataset into the cell-based format – the table paradigm Evolve reveal the underlying semantics of the Load Dataset – ‘entification’ – in an ‘entified’ dataset Mine the ontology from the ‘entified’ dataset – the Evolve ontology dataset Assimilate merge the Evolve ontology dataset into the full ontology model Reuse publish dataset in a format suitable for the reuse context collect load evolve assimilate reuse In-scope Out -of-scope 28 -- 28 of 57 -- Space-time – top-down workstream 29 -- 29 of 57 -- Formalising space-time: some historical precedents Year Author Title 1919 A. N. Whitehead An enquiry concerning the principles of natural knowledge 1928 R. Carnap The Logical Structure of the World 1936 B. Russell On order in time 1939 J. H. Woodger The Technique of Theory Construction 1981 P. Needham Temporal Intervals and Temporal Order 1982 C. Lejewski Ontology: What's Next? Opportunity to build upon previous work: 30 -- 30 of 57 -- A recurring theme: worldlines: the first example Illustration: Minkowski, Hermann (1909), "Raum und Zeit", Physikalische Zeitschrift, 10: 75–88 31 In Minkowski’s formalization: • the worldline is the path that a particle traces in 4-dimensional spacetime • worldlines are primitive • so ‘points’ are derived as the intersections of the worldlines -- 31 of 57 -- Worldlines: a logical (ontological) primitive Year Author Title 1909 H. Minkowski Raum und Zeit 1958 R. Carnap Introduction to symbolic logic and its applications 1972 P. Suppes Some open problem in the philosophy of space and time 2008 T. Benda A formal construction of the spacetime manifold “49. ASs OF SPACE-TIME: TOPOLOGY: 2. THE Wlin-SYSTEM The present second form is called the Wlin-system. Its single primitive sign is ‘Wlin’.” Rudolf Carnap (1958) Introduction to symbolic logic and its applications. Selected examples 32 -- 32 of 57 -- Worldlines: a more physical primitive (selected examples) Year Author(s) Title 1972 R. Penrose Techniques of differential topology in relativity 1973 S.W. Hawking G. F. R. Ellis The large scale structure of space-time 1976 S.W. Hawking A. R. King P. J. McCarthy A new topology for curved space-time which incorporates the causal, differential, and conformal structures 1977 D. Malament The class of continuous timelike curves determines the topology of spacetime 33 -- 33 of 57 -- Why worldlines? Proposed ‘space-time’ scope: • a formalized spatio-temporal account of spatial objects and locations Worldlines handle the full span of the challenges • we’ve noted that they have been used to characterise space-time • we now show they do this in a way that also naturally characterizes rigid (that is, spatial) objects and their locations (and so also coordinate systems) 34 See: DiSalle, Robert, "Space and Time: Inertial Frames", The Stanford Encyclopedia of Philosophy (Winter 2020 Edition), Edward N. Zalta (ed.), Partridge, C.: An Information Model for Geospatial and Temporal Reference. 2011 -- 34 of 57 -- Worldlines: the same ‘spatial’ place Being in the same ‘spatial’ place is remaining on the same ‘coordinate’ reference worldline s1 s2 s3 (x, y, z) (x’, y’, z’) (x”, y”, z”) 35 time -- 35 of 57 -- Worldlines: different reference worldframes s1 s2 s3 f2 f1 f2 worldframe s1 s2 s3 f2 f1 f1 worldframe There are different reference worldframes and so different ‘same’ places These reference frames are sets of ‘mutually at rest’ worldlines: worldframes Galilean relativity 36 time -- 36 of 57 -- Worldframes: the same ‘spatial’ object Being the same ‘spatial’ object (no spatial change – or deformation) involves one’s parts staying on worldlines that are mutually at rest – equidistant – so in the same reference worldframe 37 rigid rod: stays on the same reference worldlines reference worldframe worldlines -- 37 of 57 -- Worldlines: how many different reference worldframes Each set of ‘mutually at rest’ worldlines marks out a (reference) worldframe If the two curves circulating around p1 (and correspondingly p2) are correctly aligned, then they are also mutually at rest