Extracting ALEQR Self-Knowledge Bases from Graphs

Francesco Kriegel

Institute of Theoretical Computer Science, TU Dresden, Germany [email protected]

http://lat.inf.tu-dresden.de/˜francesco

Abstract. A description graph is a directed graph that has labeled vertices and edges. This document proposes a method for extracting a knowledge base from a description graph. The technique is presented for the description logicALEQR^Self, which allows for conjunctions, primitive negations, existential restrictions, value restrictions, qualified number restrictions, existential self re- strictions, general concept inclusions, and complex role inclusions. Furthermore, also sublogics may be chosen to express the axioms in the knowledge base.

The extracted knowledge base entails exactly all those statements that can be expressed in the chosen description logic and are encoded in the input graph.

Keywords: Description Logics·Formal Concept Analysis·Terminological Learning·Knowledge Base·General Concept Inclusion·Canonical Base·Most- Specific Concept Description·Interpretation·Description Graph·Folksonomy

·Social Network

1 Introduction

There have been several approaches towards the combination of description logics [5] and formal concept analysis [12] for knowledge acquisition, knowledge exploration, and knowledge completion. Rudolph [19] invented a method for the exploration of con- cept inclusions holding in anFLE-interpretation. Baader, Ganter, Sattler, and Sertkaya, [3, 4, 20] provided a technique for completion of knowledge bases. Furthermore, Baader and Distel [1, 2, 9] gave a method for computing a finite base of all concept inclusions holding in a finiteEL-interpretation by means of the Duquenne-Guiges-base [13] of a so-called induced formal context. Finally, Borchmann [6–8] extended the results by defining the notion of confidence for concept inclusions, and utilized the Luxenburger-base [16–18] of the induced formal context to formulate a base for the concept inclusions whose confidence exceeds a given threshold.

In the following text we provide a method to compute a knowledge base for concept and role inclusions holding in anALEQR^Self-interpretation or description graph, re- spectively, which entails all knowledge that is encoded in the interpretation/graph and can be expressed inALEQR^Self. For this purpose we need the notion of a model-based most-specific concept description. It is defined as a concept description which describes a given individualx, i.e., the individual is an instance of the concept, and is most spe- cific w.r.t. this property, i.e., for all concept descriptionsCthat havexas an individual, the most-specific concept is subsumed byC. Since we do not want to use greatest fixpoint semantics here, we restrict the role-depth to ensure existence of most-specific

concepts. The description logicALEQR^Selfis chosen for knowledge representation here, since it is an expressive description logic that does not allow for disjunctions (like ALC), and hence will not model the examples in the input graph too exactly.

We start with a short introduction of the description logicALEQR^Self. Then we de- fine description graphs, and show their equivalence to interpretations. Furthermore, we then present model-based most-specific concept descriptions, and their relationships to formal concept analysis. We then continue with induced concept contexts and induced role contexts, and eventually utilize them to construct the desired knowledge base.

Please note that many of the results on model-based most-specific concept descrip- tions, induced concept contexts, and bases of general concept inclusions, have already been observed and proven by Baader and Distel [1, 2, 9] for the light-weight descrip- tion logicEL^⊥w.r.t. greatest fixpoint semantics, which allows for the bottom concept, conjunctions, existential restrictions, and general concept inclusions. Their results are extended to the additional concept constructors ofALEQR^Self, and furthermore we also take complex role inclusions into account.

2 The Description Logic ALEQR

Let(N_C,N_R)be asignature, i.e.,N_Cis a set ofconcept names, andN_Ris a set ofrole names, such thatN_CandN_Rare disjoint. We stick to the usual notations and hence concept names are written as upper-case latin letters, e.g.,AandB, and role names are written as lower-case latin letters, e.g.,rands. Aninterpretationover(N_C,N_R)is a tupleI = (∆^I,·^I)where∆^I is a non-empty set, calleddomain, and·^I is anextension functionthat maps concept namesA∈NCto subsetsA^I ⊆∆^Iand role namesr∈NR

to binary relationsr^I ⊆_∆^I×_∆^I.

The set of allALEQR^Self-concept descriptionsis denoted byALEQR^Self(N_C,N_R), and is inductively defined as follows. Every concept nameA ∈ N_C, thebottom concept

⊥, and thetop concept>, is anatomicALEQR^Self-concept description. IfA ∈ N_C is a concept name,r∈ NRis a role name,C,D∈ ALEQR^Self(NC,NR)are concept descriptions, andn∈N+is a positive integer, then¬A,CuD,∃r.C,∀r.C,≥n.r.C,

≤n.r.C, and∃r.Self, arecomplexALEQR^Self-concept descriptions. The extension function of an interpretationI is canonically extended to all ALEQR^Self-concept descriptions as shown in the semantics column of Figure 1.

