Deciding Fuzzy Description Logics by Type Elimination

Deciding Fuzzy Description Logics by Type Elimination

Uwe Keller¹and Stijn Heymans²

1 Semantic Technology Institute (STI) Innsbruck, University of Innsbruck, Austria.

eMail: uwe.keller@sti2.at

2 Knowledge-based Systems Group, Institute of Information Systems, Vienna University of Technology, Austria. eMail: heymans@kr.tuwien.ac.at

Abstract. We present a novel procedureFixIt(ALC)for deciding knowledge base satisfiability in the Fuzzy Description Logic (FDL)ALC.FixIt(ALC) does not search for tree-structured models as in tableau-based proof procedures, but embodies a fixpoint-computation of canonical models that are not necessar- ily tree-structured. Conceptually, the procedure is based on a type-elimination process. Soundness, completeness and termination are proven. To the best of our knowledge it is the firstfixpoint-baseddecision procedure for FDLs, hence intro- ducing a new class of inference procedures into FDL reasoning.

1 Introduction

Description Logics (DLs) [1] are a popular family of formally well-founded and decid- able knowledge representation languages. DLs have a wide range of applications, e.g., they form the basis for Semantic Web (SW) ontology languages used such as OWL [7].

Fuzzy Description Logics (FDLs) [13] extend DLs to represent vagueconcepts and relations, and as such are very well suited to cover for representing and reasoning with uncertainty, a requirement that naturally arises in many practical applications of knowledge-based systems, in particular the SW.

So far, reasoning in Fuzzy DLs is mainly based on tableau-methods (e.g. [13,12,4,11,16,3]).

Further, [14] demonstrates how to use inference procedures for classical DLs to perform reasoning in (some) FDLs. Still, reasoning in FDLs is at least as hard as reasoning in classical (crisp) DLs. Even in DLs of modest expressivity (e.g. ALC[13,14,12] the fuzzy variant of the DLALC [10]) the worst-case complexity of reasoning is signif- icant even in restricted cases [13]. Therefore, it is clear that there can not be asingle inference method that works well onallproblems.

Consequently, our goal is to enrich the range of available methods for reasoning with FDLs with a fundamentally different approach. In practical applications of DLs (and hence FDLs) a particularly important feature for representing domain models is the support of so-calledgeneral terminologies(see e.g. [12]), i.e., the possibility to cap- ture (potentially recursive) interdependencies between complex concepts in a domain

!This work has been partially funded by the European Commission under the LarKC project (FP7 - 215535). Stijn Heymans is supported by the Austrian Science Fund (FWF) under projects P20305 and P20840.

model. However, besides the tableau-based methods for DLs (e.g [12,4,16,3]) there are at present no other FDL inference methods which can deal with general terminologies.

We want to provide an alternative to tableau-based methods that can deal with general terminologies.

The main contributions of the paper are as follows:

− We present a novel procedureFixIt(ALC)(cf. Section 3) for deciding knowledge base (KB) satisfiability in the FDLALC(cf. Section 2).

− We formally prove soundness, completeness and termination of the algorithm (cf.

Section 3).

− FixIt(ALC)generalizes a type-elimination-based decision procedure [8] for the (classical) modal logicK(i.e.KBDD[6]) to the FDLALC. Additionally we inte- grate (fuzzy) ABoxes and general TBoxes which are not dealt with inKBDD.

− To the best of our knowledge it is the firstfixpoint-baseddecision procedure that has been proposed for FDL introducing anew class of inference proceduresinto FDL reasoning.

− Besides the tableau-based methods in [12,4,16,3], it is the only approach to inte- grate general terminologies in FDL reasoning and the first non-tableau-based one that we are aware of. General terminologies are handled in a fundamentally differ- ent way than in standard tableau-based method such as [12,4].

Our method is interesting especially regarding the last aspect since the handling of gen- eral terminologies in standard tableau-based methods (e.g. [12,4]) is amajorsource of non-determinism and thus computational inefficiency. In our case no non-deterministic choice is introduced by terminologies.

2 Preliminaries

We introduceALC[13], the fuzzy variant of the Description LogicALC[10] (the latter can be seen as a syntactic variant of the multi-modal logicK(m)[9]).ALCprovides the starting point for more expressive FDLs [15] that have been proposed to fuzzify major fragments of OWL [7].

Syntax. Concept expressions are constructed from a signatureΣ = (C,R,I) with concept names C, role namesR, and individual namesI. The set of concept expres- sions C(Σ)over Σ is defined as the smallest set of expressions that contains C, "

and is closed under the application of the concept constructorsC#D (intersection), C$D(union),¬C(complement), and∀R.C (universal role restriction) forR ∈ R andC, D ∈ C(Σ). We allow expressions∃R.C forC ∈ C(Σ), R ∈ Rand⊥and treat them as shortcuts for¬∀R.¬C and¬"respectively. A TBox axiom (or general concept inclusion axiom (GCI)) is an expression of the formC)Ds.t.C, D∈ C(Σ).