and so mark out a (reference) worldframe s1 s2 s3 38 o1 o2 p1 p2o3 o4 o5 o6 relatively at rest relatively uniform motion -- 38 of 57 -- Single Platform Multi-Platform Requirements for multiple coordinate systems 39 -- 39 of 57 -- Tracing out higher-dimensional objects point surface line worldvolume worldsheet worldline 40 Objects of higher dimension can trace out shapes in space-time -- 40 of 57 -- space-time Decomposing the space-time ‘module’ using worldlines worldlines From the perspective of the space-time ‘module’, there is an opportunity for horizontal slicing – modularization: • a decomposition into two sub-components based upon the 7 circles: • from core to worldlines • from worldlines to spatial objects and locations Foundation Data Model Top Level Ontology core foundation 41 -- 41 of 57 -- Space-time: Foundation Data Model from worldlines to spatial objects and locations 42 -- 42 of 57 -- space-time Space-time: foundation data model worldlines Foundation Data Model from worldlines to spatial objects and locations 43 -- 43 of 57 -- Visualising the coordinate systems (in 3D + time) Coordinate systems are characterised in terms of surfaces – which intersect at points intersecting surfaces uniquely identify a point Cartesian Cylindrical Spherical 44 NB: coordinate systems AKA spatial locations -- 44 of 57 -- Three common coordinate (point-labelling) systems Coordinate System Surface Types Cartesian 3 × planes Spherical sphere, cone and half-plane Cylindrical cylinder, half-plane and plane • Each decompose into three reusable components: • sets of co-oriented coordinate surfaces • What distinguishes the systems is the types of the surface 45 -- 45 of 57 -- Spheres Planes Cones Visualising the types of sets of co-oriented surfaces 46 Note: the surfaces have an orientation only • no notion of axes or origin -- 46 of 57 -- space-time Coordinate systems – built on worldframes fixing how points are labelled fixing what it means to be at rest coordinate systems Cartesian coordinate systems spherical coordinate systems cylindrical coordinate systems worldlines 47 worldframes -- 47 of 57 -- space-time One example: cartesian coordinate systems fixing what it means to be at rest coordinate systems Cartesian coordinate systems worldlines 48 worldframes -- 48 of 57 -- Cartesian coordinate systems: fix the dimensions Cartesian coordinate systems fixing the dimensions worldlines fixing what it means to be at rest xyz-axes dimension worldvolumes triple Planes 49 NB: dimension worldvolumes (sets of planes, in this case) can be derived from axes – but not vice versa worldframes -- 49 of 57 -- Cartesian coordinate systems: fix origin worldline Cartesian coordinate systems worldlines origin worldline xyz-axes dimension worldvolumes triple Cartesian 50 NB1: The visualisation is 3D – so the worldline is a 3D point NB2: From the origin and dimension worldvolumes one can infer the axes worldframes -- 50 of 57 -- coordinate systems Cartesian coordinate systems metric Cartesian coordinate system – components 51 worldlines xyz-axes dimension worldvolumes triple origin worldline xyz-axes dimension worldvolumes orderings xyz-axes worldvolumes coordinate labels worldframes fixing how points are labelled See: • Partridge, C.: An Information Model for Geospatial and Temporal Reference. 2011 • Partridge, C.: Geospatial and Temporal Reference – A Case Study Illustrating (Radical) Refactoring. ONTOBRAS-2013 6th Ontology Research Seminar in Brazil, 2013 -- 51 of 57 -- Common building process Order Stage Description 1 Surface Orientation selecting the set of co-oriented surfaces 2 Solid Ordering building a mereological ordering for the surfaces – the process varies by surface 3 Ratio Scaling in these three systems, shifting down one or two dimensions to distance and angle ratios 4 Unitising selecting the unitised distance or angle ratios – based upon the selection of unit 5 Labelling labelling the unit ratios Devised a common process, with variations, for building up the coordinate systems 52 