Classical ontologies are not suitable to represent imprecise or vague pieces of information, which has led to fuzzy extensions of Description Logics. In order to support an early acceptance of the OWL 1.1 ontology language, we present DeLorean, the first reasoner that supports a fuzzy extension of the Description Logic SROIQ, closely equivalent to it. It implements some interesting optimization techniques, whose usefulness is shown in a preliminary empirical evaluation.
Ontologies are a core element in the layered architecture of the Semantic Web.
The current standard language for ontology representation is the Web Ontology
Language (OWL). However, since its first development, several limitations on the
expressiveness of OWL have been identified, and consequently several extensions
to the language have been proposed. Among them, the most significant is OWL
1.1 [
It has been widely pointed out that classical ontologies are not appropriate
to deal with imprecise and vague knowledge, which is inherent to several
realworld domains. Since fuzzy logic is a suitable formalism to handle these types
of knowledge, several fuzzy extensions of DLs have been proposed [
The broad acceptance of the forthcoming OWL 1.1 ontology language will
largely depend on the availability of editors, reasoners, and other numerous tools
that support the use of OWL 1.1 from a high-level/application perspective [
This paper is organized as follows. Section 2 describes the fuzzy DL SROIQ, which is equivalent to the fuzzy language supported by DeLorean. Then, Section 3 describes the reasoning algorithm, based on a reduction into crisp SROIQ. Section 4 presents our implementation, some of the implemented optimizations (including handling of superfluous elements), and a preliminary evaluation. Finally, Section 6 sets out some conclusions and ideas for future work.
In this section we recall the definition of fuzzy SROIQ [
Syntax. In fuzzy SROIQ we have three alphabets of symbols, for concepts
(C), roles (R), and individuals (I). The set of roles is defined by RA ∪ U ∪
{R−|R ∈ RA}, where RA ∈ R, U is the universal role and R− is the inverse
of R. Assume A ∈ C, R, S ∈ R (where S is a simple role [
ABox
TBox !a : C ≥ α#, !a : C > β#, !a : C ≤ β#, !a : C < α# !(a, b) : R ≥ α#, !(a, b) : R > β#, !(a, b) : R ≤ β#, !(a, b) : R < α# !a %= b# !a = b# !C & D ≥ α#, !C & D > β# C ≡ D, equivalent to {!C & D ≥ 1#, !D & C ≥ 1#}
RBox
Fuzzy RIA !R1R2 . . . Rn & R ≥ α#, !R1R2 . . . Rn & R > β#
Transitive role axiom trans(R)
Disjoint role axiom dis(S1, S2)
Reflexive role axiom ref (R)
Irreflexive role axiom irr(S)
Symmetric role axiom sym(R)
Asymmetric role axiom asy(S)
Semantics. A fuzzy interpretation I is a pair (ΔI , ·I ), where ΔI is a non empty
set (the interpretation domain) and ·I a fuzzy interpretation function mapping:
(i) every individual a onto an element aI of ΔI ; (ii) every concept C onto a
function CI : ΔI → [
Irreflexive, transitive and symmetric role axioms are syntactic sugar for every R-implication (and consequently it can be assumed that they do not appear in fuzzy KBs) due to the following equivalences: irr(S) ≡ 3& 5 ¬∃S.Self ≥ 14, trans(R) ≡ 3RR 5 R ≥ 14 and sym(R) ≡ 3R 5 R− ≥ 14.
In the rest of the paper we will only consider fuzzy KB satisfiability, since
(as in the crisp case) most inference problems can be reduced to it [
In this section we show how to reduce a fuzzy Z SROIQ fuzzy KB into a
crisp KB (see [
Now, for each α, β ∈ N K with α ∈ (0, 1] and β ∈ [0, 1), for each A ∈ AK and for each RA ∈ RK, two new atomic concepts A≥α, A>β and two new atomic roles R≥α, R>β are introduced. A≥α represents the crisp set of individuals which are instance of A with degree higher or equal than α i.e the α-cut of A.