A terminology (or TBox)T is a finite set of TBox axioms. Syntactically, the vagueness of descriptions becomes explicit only when describing specific instances and their inter- relations: a (fuzzy) ABox axiom is either a*i:C "# d+or a*R(i, i^!)≥d+s.t.i, i^! ∈I, d ∈[0,1], and"#∈ {≤,≥,=}. An ABoxAis a finite set of ABox axioms. Finally, a knowledge baseK= (T,A)consists of a TBoxT and an ABoxA. LetInd_A⊆Ide- note the individual names that occur inA. We denote the set of all concept expressions that occur as subexpressions inKbysub(K).

Semantics. Semantically, vagueness is reflected in the use of fuzzy sets and relations when interpreting concepts and roles: an interpretationI = (∆^I,·^I)consists of a non- empty set ∆^I called the domain, and a function ·^I which maps each concept name C ∈ Cto a fuzzy setC^I : ∆^I → [0,1], each role nameR ∈ Rto a fuzzy relation R^I :∆^I×∆^I →[0,1]and each individual namei∈Ito an elementi^I ∈∆^I. The interpretation function·^I is extended to arbitrary concept expressionsC ∈ C(Σ)as follows:

– (C#D)^I(o) =min(C^I(o), D^I(o)) – (C$D)^I(o) =max(C^I(o), D^I(o)) – (¬C)^I(o) = 1−C^I(o)

– (∀R.C)^I(o) =inf_o!∈∆^I{max(1−R^I(o, o^!), C^I(o^!))} – "^I(o) = 1

for allo∈∆^I, C, D∈ C(Σ),R∈R.

An interpretation I satisfies a TBox axiomα = C ) D iff. for allo ∈ ∆^I it holds thatC^I(o) ≤D^I(o), i.e.C is a fuzzy subset ofD.I satisfies an ABox axiom α=*i:C "# d+iff.C^I(i^I)"# d.I satisfies an ABox axiomα=*R(i, i^!)≥d+iff.

R^I(i^I, i^!I)≥d. In all these cases, we writeI |=α.Isatisfies a TBoxT (or is a model ofT) iff.I |=αfor allα∈ T.I satisfies an ABoxA(or is a model ofA) iff.I |=α for allα∈ A. Finally,Isatisfies a knowledge baseK = (T,A)(or is a model ofK) iff.I |=T andI |=A.

Reasoning inALC.Given a fuzzy KBK= (T,A), fuzzy ABox axioms or GCIsαand concept expressionsC, D ∈ C(Σ), we can analyze particular semantic characteristics and interdependencies: We say thatKissatisfiable(or consistent) iff. there is a model IforK.Kentailsα(denoted asK |=α) iff. all modelsIofKsatisfyα. ConceptCis subsumed by conceptD(wrt. a KBK) iff.K |=C )D. Two conceptsCandDare calledequivalent(wrt. a KBK) iff. for any modelIofKit holds thatC^I(o) =D^I(o) for allo ∈ ∆^I. Two conceptsC andD are calleddisjoint(wrt. a KBK) iff. for any modelI ofKit holds that there does not exists ano ∈∆^Isuch thatC^I(o)> 0and D^I(o)>0. A conceptCis called satisfiable (wrt. a KBK) iff. there exists a modelIof T such thatC^I(o)>0for someo∈∆^I. Further, one might want to compute the truth value bounds for a given ABox assertionαwrt.Kto determine the possibility interval that is enforced forαby the background knowledge inK: Thegreatest lower boundof αwrt.Kis defined asglb(α,K) :=sup{d∈[0,1]| K |=*α≥d+}and theleast upper boundofαwrt.Kis defined aslub(α,K) := inf{d∈[0,1]| K |=*α≤d+}(where sup∅=inf∅= 0). Computingglb(α,K)andlub(α,K)is usually called thebest truth value bounds(BTVB) problem.

One of the most fundamental reasoning problems is to determine whether a given fuzzy KBKis satisfiable. A lot of other reasoning tasks (e.g., checking for concept satisfi- ability wrt. a TBox, entailment of fuzzy ABox assertions, or the BTVB problem) can be reduced to KB satisfiability checking (cnf. [13]) and therefore solved by a respec- tive decision procedure. For this reason, we consider KB satisfiability as the reasoning problem to be solved.

3 Fixpoint-based Decision Procedure

We present a decision procedure for KB satisfiability inALCwhich does not rely on systematic search in the first place (as e.g. tableau-based methods), but instead con- structs a canonical interpretation by means of a fixpoint construction. The so-constructed (canonical) interpretation (if non-empty) satisfies the TBox of a KB and allows to de- rive a model for the given knowledge baseKiff.Kis satisfiable. In contrast to tableau- based procedures a canonical interpretation is in generalnottree-shaped. Further, it can be shown that the number of iterations required to reach a fixpoint islinearin the modal depth ofK.