See: Partridge, C., A. Mitchell, M. Loneragan, et al. (2019). “Coordinate Systems: Level Ascending Ontological Options”. In: 2019 - MODELS-C. url:https://www.academia.edu/40354620. -- 52 of 57 -- Space-time: top-level-ontology from core to worldlines 53 -- 53 of 57 -- space-time Space-time: top-level-ontology worldlines from core to worldlines Top Level Ontology core foundation 54 -- 54 of 57 -- Modularising TLO Space-Time 55 space-time core mereotopology chronology worldlines curveology congruence legend space-time (possible) worlds worldframes core pre-time atomic worlds Building worldlines out of smaller modules -- 55 of 57 -- Questions 56 -- 56 of 57 -- -- 57 of 57 -- A 4-Dimensionalist Top Level Ontology (TLO): Mereotopology and Space-Time 4-Dimensionalism in Large Scale Data Sharing and Integration Newton Gateway to Mathematics Chris Partridge – BORO Solutions Team effort: the BORO Team http://www.borosolutions.net/who-we-are 2 Top Level Ontology Foundation Data Model Industry Data Models – Reference Data Integration Architecture Process Model based Information Requirements Information Quality Management Core Constructional Ontology Where in the Seven Circles of Information Management 3 Preliminaries - overall approach How, broadly speaking, do we develop the ontology? 4 Target structure A modular, component-based architecture – motivated by the usual reasons: complexity management understandability encapsulation simpler substitution/replacement recombinability expandability resilience 5 coupling cohesion Modular: loosely coupled, highly cohesive modules coupling about the relations between the modules cohesion about the relations between the elements within the module loose coupling, high cohesion aims for a structure where relations cluster inside the modules looser higher tighter lower 6 ‘Agile’ development: an iterative, adaptive approach At each stage, the ‘product’ needs to be: something useful and usable a step towards the target structure minimum viable 7 mereotopology and space-time Situating 4D in ontological space A requirement for space-time is central 8 Reminder: 4D ontologies’ choices {5C22544A-7EE6-4342-B048-85BDC9FD1C3A} category type choice 4D Ontologies horizontal aspect spacetime unifying or separating unifying locations unifying or separating unifying properties unifying or separating unifying endurants unifying or separating unifying immaterial unifying or separating unifying 4D ontologies are maximally unifying Motivation: the perceived benefits of parsimony and cost of separation the easy fit with plenitude 9 https://digitaltwinhub.co.uk/a-survey-of-top-level-ontologies/#a_survey_of_TLOs_contents Two (of many) levels of unifying temporal locations spatial locations spatio -temporal locations more stratified more unified happens at locations in space and locations in time are unified as locations in space-time supersubstantival objects spatio -temporal locations material entities supersubstantival objects located at matter and the space-time it is located in are unified as supersubstantival objects – matter is then a way space-time can be 10 Broad modularisation context 11 Data perspective: four coarse grained layers Reference Data Foundation Data Model Top Level Ontology Core Constructional Ontology scope 12 Basis: Core Constructional Ontology Provides the ‘base’ for the whole ontology: object completeness – builds all the objects categorical completeness – establishes all the (ontological) categories of objects extensional identity criteria – establishes an extensional criteria of identity for the objects 13 First iteration: scope What should the scope of the first ‘MVP’ be? 14 Industry Data Model survey provides a context ISO 10303 Part 42 INSPIRE OS Open Names A core set of standards and associated data 15 These cover ‘geometry’ from CAD and geographical perspective https://digitaltwinhub.co.uk/a-survey-of-idms-and-rdls-intro/ Geometry - CAD – examples from BuildingSMART https://standards.buildingsmart.org/IFC/RELEASE/IFC2x/FINAL/HTML/ifcproductextension/lexical/ifcspace.html