Formal Model for Simple OML

~ Under Construction ~

Overview

On this page we describe the formal model for Simple OML that is based upon the classification projection diagram. We relate this to the formal model for RDF.

Formal Model for RDF

RDF is a graph model for information, largely based upon the notion of a semantic net from artificial intelligence. In defining the formal model for RDF, not only do we use the Resource Description Framework (RDF) Model and Syntax Specification, but we also use the Algebraic Specification for RDF Models by Sergey Melnik.

The formal model M = á R, L, S, P ñ for RDF (ignoring containers) is described as follows.

Data

1. R is a set called Resources.

2. L is a set called Literals.

3. R + L is the disjoint union of resources and literals.

4. P is a distinguished subset of resources R called Properties, which contains the four basic properties

{type, predicate, subject, object} Í P.

5. S Í P ´ R ´ (R + L) is a set called Statements.

Axioms

Axioms for type

1. A resource cannot be typed using a literal

(type, s, o) Î S implies o Î R

Axioms for reification

2. Reification of a statement (p, s, o) Î S is a resource r Î R that represents the statement. In addition, there are four statements of the form:

a. (type, r, Statement) Î S

b. (predicate, r, p) Î S

c. (subject, r, s) Î S

d. (object, r, o) Î S

3. If r is a resource of type statement, (type, r, Statement) Î S, then there exists exactly one statement (p, s, o) Î S satisfying a – d above.

Notes

1. The statement set could be extended symmetrically to include literals: S Í P ´ (R + L) ´ (R + L). Then the type axioms would be

a. (type, s, o) Î S and s Î R implies o Î R, where s is a resource (object) instance and o is a resource (object) type.

b. (type, s, o) Î S and s Î L implies o Î L, where s is a literal (data) instance (value) and o is a literal (data) type.

Table of Analogs between RDF and Simple OML

RDF	Simple OML
resource	object
R	Instance.Object
literal	value in a (builtin or defined) data type
L	Instance.Data
resource or literal	entity
R + L	Instance.Entity
property	binary relation type
P	Type.BinaryRelation
statement	binary relation instance
S	Instance.BinaryRelation

There are several things to note.

· RDF resources correspond to Simple OML objects and RDF properties correspond to Simple OML binary relations. But we do not assume that everything is a resource. In particular, we do not assume that Simple OML binary relation instances and types are Simple OML objects. Much like the Entity-Relationship model from database theory, Simple OML distinguishes between entities and the connections between entities called (binary) relations. In an abstract graph there is no requirement that the edge set (or even edge-label set) be contained in the node set. The edges and nodes are kept distinct. This is also the case in category theory. Simple OML follows these traditions. The requirement that everything be a resource makes the set Resource resemble the Simple OML type Thing. So in one sense, Simple OML splits and refines the RDF resource set into the all-encompassing Simple OML type Thing and the Simple OML type Object.

· The RDF type property corresponds to the Simple OML classification relation in the generic sense. The Simple OML classification relation also corresponds to the RDF predicate property in the specific sense. Since Simple OML is a subset of CKML, and follows the philosophy of Information Flow, it treats the classification relation specially.

· RDF statements are typed binary relations of the form (p, s, o) for property p, resource s and resource or literal o. The three individual parts of a statement are called: the subject, the predicate and the object. The predicate part identifies the specific type of the binary relation. Typed binary relations are natural linguistically since the verb in a sentence corresponds to the property in an RDF statement. They are also natural mathematically and epistemologically. Relations exist in hierarchies of generality: For example, an officer for a bank is a person who works for an organization, and such a person is an entity connected with an organization. Entities can be related in multiple ways: An officer for a bank could also be an investor in that bank and a customer of the bank, all at the same time. The connection between the pair (untyped relation) (s, o) and the property p is one of classification.

· The disjoint union (type sum) of the set of RDF resources and the set of RDF Literals corresponds to the set of Simple OML entities, and the set of RDF statements corresponds to the Simple OML binary relations. However, Simple OML follows the note above and completes the picture for the statement analog, by specifying the signature of binary relations to be from (source) entity to (target) entity.

Classification Projection Diagram

· The part within the green background corresponds to RDF(without schemas).

· The symbol ⊨ represents a classification relation between instances and types.

· The symbol ⊢ represents subtyping. This is just one part of the structure represented by sequents.

Formal Model for Simple OML

We paraphrase the formal model for RDF. All data, axioms and notes are made in references to the fundamental classification projection diagram above.

Data (corresponding to RDF)

1. Instance.Object is a set called Object Instances.

2. Instance.Data is a set called Literals or Values or Data Instances.

3. Instance.Entity = Instance.Object + Instance.Data is the disjoint union of objects and values called Entity Instances.

4. Type.BinaryRelation is a set called Binary Relation Types (disjoint from entities), which contains the one basic binary relation and two basic functions

{classification.BinaryRelation, source.Instance, target.Instance} Í Type.BinaryRelation.

5. Instance.BinaryRelation is a set called Instance.BinaryRelations.

Data (in addition to RDF)

6. Type.Object is a set called Object Types.

7. Type.Data is a set called Data Types.

8. Type.Entity = Type.Object + Type.Data is the disjoint union of objects and values called Entity Types.

9. Type.BinaryRelation additionally contains the a second basic binary relation and two other basic functions

{classification.Entity, source.Type, target.Type} Í Type.BinaryRelation.

Axioms

Classification Projection Axiom

1 If a relation instance is classified by a relation type, then the source instance of the relation instance is classified by the source type of the relation type, and (dually) the target instance of the relation instance is classified by the target type of the relation type. More formally, suppose that r Î Instance.BinaryRelation is any relation instance classified classification(r,R) by type R Î Type.BinaryRelation. Then,

i s = source.Instance(r) and S = source.Type(R) implies classification(s,S); and

ii t = target.Instance(r) and T = target.Type(R) implies classification(t,T).

Abstract Relation Axiom

2 The relation instance r in the axiom above is called an abstract relation instance. An (abstract) relation instance (member of the set Instance.BinaryRelation) must have exactly one source.Instance, one target.Instance, and one classification. This means that we can define a injective (one-one) function from instances of binary relations to triples:

i Instance.BinaryRelation ® Type.BinaryRelation ´ Instance.Entity ´ Instance.Entity

This means that we can treat binary relation instances as triples

ii Instance.BinaryRelation Í Type.BinaryRelation ´ Instance.Entity ´ Instance.Entity

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