In this paper, we present a satisfiability preserving transformation of the fuzzy Description Logic ALCFL into the classical Description Logic ALCH. We can use the already existing DL systems to do the reasoning of ALCFL by applying the result of this paper. This work is inspired by Straccia, who has transformed the fuzzy Description Logic fALCH into the classical Description Logic ALCH.
1 Please visit http://www.w3.org/TR/owl-guide/ for more details. are called hedges in fuzzy DLs. Some approaches to handling uncertainty and vagueness in DL for the Semantic Web are described in [10].
A well known feature of DLs is the emphasis on reasoning as a central service. Some reasoning procedures for fuzzy DLs have been proposed in [16]. A transformation of fALCH into ALCH has been presented in [17]. This approach, however, only works for DLs where modifier concepts are not allowed.
In this paper we consider the fuzzy linguistic description logic ALCFL [7] which is an instance of the description logic framework L − ALC with the certainty lattice characterized by a hedge algebra and allows the modification by hedges. Because the certainty lattice is characterized by a HA, the modification by hedges becomes more natural than that in ALCFH [8] and ALCFLH [14] which extend fuzzy ALC by allowing the modification by hedges of HAs. We will present a satisfiability preserving transformation of ALCFL into ALCH which makes the reuse of the technical results of classical Dls for ALCFL feasible.
The remaining part of this paper is organized in the following way. First we state some preliminaries on ALCH, hedge algebra and ALCFL. Then we present the transformation of ALCFL into ALCH. Finally we discuss the main result of the paper and identify some possibilities for further work. 2
ALCH We consider the language ALCH (Attributive Language with Complement and role Hierarchy). In abstract notation, we use the letters A and B for concept names, the letter R for role names, and the letters C and D for concept terms. Definition 1. Let NR and NC be disjoint sets of role names and concept names. Let A ∈ NC and R ∈ NR. Concept terms in ALCH are formed according to the following syntax rule:
A|⊤|⊥|C ⊓ D|C ⊔ D|¬C|∀R.C|∃R.C The semantics of concept terms are defined formally by interpretations. Definition 2. An interpretation I is a pair (ΔI , ·I), where ΔI is a nonempty set ( interpretation domain) and ·I is an interpretation function which assigns to each concept name A a set AI ⊆ ΔI and to each role name R a binary relation RI ⊆ ΔI × ΔI . The interpretation of complex concept terms is extended by the following inductive definitions: ⊤I = ΔI ⊥I = ∅ (C ⊓ D)I = CI ∩ DI (C ⊔ D)I = CI ∪ DI
(¬C)I = ΔI \ CI (∀R.C)I = {d ∈ ΔI | ∀d′.(d, d′) ∈/ RI or d′ ∈ CI } (∃R.C)I = {d ∈ ΔI | ∃d′.(d, d′) ∈ RI and d′ ∈ CI } A concept term C is satisfiable iff there exists an interpretation I such that CI 6= ∅, denoted by I |= C. Two concept terms C and D are equivalent (denoted by C ≡ D) iff CI = DI for all interpretation I.
We have seen how we can form complex descriptions of concepts to describe classes of objects. Now, we introduce terminological axioms, which make statements about how concept terms and roles are related to each other respectively.
In the most general case, terminological axiom have the form C ⊑ D or R ⊑ S, where C, D are concept terms, R, S are role names. This kind of terminological axioms are also called inclusions. A set of axioms of the form R ⊑ S is called role hierarchy. An interpretation I satisfies an inclusion C ⊑ D (R ⊑ S) iff CI ⊆ DI (RI ⊆ SI ), denoted by I |= C ⊑ D (I |= R ⊑ S).
A terminology, i.e., TBox, is a finite set of terminological axioms. An interpretation I satisfies (is a model of) a terminology T iff I satisfies each element in T , denoted by I |= T .