Preprocessing. Without loss of generality, we can restrict ourselves to normalized knowledge bases, i.e. knowledge bases which contain only fuzzy ABox assertions of the form*α≥d+[12]. Further, we can assume that all axioms inKare in box normal form (BNF) [6] (i.e. the only negative concept subexpressions are of the form¬∀R.C or negated atomic concept names¬C).

3.1 Basic Notions and Intuition

Types. LetK= (T,A)denote a normalizedALCknowledge base in BNF. Letsub(K) denote the set of all concept expressions that occur as subexpressions somewhere in an axiom inK. Theclosure of a knowledge basecl(K)is defined as the smallest set of concept expressions such that for all C ∈ sub(K), ifC is not of the form¬D, then {C,¬C} ⊆ cl(K). Further, let PossDeg(K)denote the set of all relevant possibility degrees that can be derived from K, i.e.PossDeg(K) = {0,0.5,1} ∪ {d|*α ≥ d+ ∈ A} ∪ {1−d|*α ≥ d+ ∈ A}. It has been shown in [13,14] that ifK is satisfiable, then there is as well a model ofK which assigns possibility degrees in PossDeg(K) only. Hence, for our purposes we do not need to consider arbitrary possibility degrees d∈[0,1], but only thefinitesetPossDeg(K)that can be derived fromK.

The closurecl(K)and the relevant possibility degreesPossDeg(K)together give us the basic vocabulary to describe individuals and their (fuzzy) properties in inter- pretations for K. More specifically, the notion of a type allows to represent individ- uals of an interpretation in a syntactic way: a fuzzyK-typeτ is a maximal subset of cl(K)×PossDeg(K)such that:

1. if*C, d+ ∈τand*C, d^!+ ∈τthend=d^! 2. ifC=¬C^!then*C, d+ ∈τiff.*C^!,1−d+ ∈τ

3. ifC = C^! #C^!! then*C, d+ ∈ τ iff. *C^!, d^!+ ∈ τ and*C^!!, d^!!+ ∈ τ andd = min(d^!, d^!!)

4. ifC = C^! $C^!! then*C, d+ ∈ τ iff. *C^!, d^!+ ∈ τ and*C^!!, d^!!+ ∈ τ andd = max(d^!, d^!!)

5. for allC)C^!∈ T: if*C, d+ ∈τand*C^!, d^!+ ∈τthend≤d^! 6. ifC="then*C,1+ ∈τ

Sincecl(K)andPossDeg(K)are both finite sets, there are at most2|cl(D)|·|PossDeg(K)|

differentK-types. Each typeτcan be seen as an individual and syntactically represents all(fuzzy) properties that can be observed about that individual:*C, d+ ∈τ represents the statement that the respective individualτbelongs to conceptCwith the possibility degreed. Hence, the set of allK-types (or simply types) provides enough vocabulary to let us describe all kinds of interpretations forKsimply be fixing how to interconnect individuals (and therefore types).

Canonical Model. It turns out that it is possible to connect types in a fixed (or canon- ical) way, such that the interconnection defined is consistent withalmostall properties specified syntactically in the type. The interconnections can be derived from the types themselves:

For a set of typesT we can define for each roleRacanonical accessibility relation

∆R : T ×T → PossDeg(K)that “maximally” interconnects typesτ, τ^! ∈ T with possibility degreed∈PossDeg(K): Letδ(d, d^!) := 1ifd≤d^!andδ(d, d^!) := 1−dif d > d^!. Then, we can define∆Rby

∆R(τ, τ^!) :=min{δ(d, d^!)|*∀R.C, d+ ∈τ,*C, d^!+ ∈τ^!} if∀R.C∈cl(K)for someC∈C, and∆R(τ, τ^!) := 1otherwise.

This way, we can construct a canonical interpretationI^T for any given set of typesT using the canonical interconnection of types by∆Ras follows:I^T = (T,·^IT)with (i) for any (atomic) concept nameCinKand anyτ∈Twe setC^IT(τ) =dif*C, d+ ∈τ, and (ii)R^IT(τ, τ^!) = ∆R(τ, τ^!)for any roleRinKand anyτ, τ^! ∈ T. Please note, that by our definition ofK-types,I^Tis well-defined for any concept name or role name.

However, our definition deliberately leaves open the interpretation of individuals. We therefore define in fact a class of canonical interpretations, each of which fixes a specific way of how to interpret the individuals in a KBK.

