1. Constituent ontologies and granular partitions Thomas Bittner and Barry Smith IFOMIS – Leipzig and Department of Philosophy, SUNY Buffalo

2. Overview Constituent ontologies Levels of ontological theory The hierarchical structure of constituent ontologies The projective relation of constituent ontologies and reality Relations between constituent ontologies Types of constituent ontologies

3. The method of constituent ontology: to study a domain ontologically –is to establish the parts and moments of the domain and –then to establish the interrelations between them

4. Examples of constituent ontologies

5. Constituent ontologies I M W ND SD N I M W

6. Constituent ontologies Database tablesCategory trees

7. Nice properties Very simple structure Very simple reasoning Corresponds to the way people represent domains –In databases –Spreadsheets –Maps

8. Meta-level relations between constituent ontologies

9. Meta level (sub-ontologies) I M W I M W ND SD N  x  y x is sub-constituent-ontology of y

10. Meta-level (granularity)

11. Alabama Alaska Arkansas Arizona … Wyoming West Midwest Northeast South

12. Levels of granularity Alabama Alaska Arkansas Arizona … Wyoming West Midwest Northeast South USA Coarse IntermediateFine

13. Meta-level (themes) USA physical Mountains Rivers Planes

14. Meta-level (themes) USA physical Mountains Rivers Planes USA political Federal states

15. Levels of ontological theory Constituent ontology 1 Constituent ontology 2 Constituent ontology n

16. Levels of ontological theory Level of foundation Formal relations: mereology, topology, location Space and time Basic categories: entities, regions, perdurants, endurants, … Constituent ontology 1 Constituent ontology 2 Constituent ontology n

17. Levels of ontological theory Object-level (Taxonomies, partonomies) Formal relations: mereology, topology, location Space and time Basic categories: entities, regions, perdurants, endurants, … Meta-level Granularity and selectivity (Theory of granular partitions) Relations between constituent ontologies Negation, Modality Constituent ontology 1 Constituent ontology 2 Constituent ontology n

18. Object-level

19. Levels of ontological theory Object-level Formal relations: mereology, topology, location Space and time Basic categories: entities, regions, perdurants, endurants Meta-level Granularity and selectivity (Theory of granular partitions) Relations between constituent ontologies Constituent ontology 1 Constituent ontology 2 Constituent ontology n

20. Formal relations Mereology (part-of) -- Partonomy Mereotopology (is-connected-to) Location (is-located-at) Dependence (depends-on) Subsumption (is-a) -- Taxonomy

21. Constituent ontologies A constituent ontology is an abstract entity Has constituents as parts Constituents are abstract entities that project onto something that is not a constituent itself

22. Constituent ontologies as granular partitions

23. Levels of ontological theory Level of foundation Formal relations: mereology, topology, location Space and time Basic categories: entities, regions, perdurants, endurants, … Constituent ontology 1 Constituent ontology 2 Constituent ontology n Meta-level Granularity and selectivity (Theory of granular partitions)

24. Constituent ontologies have a simple hierarchical structure Database tables Category trees Maps Granular partitions: Theory A

25. Cell structures as Venn diagrams and trees Animal Bird Fish Canary Ostrich Shark Salmon

26. Constituent structures (1) minimal cells: H, He, … non-minimal cells: orange area, green area, yellow area (noble gases)... one maximal cell: the periodic table (PT)

27. Cell structures (2)  - subcell relation He  noble_gases (NG) NG  PT Partial ordering

28. Remember: Constituent ontologies A constituent ontology is an abstract entity Has constituents as parts Constituents are abstract entities that project onto something that is not a constituent itself Granular partitions: Theory B

29. Projective relation to reality

30. Constituents project like a flashlight onto reality P(c, bug)

31. A constituent ontology is like an array of spotlights

32. Pets in your kitchen Bug 1 Bug 2Bug 3Bug 4 Constituent 1Constituent 2Constituent 3Constituent 4

33. Pets in your kitchen Constituent 1 Constituent 2 Constituent 3 Constituent 4 Constituent ontology Reality Projection Bug 1 Bug 2 Bug 3 Bug 4

34. Projection of constituents constituent ontology Targets in reality Hydrogen Lithium Projection

35. Projection of constituents (2) … Wyoming Idaho Montana … Constituent ontology North America Projection

36. Multiple ways of projecting County partition Highway partition Big city partition

37. I shall now use the notions cell and constituent synonymously! I shall also use the notions constituent ontology and granular partition synonymously!