Assertions define how individuals relate with each other and how individuals relate with concept terms. Let NI be a set of individual names which is disjoint to NR and NC . An assertion α is an expression of the form a : C or (a, b) : R, where a, b ∈ NI , R ∈ NR and C ∈ NC . A finite set of assertions is called ABox. An interpretation I satisfies a concept assertion a : C iff aI ∈ CI , denoted by I |= a : C. I satisfies a role assertion (a, b) : R iff (aI , bI ) ∈ RI , denoted by I |= (a, b) : R. An interpretation I satisfies (is a model of) an ABox A iff I satisfies each assertion in A, denoted by I |= A.
A knowledge base is of the form hT , Ai where T is a TBox and A is an ABox. An interpretation I satisfies (is a model of, denoted by I |= K) a knowledge base K = hT , Ai iff I satisfies both T and A. We say that a knowledge base K entails an assertion α, denoted K |= α iff each model of K satisfies α. Furthermore, let T be a TBox and let C, D be two concept terms. We say that D subsumes C with respect to T (denoted by C ⊑T D) iff for each model of T , I |= CI ⊆ DI .
The problem of determining whether K |= α is called entailment problem; the problem of determining whether C ⊑T D is called subsumption problem; and the problem of determining whether K is satisfiable is called satisfiability problem. Entailment problem and subsumption problem can be reduced to satisfiability problem.
In this section, we introduce linear symmetric Hedge Algebras (HAs). For general HAs, please refer to [12, 11, 13].
Let us consider a linguistic variable TRUTH with the domain dom(TRUTH ) = {True, False, VeryTrue, VeryFalse, MoreTrue, MoreFalse, PossiblyTrue, . . .}. This domain is an infinite partially ordered set, with a natural ordering a < b meaning that b describes a larger degree of truth if we consider T rue > F alse. This set is generated from the basic elements (generators) G = {True, False} by using hedges, i.e., unary operations from a finite set H = {Very, Possibly, More }. The dom(TRUTH ) which is a set of linguistic values can be represented as X = {δc | c ∈ G, δ ∈ H∗} where H∗ is the Kleene star of H, From the algebraic point of view, the truth domain can be described as an abstract algebra AX = (X, G, H, >).
To define relations between hedges, we introduce some notations first. We define that H(x) = {σx | σ ∈ H∗} for all x ∈ X. Let I be the identity hedge, i.e., ∀x ∈ X.Ix = x. The identity I is the least element. Each element of H is an ordering operation, i.e., ∀h ∈ H, ∀x ∈ X, either hx > x or hx < x. Definition 3. [12] Let h, k ∈ H be two hedges, for all x ∈ X we define: – h, k are converse if hx < x iff kx > x; – h, k are compatible if hx < x iff kx < x; – h modifies terms stronger or equal than k, denoted by h ≥ k if hx ≤ kx ≤ x or hx ≥ kx ≥ x; – h > k if h ≥ k and h 6= k; – h is positive wrt k if hkx < kx < x or hkx > kx > x; – h is negative wrt k if kx < hkx < x or kx > hkx > x.
ALCFL only considers symmetric HAs, i.e., there are exactly two generators as in the example G = {True, False}. Let G = {c+, c−} where c+ > c−. c+ and c− are called positive and negative generators respectively. Because there are only two generators, the relations presented in Definition 3 divides the set H into two subsets H+ = {h ∈ H | hc+ > c+} and H− = {h ∈ H | hc+ < c+}, i.e., every operation in H+ is converse w.r.t. any operation in H− and vice-versa, and the operations in the same subset are compatible with each other. Definition 4. [7] An abstract algebra AX = (X, G, H, >), where H 6= ∅, G = {c+, c−} and X = {σc | c ∈ G, σ ∈ H∗} is called a linear symmetric hedge algebra if it satisfies the properties (A1)-(A5). (A1) Every hedge in H+ is a converse operation of all operations in H−. (A2) Each hedge operation is either positive or negative w.r.t. the others, including itself. (A3) The sets H+ ∪ {I} and H− ∪ {I} are linearly ordered with the I. (A4) If h 6= k and hx < kx then h′hx < k′kx, for all h, k, h′, k′ ∈ H and x ∈ X. (A5) If u ∈/ H(v) and u ≤ v (u ≥ v) then u ≤ hv (u ≥ hv), for any hedge h and u, v ∈ X.