The canonical interconnection inI^T is chosen in such a way that all assignments of possibility degrees to concepts of the formC = ∀R.C ∈ τ are lower bounds for the possibility degrees that are in fact assigned by a canonical interpretationI^T. Hence, such a canonical interpretation isalmostimmediately a (canonical) model for the ter- minologyT, i.e. it satisfies that

C^IT(τ) =diff.*C, d+ ∈τ (∗) for almostallC ∈ cl(K)it holds and thereforeI^T |=C ) C^! for allC )C^! ∈ T by clause (5) in our definition ofK-types. That (∗) is satisfied byI^T is straightforward for the cases of concept names C, or complex concepts of the formC = C^! #C^!!, C =C^! $C^!!,C =¬C^! and theC^IT(τ)≥dcase forC =∀R.C by our definition of types and the definition of ∆R. The only cases where (∗) can be violated by I^T is for typesτcontaining universally role restricted concepts∀R.C that are assigned a possibility degree which istoo small(wrt. theR-successor typesτ^!inI^T) to properly reflect the semantics of∀R.CinALC, i.e. to coincide with thegreatestlower bound of the set

{max(1−R^IT(τ, τ^!), C^IT(τ^!))|τ^!∈T}

Typesτin which the possibility degree assigneddto∀R.Cis too small to be consis- tent with the semantics ofALCare calledbad types. Bad typesτ ∈T can be detected

easily, since they satisfy that there exist R ∈ R, C ∈ C(Σ), d ∈ PossDeg(K) s.t.

*∀R.C, d+ ∈τand for allτ^!∈T: if*C, d^!+ ∈τ^!thenmax(1−∆R(τ, τ^!), d^!)> d.

This suggest the the following simple algorithm (which uses afuzzy type elimina- tionprocess as its core): in order to compute a maximal interpretation that satisfies all terminological axioms, we start off with the maximal set of types (i.e allK-types) and iteratively fix all problems that prevent(∗)from being satisfied by removing bad types.

This way, we must eventually reach a fixpoint after finitely many steps. If the resulting set of types is non-empty, we know that (∗) must hold (since all problems have been fixed) and therefore we can be certain that the corresponding canonical interpretation satisfies T (and covers all other possible models ofT at the same time). Hence, we eventually need to check if all ABox axioms are satisfied by the canonical interpreta- tion. If this is the case, we have found a model forK, otherwise, we know that there can not be any interpretation that satisfies bothT andAat the same time. In other words, Kis not satisfiable.

Algorithm. The type elimination process sketched above can be formalized as shown in Fig. 1. Note that the emptiness test for the fixpointT is covered implicitly: if the fixpointTis empty, then the test in the if-statement fails trivially.

proceduresatisfiable(K): boolean T :={τ|τis aK-type};

repeat T^!:=T;

T:=T^!\badtypes(T^!);

untilT =T^!;

if there exists a total functionπ:Ind_A→T s.t."C, d^!# ∈π(o)andd≤d^!for each

"o:C≥d# ∈ A, and∆R(π(o), π(o^!))≥dfor each"R(o, o^!)≥d# ∈ Athen returntrue;

endreturnfalse;

functionbadtypes(T) : 2^T

return{τ∈T|"∀R.C, d# ∈τand for allτ^!∈T: if"C, d^!# ∈τ^!then max(1−∆R(τ, τ^!), d^!)> d};

Algorithm 1: The Type Elimination-based Decision ProcedureFixIt(ALC)

3.2 Soundness, Completeness and Termination

The termination, soundness, and completeness of our algorithm can be proven formally.

Theorem 1 (Termination).For anyALCknowledge baseK= (T,A)the algorithm FixIt(ALC)terminates after finitely many steps with either true or falseas return value.

Proof. The initialization step of the algorithm takes finitely many steps since the number ofK-types is finite. The repeat-loop must terminate after finitely many steps, since we start with a finite set of typesT in the beginning: if we do not remove any type in an iteration (i.e.badtypes(T^!) = ∅) we haveT =T^! at the end of the loop (i.e. reaching a fixpoint) and therefore terminate the loop. On the other hand, ifbadtypes(T^!)4=∅in an iteration, at least one type is removed fromT^!and henceT ⊂T^!. This means, that the input set of typesT for the next iteration is finite and strictly smaller. Clearly, the empty set is a fixpoint ofbadtypes(·)too, i.e.badtypes(∅) =∅. Hence, we can repeat the loop only finitely many times until we finally will reach a fixpoint. Since this fixpoint T is a subset of the finite set of the initial set of types and there are only finitely many possible mappingsπto consider, deciding the criterion in the if-statement (based onT) takes as well only finitely many steps. Therefore, the algorithm terminates with one of the return-statements that give as a result eithertrueorfalse.

The following lemma is a key element of the soundness and completeness proof and shows that by successively removing bad types we can indeed ensure that types encode possibility degree assignments to concepts that coincide with the canonical interpreta- tion, and that any such canonical interpretation is a model of theT.