Note that every individual without anyr-successors in the interpretationIat all is an element of the extension of every value restriction∀r.Cfor arbitrary concept descriptionsC. We use the usual notation_X

k

for the set of all subsets ofXwith exactlykelements. It is well-known that

X k

=^|X|_k.

Furthermore,ALEQR^Selfallows to express the followingterminological axioms. If Ais a concept name, andC,Dare concept descriptions, thenC v Dis a(general) concept inclusion(abbr. GCI), andA≡Cis aconcept definition. Of course, every concept definitionA≡Ccan be simulated by two concept inclusionsAvCandCvA. If r,r1, . . . ,rn,sare role names, thenrvsis asimple role inclusion, andr1◦. . .◦rn vs is acomplex role inclusion, also calledrole inclusion axiom(abbr. RIA). We then say that an interpretationIis amodelof an axiomα, denoted asI |=α, if the condition in the

name syntax C semantics C^I

bottom concept ⊥ _∅

top concept > ∆^I

primitive negation ¬A ∆^I\A^I

conjunction CuD C^I∩D^I

existential restriction ∃r.C {x∈_∆^I| ∃y∈_∆^I:(x,y)∈r^I∧y∈C^I} value restriction ∀r.C {x∈_∆^I| ∀y∈_∆^I:(x,y)∈r^I→y∈C^I} qualified number ≥n.r.C {x∈∆^I| ∃Y∈^∆_n^I: {x} ×Y⊆r^I∧Y⊆C^I}

restriction ≤n.r.C {x∈_∆^I| ∀Y∈_n+1^∆I: {x} ×Y⊆r^I→Y6⊆C^I} self restriction ∃r.Self {x∈_∆^I|(x,x)∈r^I}

Fig. 1.Concept Constructors ofALEQR^Self

semantics column of Figure 2 is satisfied. An axiom isgenerally validif all interpretations are models of it. IfCvDis generally valid, then we denote this byCvD, too, and say thatCissubsumedbyD,Cis asubsumeeofD, andDis asubsumerofC.

ATBoxis a set of concept inclusions and concept definitions, and aRBoxis a set of role inclusions.Iis amodelof a TBoxT, denoted asI |=T, ifIis a model of all axiomsα∈ T, and analogously for RBoxesR. Aknowledge baseKis a pair(T,R)of a TBoxT and a RBoxR.

name syntaxα semanticsI |=α

concept inclusion CvD C^I ⊆D^I

concept definition A≡C A^I=C^I

simple role inclusion rvs r^I⊆s^I

complex role inclusion r₁◦r₂◦. . .◦rnvs r^I₁◦r^I₂◦. . .◦r_n^I⊆s^I

Fig. 2.Axiom Constructors ofALEQR^Self

◦denotes theproductoperator whereR◦S:={(x,z)| ∃y:(x,y)∈R∧(y,z)∈S}. Definition 1 (Knowledge Base).LetIbe an interpretation. Aknowledge baseforIis a knowledge baseKthat has the following properties.

(sound) All axioms inKhold inI, i.e.,I |=K.

(complete) All axioms that hold inI, are entailed byK, i.e.,I |=α⇒ K |=α.

(irredundant) None of the axioms inKfollows from the others, i.e.,K \ {α} 6|=αfor all α∈ K.

3 Graphs

The semantics ofALEQR^Selfcan also be characterized by means of description graphs, which are cryptomorphic to interpretations. Adescription graphover(NC,NR)is a tupleG = (V,E,`), such that the following conditions hold.

1. (V,E)is a directed graph, i.e.,Vis a set ofvertices, andE⊆ V×Vis a set of directed edgesonV. For an edge(v,w)∈Ewe say thatvandware connected,v is thesource vertex, andwis thetarget vertexof(v,w).

2. `=`_V∪˙ `<sub>E</sub>is alabeling functionwhere`_V:V→2^NCmaps each vertexv∈Vto a label set`<sub>V</sub>(v)⊆NC, and`_E:E→2^NRmaps each edge(v,w)∈Eto a label set`_E(v,w)⊆NR.

The vertices of the graphG are labeled with subsets ofNCto indicate the concept names they belong to. Analogously, the edges are labeled with subsets ofNRto allow multiple (named) relations between the same two vertices in the graph. Usually, one would also specify a root vertexv0∈Vfor description graphs, but this is not necessary for our purposes here.