multiple coordinate systems multiple spatial objects 16 Geometry - CAD – examples from STEP (ISO 10303 Part 42) coordinate systems multiple spatial objects ISO 10303 Part 42 17 Geometry - Geography – examples from TC211/INSPIRE coordinate systems multiple spatial objects INSPIRE OS Open Names 18 Two (three) clear candidates emerge {5C22544A-7EE6-4342-B048-85BDC9FD1C3A} spatial objects one, two or three (spatial) dimensional objects either eternal or recurring (at each snapshot) technically rigid, in the sense of no (or practically no) deformation over time spatial locations where the spatial objects are located typically expressed as a (2/3D) coordinate system note: these have the same characteristics as spatial objects (eternal/recurring and rigid) (names things, including the objects and locations have names) Current situation: space appears to be Euclidean no clear way of dealing with time (mostly, not even mentioned) no deep formalization 19 Some examples ISO 10303 Part 42 INSPIRE OS Open Names {5C22544A-7EE6-4342-B048-85BDC9FD1C3A} places spatial objects geography geometry roads spatial objects geography geometry solids spatial objects CAD geometry surfaces spatial objects CAD geometry points spatial objects CAD geometry Eastings and Northings spatial locations geography geometry coordinates (2D and 3D) spatial locations CAD geometry examples 20 Proposed scope: ‘space-time’ Proposed scope: a formalized spatio -temporal account of spatial objects and locations hence named ‘space-time’ Spatio -temporal challenge: provide a clear way of dealing with space-time, space and time provide a deep formalization of this Spatial challenge: provide, an account of spatial objects and locations in space-time there is a nexus of things needed for this, including: rigid objects and being relatively at rest 21 space-time The space-time ‘module’ in context relations (foundation extension) names 22 ST: ISO 10303:42 ST: IFCs ST: INSPIRE core foundation focus of this presentation for another time next presentation needed for some of the space-time data Top-down and bottom-up approach 23 Approach – two workstreams Top-down workstream: (where top is the more general or abstract) builds the formal ontological skeleton discuss in following sections Bottom-up workstream: (where bottom is the more concrete) data driven mine an ontology from current industry standard schemas and data Evidence-based Ontological Requirements Elicitation also called: bCLEARer 24 Two mutually supporting workstreams The two workstreams validate and inform each other top-down workstream guides the bottom-up workstream bottom-up workstream validates the top-down workstream; its completeness and fit early de-risking t op-down workstream b ottom-up workstream 25 Bottom up workstream – breakdown - visualisation CAD Geometry ISO 10303 Part 42 buildingSMART IFC geographic information INSPIRE – OS Open Names geoSPARQL Bottom up Workstream Spatial Objects Coordinates Names 26 NB: As noted earlier, coordinates are a form of spatial location Bottom-up bCLEARer process: visualisation data collect collect collect increasing semantic maturity reuse reuse Foundational ontology A repeated sequence of processes: increasing semantic maturity 27 Bottom-up workstream – stages {21E4AEA4-8DFA-4A89-87EB-49C32662AFE0} b(e) Collect collect the datasets in scope in order to establish the broad scope of the process – establishing a bCLEARer master dataset Load define the detailed scope by selecting from the Collect dataset the data in scope Translate the dataset into the cell-based format – the table paradigm Evolve reveal the underlying semantics of the Load Dataset – ‘entification’ – in an ‘ entified ’ dataset Mine the ontology from the ‘ entified ’ dataset – the Evolve ontology dataset Assimilate merge the Evolve ontology dataset into the full ontology model Reuse publish dataset in a format suitable for the reuse context collect load evolve assimilate reuse In-scope Out-of-scope 28 