Atomic and Negated Atomic Concepts

Given a downward refinement operator ρ, a conjunctive step u is defined as refine-

ment of the kind:

ρ(C) u CuD

for concept expressions C and D. For this refinement step to be proper we require that C 6v D, for it to be non-redundant we require that D 6v C, and for CuD to be satisfiable we require CuD 6v ⊥. In this section, we consider the case when conceptsCandD are both either atomic or negated atomic concepts. The definition ofρ_B includes the following relevant cases for refinement of a conceptC:

{A0 |A0 ∈ sh↓(A)} ∪ {AuD|D∈ρ_B(>)} ifC= A(A∈NC)

{¬A0 | A0 ∈sh↑(A)} ∪ {¬AuD|D∈ρ_B(>)} ifC=¬A(A∈ NC)

These cases describe the conjunction of any atomic term with another in the con- text of the role range concept B. Assume the concepts {A,B,C,D} ∈ NC and ap-

ply in the context of B, and where {D v C,C v B,AuD v ⊥} ⊆ T such that {A,¬A,C,¬C,D,¬D} ∈ MB. Then, the rules above permit the construction of the

following refinement chains:

1. A AuA(asA∈ρ_>(>), however A≡ A)

2. A Au ¬A(as¬A∈ ρ_B(>), however Au ¬Av ⊥)

3. A AuC(asC∈ρ_>(>))

We can simply explicitly avoid the first two cases by recognising them syntactically, however we cannot do so in the last case. WhileAuCmay be satisfiable with respect toT, as we discussed in Section 4.2 this concept relative to a particular subexpression contextλ may be unsatisfiable where Tλ |= AuB v ⊥. Because ρB is not aware of

context-specific information about the satisfiability of such concepts, it will otherwise always permit AuC. Furthermore, consider the refinement case of ρ_B for handling

conjunctive expressionsC1u. . .uCn forn≥2:

{C1u. . .uCi−1uDuCi+1u. . .uCn|D∈ρ_B(C_i), 1≤i≤n}

If the concept chosen to refine within the expression Ci is atomic, then the atomic

refinement case above applies, but does not take into account any of the other ex- pressions in the conjunction. In this case, such a refinement step would be permitted:

§4.3 Building a Context-Specific Refinement Operator 81

The operatorρ_B permitted refinement to the unsatisfiable concept AuBuD in this

case becauseCwas refined in isolation of the remainder of the conjunctive expression and ignored the other conjuncts. To motivate a tighter definition for a refinement operator over atomic and negated concepts within any particular local context λ,

consider the following example.

Figure 4.7: A diagram representing subsumption relationships between an example set of concept namesNC = A,B,C,D,Emodelled by local axiomsTλrelative to some subexpression contextλ.

Example 4.3.1. Figure 4.7 represents the entirety of a local domain ∆λ with the following

two local equivalence groups:

{B,¬A} {A,¬B}

For the purposes of refinement, we may exclude concepts¬A and¬B in preference for their simpler equivalent concepts B and A. The remaining pairwise relationships between the reduced set of nine concepts {>,A,B,C,¬C,D,¬D,E,¬E}as captured by T_λ along with overlapping concepts which are not related by axioms inT_λ are as follows:

Strict subsumption Overlap Disjoint

E<B<¬D<> A:C,¬C,¬D A: B,E D<A<¬E<> B:C,¬C,¬E B: A,D D<C<> C: A,B,E,¬D,¬E C:¬C ¬C<> D:(none) D:B,E,¬D E:C,¬C E: A,D,¬E ¬C: A,B,E,¬E ¬C :C ¬D: A,C,¬E ¬D:D ¬E:B,C,¬C,¬D ¬E:E

The strict subsumption and disjointness cases describe pairs of concepts which cannot ap- pear in conjunction together as they will result in a non locally minimal expression or an unsatisfiable one. The only conjunctive expressions which are minimal and satisfiable are the overlapping pairs, such as AuC. For conjunctions with more than two concepts, each conjunct must overlap with all others for the full expression to be locally minimal, that is for the conjunctive concept C1u. . .uCn, all pairs Ci,Cj where1 ≤ i < j ≤ n, we have that CJλ

i and C

Jλ

j overlap. From Figure 4.7, the full set of minimal conjunctions where n ≥3in this example are:

AuCu ¬D BuCu ¬E Bu ¬Cu ¬E Cu ¬Du ¬E

Therefore, there are 25 possible minimal conjunctive expressions for this example (the single top concept, 8 named atomic and atomic negated concepts, 12 overlapping minimal conjuncts with 2 atoms and 4 overlapping minimal conjunctions of 3 atoms), compared to a total of

29−1=511possible atomic and conjunctive expressions, most of which are unsatisfiable or not locally minimal.

