1 Introduction
It is wellknown that one of the main requirements for the development of an intelligent application is a formalism capable of representing and handling knowledge without ambiguity. Description Logics (DLs) are a wellstudied family of knowledge representation formalisms [Baader et al.2007]. They constitute the logical backbone of the standard Semantic Web ontology language OWL 2,^{1}^{1}1http://www.w3.org/TR/owl2overview/ and its profiles, and have been successfully applied to represent the knowledge of many and diverse application domains, particularly in the biomedical sciences.
DLs describe the domain knowledge using concepts (such as Patient) that represent sets of individuals, and roles (hasRelative) that represent connections between individuals. Ontologies are collections of axioms formulated over these concepts and roles, which restrict their possible interpretations. The typical axioms considered in DLs are assertions, like , providing knowledge about specific individuals; and general concept inclusions (GCIs), such as , which express general relations between concepts. Different DLs are characterized by the constructors allowed to generate complex concepts and roles from atomic ones. [SchmidtSchauß and Smolka1991] is a prototypical DL of intermediate expressivity that contains the concept constructors conjunction (), negation (), and existential restriction ( for a role ). If additionally qualified number restrictions ( for ) are allowed, the resulting logic is denoted by . Common reasoning problems in , such as consistency of ontologies or subsumption between concepts, are known to be complete [Schild1991, Tobies2001].
Fuzzy Description Logics (FDLs) have been introduced as extensions of classical DLs to represent and reason with vague knowledge. The main idea is to consider all the truth values from the interval instead of only true and false. In this way, it is possible give a more finegrained semantics to inherently vague concepts like LowFrequency or HighConcentration, which can be found in biomedical ontologies like Snomed CT,^{2}^{2}2http://www.ihtsdo.org/snomedct/ and Galen.^{3}^{3}3http://www.opengalen.org/ The different members of the family of FDLs are characterized not only by the constructors they allow, but also by the way these constructors are interpreted.
To interpret conjunction in complex concepts like
a popular approach is to use socalled tnorms [Klement et al.2000]. The semantics of the other logical constructors can then be derived from these tnorms in a principled way, as suggested by Haje01 Haje01. Following the principles of mathematical fuzzy logic, existential restrictions are interpreted as suprema of truth values. However, to avoid problems with infinitely many truth values, reasoning in fuzzy DLs is often restricted to socalled witnessed models [Hájek2005], in which these suprema are required to be maxima; i.e., the degree is witnessed by at least one domain element.
Unfortunately, reasoning in most FDLs becomes undecidable when the logic allows to use GCIs and negation under witnessed model semantics [Baader and Peñaloza2011, Cerami and Straccia2013, Borgwardt et al.2015]. One of the few exceptions known are FDLs using the Gödel tnorm defined as to interpret conjunctions [Borgwardt et al.2014]. Despite not being as wellbehaved as finitely valued FDLs, which use a finite total order of truth values instead of the infinite interval [Borgwardt and Peñaloza2013], it has been shown using an automatabased approach that reasoning in Gödel extensions of exhibits the same complexity as in the classical case, i.e. it is complete. A major drawback of this approach is that it always has an exponential runtime, even when the input ontology has a simple form.
In this paper, we extend the results of [Borgwardt et al.2014] to deal with qualified number restrictions, showing again that the complexity of reasoning remains the same as for the classical case; i.e., it is complete. To this end, we focus only on the problem of local consistency, which is a generalization of the classical concept satisfiability problem. We follow a more practical approach that combines the automatabased construction from [Borgwardt et al.2014] with reduction techniques developed for finitely valued FDLs [Straccia2004, Bobillo et al.2009, Bobillo and Straccia2013]. We exploit the forest model property of classical [Kazakov2004] to encode order relationships between concepts in a fuzzy interpretation in a manner similar to the Hintikka trees from [Borgwardt et al.2014]. However, instead of using automata to determine the existence of such trees, we reduce the fuzzy ontology directly into a classical ontology whose local consistency is equivalent to that of the original ontology. This enables us to use optimized reasoners for classical DLs. In addition to the cutconcepts of the form for a fuzzy concept and a value , which are used in the reductions for finitely valued DLs [Straccia2004, Bobillo et al.2009, Bobillo and Straccia2013], we employ order concepts expressing relationships between fuzzy concepts. Contrary to the reductions for finitely valued Gödel FDLs presented by BDGSIJAR09 BDGSIJAR09,BDGSIJUF12, our reduction produces a classical ontology whose size is polynomial in the size of the input fuzzy ontology. Thus, we obtain tight complexity bounds for reasoning in this FDL [Tobies2001]. An extended version of this paper appears in [Borgwardt and Peñaloza2015].
2 Preliminaries
For the rest of this paper, we focus solely on vague statements that take truth degrees from the infinite interval , where the Gödel tnorm, defined by , is used to interpret logical conjunction. The semantics of implications is given by the residuum of this tnorm; that is,
We use both the residual negation and the involutive negation in the rest of this paper.
We first recall some basic definitions from [Borgwardt et al.2014], which will be used extensively in the proofs throughout this work. An order structure is a finite set containing at least the numbers , , and , together with an involutive unary operation such that for all . A total preorder over is a transitive and total binary relation . For , we write if and . Notice that is an equivalence relation on . The total preorders considered in [Borgwardt et al.2014] have to satisfy additional properties; for instance, that and are always the least and greatest elements, respectively. These properties can be found in our reduction in the axioms of (see Section 3 for more details).