LetTbe the set of types that is computed as the fixpoint in the algorithmFixIt(ALC), i.e.badtypes(T) =∅and letI^T=(T,·^IT)be a canonical interpretation forTas defined above.

Lemma 1. For eachK-typeτ, conceptC ∈cl(K)andd∈PossDeg(K)it holds that C^IT(τ) =diff.*C, d+ ∈τ. Further,I^T |=T.

Proof. For the first part of the lemma, letτbe anyK-type andd∈PossDeg(K)be any relevant possibility degree.

We show by induction over the structure of conceptsC ∈ cl(K)that*C, d+ ∈ τ iff.C^IT(τ) =d: the base case (i.e.C ∈cl(K)is an atomic concept nameC ∈Cor C=") is trivially satisfied by our definition ofI^T. For the induction step, we consider the different cases of compound concept expressionsC∈cl(K)one-by-one:

1. for C = C1#C2 ∈ cl(K), we know that C1, C2 ∈ cl(K). By clause (3) in our definition of types, we know that*C, d+=*C1#C2, d+ ∈τiff.*C1, d1+ ∈τ and

*C2, d2+ ∈τandd=min(d1, d2). Applying the induction hypothesis toC1andC2, we know that this is the case iff.C1IT(τ) = d1andC2IT(τ) =d2andd=min(d1, d2) iff.d=min(C1IT(τ), C2IT(τ)) iff.d= (C1#C2)^IT =C^IT by the by the semantics of#inALC.

2. forC=C1$C2∈cl(K)the proof is analogous.

3. for C = ¬D ∈ cl(K), we know that D ∈ cl(K) by the definition ofcl(K).

Because of clause (2) and the maximality requirement in our definition ofK-types, we know that*C, d+=*¬D, d+ ∈τiff.*D,1−d+ ∈τ. Applying the induction hypothesis forD, we know that this holds iff.D^IT(τ) = 1−diff.(¬D)^IT(τ) =C^IT(τ) =dby the semantics of concept negation inALC.

4. for C = ∀R.D ∈ cl(K),D ∈ sub(K) holds and henceD ∈ cl(K) by the definition ofcl(K).

First, we show one direction, i.e. that C^IT(τ) = dif *C, d+ ∈ τ: Assume that

*C, d+=*∀R.D, d+ ∈τ. According to the semantics of the universal role restriction in ALCand our definition ofI^T, we haveC^IT(τ) = (∀R.D)^IT(τ) =inf_τ!∈T{max(1− R^IT(τ, τ^!), D^IT(τ^!))}=inf_τ!∈T{max(1−∆R(τ, τ^!), D^IT(τ^!))}. We show thatdis a lower bound for{max(1−∆R(τ, τ^!), D^IT(τ^!))|τ^!∈T}: Assume there exists aτ^!∈ T s.t.d > max(1−∆R(τ, τ^!), D^IT(τ^!)). LetD^IT(τ^!) = d^!. Applying the induction hypothesis to D ∈ cl(K), we know*D, d^!+ ∈ τ^!. Hence, both d > 1−∆R(τ, τ^!) and d > d^! must hold. Hence∆R(τ, τ^!) > 1−d. But, since*∀R.D, d+ ∈ τ this is not possible by our by our definition of∆R, because∆R(τ, τ^!) ≤ 1−d. From the contradiction we can conclude thatdis in fact a lower bound for the considered set.

Therefore,d≤inf_τ!∈T{max(1−∆R(τ, τ^!), D^IT(τ^!))}

Next, we show thatinf_τ!∈T{max(1−∆R(τ, τ^!), D^IT(τ^!))} ≤ dtoo, by proving that there exists aτ^!∈Ts.t.max(1−∆R(τ, τ^!), D^IT(τ^!))≤d. Assume, the contrary, i.e. for allτ^! ∈T:max(1−∆R(τ, τ^!), D^IT(τ^!))> d(‡). By applying our induction hypothesis toDandτ^!, we know that this is the case iff. for allτ^! ∈T: if*D, d^!+ ∈τ^! thenmax(1−∆R(τ, τ^!), d^!)> d. But then,τ would be a bad type which contradicts the fact that T the computed fixpoint which can not contain any bad types (i.e.τ ∈ badtypes(T) = ∅). Hence our assumption (‡) must be wrong, and we can conclude thatinf_τ!∈T{max(1−∆R(τ, τ^!), D^IT(τ^!))} ≤d. Therefore,d=inf_τ!∈T{max(1−

∆R(τ, τ^!), D^IT(τ^!))}, and henced=C^IT(τ).

The other direction of the induction hypothesis (i.e. that*∀R.D, d+ ∈τif(∀R.D)^IT(τ) = d) can now be proven as follows: Assume that(∀R.D)^IT(τ) =d(†) but*∀R.D, d+ 4∈τ.

