## General Role Inclusion

Created on March 12, 2013, 1:07 a.m. by Hevok & updated by Hevok on May 2, 2013, 5:33 p.m.

General Role Inclusion means that you will be able to construct complex Roles/Properties from simplier atomic Properties. For example the Uncle is the Brother of my Parents.

So one has `isParents` and `hasBrother` as atomic or simple Properties and by connecting them one creates a new Property which is then the uncle.

There are some Restriction that hold to keep such definitions decidable and computable.

In general complex Roles/Properties means that one constructs them from simple Roles via simply connecting them with each other.

A statement like "The friends of my friends are also my friends" can simple be expressed by defining `isFriendOf` or `hasFriend` as a Transitive Property. However if one wants to state the "The foes of my friends are also my foes.", then as there are two different Properties are connected, one need General Role Inclusion.

In First Order Logic is easily expressed where one states that for all x, y and z it holds that if x has the friend y and y has the foe z, then x also has the foe z.

General Property Inclusion works so that one has an R-Box expression that can connect several atomic Roles another more complex Role S. For example one can connect `hasFriend` with `hasFoe` which is subclass or includes `hasFriendsFoe`. So one has combined or declared a new complex Property that says the Foes of my friends which is composed of the two atomic Roles `hasFriend` and `hasFoe`.

In general one has R1 connected by R2, R2 connected by R3 and so on up to Rn, they constitute the Role S.

Then it holds that x0 (the very first one) and xn (the very last one) both belong to the Interpretation of the complex Role S. The Semantics of it expressed by x0 and x1 belong to the Interpretation of Role number 1 and x1 and x2 belong to the Interpretation of R2 and so on until for xn-1 and xn belong to the Interpretation of Rn, then it also holds that x0 and xn both belong to the Interpretation of the complex Role S.

• Complex Roles / Properties can be constructed from simple Roles / Properties (R-Box)

• "The friends of my friends are also my friends."

• can be expressed as SHOIN(D) Transitive Property
• But: "The foes of my friends are also my foes."

• cannot be expressed as SHOIN(D)
• In FOL expressed as a Rule (Axiom):

``````∀x,y,z:hasFried(x,y)^hasFoe(y,x) -> hasFriendsFoe(x,z)
``````
• SHROIQ(D) enables the construction of complex Roles

• R-Box expressions of the form R1oR2oR3o...oRn⊑S e.g.: hasFriend o hasFoe ⊑ hasFriendFoe

• Semantics: if (x0,x1)∈R1I, (x1,x2)∈R2I ... (xn-1, xn)∈RnI, then it also holds that (x0m xn)∈SI

E.g. (x0,x1)∈hasFriendIand (x1,x2)∈hasFoeI,then it also holds (x0,x2)∈hasFriendFoeI

Tags: construction, complex, ontology, property
Categories: Concept
Parent: Properties
Children: Expressivity of General Property Inclusion, OWL 2 General Property Inclusion, Restrictions for Simple Properties

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