Let AX = (X, G, H, >) be a linear symmetric hedge algebra and c ∈ G. We define that, c¯ = c+ if c = c− and c¯ = c− if c = c+. Let x ∈ X and x = σc, where σ ∈ H∗. The contradictory element to x is y = σc¯ written y = −x.
[12] gave us the following proposition to compare elements in X.
Proposition 5 Let AX = (X, G, H, >) be a linear symmetric HA, x = hn · · · h1u and y = km · · · k1u are two elements of X where u ∈ X. Then there exists an index j ≤ min{n, m} + 1 such that hi = ki for all i < j, and (i) x < y iff hjxj < kjxj , where xj = hj−1 · · · h1u; (ii) x = y iff n = m = j and hjxj = kj xj.
In order to define the semantics of the hedge modification, we only consider monotonic HAs defined in [7] which also extended the order relation on H+ ∪{I} and H− ∪ {I} to one on H ∪ {I}. We will use “hedge algebra” instead of “linear symmetric hedge algebra” in the rest of this paper.
Fuzzy description logics represent the assessment “It is true that Tom is very old” by
(VeryOld )I (Tom)I = True.
In a fuzzy linguistic logic [19–21], the assessment “It is true that Tom is very
old” and the assessment “It is very true that Tom is old” are equivalent, which
means
(Old )I (Tom)I = VeryTrue,
(
RT 1 : (hC)I (d) = σc → (C)I (d) = σhc
RT 2 : (C)I (d) = σhc → (hC)I (d) = σc. where C is a concept term and d ∈ ΔI .
Definition 6. [7] Consider a monotonic HA AX = (X, {c+, c−}, H, >) and a h ∈ H. A mapping h− : X → X is called an inverse mapping of h iff it satisfies the following two properties, 1. h−(σhc) = σc. 2. σ1c1 > σ2c2 ⇔ h−(σ1c1) > h−(σ2c2). where c, c1, c2 ∈ G, h ∈ H and σ1, σ2 ∈ H∗.
ALCFL ALCFL is a Description Logic in which the truth domain of interpretations is represented by a hedge algebra. The syntax of ALCFL is similar to that of ALCH except that ALCFL allows concept modifiers and does not include role hierarchy. Definition 7. Let H be a set of hedges. Let A be a concept name and R a role, complex concept terms denoted by C, D in ALCFL are formed according to the following syntax rule:
A|⊤|⊥|C ⊓ D|C ⊔ D|¬C|δC|∀R.C|∃R.C where δ ∈ H∗.
In [13], HAs are extended by adding two artificial hedges inf and sup defined as inf(x) = infimum(H(x)), sup(x) = supremum(H(x)). If H = ∅, H(c+) and H(c−) are infinite, according to [13] inf(c+) = sup(c−). Let W = inf (True) = sup (False) and let sup(True) and inf(False) be the greatest and the least elements of X respectively.
The semantics is based on the notion of interpretations.
Definition 8. Let AX be a monotonic HA such that AX = (X, {True, False}, H, > ). A fuzzy interpretation (f-interpretation) I for ALCFL is a pair (ΔI , ·I ), where ΔI is a nonempty set and ·I is an interpretation function mapping: – individuals to elements in ΔI ; – a concept C into a function CI : ΔI → X ; – a role R into a function RI : ΔI × ΔI → X .
For all d ∈ ΔI the interpretation function satisfies the following equations ⊤I (d) = sup(True), ⊥I (d) = inf(False), (¬C)I (d) = −CI (d), (C ⊓ D)I (d) = min(CI (d), DI (d)), (C ⊔ D)I (d) = max(CI (d), DI (d)),
(δC)I (d) = δ−(CI (d)), (∀R.C)I (d) = infd′∈ΔI {max(−RI (d, d′), CI (d′))}, (∃R.C)I (d) = supd′∈ΔI {min(RI (d, d′), CI (d′))}, where −x is the contradictory element of x, and δ− is the inverse of the hedge chain δ.