A description graph may also be calledfolksonomyorsocial networkhere. For example, the setNRof role names in the signature may contain a relationfriendthat connects friends in a social network (graph). Other relations are for exampleisMarriedWith, sentFriendrequestTo,likes,follows, andhasAttendedEvent, with their obvious meaning.

The vertices in a social network are of course the users (and possibly other objects).

The vertex labels in the setN_Cof concept names can be used to categorize the users in a social network, e.g., by nationality, sex, marital status, profession, etc.

For each description graphG= (V,E,`)we define a canonical interpretationI_G that contains all information that is provided inGas follows. The domain is just the vertex set, i.e.,∆^IG :=V, and the extensions of concept namesA∈NC, and of role namesr∈NR, respectively, are given as follows.

A^IG :=`<sup>−1</sup><sub>V</sub> (A) ={v∈V|A∈`(v)} r^IG :=`<sup>−1</sup><sub>E</sub> (r) ={(v,w)∈E|r∈`(v,w)}

Furthermore, we can easily construct a description graphG_Ifrom an interpretation I = (_∆^I,·^I)by settingG_I := (V,E,`)where

V:=∆^I E:= ^[

r∈N_R

r^I

`_V(v):=ⁿA∈N_C

v∈A^Io

`_E(v,w):=ⁿr∈ N_R

(v,w)∈r^Io .

It can be readily verified that both transformations are mutually inverse, i.e.,IG_I =I for all interpretationsI, andGI_G =Gfor all description graphsG.

As a consequence, we do not have to distinguish between interpretations and descrip- tion graphs, and we may also compute model-based most-specific concept descriptions (which are usually defined for individuals of an interpretation, cf. next section) for vertices in description graphs. In the following we want to propose a method to com- pute a knowledge baseK= (T,R)from a given description graphGthat entails all knowledge that is encoded inGand is expressible in the description logicALEQR^Self.

4 Model-Based Most-Specific Concept Descriptions

Therole depthrd(C)of a concept descriptionCis defined as the greatest number of roles in a path in the syntax tree ofC. Formally, we inductively define the role depth as follows.

1. Every atomic concept descriptionA,⊥,>, and every primitive negation¬A, has role depth 0.

2. The role depth of a conjunction is the maximum of the role depths of the conjuncts, i.e.,rd(CuD):=rd(C)∨rd(D)for all concept descriptionsCandD.

3. The role depth of a restriction is the successor of the role depth of the concept description in the restriction’s body, i.e.,rd(Q r.C):=1+rd(C)for all quantifiers Q∈ {∃,∀,≥n,≤n}, role namesr∈ N_R, and concept descriptionsC.

4. The role depth of a self restriction is just defined as 1, i.e.,rd(∃r.Self):=1.

It is easy to see that the role-depth of a concept description is well-defined. However, equivalent concept descriptions do not necessarily have the same role depth. For example the concept description⊥and∃r.⊥are equivalent, but the former concept description has role depth 0 and the latter has role depth 1.

Definition 2 (Model-Based Most-Specific Concept Description).Let(NC,NR)be a signature,I = (∆^I,·^I)an interpretation over(NC,NR),δ∈Na role-depth bound, and X⊆_∆^I a subset of the interpretation’s domain. Then anALEQR^Self-concept description C is called amodel-based most-specific concept description(abbr.mmsc) of X w.r.t.Iandδif it satisfies the following conditions.

1. C has a role depth of at most d, i.e.,rd(C)≤δ.

2. All elements of X are in the extension of C w.r.t.I, i.e., X⊆C^I.

3. For all concept descriptions D withrd(D)≤δand X⊆D^I it holds that CvD.

Since all model-based most-specific concept descriptions ofXw.r.t.I andδare unique up to equivalence, we speak ofthemmsc, and denote it byX^Iδ.

Lemma 3. LetIbe an interpretation over the signature(N_C,N_R). Then the following state- ments hold for all subsets X,Y⊆∆^I, and concept descriptions C,D∈ ALEQR^Self(NC,NR) with a role-depth≤δ.

1. X⊆C^I if, and only if, X^IδvC.

2. X⊆Y implies X^Iδ wY^Iδ. 4. X⊆X^IδI.

6. X^Iδ ≡X^IδIIδ.

3. CvD implies C^I ⊆D^I. 5. CwC^IIδ.

7. C^I =C^IIδI.

It then follows that·^IIδ is a closure operator on the concept description poset (ALEQR^Self(N_C,N_R),w)factorized by concept equivalence, and a concept inclusion CvDholds inIif, and only if, the implicationC→Dholds in the closure operator

·^IIδ. It follows that there is a (finite) canonical base of concept inclusions holding in a (finite) interpretationI.