By the maximality requirement in our definition of K-types there must hence exist a d^! ∈PossDeg(K)s.t.*∀R.D, d^!+ ∈τandd^! 4=d. Using the same argument as for the if-direction in this case, we can therefore conclude that(∀R.D)^IT(τ) =d^! 4=dwhich contradicts (†). Hence, our assumption must be wrong and*∀R.D, d+ ∈ τ must hold whenever(∀R.D)^IT(τ) =d.

For the second part of the lemma, to show thatI^T |= T, assume that for some α=C )C^! ∈ T and someτ ∈T it holds thatC^IT(τ)> C^!IT(τ), in other words, if C^IT(τ) = dand C^!IT(τ) = d^! thend > d^!. Thus, we can deduce (from the first part of this lemma) that, if*C, d+ ∈τand*C^!, d^!+ ∈ τthend > d^!. However, by our definition ofK-type (i.e. clause (5)), we also know that in this cased≤d^!must hold, which is contradictive. Hence, our assumption must be wrong andI^T |= αfor each α∈ T which means thatI^T |=T.

Theorem 2 (Soundness).IfFixIt(ALC)returns true for aALCknowledge baseK= (T,A), thenKis satisfiable.

Proof. We show that a canonical interpretationI^T for the computed fixpointT can be extended to a model ofK. By Lemma 1, we already know thatI^T |=T. We now show, thatI^T can be extended such thatI^T |=Atoo, which completes the proof: Since the algorithm returns true, there exist a total functionπ: Ind_A → T s.t.*C, d^!+ ∈ π(o) and d ≤ d^! for each *o : C ≥ d+ ∈ A ()), and ∆R(π(o), π(o^!)) ≥ d for each

*R(o, o^!)≥d+ ∈ A(†). We extend the definition ofI^T to the ABoxAas follows: for all Abox individual names o ∈ Ind_A, we seto^IT := π(o) ∈ T. First, consider an ABox axiom of the formα = *o : C ≥d+ ∈ A. Then,I^T |= αiff.C^IT(o^IT) ≥ d,

iff.C^IT(π(o))≥diff. there exists ad^! ≥ds.t.C^IT(π(o)) =d^!. By Lemma 1 this is the case iff. there exists ad^! ≥ds.t.*C, d^!+ ∈ π(o)which is satisfied since ()) holds.

Second, consider an ABox axiom of the formα=*R(o, o^!)≥d+ ∈ A. Then,I^T |=α iff.R^IT(o^IT, o^!IT) ≥ diff.∆R(π(o), π(o^!)) ≥ d(by Def. of the extendedI^T). The latter is satisfied because of (†).

An second key element for the completeness proof is the following lemma that shows that our canonical way of interconnecting types (in the fixpoint set) is maximal or the strongest possible one in the following sense: the interconnectionRof individuals o, o^!defined by any modelI ofKis covered by the canonical interconnection∆R of the respective typesτ(o), τ(o^!)representingo, o^!inI.

Lemma 2. LetI= (∆^I,·^I)be any model ofK= (T,A). For each individualo∈∆^I we define its corresponding typeτ(o) :={*C, d+ ∈cl(K)×PossDeg(K)|C^I(o) =d}. Then,∆R(τ(o), τ(o^!))≥R^I(o, o^!)for allo, o^!∈∆^I.

Proof. Assume that there existo, o^! ∈ ∆^I s.t.∆R(τ(o), τ(o^!)) < R^I(o, o^!). By our definition of∆R, we then know thatδ(d, d^!)< R^I(o, o^!)(∗) for some*∀R.C, d+ ∈τ(o) and*C, d^!+ ∈τ(o^!). From the definition ofτ(·), we know thatδ(d, d^!)< R^I(o, o^!)for someo, o^! ∈∆^Is.t.(∀R.C)^I(o) =dandC^I(o^!) =d^!. From the semantics of∀R.Cin ALC, we derived=inf_o!!∈∆^I{max(1−R^I(o, o^!!), C^IT(o^!!))}. Hence, in particular d ≤ max(1−R^I(o, o^!), C^IT(o^!)) = max(1−R^I(o, o^!), d^!)(†). We consider two cases: first,d^! < d, then in order to satisfy (†),max(1−R^I(o, o^!), d^!) = 1−R^I(o, o^!) must hold and (†) simplifies tod≤1−R^I(o, o^!)iff.R^I(o, o^!)≤1−d. Sinced^!< d, (∗) simplifies to1−d < R^I(o, o^!), hence,1−d < R^I(o, o^!)≤1−dwhich is contradictory.

In the second case, we assume thatd^! ≥d. Hence,δ(d, d^!) = 1. Then, (†) simplifies to 1 < R^I(o, o^!), which is contradictory, since R^I(o, o^!) ∈ PossDeg(K) and 1 is the maximum possibility degree inPossDeg(K). Therefore, in both cases we reach a contradiction and can conclude that our assumption must be wrong. This concludes the proof.