Definition 9. A fuzzy assertion (fassertion) is an expression of the form hα ⊲⊳ σci where α is of the form a : C or (a, b) : R, ⊲⊳ ∈ {≥, >, ≤, <} and σc ∈ X .
Formally, an f-interpretation I satisfies a fuzzy assertion ha : C ≥ σci (respectively h(a, b) : R ≥ σci) iff CI (aI ) ≥ σc (respectively RI (aI , bI ) ≥ σc). An f-interpretation I satisfies a fuzzy assertion ha : C ≤ σci (respectively h(a, b) : R ≤ σci) iff CI (aI ) ≤ σc (respectively RI (aI , bI ) ≤ σc). Similarly for > and <.
Concerning terminological axioms, an ALCFL terminology axiom is of the form C ⊑ D, where C and D are ALCFL concept terms. From a semantics point of view, a f-interpretation I satisfies a fuzzy concept inclusion C ⊑ D iff ∀d ∈ ΔI .CI (d) ≤ DI (d). Two concept terms C, D are said to be equivalent, denoted by C ≡ D iff CI = DI for all f-interpretations I. Some properties concerning the hedge modification are showed in the following proposition [7]. Proposition 10 We have the following semantical equivalence: δ(C ⊓ D) ≡ δ(C) ⊓ δ(D) δ(C ⊔ D) ≡ δ(C) ⊔ δ(D) δ1(δ2C) ≡ (δ1δ2)C.
A fuzzy knowledge base (fKB) is hT , Ai, where T and A are finite sets of terminological axioms and fassertions respectively.
Example 11 A fKB fK = h{A ⊑ ∀R.¬B}, {a : ∀R.C ≥ VeryTrue}i. An f-interpretation I satisfies (is a model of) a TBox T iff I satisfies each element in T . I satisfies (is a model of) an ABox A iff I satisfies each element in A. I satisfies (is a model of) a fKB fK = hT , Ai iff I satisfies both A and T . Given a fKB fK and a fassertion fα. We say that fK entails fα (denoted fK |= fα) iff each model of fK satisfies fα. 3
Transforming ALCFL into ALCH We will introduce a satisfiability preserving transformation from ALCFL into ALCH in this section. First, we illustrate the basic idea which is similar to the one in [17] which is the first efforts in this direction. There is also other more efficient representation in [3].
Consider a monotonic HA AX = (X, {True, False}, H, >). In the following, we assume that c ∈ {c+, c−} where c+ = True, c− = False, σ ∈ H∗, σc ∈ X and ⊲⊳ ∈ {≥, >, ≤, <}. Assume we have an ALCFL knowledge base, fK = hT , Ai, where A = {fα1, fα2, fα3, fα4} and fα1 = ha : A ≥ Truei, fα2 = hb : A ≥ VeryTruei, fα3 = ha : B ≤ Falsei, and fα4 = hb : B ≤ VeryFalsei where A, B are concept names. We introduce four new concept names: A≥True , A≥VeryTrue , B≤False and B≤VeryFalse . The concept name A≥True represents the set of individuals that are instances of A with degree greater and equal to True. The concept name B≤VeryFalse represents the set of individuals that are instances of B with degree less and equal to VeryFalse. We can map the fuzzy assertions into classical assertions: ha : A ≥ Truei → ha : A≥True i, hb : A ≥ VeryTruei → hb : A≥VeryTrue i, ha : B ≤ Falsei → ha : B≤False i, hb : B ≤ VeryFalsei → hb : B≤VeryFalse i.
We also need to consider the relationships among the newly introduced concept names. Because VeryTrue > True, it is easy to get if a truth value σc ≥ VeryTrue then σc ≥ True. Thus, we obtain a new inclusion A≥VeryTrue ⊑ A≥True . Similarly for B, because VeryFalse < False, a truth value σc ≤ VeryFalse implies σc ≤ False too. Then the inclusion B≤VeryFalse ⊑ B≤False is obtained.
Now, let us proceed with the mappings. Let fK = hT , Ai be an ALCFL knowledge base. We are going to transform fK into an ALCH knowledge base K. We assume σc ∈ [inf(False), sup(True)] and ⊲⊳ ∈ {≥, >, ≤, <}.