The semantics of these newly introduced atomic concepts and roles is preserved by some terminological and role axioms. For each 1 ≤ i ≤ |N K| − 1, 2 ≤ j ≤ |N K|−1 and for each A ∈ AK, T (N K) is the smallest terminology containing these two axioms: A≥γi+1 5 A>γi , A>γj 5 A≥γj . Similarly, for each RA ∈ RK, R(N K) contains these axioms: R≥γi+1 5 R>γi , R>γi 5 R≥γi .
Example 1. Consider the fuzzy KB K = {τ }, where τ = 3StGenevieveTexasWhite : WhiteWine ≥ 0.754}. We have that N K = {0, 0.25, 0.5, 0.75, 1} and T (N K) = {WhiteWine≥0.25 5 WhiteWine>0, WhiteWine>0.25 5 WhiteWine≥0.25, WhiteWine≥0.5 5 WhiteWine>0.25, WhiteWine>0.5 5 WhiteWine≥0.5, WhiteWine≥0.5 5 WhiteWine>0.25, WhiteWine>0.5 5 WhiteWine≥0.5, WhiteWine≥0.75 5 WhiteWine>0.5, WhiteWine>0.75 5 WhiteWine≥0.75, WhiteWine≥1 5 WhiteWine>0.75}. () Mapping fuzzy concepts, roles and axioms. Concept and role expressions are reduced using mapping ρ, as shown in the first part of Table 1. Given a fuzzy concept C, ρ(C, ≥ α) is a crisp set containing all the elements which belong to C with a degree greater or equal than α (the other cases are similar). For instance, the 1-cut of the fuzzy concept ∀madeFromFruit.(NonSweetFruit ) SweetFruit) is ρ(∀madeFromFruit.(NonSweetFruit ) SweetFruit), ≥ 1) = ∀madeFromFruit>0.NonSweetFruit≥1 ) SweetFruit≥1.
Finally, we map the axioms in the ABox, TBox and RBox. Axioms are
reduced as shown in the second part of Table 1, where σ maps fuzzy axioms into
crisp assertions, and κ maps fuzzy TBox (resp. RBox) axioms into crisp TBox
(resp. RBox) axioms. Recall that we are assuming that irreflexive, transitive
and symmetric role axioms do not appear in the RBox. For example, assuming
N K = {0, 0.25, 0.5, 0.75, 1}, the reduction of the fuzzy GCI 3Port 5 RedWine ≥ 14
is κ(3Port 5 RedWine ≥ 14) = {Port>0 5 RedWine>0, Port≥0.25 5 RedWine≥0.25,
Port>0.25 5 RedWine>0.25, Port≥0.5 5 RedWine≥0.5, Port>0.5 5 RedWine>0.5,
Port≥0.75 5 RedWine≥0.75, Port>0.75 5 RedWine>0.75, Port≥1 5 RedWine≥1}.
Properties. Summing up, a fuzzy KB K = 3A, T , R4 is reduced into a KB
crisp(K) = 3σ(A), T (N K) ∪ κ(K, T ), R(N K) ∪ κ(K, R)4. The following theorem
shows that the reduction preserves reasoning:
Theorem 1. A Z SROIQ fuzzy KB K is satisfiable iff crisp(K)) is [
The resulting KB is quadratic because it depends on the number of relevant degrees |N K|, or linear if we assume a fixed set. An interesting property is that the reduction of an ontology can be reused when adding a new axiom. If the new axioms does not introduce new atomic concepts, atomic roles nor a new degree of truth, we just need to add the reduction of the axiom. 4
This section describes the prototype implementation of our reasoner, which is called DeLorean (DEscription LOgic REasoner with vAgueNess).