Definition 4 (Least Common Subsumer).Let C,D beALEQR^Self-concept descriptions w.r.t. the signature(N_C,NR). Then a concept description E∈ ALEQR^Self(N_C,NR)is called aleast common subsumer(abbr.lcs) of C and D if the following conditions are fulfilled.

1. E subsumes both C and D, i.e., CvE and DvE.

2. Whenever F is a common subsumer of C and D, then F subsumes E, i.e., Cv F and DvF implies EvF for all concept descriptions F∈ ALEQR^Self(N_C,N_R).

It follows that least common subsumers are always unique up to equivalence. Hence, we can speak ofthelcs of two concept descriptions, and furthermore we denote it bylcs(C,D)orC t D. The definition can be canonically extended to an arbitrary number of concept descriptions, and we then writelcs(C₁, . . . ,Cn)or^Fn_i=1C_ifor the least common subsumer of the concept descriptionsC₁, . . . ,Cn.

C

D

CtD E

v v

v

v v

Fig. 3.The least common subsumer is a pullback in the category, whose objects are concept descriptions and whose morphisms are subsumptions.

Lemma 5. Let(Xt)_t∈Tbe a family of subsets Xt ⊆_∆^I, and(Cs)_s∈S a family of concept descriptions Cs∈ ALEQR^Self(N_C,NR). Then the following statements hold.

1. (^S_t∈TXt)^Iδ ≡^F_t∈TX_t^Iδ 2. (^d_s∈SCs)^I =^T_s∈SC_s^I

Lemma 6. If C v D holds inI, and both C and D have a role depth≤ δ, then also CvC^IIδholds inI, and CvD follows from CvC^IIδ.

Beforehand we have observed a pair of mappings that has similar properties like the well-known galois connection which is induced by a formal context. More specifically, the pair(·^Iδ,·^I)is an adjunction. Consequently, we adapt the notions of a formal concept and a formal concept lattice as follows.

Definition 7 (Description Concept).LetIbe a finite interpretation over the signature (N_C,NR), andδ∈_Na role-depth bound.

Adescription conceptofIandδis a pair(X,C)that consists of a subset X⊆_∆^I, and anALEQR^Self-concept description over(N_C,N_R), such that X is the extension C^I, and C is the model-based most-specific concept description X^Iδ. Furthermore, we call X theextent, and C theintentof(X,C). The set of all description concepts ofIandδis denoted asB(I,δ). Analogously,Ext(I,δ)andMmsc(I,δ)denote the sets of all extents and intents, respectively.

To ensure formal correctness, we require thatB(I,δ)only contains at most one description concept with the extentX. This is no limitation as we will see in the next lemma that all description concepts with the same extent have equivalent intents.

Definition 8 (Subconcept, Superconcept, Description Concept Lattice).Let(X,C) and(Y,D)be two description concepts. Then(X,C)is asubconceptof(Y,D)if X⊆Y holds. We then also write(X,C)≤(Y,D), and call(Y,D)asuperconceptof(X,C).

Additionally, the pairB(I,δ):= (B(I,δ),≤)is calleddescription concept latticeof Iandδ.

Lemma 9 (Order on Description Concepts).LetI be a finite interpretation over the signature(N_C,NR), andδ∈_Na role-depth bound.

1. For two description concepts(X,C)and(Y,D)it is true that (X,C)≤(Y,D)⇔X⊆Y⇔CvD.

2. The relation≤is an order onB(I,δ).

We may furthermore observe that the set of all description concepts with the given order≤is a complete lattice.

Definition 10 (Description Lattice).LetI be a finite interpretation over the signature (N_C,NR), andδ∈_Na role-depth bound. ThenB(I,δ)is a complete lattice whose infima and suprema are given by the following equations.

^

t∈T

(Xt,Ct) =





\

t∈T

Xt, l

t∈T

Ct

!II_δ



_

t∈T

(Xt,Ct) =



 [

t∈T

Xt

!I_δI

,^G

t∈T

Ct





A description lattice is a nice visualization of the information provided in a descrip- tion graph or in an interpretation, respectively. Since interpretations and description graphs are cryptomorphically defined, we do not need to further distinguish between them. One can think of description lattices as a natural generalization of concept lattices which do not only allow conjunctions of attributes as intents, but also more complex concept descriptions that can be expressed in the underlying description logic. Of course, if the chosen description logic isL₀, i.e., only allows for conjunctions u, then the concept lattices and description lattices w.r.t.L₀coincide. However, for more complex description logics likeELorFLEor extensions thereof, we can further involve roles in the intents of the description concepts which adds further expressivity.