In order to transform A, we define two mappings θ and ρ to map all the assertions in A into classical assertions. Notice that we do not allow assertions of the forms (a, b) : R < σc and (a, b) : R ≤ σc although they are legal forms of assertions in ALCFL because they related to ‘negated role’ which is not part of classical ALCH.
We use the mapping ρ to encode the basic idea we present at the beginning of this section. The mapping ρ combines the ALCFL concept term, the ⊲⊳ and the fuzzy value σc together into one ALCH concept term.
Let A be a concept name, C, D be concept terms and R be a role name. For roles we have simply
ρ(R, ⊲⊳ σc) = R⊲⊳σc.
For concept terms, the mapping ρ is inductively defined on the structures of concept terms: For ⊤, For ⊥, ⊤ if ⊲⊳ σc = ≥ σc ⊤ if ⊲⊳ σc = > σc, σc < sup(c+) ρ(⊤, ⊲⊳ σc) = ⊥ if ⊲⊳ σc = > sup(c+)
⊤ if ⊲⊳ σc = ≤ sup(c+) ⊥ if ⊲⊳ σc = ≤ σc, σc < sup(c+) ⊥ if ⊲⊳ σc = < σc. ⊤ if ⊲⊳ σc = ≥ inf(c−) ⊥ if ⊲⊳ σc = ≥ σc, σc > inf(c−) ρ(⊥, ⊲⊳ σc) = ⊥ if ⊲⊳ σc = > σc
⊤ if ⊲⊳ σc = ≤ σc ⊤ if ⊲⊳ σc = < σc, σc > inf(c−) ⊥ if ⊲⊳ σc = < inf(c−).
ρ(A, ⊲⊳ σc) = A⊲⊳σc.
ρ(C ⊓ D, ⊲⊳ σc) = ρ(C, ⊲⊳ σc) ⊓ ρ(D, ⊲⊳ σc) if ⊲⊳ ∈ {≥, >} ρ(C, ⊲⊳ σc) ⊔ ρ(D, ⊲⊳ σc) if ⊲⊳ ∈ {≤, <}. ρ(C ⊔ D, ⊲⊳ σc) = ρ(C, ⊲⊳ σc) ⊔ ρ(D, ⊲⊳ σc) if ⊲⊳ ∈ {≥, >} ρ(C, ⊲⊳ σc) ⊓ ρ(D, ⊲⊳ σc) if ⊲⊳ ∈ {≤, <}. ρ(¬C, ⊲⊳ σc) = ρ(C, ¬ ⊲⊳ σc¯), where ¬ ≥ = ≤, ¬ > = <, ¬ ≤ = ≥, ¬ < = >.
ρ(δC, ⊲⊳ σc) = ρ(C, ⊲⊳ σδc).
ρ(∃R.C, ⊲⊳ σc) =
∃ρ(R, ⊲⊳ σc).ρ(C, ⊲⊳ σc) if ⊲⊳ ∈ {≥, >} ∀ρ(R, − ⊲⊳ σc).ρ(C, ⊲⊳ σc) if ⊲⊳ ∈ {≤, <}, where − ≤ = > and − < = ≥.
For universal quantification ∀R.C, ρ(∀R.C, ⊲⊳ σc) = ∀ρ(R, + ⊲⊳ σc¯).ρ(C, ⊲⊳ σc) if ⊲⊳ ∈ {≥, >} ∃ρ(R, ¬ ⊲⊳ σc¯).ρ(C, ⊲⊳ σc) if ⊲⊳ ∈ {≤, <}, where + ≥ = > and + > = ≥.
θ maps fuzzy assertions into classical assertions using ρ. Let fα be a fassertion in A, we define it as follows.
θ(fα) =
a : ρ(C, ⊲⊳ σc) if fα = ha : C ⊲⊳ σci (a, b) : ρ(R, ⊲⊳ σc) if fα = h(a, b) : R ⊲⊳ σci.
