Change - General Role Inclusion

Created on March 12, 2013, 1:07 a.m. by Hevok & updated on March 12, 2013, 1:11 a.m. by Hevok

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. ¶
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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. ¶
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There are some Restriction that hold to keep such definitions decidable and computable. ¶
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In general complex Roles/Properties means that one constructs them from simple Roles via simply connecting them with each other. ¶
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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. ¶
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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. ¶
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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. ¶
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In general one has R1 connected by R2, R2 connected by R3 and so on up to Rn, they constitute the Role S. ¶
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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. ¶
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Complex Roles / Properties can be constructed from simple Roles / Properties (R-Box) ¶
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"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) ¶
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In FOL expressed as a Rule (Axiom): ¶
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∀x,y,z:hasFried(x,y)^hasFoe(y,x) -> hasFriendsFoe(x,z) ¶
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SHROIQ(D) enables the construction of complex Roles ¶
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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 ¶
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E.g. (x0,x1)∈hasFriendIand (x1,x2)∈hasFoeI,then it also holds (x0,x2)∈hasFriendFoeI ¶
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Summary ¶
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In order to ensure Decidability the following structural restrictions must hold for the Description Logics SHOIQ(D). ¶
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The first thing is Regularity, which means for the RBox Axioms there are certain Restrictions concerning the Interactions of RBox Axioms that deal with Regularity. There are only certain forms/schemata of Role Inclusions allowed and there must be a strict order to be maintained. On the second hand one has the simplicity of Properties that must also be maintained because there are several Restrictions on which Properties, e.g. in Number Restrictions are only applied on which other simple Procedures and other Things are applied, only on simple Properties and not on Properties that relay on General Property Inclusion, otherwise one would lose the Property of decidability. ¶
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One has to be carefully as it might be that the Union of several Knowledge Bases violate these Restrictions, although each single Knowledge Base does comply to the Restrictions. ¶
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To ensure decidability the following structural Restrictions must hold for SHROIQ(D): ¶
- Regularity: ¶
Restriction of potential interaction of RBox axioms ¶
- Simplicity of Properties: ¶
Restrictions of how to apply Properties in number Restrictions ¶
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Therefore a number of Restrictions arise for the overall Structure of the Knowledge Base that have to be considered for all axioms. ¶
Attention: The Union of several SHROIQ(D) Knowledge Bases might violate these Restrictions, although each underlying Knowledge Base does not comply to the Restrictions!