There is also a strong correspondence to the pattern structures and their lattices that have been introduced by Ganter and Kuznetsov [11]. Of course, the set of patterns consists of all concept descriptions that are expressible in the underlying description logic w.r.t. the given signature(NC,NR). The similarity operation is simply given by the least common subsumer mappingtwhich is the infimum in the lattice of all concept descriptions.

5 Induced Concept Contexts

Definition 11 (Induced Context).LetI be an interpretation, andMa set of concept descriptions, both over the signature(N_C,N_R). Then theinduced contextofI andMis defined as the formal contextKI,M := _∆^I,M,I

, where the incidence I is defined via (x,C)∈I if, and only, if x∈C^I. For a concept description C over(NC,NR)itsprojection toMis defined asπM(C):={D∈ M |CvD}. A concept description C isexpressible in terms ofMif C≡^dU for a subset U⊆ M. We haved∅=>andd

U=^d_C∈UC for all subsets∅6=U⊆ M.

Lemma 12. LetIbe an interpretation, andMa set of concept descriptions. Then the following statements hold for all subsets X,Y⊆ M, and all concept descriptions C,D.

1. X⊆π_M(C)if, and only if, Cv^dX.

2. X⊆Y impliesd

Xw^dY.

4. X⊆πM(^dX) 6. d

X≡^dπM(^dX).

3. CvD impliesπM(C)⊇πM(D). 5. Cv^dπM(C).

7. πM(C) =πM(^dπM(C)).

Lemma 13. LetKI,Mbe an induced context. Then the following statements hold for all concept descriptions C over(NC,NR), all subsets U⊆ M, and X⊆∆^I.

1. π_M(X^I) =X^I 2. (^dU)^I =U^I 3. C^I ⊆πM(C)^I 4. πM

(^dU)^II=U^II

5. C≡^dπM(C)if C is expressible in terms ofM. 6. C^I =πM(C)^Iif C is expressible in terms ofM. 7. U =πM(^dU)if U is an intent ofKI,M.

The next lemma tells us that we can directly decide in the induced contextKI,M, whether a concept inclusion between conjunctions of concept descriptions ofMholds in the given interpretationI.

Lemma 14 (Implications and concept inclusions).LetIbe an interpretation, andMa set of concept descriptions, both over the signature(N_C,NR). Then for all subsets X,Y⊆ M, the concept inclusiond

Xv^dY holds inIif, and only if, the implication X→Y holds in KI,M.

Definition 15 (Approximation).LetIbe an interpretation over the signature(NC,NR), δ ∈_Na role-depth bound, and C∈ ALEQR^Self(N_C,NR)a concept description with its normal formd

A∈UAu^d_(Q,r,D)∈ΠQ r.D. Then theapproximationof C w.r.t.I andδis defined as the concept description

bCc_I,δ:= ^l

A∈U

Au ^l

(Q,r,D)∈Π

Q r.D^IIδ.

Lemma 16. For all concept descriptions C,D, and role names r, the following statements hold.

1. (C^IIδuD)^I = (CuD)^I.

2. (Q r.C^IIδ)^I = (Q r.C)^I for all quantifiers Q∈ {∃,∀,≥n,≤n}.

Lemma 17. For every interpretationI, and every concept description C it holds that C^IIδv bCc_I,δ vC.

Lemma 18. LetIbe an interpretation, andδ∈_Na role-depth bound. Define

M_I,δ:={ ⊥ } ∪ {A,¬A|A∈N_C} ∪



















∃r.X^Iδ−1,

∀r.X^Iδ−1,

≥n.r.X^Iδ−1,

≤m.r.X^Iδ−1,

∃r.Self

r∈NR,

1≤m<n≤_∆^I,

∅6=X⊆_∆^I

















 .

Then every model-based most specific concept description ofIwith role-depth≤δis expressible in terms ofM_I,δ. Furthermore, theinduced contextofIandδis defined as the induced contextK^C_I,δ:=_KI,MI,δofIandM_I,δ.

Lemma 19 (Intents and MMSCs).LetIbe an interpretation over(N_C,N_R), andKI,δ

its induced context w.r.t. the role-depth boundδ∈N. Then the following statements hold for all subsets U⊆ MI,δ, and concept descriptions C over(NC,NR).