The syntax of the FDL  is the same as that of classical , with the addition of the implication constructor (denoted by the use of at the beginning of the name). This constructor is often added to FDLs, as the residuum cannot, in general, be expressed using only the tnorm and negation operators, in contrast to the classical semantics. In particular, this holds for the Gödel tnorm and its residuum, which is the focus of this work. Let now , , and be mutually disjoint sets of concept, role, and individual names, respectively. Concepts of  are built using the syntax rule
where , , are concepts, and . We use the abbreviations
Notice that BDGSIJUF12 consider a different definition of atmost restrictions, which uses the residual negation; that is, they define BDGSIJUF12. This has the strange side effect that the value of is always either or (see the semantics below). However, this discrepancy in definitions is not an issue since our algorithm can handle both cases.
The semantics of this logic is based on interpretations. A interpretation is a pair , where is a nonempty set called the domain, and is the interpretation function that assigns to each individual name an element , to each concept name a fuzzy set , and to each role name a fuzzy binary relation . The interpretation of complex concepts is obtained from the semantics of firstorder fuzzy logics via the wellknown translation from DL concepts to firstorder logic [Straccia2001, Bobillo et al.2012], i.e. for all ,
Recall that the usual duality between existential and value restrictions that appears in classical DLs does not hold in .
A classical interpretation is defined similarly, with the set of truth values restricted to and . In this case, the semantics of a concept is commonly viewed as a set
instead of the characteristic function
.In the following, we restrict all reasoning problems to socalled witnessed interpretations [Hájek2005], which intuitively require the suprema and infima in the semantics to be maxima and minima, respectively. More formally, the interpretation is witnessed if, for every , , , and concept , there exist (where are pairwise different) such that
The axioms of  extend classical axioms by allowing to compare degrees of arbitrary assertions in socalled ordered ABoxes [Borgwardt et al.2014], and to state inclusions relationships between fuzzy concepts that hold to a certain degree, instead of only . A classical assertion is an expression of the form or for , , and a concept . An order assertion is of the form or where , are classical assertions, and . A (fuzzy) general concept inclusion axiom (GCI) is of the form for concepts and . An ordered ABox is a finite set of order assertions, a TBox is a finite set of GCIs, and an ontology consists of an ordered ABox and a TBox . A interpretation satisfies (or is a model of) an order assertion if (where , , and ); it satisfies a GCI if holds for all ; and it satisfies an ordered ABox, TBox, or ontology if it satisfies all its axioms. An ontology is consistent if it has a (witnessed) model.
Given an ontology , we denote by the set of all role names occurring in and by the closure under negation of the set of all subconcepts occurring in . We consider the concepts and to be equal, and thus the latter set is of quadratic size in the size of . Moreover, we denote by the closure under the involutive negation of the set of all truth degrees appearing in , together with , , and . This set is of size linear on the size of . We sometimes denote the elements of as .
We stress that we do not consider the general consistency problem in this paper, but only a restricted version that uses only one individual name. An ordered ABox is local if it contains no role assertions and there is a single individual name such that all order assertions in only use . The local consistency problem, i.e. deciding whether an ontology with a local ordered ABox is consistent, can be seen as a generalization of the classical concept satisfiability problem [Borgwardt and Peñaloza2013].
Other common reasoning problems for FDLs, such as concept satisfiability and subsumption can be reduced to local consistency [Borgwardt et al.2014]: the subsumption between and to degree w.r.t. a TBox is equivalent to the (local) inconsistency of , and the satisfiability of to degree w.r.t. is equivalent to the (local) consistency of .
In the following section we show how to decide local consistency of a  ontology through a reduction to classical ontology consistency.
3 Deciding Local Consistency
Let be a  ontology where is a local ordered ABox that uses only the individual name . The main ideas behind the reduction to classical are that it suffices to consider treeshaped interpretations, where each domain element has a unique role predecessor, and that we only have to consider the order between values of concepts, instead of their precise values. This insight allows us to consider only finitely many different cases [Borgwardt et al.2014].
To compare the values of the elements of at different domain elements, we use the order structure
where , , , and , for all concepts . The idea is that total preorders over describe the relationships between the values of and at a single domain element. The elements of allow us to additionally refer to the relevant values at the unique role predecessor of the current domain element (in a treeshaped interpretation). The value represents the value of the role connection from this predecessor. For convenience, we define for all .
In order to describe such total preorders in a classical ontology, we employ special concept names of the form for . This differs from previous reductions for finitely valued FDLs [Straccia2004, Bobillo and Straccia2011, Bobillo et al.2012] in that we not only consider cutconcepts of the form with , but also relationships between different concepts.^{4}^{4}4For the rest of this paper, expressions of the form denote (classical) concept names. For convenience, we introduce the abbreviations , , and similarly for and . Furthermore, we define the complex expressions