Initially, we developed a first version based on Jena API2 [
With the aim of augmenting the expressivity of the logic, in the current
version we have changed the subjacent API to OWL API for OWL 24 [
Since DIG interface does not currently allow the full expressivity of OWL
1.1, our solution was to integrate directly DeLorean with a concrete crisp
ontology reasoner: Pellet [
Figure 1 illustrates the architecture of the system: – The Parser reads an input file with a fuzzy ontology and translates it into an internal representation. The point here is that we can use any language to encode the fuzzy ontology, as long as the Parser can understand the representation and the reduction is properly implemented. Consequently we will not get into details of our particular choice.
In order to make the representation of fuzzy KBs easier, DeLorean also allows the possibility of importing OWL 1.1 ontologies. These (crisp) ontologies are saved into a text file which the user can edit and extend, for example adding membership degrees to the fuzzy axioms or specifying a particular fuzzy operator (Zadeh or G¨odel family) for some complex concept. – The Reduction module implements the reduction procedures described in the previous section, building an OWL API model with an equivalent crisp ontology which can be exported to an OWL file. The implementation also takes into account all the optimizations already discussed along this document. – The Inference module tests this ontology for consistency, using either Pellet or any crisp reasoner through the DIG interface. Crisp reasoning does not take into account superfluous elements as we explain below. – Inputs (the path of the fuzzy ontology) and outputs (the result of the reasoning and the elapsed time) are managed by an User interface.
The reasoner implements the following optimizations:
Optimizing the number of new elements and axioms. Previous works
use two more atomic concepts A≤β , A<α and some additional axioms A<γk 5
A≤γk , A≤γi 5 A<γi+1 , A≥γk (A<γk 5 ⊥, A>γi (A≤γi 5 ⊥, & 5 A≥γk )A<γk , & 5
A>γi ) A≤γi , 2 ≤ k ≤ |N K|. In [
Allowing crisp concepts and roles. Suppose that A is a fuzzy concept. Then, we need N K − 1 concepts of the form A≥α and another N K − 1 concepts of the form A>β to represent it, as well as 2 · (|N K| − 1) − 1 axioms to preserve their semantics. Fortunately, in real applications not all concepts and roles will be fuzzy. If A is declared to be crisp, we just need one concept to represent it and no new axioms. The case for fuzzy roles is exactly the same. Of course, this optimization requires some manual intervention.
Reasoning ignoring superfluous elements. Our reduction is designed to promote reusing. For instance, consider the fuzzy KB K in Example 1. The reduction of K contains σ(τ ) = StGenevieveTexasWhite : WhiteWine≥0.75, but also the axioms in T (N K). It can be seen that the concepts WhiteWine>0, WhiteWine≥0.25, WhiteWine>0.25, WhiteWine≥0.5, WhiteWine>0.5, WhiteWine>0.75, WhiteWine≥1 are superfluous in the sense that cannot cause a contradiction. Hence, for a satisfiability test of crisp(K), we can avoid the axioms in T (N K) where they appear.
But please note that if additional axioms are added to K, crisp(K) will be different and previous superfluous concept and roles may not be superfluous any more. For example, if we want to check if K ∪ 3StGenevieveTexasWhite : WhiteWine ≥ 0.54 is satisfiable, then the concept WhiteWine≥0.5 is no longer superfluous. In this case, it is enough to consider T %(N K) = {WhiteWine≥0.75 5 WhiteWine≥0.5}. The case of atomic roles is similar to that of atomic concepts. 5
This section considers a concrete use case, a fuzzy extension of the well-known
Wine ontology5, a highly expressive ontology (in SHOIN (D)). Some metrics of
the ontology are shown in the first column of Table 2. In an empirical evaluation
of the reductions of fuzzy DLs to crisp DLs, P. Cimiano et al. wrote that “the
Wine ontology showed to be completely intractable both with the optimized
and unoptimized reduction even using only 3 degrees” [
A fuzzy extension of the ontology. We have defined a fuzzy version of the Wine ontology by adding a degree to the axioms. Given a variable set of degrees N K, the degrees of the truth for fuzzy assertions is randomly chosen in N K. In the case of fuzzy GCIs and RIAs, the degree is always 1 in special GCIs (namely concept equivalences and disjointness, domain, range and functional role axioms) or if there is a crisp element in the left side; otherwise, the degree is 0.5.
