Higher Order Simple OML

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First Order Simple OML

Higher Order Object Types

The diagram above indicates how to extend Simple OML to higher-order object types. The metatypes Entity and BinaryRelation are type sums (disjoint unions) of their type and individual parts.

Entity	=	type(Entity) + individual(Entity)
BinaryRelation	=	type(BinaryRelation) + individual(BinaryRelation)

As in the first-order case, the instance(BinaryRelation) metatype is the same as individual(BinaryRelation), however the instance(Entity) metatype has changed to the sum Entity metatype. The entity classification relation has been extended to include types for its source. This means that we can classify types with other higher-order types, ad infinitem. The source and target of individual binary relations have also been extended to include types.

Example

An example of a second order type is

Color = { Red, Orange, Yellow, Green, Blue, Indigo, Violet }

which has colors, such as Red, as instances.

Now consider the following example taken from the book Knowledge Representation : Logical, Philosophical, and Computational Foundations
by John F. Sowa. The following conceptual graph represents the English phrase a red ball.

Here the characteristic relation (Chrc) links the concept of a ball to the concept [Color: Red] whose type label is the second-order the Color and whose referent is the first-order type Red. The graph maps to the following logical formula.

($x:Ball)(color(Red) Ù chrc(x,Red)).

In Simple OML this can be represented as follows.

<Ontology>

<Type.Object name=“Color”/>

<Type.Object name=“Red”/>

· · ·

<classification instance=“Red” type=“Color”/>

<Type.Object name=“Ball”/>

<Type.BinaryRelation name=“chrc” source.Type=“Ball” target.Type=“Color”/>

</Ontology>

/* specific style */

<Ball>

<chrc target.Instance=“Red”/>

</Ball>

There are three things that are new here. An instance of the classification relation has been placed inside an ontology. The instance attribute of this classification refers to a type. The target attribute of the individual characteristic relation refers to a type.

Higher Order Classification Projection Diagram

Since the BinaryRelation metatype is a sum, the source and target functions are defined as copairings with the following definitions.

source	=	[ type(source) ◦ incl, individual(source) ]
target	=	[ type(target) ◦ incl, individual(target) ]

In addition, some explanation should be given for the definition of the classification relation for binary relations, that has now been lifted to types. This relation is the copairing of the following two binary relations.

⊨BinaryRelation	׃	type(BinaryRelation) ® type(BinaryRelation)
⊨BinaryRelation	׃	individual(BinaryRelation) ® type(BinaryRelation)

The latter is the usual first-order classification relation, where we identify individuals with instances (in that case). The former classification relation between relational types is new. There is a classification-projection requirement here that is similar to the classification-projection requirement for individuals.

Suppose that relational type R is an instance of relational type S. If R has source type A and target type B and S has source type C and target type D, then A is an instance of C and B is an instance of D. As an example how this might occur, let entity types C and D be any two second level types, and define a second-level binary relation S between C and D to be those first-level binary relations between first-level entity type instances of C and D.

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Please send questions, comments and suggestions about this page to: Robert E. Kent rekent@ontologos.org

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