A context-specific interpretation is an interpretation of some concept expression rel- ative to a context λ and local domain ∆λ. The motivation behind defining such
an interpretation is to provide a means to reveal the subsumption relationships be- tween concepts relative to a local domain, which as we have demonstrated in the last section may be different to the relationships implied under the closed-world inter- pretation (I,U). We intend to utilise such concept subsumption information in the definition of a refinement operator which modifies concept subexpressions, so that it can recognise when to avoid generating certain refinement steps which result in producing concepts which are not useful to the search.
Definition 4.2.8. (Context-Specific Interpretation) Given a closed-world interpretation
(I,U), acontext-specific interpretationJλis defined as the tupleJλ = (·Jλ,∆
λ1, . . . ,∆λk) for subexpression contextsλj for1≤ j≤ k where each∆λj is a local domain for contextλj, and where·Jλ is a function which maps atomic concepts A relative to any contextλto subsets
of∆λ, and roles r relative to any contextλto subsets of∆λ×∆
(I,U) as follows:
AJλ = A(I,U)∩∆ λ
rJλ = {hi,ji | hi,ji ∈r(I,U) s.t. i∈∆ λ}
The interpretation of complex concepts byJλ relative to any contextλis defined similarly to the IC-interpretation as per Definition 3.5.2 as follows, where
>Jλ = ∆ λ ⊥Jλ = ∅ (¬C)Jλ = ∆ λ\C Jλ (CuD)Jλ = CJλ ∩DJλ (CtD)Jλ = CJλ ∪DJλ {i}Jλ = {i|i∈∆ λ} (∃r.C)Jλ = {i|i∈∆ λs.t. ∃j.hi,ji ∈r Jλ ∧j∈C(I,U)} (∀r.C)Jλ = {i|i∈∆ λs.t. ∀j.hi,ji ∈rJλ → j∈C (I,U)} (>nr.C)Jλ = {i|i∈∆ λs.t. ]{j.hi,ji ∈rJλ ∧j∈C(I,U)}>n} (6nr.C)Jλ = {i|i∈∆ λs.t. ]{j.hi,ji ∈r Jλ ∧j∈C(I,U)} 6n}
§4.2 Structuring the Interpretation for Learning 61
This context-specific interpretation is defined in such a way that, for any concept C, it will be the case that CJλ ⊆C(I,U)as CJλ is the(I,U)interpretation of C relative to a local domain ∆λin subexpression contextλ, and where each∆λ ⊆∆(I,U)for any contextλ.
A context-specific interpretationJλ relative to some contextλhas some interest-
ing properties which make it useful for inducing concepts with refinement operators. Most importantly, the set of inclusionsCvDrelative to a local domain∆λmodelled byJλ denoted C vJλ D may be tighter than those axioms implied by a TBoxT for the whole knowledge-baseK as illustrated in Example 4.2.7. We therefore associate each contextλwith a number of inclusions which we denotelocal axioms, as follows.
Definition 4.2.9. (Local Axioms) Given two concepts C,D relative to a subexpression context λ and a closed-world interpretationJλ,local axioms denotedTλ are the set of all
axioms of the form:
• Equivalence: C ≡Jλ D where C
Jλ = DJλ • Strict subsumption: C<Jλ D where CJλ ⊂ DJλ • Disjointness: CuDvJ
λ ⊥where C
Jλ ∩DJλ =∅
Example 4.2.10. Consider concepts A,B in a knowledge base K = (T,A) where B v
A ∈ T, and where A 6= ∅ and K is consistent. Therefore, all interpretations I which are models of K have BI ⊆ AI. Consider the closed-world interpretation (I,U) which models B(I,U) ⊆ A(I,U) over K, and a context-specific interpretation J
λ relative to some
local domain∆λ. Under Jλ, it is possible that any of the following may hold:
• Tλ |=B≡ A where AJλ =BJλ
• Tλ |=BuAv ⊥ where BJλ =∅(B is unsatisfiable in local domain ∆ λ)
• Tλ |=B<A where BJλ∩AJλ =BJλ
Note while each of these interpretations are consistent with B(I,U) ⊆ A(I,U), the first case recognises that, in the context ofλ, a refinement chain A B is improper. Similarly, the second case where B ≡J
λ ⊥ can be used to avoid expressions containing B in the context of λif it will result in the production of an unsatisfiable concept.