Example 12 Let fα = ha : V ery(A ⊓ B) ≤ LessF alsei, then θ(fα) = a : ρ(V ery(A ⊓ B), ≤ LessF alse) = a : ρ((A ⊓ B), ≤ LessV eryF alse) = a : ρ(A, ≤ LessV eryF alse) ⊔ ρ(B, ≤ LessV eryF alse) = a : A≤LessV eryF alse ⊔ B≤LessV eryF alse.
4
The new TBox is a union of two terminologies. One is the newly introduced TBox (denoted by T (N fK) which is the terminology relating to the newly introduced concept names and role names. The other one is κ(fK, T ) which is reduced by a mapping κ from the TBox of an ALCFL knowledge base.
Many new concept names and new role names are introduced when we transform an ABox. We need a set of terminological axioms to define the relationships among those new names.
We need to collect all the linguist terms σc that might be the subscript of a concept name or a role name. It means that not only the set of linguistic terms that appears in the original ABox but also the set of new linguist terms which are produced by applying the ρ for modifier concepts should be included. Let A be a concept name, R be a role name.
XfK = {σc | hα ⊲⊳ σci ∈ A} ∪ {σδc | ρ(δC, ⊲⊳ σc) = ρ(C, ⊲⊳ σδc)}. such that δC occurs in fK.
We define a sorted set of linguistic terms, N fK = {inf (False), W, sup (True)} ∪ XfK ∪ {σc¯ | σc ∈ XfK} = {n1, . . . , n|NfK|} where ni < ni+1 for 1 ≤ i ≤ |N fK| − 1 and n1 = inf (False), n|NfK| = sup (True).
Let T (N fK) be the set of terminological axioms relating to the newly introduced concept names and role names.
Definition 13. Let AfK and RfK be the sets of concept names and role names occurring in fK respectively. For each A ∈ AfK, for each R ∈ RfK, for each 1 ≤ i ≤ |N fK| − 1 and for each 2 ≤ j ≤ |N fK |, T (N fK) contains A≥ni+1 ⊑A>ni , A>ni⊑A≥ni,
A<nj ⊑A≤nj , A≤ni⊑A<ni+1, A≥nj ⊓ A<nj ⊑⊥ , ⊤⊑A≥nj ⊔ A<nj , A>ni ⊓ A≤ni ⊑⊥ , ⊤⊑A>ni ⊔ A≤ni,
R≥ni+1 ⊑R>ni , R>ni⊑R≥ni. where n ∈ N fK.
ni+1 > ni because N fK is a sorted set. Then if an individual is an instance of a concept name with degree ≥ ni+1 then the degree is also > ni. The first terminological axiom shows that if an individual is an instance of A≥ni+1 then it is an instance of A>ni as well. Similarly, if an individual is an instance of a concept name with degree ≤ ni then the degree is also < ni+1. The third terminological axiom shows that if an individual is an instance of A≤ni then it is also an instance of A<ni+1 . A≥nj ⊓ A<nj ⊑ ⊥ because there is no individual such that it is an instance of a concept name with degree ≥ nj and with degree < nj at the same time.
T (N fK) contains 8|AfK|(|N fK| − 1) plus 2|RfK|(|N fK| − 1) terminological axioms.
The mapping κ κ maps the fuzzy TBox into the classical TBox.
Definition 14. Let C, D be two concept terms and C ⊑ D ∈ T . For all n ∈ N fK κ(fK, C ⊑ D) = Sn∈NfK,⊲⊳∈{≥,>}{ρ(C, ⊲⊳ n) ⊑ ρ(D, ⊲⊳ n)}
Sn∈NfK,⊲⊳∈{≤,<}{ρ(D, ⊲⊳ n) ⊑ ρ(C, ⊲⊳ n)} (3) We extend κ to a terminology T point-wise. For all τ ∈ T
κ(fK, T ) = ∪τ∈T κ(fK, τ ).
Now we can define the reduction of fK into an ALCH knowledge base, denoted K(fK),
K(fK) = hT (N fK) ∪ κ(fK, T ), θ(A)i.
The transformation can be done in polynomial time. The soundness and completeness of the algorithm is guaranteed by the following satisfiability preserving theorem.
