Handling of Default Inheritance Atoms

The second part of the hybrid reasoning process consists of the evaluation of default inheritance atoms. Default inheritance and its application to Descrip-tion Logics has been analysed in SecDescrip-tion 2.5. Although Florid is capable of evaluating default inheritance atoms by inheritance triggers, this implementa-tion is not suitable here. The semantics of default inheritance states that a slot (property) is to be filled if it cannot be filled in any other way.

Also the evaluation of derivation rules may cause the addition of properties to individuals (objects). Therefore, the evaluation of default inheritance atoms

3Object creation is a feature which is not available in OWL. DL reasoning allows to derive new properties for existing instances, but only from an existing set of properties as defined in the TBox of the knowledge base. Moreover, it is not possible toderivenew entities, neither in the TBox nor in the ABox.

has to be postponed until the alternating fixpoint between F-Logic and OWL is reached. At this point, there are two possibilities for the handling of the default inheritance atoms:

• Translate the OWL knowledge base to F-Logic as usual, with the only difference that the set of default inheritance atoms are now added to the resulting F-Logic programme instead of the derivation rules. If Florid inferes any new facts, a new alternating fixpoint has to be computed.

• The default inheritance atoms are evaluated by the hybrid reasoning core in Swan. The defaults are directly applied to the OWL knowledge base.

The disadvantage of the first method is that the whole knowledge base has to be translated and transmitted toFlorid. That is because there is no possibility to drop just parts of an F-Logic programme (e.g. the inheritance rules). This continuous re-shipping can become a severe restriction when dealing with large knowledge bases.

F-Logic

Florid

new facts

DL

P el le t

knowledge-base

Default Inheritance Evaluation

new facts

defaults

h y b ri d co re

Translation

Interface Interface Updates,Queries OWL Ontology, F-Logic Programme

Answers

Figure 9.2: Handling of Default Inheritance Atoms

Therefore, the default inheritance atoms are evaluated by the hybrid reason-ing core in this hybrid reasonreason-ing implementation (see Figure9.2). The default inheritance atoms are still given in F-Logic syntax. Because of the strong simi-larities between the semantics of default inheritance for DL-knowledge bases and the semantics of default inheritance in F-Logic (see Section2.5) these atoms can be evaluated easily on an OWL knowledge base.

Cautious default inheritance application. In Section2.5the notion of cau-tious inflationary extensions has been explained in detail. Default inheritance

as implemented with the hybrid reasoning engine allows for the computation of exactly the kind of extensions which are described by this formalisation.

Default Reasoning Process. Default reasoning in the hybrid reasoning core consists of the following steps:

• Analyse the default inheritance atoms. They have to be evaluated in correct order so that individuals inherit a default value from their most specific super-class.

• For each default inheritance atom a set of inherited statements is calcu-lated (by checking class memberships).

• The sets of inherited statements are added to the knowledge base one by one. Sets that cannot be added consistently to the knowledge base are reverted.

• The statements of a reverted set are added separately. Additions of state-ments that cannot be added consistently are reverted.

Sets of inherited statements might be reverted because they contain mutual exclusive statements. By adding the statements of reverted sets separately in the last step at least some of the statements can be added to the knowledge base given that a consistent subset of applicable statements exists.

Consider the following example where the knowledge base contains :Sarah, :Abraham, and:Isaacas instances of :Person. The F-Logic programme contains three default inheritance atoms, all of which define the inheritable property :believes infor instances of:Person.

Example 9.11

The hybrid knowledge base consists of an OWL ontology. . .

@ p r e f i x : <h t t p : / / e x a m p l e . o r g#> .

Every inherited statement (x :believes in :Baal) violates the definition of the be-lieves inproperty because only instances of:Godare in the defined range of that property. :Baal, however, is an instance of :OldGodwhich is disjoint with:God.

:Isaac is not considered for default inheritance at all because this individual al-ready has a:believes inproperty. :Sarah cannot inherit the :believes inproperty because every inherited statement (:Sarah :believes in y) would be inconsistent with the definition of the domain of the:believes inproperty. As a consequence, there exists no set of inherited statements that can be added as a whole to the knowledge base. However, applying the inherited statements (:Sarah :believes in :Jachwe) and (:Abraham :believes in :Jachwe) one by one, the former statement will be reverted for the known reasons whereas the latter statement can be added

consistently. ₂

Depending on the order in which inherited statements and the default in-heritance atoms are evaluated there can be different extensions to an OWL knowledge base. Hence, this evaluation strategy is non-deterministic (as was already shown for defaults in general in Section 2.5). For example, :Abraham will believe in either:Jachweor :Ra.

Despite of the non-determinism of default inheritance it can be utilised in a meaningful way. Consider the following example where the extension of the OWL knowledge base depends on the order in which the inherited statements are applied:

Example 9.12

@ p r e f i x o w l : <h t t p : / /www . w3 . o r g /2002/07/ o w l#>.

@ p r e f i x :<h t t p : / / e x a m p l e . o r g #>.

@ p r e f i x o w l 1 1 : <h t t p : / /www . w3 . o r g /2006/12/ o w l 1 1#>.

: p e t e r a : Human .

: p a u l a : Human ; o w l : d i f f e r e n t F r o m : p e t e r . : Pope o w l : e q u i v a l e n t C l a s s

[ a o w l : R e s t r i c t i o n ;

o w l : o n P r o p e r t y : h a s A n u l u s P i s c a t o r i s ; o w l : m i n C a r d i n a l i t y 1 ] .

: h a s A n u l u s P i s c a t o r i s a o w l : I n v e r s e F u n c t i o n a l P r o p e r t y .

The F-Logic programme contains the following default inheritance atom:

” h t t p : / / e x a m p l e . o r g#Human ” [

” h t t p : / / e x a m p l e . o r g#h a s A n u l u s P i s c a t o r i s ” ∗−> t r u e ] .

The property:hasAnulusPiscatorisis an inverse functional property, i.e. only one :Human can be subject to that property (and therefore be owner of the ring).

If the extension is computed using GD⁺_caut (see the definition on page 30) it is possible to apply the default to all instances separately, which leads to the situation where either:Peteror:Paulwill be the ring bearer.

Translated into commsense knowledge the default together with the on-tology reads likethe person that first catches the fisherring, thrown into the air, will be the next pope⁴.

The main characteristics of the hybrid reasoning process inSwanhave been described. For better performance, this process is optimised in several ways.

4Of course this is not likely to be a new strategy for the Vatican, although it would be a considerably faster method compared to the current election procedure.