1. (^dU)^IIδ ≡^dU^II.

2. If U is an intent ofKI,δ, thend

U is a mmsc ofIwith role-depth≤δ.

3. If C is a mmsc ofIwith role-depth≤δ, thenπ_MI,δ(C)is an intent ofKI,δ. Consequently, the mappingd

:MI,δ→ ALEQR^Self(NC,NR)is an isomorphism from the intent-lattice(Int(KI,δ),∩)to the mmsc-lattice(Mmsc(I,δ),t), and has the inverseπM_I,δ. This shows the strong correspondence between the formal concept lattice ofKI,δand the description concept lattice ofIw.r.t. role depth≤δ. We can infer the following corollary from Lemmata 12 and 19.

Corollary 20. The intent lattice ofKI,δis isomorphic to the mmsc lattice ofI,δ.

We can further observe that the concept inclusions holding inIand the implications holding inKI,δare also in a strong correspondence. We can show that whenever the implicationU→Vholds inKI,δ, then also the concept inclusiond

Uv^dVholds inI. Furthermore, since every mmsc ofIwith a role depth≤δis expressible in terms ofM_I,δ, and conjunctions of intents ofKI,δare exactly the mmscs ofI, and every concept inclusionCvDholding inIis entailed by the concept inclusionCvC^II, we can deduce that indeed every concept inclusion holding inI is entailed by the transformation of the canonical implicational base ofKI,δ, which consists of all GCIs that have a conjunction of a pseudo-intent as premise and the conjunction of the closure of the pseudo-intent as conclusion.

Lemma 21. LetI be an interpretation over the signature(NC,NR),δ∈ Na role-depth bound, and CvD a concept inclusion, such that both concepts C,D have a role-depth≤δ.

Ext(_KI,δ)

Int(_KI,δ) Mmsc(I,δ)

B(_KI,δ) B(I,δ)

·^I

·^I

πM

d

·^Iδ ·^I π₁

(id,·^I)

π2 (·^I,id)

π1

(id,·^Iδ)

π2

(·^I,id)

Fig. 4.Overview on the isomorphisms between the extent lattice, intent lattice, and mmsc lattice ofKI,δandI,δ, respectively. Note thatExt(_KI,δ) =Ext(I,δ)holds.

1. If D is expressible in terms ofM_I,δ, and the implicationπ_MI,δ(C)→π_MI,δ(D)holds inKI,δ, then the concept inclusion CvD holds inI.

2. If C is expressible in terms ofM_I,δ, and the concept inclusion CvD holds inI, then the implicationπM_I,δ(C)→πM_I,δ(D)holds inKI,δ.

Corollary 22 (Concept Inclusion Base).LetI be an interpretation over the signature (N_C,NR), andδ∈_Na role-depth bound. Then the following statements hold:

1. For all subsets X,Y⊆ M_I,δ, the implication X→Y holds inKI,δif, and only if, the concept inclusiond

Xv^dY holds inI.

2. The intents ofKI,δare exactly the model-based most-specific concept descriptions ofI with role-depth bound≤δ.

3. IfLis an implicational base forKI,δ, thend

L:={^dXv^dY|X→Y∈ L }is a sound and complete TBox for all concept inclusions holding inI,δ. Especially this holds for the following TBox.

nl

Pv^lP^II

P is a pseudo-intent ofKI,δ

o

6 Induced Role Contexts

Role contexts have been introduced by Zickwolff [21], and have been used by Rudolph [19] for gaining knowledge on binary relations or roles (that are interpreted as binary relations). We use their definition here for the deduction of complex role inclusions holding in an interpretation.

Definition 23 (Induced Role Context).LetI be an interpretation over the signature (NC,NR), andδ∈_Na role depth bound. Furthermore, assume that X={x0,x1, . . . ,x_δ} is a set ofδ+1 variables. Then theinduced role contextforIandδis defined as

K^R_I,δ:=

∆^IX

,X×NR×X,J

where(f,(x,r,y))∈J if, and only if,(f(x),f(y))∈r^I.

Lemma 24 (Role Inclusions and Implications).LetIbe an interpretation over(NC,NR), δ∈_Na role-depth bound, and n≤δ. Then the complex role inclusion r1◦r2◦. . .◦rn vs holds inIif, and only if, the implication{(x0,r1,x1),(x1,r2,x2), . . . ,(xn−1,rn,xn)} → {(x0,s,xn)}holds in the induced role contextK^R_I,δ.