Theorem 15 Let fK be an ALCFL knowledge base. Then fK is satisfiable iff the ALCH knowledge base K(fK) is satisfiable.
Proof. Please refer to my thesis [18] which can be download from my homepage.2 5
In this paper, we have presented a satisfiability preserving transformation of ALCFL into ALCH which is with general TBox and role hierarchy. Since all other reasoning tasks such as entailment problem and subsumption problem can be reduced to satisfiability problem, this result allows for algorithms and complexity results that were found for ALCH to be applied to ALCFL.
As for the complexity of the transformation, we know the fact that |θ(A)| is linearly bounded by |A|, |T (N fK)| = 8|AfK|(|N fK| − 1) + 2|RfK|(|N fK| − 1) and κ(fK, T ) contains at most 4|T ||N fK|. Therefore, the resulted classical knowledge base (at most polynomial size) can be constructed in polynomial time.
There exist some reasoners for fuzzy DLs, e.g. FiRE [15], GURDL [5], DeLorean [2], GERDS [6], YADLR [9] and fuzzyDL [4]. Among them, fuzzyDL allows modifiers defined in terms of linear hedges and triangular functions and DeLorean supports triangularly-modified concept. So if we can transform variety of fuzzy DLs into classical DLs then we can use the already existing DL systems to do the reasoning of fuzzy DLs. 6
Thank Pascal Hiltzler and Martin Caminada for their comments on this paper. 2 http://icr.uni.lu/ yining 3. Fernando Bobillo, Miguel Delgado, Juan G´omez-Romero, and Umberto Straccia. Fuzzy description logics under g¨odel semantics. Int. J. Approx. Reasoning, 50(3):494–514, 2009. 4. Fernando Bobillo and Umberto Straccia. fuzzyDL: An expressive fuzzy description logic reasoner. In 2008 International Conference on Fuzzy Systems (FUZZ-08).
IEEE Computer Society, 2008. 5. Volker Haarslev, Hsueh-Ieng Pai, and Nematollaah Shiri. Optimizing tableau reasoning in alc extended with uncertainty. In Description Logics, 2007. 6. Hashim Habiballa. Resolution strategies for fuzzy description logic. In EUSFLAT
Conf. (
Hedge algebras, linguistic-valued logic and their application to fuzzy reasoning. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 7(4):347–361, 1999. 12. Cat-Ho Nguyen and W. Wechler. Hedge algebras: an algebraic approach to structure of sets of linguistic truth values. Fuzzy Sets and Systems, 35(3):281–293, 1990. 13. Cat-Ho Nguyen and W. Wechler. Extended hegde algebras and their application to fuzzy logic. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 52:259–281, 1992. 14. H.-N. Nguyen S. H¨olldobler and D.-K. Tran. The fuzzy description logic ALCFLH.
In Proc. 9th IASTED International Conference on Artificial Intelligence and Soft Computing, pages 99–104, 2005. 15. N. Simou and S. Kollias. Fire : A fuzzy reasoning engine for impecise knowledge.
K-Space PhD Students Workshop, Berlin, Germany, 14 September 2007, 2007.
16. Umberto Straccia. Reasoning within fuzzy description logics. JAIR, 14:137–166,
2001.
17. Umberto Straccia. Transforming fuzzy description logics into classical description
logics. In Proceedings of the 9th European Conference on Logics in Artificial
Intelligence (JELIA-04), number 3229 in Lecture Notes in Computer Science, pages
385–399, Lisbon, Portugal, 2004. Springer Verlag.
18. Yining Wu. Transforming fuzzy description logic ALCFL into classical description
logic ALCH. Master Thesis, 2007.
19. Lotfi A. Zadeh. The concept of a linguistic variable and its application to
approximate reasoning - I. Information Sciences, 8(3):199–249, 1975.
20. Lotfi A. Zadeh. The concept of a linguistic variable and its application to
approximate reasoning - II. Information Sciences, 8(4):301–357, 1975.
21. Lotfi A. Zadeh. The concept of a linguistic variable and its application to
approximate reasoning- III. Information Sciences, 9(