Proposition 4.2.11. For all inclusion axioms φ in the set of local axioms Tλ by Defini-
tion 4.2.9 for some contextλ, ifT |=φthenTλ |=φfor the TBoxT which models inclusion
axiomsφrelative to the closed-world interpretation(I,U).
Proof. We prove Proposition 4.2.11 for each form of local axiomφ∈ Tλ over any two conceptsC,Dwhich can be interpreted by(I,U)andJλby noting that each context- specific interpretation CJλ and DJλ are subsets of C(I,U) and D(I,U) respectively as
CJλ =C(I,U)∩∆
λandD
Jλ = D(I,U)∩∆ λ.
• Forφ= (C<D), we must show that the implicationC(I,U)⊂ D(I,U)→CJλ ⊂
DJλ always holds for any subexpression context λ. The implication fails only if the antecedent C(I,U) ⊂ D(I,U) holds and the consequent CJλ ⊂ DJλ does not. AsCJλ is always a subset of C(I,U) where CJλ = C(I,U)∩∆
λ, it must be the case that CJλ ⊂ D(I,U). In the case where CJλ = ∅, the consequent holds trivially. Now consider there exists an individual i ∈ CJλ, and therefore we also know that i ∈ D(I,U). The consequent fails when it can be shown that
i6∈ DJλ. If we assumei6∈ DJλ, then it must be the case thati6∈D(I,U), which is a contradiction. Therefore, it must be the case that i ∈ DJλ which means
CJλ ⊂DJλ also holds for any individuali.
• For φ = (C ≡ D), we must show that C(I,U) = D(I,U) → CJλ = DJλ for any subexpression contextλ. This holds trivially asCJλ and DJλ are the same subset in the intersectionC(I,U)∩∆λ andD(I,U)∩∆λ.
• For φ= (CuDv ⊥), we must show that C(I,U)∩D(I,U) =∅ →CJλ ∩DJλ = ∅. This also holds trivially as there are no common subsets ofC(I,U)andD(I,U) which are not disjoint.
Therefore, we conclude that for any local axiomφ in Tλ it is the case that ifT |= φ thenTλ |=φfor any contextλ.
Example 4.2.10 and Proposition 4.2.11 demonstrate that any inclusion axioms
Cv D∈ T whereC(I,U) ⊆D(I,U)are not contradicted underJλrelative to a context
λ as it is always the case that CJλ ⊆ DJλ, even if it can be shown that under J λ that C ≡J
λ D or C∩D vJλ ⊥ hold, yet this cannot be shown under (I,U). While relationships such asC v Dare typically inferred with logical reasoning algorithms under the open-world assumption relative to the standard first-order interpretation I, they can be computed explicitly over a fixed model such as (I,U) under the closed-world assumption for any two concepts by testing the relationship between sets C(I,U) and D(I,U) composed of asserted data, and similarly, under J
λ between
CJλ andDJλ for any subexpression contextλ.
Unfortunately, for any knowledge base consisting of many concept terms, role terms and individuals, it would be practically infeasible to enumerate all possible concepts expressible in a concept language likeS ROI Q(D), along with their subex- pression contexts λ, in order to pre-compute Jλ in its entirety, unlike (I,U). Fur- thermore, even if it is possible to enumerate each concept and subexpression context, the computation of each local domain ∆λ and the context-specific interpretation of all concept terms is again likely to be infeasible especially given a large data set of
§4.2 Structuring the Interpretation for Learning 63
individuals and literals.
Recall that our aim is to define and construct a context-specific interpretation to provide a refinement operator with information about which concepts to prune when performing refinements of concept subexpressions which do not aid in the search. With this aim in mind, we intend to describe how to construct a particular finite fixed context-specific interpretationJλ based on(I,U)which is small enough to compute reasonably quickly, but which may still be used to permit a refinement operator to take advantage of knowledge which Jλ affords relative to a limited set of contexts. Specifically, the contexts we will describe consist exclusively of single role expres- sions in conjunction with simple concepts including >, atomic concept names and negated atomic concept names. At the very least, such contexts will subsume more complex subexpressions composed of such fragments, and can be used to identify certain cases where refinements would lead to concepts which can clearly be avoided. As we will describe in the next section, such a limited interpretation still affords new knowledge which can be effectively leveraged by a refinement operator to reduce the search space of concepts. We will begin by describing a method for construct- ing such a limited context-specific interpretation, and then describe how it can be incorporated directly into the definition of a new pair of downward and upward refinement operators ρλ¯ and υλ¯ which are defined in terms of a set of applicable
contexts ¯λ.