In particular, we are only interested in implications whose premise contains a subset of the form{(x0,r₁,x₁),(x₁,r2,x2), . . . ,(xk−1,r_k,x_k)}. Hence, we define a constrain- ing closure operatorφRon the attribute setX×NR×Xof the induced role context as follows.

φ_R(B):=









 B

if∃k∈_N+∃r₁,r2, . . . ,r_k∈NR

∃ {x₀,x₁, . . . ,x_k} ∈_k+1^X :

{(x0,r₁,x₁), . . . ,(xk−1,r_k,x_k)} ⊆B, B∪ {(x0,r,x1)|r∈NR} otherwise.

We shall now formulate a base of all complex role inclusions holding in an inter- pretation. For this purpose, we refer to [15] for the notions of constrained implications and their bases. Aφ-constrainedimplication overMis an implicationX→YoverM such that both premiseXand conclusionYareφ-closed. Aφ-constrainedimplicational base for a formal contextKis a set ofφ-constrained implications that is valid inK, and furthermore entails allφ-constrained implications that hold inK.

Theorem 25 (Role Inclusion Base).LetIbe an interpretation over(NC,NR). IfLis a φ_R-constrained implicational base ofKI,R, then the following RBoxR_I,δis sound, complete, and irredundant, for all complex role inclusions holding inI,δ.

R_I,δ:=



















r₁◦r₂◦. . .◦r_kvs

∃X→Y∈ L

∃r1,r2, . . . ,rk,s∈NR

∃ {x0,x1, . . . ,xk} ∈_k+1^X:

X⊇ {(x₀,r₁,x₁),(x₁,r₂,x₂), . . . ,(xk−1,r_k,x_k)} Y3(x0,s,x_k)



















7 Construction of the Knowledge Base

By means of the results of the previous Sections 5 and 6 we are now ready to formulate a knowledge base for an interpretationI, or for a description graphG, respectively.

Beforehand, it is necessary to inspect the interplay of role and concept inclusions to ensure irredundancy of the knowledge base. First, we list some trivial concept inclusions that hold in all interpretations.

Lemma 26. Let m,n∈_N+be non-negative integers with n<m, r∈ NRa role name, and C a concept description. The following general concept inclusions hold in every interpretationI.

Au ¬Av ⊥

∃r.Selfu ∀r.CvC

∃r.SelfuCv ∃r.C

∃r.SelfuCu ≤1.r.Cv ∀r.C

∃r.Cu ∀r.Dv ∃r.(CuD)

≥n.r.Cu ∀r.Dv ≥n.r.(CuD)

≤n.r.Cu ∀r.Dv ≤n.r.(CuD)

∃r.Cv ≥1.r.C

≥n.r.Cv ∃r.C

≤n.r.Cv ≤m.r.C

≥m.r.Cv ≥n.r.C

≥∆^I

.r.CvCu ∀r.Cu ∃r.Self

> v ≤∆^I .r.C

Please note that there are no direct subsumptions between existential restrictions

∃r.Cand value restrictions∀r.C, i.e., both∃r.C v ∀r.Cand∀r.C v ∃r.Cdo not hold. There is also a crossover between both constructors existential restric- tion and value restriction. The constructor is denoted by∀∃, and has the semantics (∀∃r.C)^I := (∃r.C)^I∩(∀r.C)^I, i.e., a domain element is in the extension of∀∃r.C if, and only if, there is anr-successor inC, and allr-successors are inC.

The next two lemmata show us which concept inclusions can be inferred from known role inclusions.

Lemma 27. LetI be a model of the role inclusion axiom r v s, C an arbitrary concept description, Q₁∈ { ∃,≥n}, Q2∈ { ∀,≤n}, and n∈_N+. ThenIis also a model of the following general concept inclusions.

Q1r.CvQ1s.C

∃r.Selfv ∃s.Self Q2s.CvQ2r.C

Lemma 28. Let I be a model of the complex role inclusion r₁◦r2◦. . .◦r_k v s, C an arbitrary concept description, Q₁∈ { ∃,≥n}, Q2∈ { ∀,≤n}, and n∈_N+. ThenIis also a model of the following concept inclusions.

∃r1.∃r2. . . .Q1rk.CvQ1s.C

Q2s.Cv ∀r₁.∀r2. . . .Q2r_k.C

As final step we use the trivial concept inclusions and concept inclusions that are entailed by valid role inclusions to define some background knowledge for the computation of the canonical implicational base of the induced concept context which is trivial in terms of description logics, but not for formal concept analysis due to their different semantics.