Theorem 3 (Completeness). If anALCknowledge baseK = (T,A)is satisfiable, thenFixIt(ALC)returns true forK

Proof. Let I = (∆^I,·^I)be any model of K = (T,A), i.e. I |= T and I |= A. In [13,14] it is shown that a KB in ALCis consistent iff. there is a modelI of the KB which only assigns possibility degrees that occur in the ABoxA for interpreting atomic concept or role names. Hence, without loss of generality, we can assume in the following thatIassigns possibility degrees inPossDeg(K)only.

For each individualo ∈ ∆^I we define its corresponding typeτ(o) := {*C, d+ ∈ cl(K)×PossDeg(K)|C^I(o) =d}and defineT_I :={τ(o)|o∈ ∆^I}. It is easy to see thatT_Iis a set ofK-types. Further,T_I 4=∅since∆^I4=∅.

LetT⁽ⁱ⁾denote the set of types that is computed afteriiterations of the repeat-loop in our algorithm. We first show thatT_I ⊆ T⁽ⁱ⁾for alli ≥ 0 by induction over the number of iterationsi:

In the base casei= 0, our initialization step setsT⁽⁰⁾to contain allK-types. Since T_I consists ofK-types only,T_I ⊆ T⁽⁰⁾ must hold. To proof the induction step, we

assume that T_I ⊆ T⁽ⁱ⁾butT_I 4⊆ T⁽ⁱ⁺¹⁾. Therefore, there must be aτ(o) ∈ T_I s.t.

τ(o) ∈ T⁽ⁱ⁾ but notτ(o) ∈ T⁽ⁱ⁺¹⁾. From the repeat-loop in the algorithm, we know that T⁽ⁱ⁺¹⁾ = T⁽ⁱ⁾ badtypes(T⁽ⁱ⁾). Consequently,τ(o) must be a bad type τ(o) ∈ badtypes(T⁽ⁱ⁾)and we can not have reached a fix-point yet.

From our definition of bad-types we can derive that there must exist a*∀R.C, d+ ∈ τ(o)and for allτ^!∈T⁽ⁱ⁾: if*C, d^!+ ∈τ^!thenmax(1−∆R(τ, τ^!), d^!)> d}(‡). Since T_I ⊆T⁽ⁱ⁾(‡) must hold in particular for allτ(o^!)∈T_I. Using our definition ofτ(·) we can rephrase (‡) as follows: there must exist a∀R.C ∈ cl(K)s.t. for allo^! ∈ ∆^I: max(1−∆R(τ(o), τ(o^!)), C^I(o^!))>(∀R.C)^I(o)()).

By Lemma 2, we know that∆R(τ(o), τ(o^!))≥R^I(o, o^!)for allo, o^!∈∆^I. Hence, 1 −∆R(τ(o), τ(o^!)) ≤ 1 −R^I(o, o^!) (∗) for all o, o^! ∈ ∆^I. Since max(a, b) ≤ max(a^!, b)for any a, a^!, b s.t. a ≤ a^!, we can reformulate ()) using (∗) as follows:

there must exist a∀R.C ∈cl(K)s.t. for allo^! ∈∆^I:max(1−R^I(o, o^!), C^I(o^!))>

(∀R.C)^I(o)}(*), which contradicts the fact that(∀R.C)^I(o) = inf_o!∈∆^I{max(1− R^I(o, o^!), C^I(o^!))}: SinceR^I(o, o^!) ∈ PossDeg(K)andC^I(o^!) ∈ PossDeg(K), we know that max(1−R^I(o, o^!), C^I(o^!)) ∈ PossDeg(K)for all o^! ∈ ∆^I by our def- inition ofPossDeg(K). Because∆^I 4= ∅andPossDeg(K)is a finite set, there must exist an o^∗ ∈ ∆^I for whichmax(1−R^I(o, o^∗), C^I(o^∗))is minimal, i.e.max(1− R^I(o, o^∗), C^I(o^∗)) ≤ max(1−R^I(o, o^!), C^I(o^!))for allo^! ∈ ∆^I. Hence, d^∗ :=

max(1−R^I(o, o^∗), C^I(o^∗))∈ PossDeg(K)is a lower bound for the set{max(1− R^I(o, o^!), C^I(o^!)|o^! ∈∆^I}. However, from (*) we know thatd < d^∗, hencedcan not be the greatest lower bound (i.e. the infimum) of the set{max(1−R^I(o, o^!), C^I(o^!)|o^! ∈

∆^I}, hence∀R.C)^I(o) =d^∗> dwhich is contradictive.

Therefore, our assumption thatτ(o)is a bad type must be wrong and we have com- pleted the proof of the induction step as well as the induction argument.