Space-time – top-down workstream 29 Formalising space-time: some historical precedents {5C22544A-7EE6-4342-B048-85BDC9FD1C3A} Year Author Title 1919 A. N. Whitehead An enquiry concerning the principles of natural knowledge 1928 R. Carnap The Logical Structure of the World 1936 B. Russell On order in time 1939 J. H. Woodger The Technique of Theory Construction 1981 P. Needham Temporal Intervals and Temporal Order 1982 C. Lejewski Ontology: What's Next? Opportunity to build upon previous work: 30 A recurring theme: worldlines: the first example Illustration: Minkowski, Hermann (1909), "Raum und Zeit", Physikalische Zeitschrift, 10: 75–88 31 In Minkowski’s formalization: the worldline is the path that a particle traces in 4-dimensional spacetime worldlines are primitive so ‘points’ are derived as the intersections of the worldlines Worldlines: a logical (ontological) primitive {5C22544A-7EE6-4342-B048-85BDC9FD1C3A} Year Author Title 1909 H. Minkowski Raum und Zeit 1958 R. Carnap Introduction to symbolic logic and its applications 1972 P. Suppes Some open problem in the philosophy of space and time 2008 T. Benda A formal construction of the spacetime manifold “ 49. ASs OF SPACE-TIME: TOPOLOGY: 2. THE Wlin -SYSTEM The present second form is called the Wlin -system. Its single primitive sign is ‘ Wlin ’ .” Rudolf Carnap (1958) Introduction to symbolic logic and its applications . Selected examples 32 Worldlines: a more physical primitive (selected examples) {5C22544A-7EE6-4342-B048-85BDC9FD1C3A} Year Author(s) Title 1972 R. Penrose Techniques of differential topology in relativity 1973 S.W. Hawking G. F. R. Ellis The large scale structure of space-time 1976 S.W. Hawking A. R. King P. J. McCarthy A new topology for curved space-time which incorporates the causal, differential, and conformal structures 1977 D. Malament The class of continuous timelike curves determines the topology of spacetime 33 Why worldlines? Proposed ‘space-time’ scope: a formalized spatio -temporal account of spatial objects and locations Worldlines handle the full span of the challenges we’ve noted that they have been used to characterise space-time we now show they do this in a way that also naturally characterizes rigid (that is, spatial) objects and their locations (and so also coordinate systems) 34 See: DiSalle , Robert, "Space and Time: Inertial Frames", The Stanford Encyclopedia of Philosophy (Winter 2020 Edition), Edward N. Zalta (ed.), Partridge, C.: An Information Model for Geospatial and Temporal Reference. 2011 Worldlines: the same ‘spatial’ place Being in the same ‘spatial’ place is remaining on the same ‘coordinate’ reference worldline s 1 s 2 s 3 (x, y, z) (x’, y’, z’) (x”, y”, z”) 35 time Worldlines: different reference worldframes s 1 s 2 s 3 f 2 f 1 f 2 worldframe s 1 s 2 s 3 f 2 f 1 f 1 worldframe There are different reference worldframes and so different ‘same’ places These reference frames are sets of ‘mutually at rest’ worldlines: worldframes Galilean relativity 36 time Worldframes: the same ‘spatial’ object Being the same ‘spatial’ object (no spatial change – or deformation) involves one’s parts staying on worldlines that are mutually at rest – equidistant – so in the same reference worldframe 37 rigid rod: stays on the same reference worldlines reference worldframe worldlines Worldlines: how many different reference worldframes Each set of ‘mutually at rest’ worldlines marks out a (reference) worldframe If the two curves circulating around p 1 (and correspondingly p 2 ) are correctly aligned, then they are also mutually at rest and so mark out a (reference) worldframe s 1 s 2 s 3 38 o 1 o 2 p 1 p 2 o 3 o 4 o 5 o 6 relatively at rest relatively uniform motion Single Platform Multi-Platform Requirements for multiple coordinate systems 39 Tracing out higher-dimensional objects point surface line worldvolume worldsheet worldline 40 Objects of higher dimension can trace out shapes in space-time space-time Decomposing the space-time ‘module’ using worldlines worldlines From the