Theory - Instantiation

Towards a General Ontology Theory (ONMA)

Instantiation

The smallest instantiation step is to create or update a (s, p, o) triple. There either a data property or an object property is involved. The first requires a Data Property Definition (DPD) as template, the second an Object Property Definition (OPD). Instantiation patterns specify what form the resulting triple must have. Now we define patterns for Data Property Instantiation (DPI) and Object Property Instantiation (OPI) where a ∈ A is a Data Property, adt ∈ ADT is an Atomic Data Type, u is a universal either from the set C of classes or from the set R of object properties, ^C and ^D are classes, and v is a value from the value set of attribute a:

  Data Property Instantiation (DPI):
Then each DPI must map to one of the following patterns where the symbol ⊨ relates the property definition template to its allowed property instantiations. The first rule expresses that a DPD can be used to instantiate particulars or universals. The second rule expresses that binary and n-ary relations between objects can also have data properties. We refer to the next section where we will define the function pof() to return the set of particulars of a class .
  • (u, A, adt) ⊧ (x, A, v) where (x, »pof, u) or x ∈ ∴u
  • (◊OP, A, adt) ⊧ (»OPI, A, v) for relators with (»OPI, »opi, ◊OP)
  Object Property Instantiation (OPI):
An Object Property Definition (OPD) has the short form (^C, ◊OP, ^D). The classes ^C and ^D may be identical. Then the possible Object Property Instances (OPI) for Particular-Particular Relationships (PPR) and for Class-Particular Relationships (CPR) are:
  • OPI: Let >P1 ∈ pof(^C) and >P2 ∈ pof(^D), then (>P1, »OP, >P2) is an object property instance of (^C, ◊OP, ^D).
  • PPR: (>P, »isOPOf, ^C), or also its inverse (^C, »hasOP, >P) both mean that for an OPD (^Person, ◊isDesignerOf, ^Car) the particular >P on the particulars layer is connected with ^C on the schema layer, for example, (>Ferdi_Porsche, »isDesignerOf, ^VW_Beetle) and (^VW_Beetle, »hasDesigner, >Ferdi_Porsche).
  Object Property Instantiation Rules (OPIR):
Be ◊OPOf = ◊OP-1, and >P1 ∈ pof(^C), and >P, >P2 ∈ pof(^D). For PPRs and for CPRs and their inverses we get as OP instantiation rules:
  • PPR: (^C,◊OP,^D) ⊧ (>P1,»OP,>P2); (^D,◊OPOf,^C) ⊧ (>P2,»OPOf,>P1)
  • CPR: (^C,◊OP,^D) ⊧ (^C,»OP,>P);   (^D,◊OPOf,^C) ⊧ (>P,»OPOf,^C)

Extension: deriver.app

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Source: taoke.de — Theory - Instantiation.