38. Projection and location

39. Location L(bug,c) Being located is like being in the spotlight

40. Projection does not necessarily succeed John is not located in the spotlight!  L(John, c) P(c, John) John

41. Projection does not necessarily succeed Mary is located in the spotlight! L(Mary, c) P(c, John) John Mary

42. Misprojection … Idaho Montana Wyoming … P(‘Idaho’,Montana) but NOT L(Montana,’Idaho’) Location is what results when projection succeeds

44. Transparency Transparency: L(x, c)  P(c, x) P(c1, Mary)P(c2, John) L(Mary, c1) L(John, c2)

45. Projection and location Humans Apes Dogs Mammals

46. Functionality constraints (1) Location is functional: If an object is located in two cells then these cells are identical, i.e., L(o,z 1 ) and L(o,z 2 )  z 1 = z 2 Venus Evening Star Morning Star Two cells projecting onto the same object

47. Functionality constraints (2) China Republic of China People’s Republic of China The same cell (name) for the two different things: Projection is functional: If two objects are targeted by the same cell then they are identical, i.e., P(z,o 1 ) and P(z,o 2 )  o 1 = o 2

48. Preserve mereological structure Helium Noble gases Neon Potential of preserving mereological structure

49. Partitions should not distort mereological structure Humans Apes Dogs Mammals distortion If a cell is a subcell of another cell then the object targeted by the first is a part of the object targeted by the second.

50. Mereological monotony … Helium Noble gases Neon … Helium Noble gases Neon Projection does not distort mereological structureProjection ignores mereological structure

51. Well-formed constituent ontologies are granular partitions which are such that: Projection and location are functions Location is the inverse of projection wherever defined Projection is order preserving If x  y then p(x)  p(y) If p(x)  p(y) then x  y

52. Mathematical Models for COs: (Z, P,  ) FTM Partial order Unique root Finite chain of immediate subcells between every cell and the root GEM Partial order Summation principle Extensionality P: Z   x  y  P(x)  P(y) (P(x)  P(y)  x  y))

53. Constituent ontologies are mappings Object-level Meta-level Granularity and selectivity (Theory of granular partitions) Constituent ontology 1 Constituent ontology 2 Constituent ontology n

54. Relations between constituent ontologies (COs)

55. Relations between constituent ontologies Object-level Meta-level Relations between constituent ontologies Constituent ontology 1 Constituent ontology 2 Constituent ontology n

56. Ordering relations between COs P 1 << P 2 << is sub-partition-of << is reflexive, transitive, antisymmetric  I M W I M W ND SD N

57. I M W ND SD N Ordering relations between LGPs (2)  I M W Z1Z1  Z2Z2 P1P1 P2P2 f f is one-one into order preserving if x  y then f(x)  f(y) (if f(x)  f(y) then x  y) P 1 << P 2

58. P 2 is an extension of P 1  I M W I M W ND SD N

59. P 2 is a refinement P 1 << Z1Z1  Z2Z2 P1P1 P2P2 f

60. Composition of COs  composition operation P 1  P 2 = P 3 iff –P 1 << P 3 and –P 2 << P 3 I M W ° ND SD N = I M W ND SD N

61. Composition of COs I M W = I M W ND SD N ° ND SD W N

62. Composition of COs I M W ° I M W I M W =

63. Composition (cont.)

64. The root cell (G) is created ‘on the fly’

65. Classifying constituent ontologies

66. Projective completeness www.webelements.com Empty cells In every cell there is an object located, i.e.,

67. Exhaustiveness Humans Apes Dogs Mammals Everything of kind  in the domain of the partition A is recognized by some cell in A Do the objects targeted by cells exhaust a domain ?

68. The End