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

User Ontologies for Adaptive Interactive Software Systems (with I. Nebel)

Adaptivity > -- improves performance in recall and behavior Our idea: User Ontologies, Competency Ontologies vs. Statistical Stereotyping Methods

User-Ontology vs. User-Profiles

To support adaptivity: Need for reasoning simultaneously with cross-cutting ontologies at different levels of granularity

Overview The method of constituent ontology 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

The method of constituent ontology: to study a domain ontologically is to establish the parts and features in the domain and the interrelations between them

Examples of constituent ontologies

Constituent ontologies I M W ND SD N I M W

Constituent ontologies Database tablesCategory trees

Nice properties Very simple structure Correspond to the way people represent domains –In databases –Spreadsheets –Maps

Meta-level relations between constituent ontologies

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

Meta-level (granularity)

Alabama Alaska Arkansas Arizona … Wyoming West Midwest Northeast South

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

Meta-level (themes) USA physical Mountains Rivers Planes

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

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

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

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 Relations between ontologies Negation, Modality Constituent ontology 1 Constituent ontology 2 Constituent ontology n

Object-level

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

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

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

Constituent ontologies as granular partitions

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)

Constituent ontologies have a simple hierarchical structure Database tables Category trees Maps Granular partitions

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

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)

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

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

Projective relation to reality

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

A constituent ontology is like an array of spotlights

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

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

Projection of constituents constituent ontology Targets in reality Hydrogen Lithium Projection

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

Multiple ways of projecting County partition Highway partition Big city partition

Projection and location

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

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

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

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

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

Projection and location Humans Apes Dogs Mammals

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

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

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

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

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))

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

Relations between constituent ontologies (COs)

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

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

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

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

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

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

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

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

Composition (cont.)

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

Classifying constituent ontologies

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

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 ?

The End