Theorem 29 (Knowledge Base).LetIbe an interpretation over the signature(NC,NR), andδ∈_Na role-depth bound. Furthermore, assume thatLis an implicational base of the induced concept contextK^C_I,δw.r.t. the background knowledge

S_I :=

{C} → {D}

C,D∈ MI,δ, CvD

∪ { {A,¬A} → M_I,δ|A∈N_C}

∪















n∃r.X^Iδ−1,∀r.Y^Iδ−1o

→ⁿ∃r.Z^Iδ−1o , n≥n.r.X^Iδ−1,∀r.Y^Iδ−1o

→ⁿ≥n.r.Z^Iδ−1o , n≤m.r.X^Iδ−1,∀r.Y^Iδ−1o

→ⁿ≤m.r.Z^Iδ−1o ,

r∈NR,

∅6=X,Y,Z⊆∆^I, Z^Iδ−1 ≡X^Iδ−1uY^Iδ−1, 1≤m<n≤_∆^I















∪



























n∃r.X^Iδ−1o

→ⁿ∃s.X^Iδ−1o , n∀s.X^Iδ−1o

→ⁿ∀r.X^Iδ−1o , n≥n.r.X^Iδ−1o

→ⁿ≥n.s.X^Iδ−1o , n≤m.s.X^Iδ−1o

→ⁿ≤m.r.X^Iδ−1o , { ∃r.Self} → { ∃s.Self}

r,s∈NR, rvs∈ R,

1≤m<n≤_∆^I,

∅6=X⊆_∆^I

























 .

ThenKI,δ = (TI,δ,RI,δ)is a knowledge base forI whereTI,δ :=^dLholds as in Corol- lary 22, andRI,δis defined as in Theorem 25.

8 Other Description Logics

If only a lower expressivity of the underlying description logic is necessary, then one could also useEL,FLE, or extensions thereof with role hierarchiesH, or complex role inclusionsR. All of the previous results are still valid, however one has to remove some of the used concept descriptions that are not expressible in the chosen description logic. Figure 5 gives an overview on description logics that have a lower expressivity thanALEQR^Self, and could also be used for knowledge acquisition.

8.1 Role HierarchiesHinstead of Complex Role InclusionsR

In the special case of simple role inclusions provided by the extensionHit is not necessary to use the induced role context. We can directly extract the role hierarchy from the interpretationI, or the description graphG, respectively, as follows.

First, we want to extract a minimal RBoxR_I from the interpretation that entails all role inclusion axioms holding inI. We therefore define an equivalence relation

≡_I on the role names as follows:r ≡_I sif, and only if,r^I =s^I. Then letN_R^I be a set of representatives of this equivalence relation, i.e.,

N_R^I∩[r]_≡I= 1 for all role namesr ∈ NR. Then add the following role equivalence axioms toRI: For each

constructor EL FL₀ FLE ALE Q Self H R

⊥ ×

> × × × ×

¬A ×

CuD × × × ×

∃r.C × × × ×

∀r.C × × ×

≥n.r.C ×

≤n.r.C ×

∃r.Self ×

CvD × × × ×

C≡D × × × ×

rvs × ×

r₁◦. . .◦rnvs ×

Fig. 5.Overview on various Description Logics belowALEQR^Self

representative roler∈N_R^I, add the axiomsr≡sfor alls∈[r]_≡

I\ {r}. Furthermore, define an order relationv_Ion the representativesN_R^Ibyrv_I sif, and only if,r^I ⊆s^I. Let≺_Ibe the neighborhood relation ofv_I, then add the role inclusion axiomsrvs for each pairr ≺_I sto the RBoxR_I. Obviously, the constructed RBox is minimal w.r.t. the property to entail all valid role inclusion axioms holding in the interpretation I. Eventually, the RBox inK_Iis defined as follows.

R_I :={r≡s|r∈N_R^I,s∈[r]_≡

I\ {r} } ∪ {rvs|r,s∈N_R^I,r≺_I s}

9 Conclusion

We have provided an extension of the results of Baader and Distel [1, 2, 9] for the deduction of knowledge bases from interpretations in the more expressive description logicALEQR^Selfw.r.t. descriptive semantics and role-depth bounds. Since role-depth- bounded model-based most-specific concept descriptions always exist, this technique can always be applied. Furthermore, the construction of knowledge bases has been reduced to the computation of implicational bases of formal contexts, which is a well-understood problem that has several available algorithms – for example the stan- dardNextClosurealgorithm from Ganter [10], or the parallel algorithm that has been introduced in [15] and implemented in [14]. The presented methods are prototypically implemented inConcept Explorer FX[14].

References

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