We continue the proof of the lemma as follows: sinceT =T⁽ⁱ⁾is the fixpoint that is computed in the loop in our algorithm in i steps for somei ≥ 0, we know that T_I⊆T⁽ⁱ⁾=T. Consider the mappingπ_I :Ind_A→T defined byπ_I(o) :=τ(o^I)for allo∈Ind_A. Then,π_Iis a well-defined, total function fromInd_AtoT. We now show that this specific mappingπ_I satisfies the condition that is checked in the if-statement of the algorithm:

In the first case, we consider any Abox axiomα∈ Aof the formα=*o:C ≥d+. SinceI |= A,I |= αmust hold.I |=αiff.C^I(o^I)≥ diff.C^I(o^I) = d^!for some d^! ∈ PossDeg(K)withd^! ≥ diff.*C, d+ ∈ τ(o^I)for somed^! ∈ PossDeg(K)with d^! ≥ d(by Lemma 1) iff. *C, d+ ∈ π_I(o)for some d^! ∈ PossDeg(K)with d^! ≥ d (by our definition of π_I). Hence, the respective part of the if-condition for α holds for π_I. In the second case, we consider any Abox axiom α ∈ Aof the form α =

*R(o, o^!)≥d+. Since,I |=A,I |= αmust hold.I |=αholds iff.R^I(o^I, o^!I)≥ d.

Since∆R(τ(o), τ(o^!))≥R^I(o^I, o^!I)by Lemma 2, we know that∆R(τ(o^I), τ(o^!I))≥ dand therefore∆R(π_I(o), π_I(o))≥dby our definition ofπ_I. Hence, the respective part of the if-condition for αis as well satisfied byπ_I. Consequently, the tested if- condition is satisfied byπ_Iand the algorithm returnstrue.

This leads to the main result, which is an immediate consequence of Theorems 2, 3, and 1:

Corollary 1. The algorithmFixIt(ALC)is a sound and complete decision procedure for knowledge base satisfiability inALC.

4 Related Work

Our methodFixIt(ALC)generalizes the principle (i.e. a type elimination process) un- derlying the top-down variant of theKBDDprocedure proposed in [6] for the modal logic K to the (more expressive) FDLALC. Further, our method integrates (fuzzy) ABoxes and TBoxes in the inference process both of which arenotdealt with inKBDD. So far, reasoning in Fuzzy DLs has been mostly based on tableau-methods (e.g., [13,12,4,11]).

Most of these methods do not support reasoning with general terminologies as it is pos- sible withFixIt(ALC). The first method ever to integrate GCIs into FDL reasoning is [12]. A very similar approach is presented in [4] for the fuzzy variant of a more ex- pressive DL, namelySHI. Very recently, [16] proposed a novel and elegant method for reasoning with GCIs (under a more general semantics than here) which is inspired by earlier works on tableau-based reasoning in multi-valued logics. To the best of our knowledge there is no other approach to deal with GCIs in FDLs available at present.

FixIt(ALC)therefore represents an interesting enrichment of inference calculi tool- box for FDLs, since no non-determinism is introduced by considering GCIs. A similar effect is achieved in [16] by the substantial modification of a standard tableau-based method and an extension with an MILP oracle. A very similar approach to [16] that is not fixed to a specific semantics is presented in [3].

Further, [14] demonstrates how to use inference procedures forclassicalDLs to perform reasoning in (some) FDLs. This allows to use algorithms that have been developed for classical DLs in FDL reasoning (for some FDLs) in an indirect way. Please note that the KBDDprocedure can not be used in such an indirect way to performALCreasoning, since both TBoxes and ABoxes are not supported.

5 Conclusions and Future Work

We presented a novel procedureFixIt(ALC)for deciding knowledge base (KB) sat- isfiability in the FDLALC, introducing anew classof inference procedures into FDL reasoning. Besides the tableau-based methods [12,4,16,3], it is the only (and the first non-tableau-based) approach to integrate general terminologies in FDL reasoning that we are aware of.

The main research questions that we want to address next are as follows: we will study means of implicit representation of sets of fuzzy types known from Symbolic Model Checking, in particular their implementation by means of Ordered Binary Deci- sion Diagrams (OBDDs) similar to [6], therefore addressing the main obstacle to apply the procedure in practice. A major question concerning optimization is clearly how to implement the final test of the algorithm efficiently, e.g. by heuristic search using the information in the ABox effectively to find the required mapping. The integration of optimizations such as full vs. lean representations or particle vs. types as discussed in [6] should be straightforward. We want to evaluate the effectiveness of the method

by an implementation and comparison to tableau-based systems for FDLs. Moreover, we believe that it is interesting to study a bottom-up variant ofKBDD in the context of FDLs too, and to check if the integration of ABoxes can be done more efficiently in such a variant. Finally, we would like to see to what extend the method can cover other semantics for FDLs (e.g. other t-norms) and extended constructs, such as fuzzy modifiers and concrete domains.

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