As shown in Example 4.3.1, the set of relationships determined over the set of con- cepts applicable to some contextλcan be used to inform which concept expressions

are minimal with respect to a context-specific interpretation J_λ and local axioms Tλ. To tighten the definition of ρ_B so as to restrict the construction of concepts to

avoid unwanted refinement steps, we define a new downward refinement operator denotedρ_λ¯(C)which is parameterised with the set of most applicable contexts ¯λfor

the subexpressionC.

Given a set of most applicable contexts ¯λ for some subexpression, the set of

atomic and negated atomic concepts which were found to be satisfiable in the con- text referred to by ¯λ is the intersection of all concepts which were found to be not

equivalent to⊥ in each set of local axiomsT_λ0 for each λ0 ∈ λ¯. We define the set of

such concepts asar f(λ¯)to represent allatomic role fillersas:

ar f(λ¯) =T_∀λ0_∈λ¯{A|A∈ sc(∆λ)s.t. Tλ 6|= A≡ ⊥}

Furthermore, we denote the common set of equivalence, strict subsumption and dis- jointness axiomsφover all local axioms Tλ for all λ

0 _{∈ ¯}

§4.3 Building a Context-Specific Refinement Operator 83

n≥1 with the shorthandT_λ¯ as follows:

T_λ¯ =Tλ1∩. . .∩ Tλn

By an abuse of notation we also denote the inclusion axiomsφ entailed by T_λ¯ with ·Jλ¯ such asC≡Jλ¯ DwhenTλ¯ |=C≡D.

The operatorρ_λ¯ also makes use of a binary preorder relationwhich is imposed on all concepts which can appear as conjuncts or disjuncts. Initially, we introduce how the preorder relationorders simple concepts in order to describe the first set of cases forρ_λ¯, but later in Section 4.3.2 will extend it for role expressions, then more complex concept expressions.

Definition 4.3.2. (Concept Preorder ) Given a set of concept expressions S, aconcept preorderis a binary function: S×S which imposes an order over concepts, such that for any three concepts C,D,E∈S we have:

• Reflexivity: C C

• Transitivity: If CD and D E, then C E

The concept preorder can be used by a refinement operator when generating con- junctions (disjunctions) of concepts such that C1u. . .uCn (C1t. . .tCn) satisfies Ci Cj for 1 ≤ i < j ≤ n. This ensures, for example, that for concepts A,B

where A B, that only AuB (AtB) is produced where BuA (BtA) is not, as

AuB≡ BuA(AtB≡ BtA) under any interpretation we consider, as the logical operatorsuandtare commutative.

We can now describe how the operatorρλ¯ is defined for simple concepts in conjunc- tions as follows: ρ_λ¯(C) =                                            . . . {A|A∈ ar f(λ¯),A<Jλ¯ C,¬∃A 0 _{∈ ar f(¯} λ)s.t. ifC∈ ar f(λ¯) A<J_λ¯ A 0 _< J_λ¯ C} ∪ {CuA|CuA∈ ar f(λ¯), C A,Coverlaps A} {C1u. . .uCnuA|A∈ ar f(λ¯), ifC=C1u. . .uCn Cn A,∀C1≤i≤n∀λ∈λ¯ :Ci ∈ar f(λ¯)→ Ci overlaps A} . . .

Note that these cases do not yet cover role expressions (§4.3.2), concept disjunction (§4.3.3), or concrete domains (§4.4). In contrast withρ_B, the operatorρλ¯ incorporates context-specific knowledge such as local axioms over the concepts it uses to reduce the chance of generating an improper refinement, or to generate non-minimal or unsatisfiable concepts. In doing so, ρ_λ¯ fails to be complete as it explicitly disallows the construction of certain non-minimal or unsatisfiable concepts. This is by design, however, as we wish to prune such concepts from the space under consideration by a refinement operator as they do not aid in the search for solutions for a learning problem.

Example 4.3.3. Given simple concepts and their relationships from Example 4.3.1 which are

ordered as > A B C D E ¬C ¬D ¬E, the refinement operator ρ_λ¯

traverses the concept space as follows:

> ¬C ¬Cu ¬E C Cu ¬E Cu ¬D Cu ¬Du ¬E CuE D ¬E A Au ¬D Au ¬C AuC AuCu ¬D D ¬D ¬Du ¬E B Bu ¬E Bu ¬C Bu ¬Cu ¬E BuC BuCu ¬E E Eu ¬C

This represents an exhaustive traversal of all 24 unique minimal satisfiable expressions from

>.

Despite the use of the preorderover concepts, Example 4.3.3 demonstrates that the operatorρ_¯

λ generated two refinement chains> C Dand> ¬E A

Dwhich took different paths from> to D. While this indicates that the refinement operator ρ_¯

λ isredundant, it is still designed in such a way as to reduce redundancy, as it was shown in Example 4.3.1 that there are a total of 511 possible expressions

§4.3 Building a Context-Specific Refinement Operator 85

instead of the 24 produced byρ_λ¯ in this case.