In most of the times fuzzy assertions are of the form 3τ ! β4 with β 1= 1. Clearly, this favors the use of elements of the forms C!β and R!β, reducing the number of superfluous concepts. Once again, we are in the worst case from the
point of view of the size of the resulting crisp ontology. Nonetheless, in practice we will be often able to say that an individual fully belongs to a fuzzy concept, or that two individuals are fully related by means of a fuzzy role. Crisp concepts and roles. A careful analysis of the fuzzy KB brings about that most of the concepts and the roles should indeed be interpreted as crisp. For example, most of the subclasses of the class Wine refer to a well-defined geographical origin of the wines. For instance, Alsatian wine is a wine which has been produced in the French region of Alsace: AlsatianWine ≡ Wine ( ∃locatedAt.{alsaceRegion}. In other applications there could exist examples of fuzzy regions, but this is not our case. Another important number of subclasses of Wine refer to the type of grape used, which is also a crisp concept. For instance, Riesling is a wine which has been produced from Riesling grapes: Riesling ≡ Wine ( ∃madeFromGrape.{RieslingGrape}( ≥ 1 madeFromGrape.&.
Clearly, there are other concepts with no sharp boundaries (for instance, those derived from the vague terms “dry”, “sweet”, “white” or“heavy”). The result of our study has identified 50 fuzzy concepts in the Wine ontology, namely: WineColor, RedWine, RoseWine, WhiteWine, RedBordeaux, RedBurgundy, RedTableWine, WhiteBordeaux, WhiteBurgundy, WhiteLoire, WhiteTableWine, WineSugar, SweetWine, SweetRiesling, WhiteNonSweetWine, DryWine, DryRedWine, DryRiesling, DryWhiteWine, WineBody, FullBodiedWine, WineFlavor, WineTaste, LateHarvest, EarlyHarvest, NonSpicyRedMeat, NonSpicyRedMeatCourse, SpicyRedMeat, PastaWithSpicyRedSauce, PastaWithSpicyRedSauceCourse, PastaWithNonSpicyRedSauce, PastaWithNonSpicyRedSauceCourse, SpicyRedMeatCourse, SweetFruit, SweetFruitCourse, SweetDessert, SweetDessertCourse, NonSweetFruit, NonSweetFruitCourse, RedMeat, NonRedMeat, RedMeatCourse, NonRedMeatCourse, PastaWithHeavyCreamSauce, PastaWithLightCreamSauce, Dessert, CheeseNutsDessert, DessertCourse, CheeseNutsDessertCourse, DessertWine.
Furthermore, we identified 5 fuzzy roles: hasColor, hasSugar, hasBody,
hasFlavor, and hasWineDescriptor (which is a super-role of the other four).
Measuring the importance of the optimizations. The reduction under
G¨odel semantics is still to be published [
Table 2 shows some metrics of the crisp ontologies obtained in the reduction of the fuzzy ontology after applying different optimizations. 1. Column “Original” shows some metrics of the original ontology. 2. “None” considers the reduction obtained after applying no optimizations. 3. “(NEW)” considers the reduction obtained after optimizing the number of new elements and axioms. 4. “(GCI)” considers the reduction obtained after optimizing GCI reductions. 5. “(C/S)” considers the reduction obtained after allowing crisp concepts and roles and ignoring superfluous elements. 6. Finally, “All” applies all the previous optimizations.