perspective of the space-time ‘module’, there is an opportunity for horizontal slicing – modularization: a decomposition into two sub-components based upon the 7 circles: from core to worldlines from worldlines to spatial objects and locations Foundation Data Model Top Level Ontology core foundation 41 Space-time: Foundation Data Model from worldlines to spatial objects and locations 42 space-time Space-time: foundation data model worldlines Foundation Data Model from worldlines to spatial objects and locations 43 Visualising the coordinate systems (in 3D + time) Coordinate systems are characterised in terms of surfaces – which intersect at points intersecting surfaces uniquely identify a point Cartesian Cylindrical Spherical 44 NB: coordinate systems AKA spatial locations Three common coordinate (point-labelling) systems {5C22544A-7EE6-4342-B048-85BDC9FD1C3A} Coordinate System Surface Types Cartesian 3 × planes Spherical sphere, cone and half-plane Cylindrical cylinder, half-plane and plane Each decompose into three reusable components: sets of co-oriented coordinate surfaces What distinguishes the systems is the types of the surface 45 Spheres Planes Cones Visualising the types of sets of co-oriented surfaces 46 Note: the surfaces have an orientation only no notion of axes or origin space-time Coordinate systems – built on worldframes fixing how points are labelled fixing what it means to be at rest coordinate systems Cartesian coordinate systems spherical coordinate systems cylindrical coordinate systems worldlines 47 worldframes space-time One example: cartesian coordinate systems fixing what it means to be at rest coordinate systems Cartesian coordinate systems worldlines 48 worldframes Cartesian coordinate systems: fix the dimensions Cartesian coordinate systems fixing the dimensions worldlines fixing what it means to be at rest xyz -axes dimension worldvolumes triple Planes 49 NB: dimension worldvolumes (sets of planes, in this case) can be derived from axes – but not vice versa worldframes Cartesian coordinate systems: fix origin worldline Cartesian coordinate systems worldlines origin worldline xyz -axes dimension worldvolumes triple Cartesian 50 NB1: The visualisation is 3D – so the worldline is a 3D point NB2: From the origin and dimension worldvolumes one can infer the axes worldframes coordinate systems Cartesian coordinate systems metric Cartesian coordinate system – components 51 worldlines xyz -axes dimension worldvolumes triple origin worldline xyz -axes dimension worldvolumes orderings xyz -axes worldvolumes coordinate labels worldframes fixing how points are labelled See: Partridge, C.: An Information Model for Geospatial and Temporal Reference. 2011 Partridge, C.: Geospatial and Temporal Reference – A Case Study Illustrating (Radical) Refactoring. ONTOBRAS-2013 6th Ontology Research Seminar in Brazil, 2013 Common building process {5C22544A-7EE6-4342-B048-85BDC9FD1C3A} Order Stage Description 1 Surface Orientation selecting the set of co-oriented surfaces 2 Solid Ordering building a mereological ordering for the surfaces – the process varies by surface 3 Ratio Scaling in these three systems, shifting down one or two dimensions to distance and angle ratios 4 Unitising selecting the unitised distance or angle ratios – based upon the selection of unit 5 Labelling labelling the unit ratios Devised a common process, with variations, for building up the coordinate systems 52 See: Partridge, C., A. Mitchell, M. Loneragan, et al. (2019). “Coordinate Systems: Level Ascending Ontological Options”. In: 2019 - MODELS-C. url:https://www.academia.edu/40354620. Space-time: top-level-ontology from core to worldlines 53 space-time Space-time: top-level-ontology worldlines from core to worldlines Top Level Ontology core foundation 54 Modularising TLO Space-Time 55 space-time core mereotopology chronology worldlines curveology congruence legend space-time (possible) worlds worldframes core pre-time atomic worlds Building worldlines out of smaller modules Questions 56