We have put together the optimizations of crisp and superfluous elements because in this ontology handling superfluous concepts is not always useful, due to the existence of a lot of concept definitions, as we will see in the next example. Example 2. Consider the fuzzy concept NonRedMeat. Firstly, this concept appears as part of a fuzzy assertion stating that pork is a non read meat: σ(3Pork : NonRedMeat ! α14) = Pork : NonRedMeat!α1 . Secondly, non read meat is declared to be disjoint from read meat: κ(3RedMeat ( NonRedMeat 5 ⊥ ≥ 14) = RedMeat>0 ( NonRedMeat>0 5 ⊥. Thirdly, non read meat is a kind of meat: κ(3NonRedMeat 5 Meat ≥ α24) = NonRedMeat>0 5 Meat. If these were the only occurrences of NonRedMeat, then the reduction would create only two nonsuperfluous crisp concepts, namely NonRedMeat>0 and NonRedMeat!α1 , and in order to preserve the semantics of them we would need to add just one axiom during the reduction: NonRedMeat!α1 5 NonRedMeat>0.
However, this is not true because NonRedMeat appears in the definition of the fuzzy concept NonRedMeatCourse. In fact, κ(NonRedMeatCourse ≡ MealCourse ( ∀hasFood.NonRedMeat) introduces non-superfluous crisp concepts for the rest of the degrees in N K. Consequently, for each 1 ≤ i ≤ |N K| − 1, 2 ≤ j ≤ |N K|−1, the reduction adds to T (N K) the following axioms: NonRedMeat≥γi+1 5 NonRedMeat>γi ; NonRedMeat>γj 5 NonRedMeat≥γj . ()
Note that the size of the ABox is always the same, because every axiom in the fuzzy ABox generates exactly one axiom in the reduced ontology.
The number of new GCIs and RIAs added to preserve the semantics of the new elements is much smaller in the optimized versions. In particular, we reduce from 4352 to 324 GCIs (7.44%) and from 136 to 34 RIAs (25%). This shows the importance of reducing the number of new crisp elements and their corresponding axioms, as well as of defining crisp concepts and roles and (to a lesser extent) handling superfluous concepts.
Optimizing GCI reductions turns out to be very useful in reducing the number of disjoint concepts, domain, range and functional role axioms: 152 to 19 (12.5 %), 104 to 97 (93.27 %), 80 to 10 (12.5 %), and 48 to 6 (12.5 %), respectively. In the case of domain role axioms the reduction is not very high because we need an inverse role to be defined in order to apply the reduction, and this happens only in one of the axioms.
Every fuzzy GCI or RIA generates several axioms in the reduced ontology. Combining the optimization of GCI reductions with the definition of crisp concepts and roles reduces the number of new axioms, from 1288 to 390 subclass axioms (30.28 %), from 696 to 318 concept equivalences (45.69 %) and from 40 to 33 sub-role axioms (82.5 %).
Finally, the number of inverse and transitive role axioms is reduced in the optimized version because fuzzy roles interpreted as crisp introduce one inverse or transitive axiom instead of several ones. This allows a reduction from 16 to 2 axioms, and from 8 to 1, respectively, which corresponds to the 12.5 %.
Table 3 shows the influence of the number of degrees on the size of the
resulting crisp ontology, as well as on the reduction time (which is shown in
seconds), when all the described optimizations are used. The reduction time is
small enough to allow to recompute the reduction of an ontology when necessary,
thus allowing superfluous concepts and roles in the reduction to be avoided.
This paper has presented DeLorean, the more expressive fuzzy DL reasoner
that we are aware of (it supports fuzzy OWL 1.1), and the optimizations that
it implements. Among them, the current version enables the definition of crisp
concepts and roles, as well as handling superfluous concepts and roles before
applying crisp reasoning. A preliminary evaluation shows that these
optimizations help to reduce significantly the size of the resulting ontology. In future
work we plan to develop a more detailed benchmark by relying on the
hypertableau reasoner HermiT, which seems to outperform other DL reasoners [