,

,

,

,
and extend these notions to the expressions etc., for , analogously.
For each concept , we now define the classical TBox , depending on the form of , as follows.
Intuitively, describes the semantics of in terms of its order relationships to other elements of . Note that the semantics of the involutive negation is already handled by the operator (see also the last line of the definition of below).
The reduced classical ontology is defined as follows:
We briefly explain this construction. The reductions of the order assertions and fuzzy GCIs in are straightforward; the former expresses that the individual must belong to the corresponding order concept or , while the latter expresses that every element of the domain must satisfy the restriction provided by the fuzzy GCI. The axioms of intuitively ensure that the relation “” forms a total preorder that is compatible with all the values in , and that is an antitone operator. Finally, the TBox expresses a connection between the orders of a domain element and those of its role successors.
The following lemmata show that this reduction is correct; i.e., that it preserves local consistency.
Lemma 1.
If has a classical model, then has a model.
Proof.
By [Kazakov2004], must have a tree model , i.e. we can assume that is a prefixclosed subset of , , for all , , with , the element is an predecessor of for some , and there are no other role connections. For any , we denote by the corresponding total preorder on , that is, we define iff , and by the induced equivalence relation.
As a first step in the construction of a model of , we define the auxiliary function that satisfies the following conditions for all :

[label=(P0),leftmargin=*]

for all , we have ,

for all , we have iff ,

for all , we have ,

if , then for all it holds that .
We define by induction on the structure of starting with . Let be the set of all equivalence classes of . Then yields a total order on . Since satisfies , we have
w.r.t. this order. For every , we now set . This function is welldefined by the axioms in . On all for , we now define , which ensures that 1 holds. For the equivalence classes that do not contain a value from , note that by , every such class must be strictly between and for . We denote the equivalence classes between and as follows:
For every , we set , which ensures that 2 is also satisfied. Furthermore, observe that and are also adjacent in and we have
by the axioms in . Hence, it follows from the definition of that 3 holds.
Let now be such that the function , satisfying the properties 1–4, has already been defined for . Since is a tree model, there must be an such that . We again consider the set of equivalence classes and set for all and , and for all and . To see that this is welldefined, consider the case that , i.e. . From the axioms in and the fact that , it follows that , and thus . Since 2 is satisfied for , we get . The same argument shows that implies . For the remaining equivalence classes, we can use a construction analogous to the case for by considering the two unique neighboring equivalence classes that contain an element of (for which has already been defined). This construction ensures that 1–4 hold for .
Based on the function , we define the interpretation over the domain , where ;
We show by induction on the structure of that
(1) 
For concept names, this holds by the definition of . For , we know that by the definition of and 2. For , we have
by the induction hypothesis and 3. For conjunctions , we know that
by the definition of and 2. Implications can be treated similarly.
Consider a value restriction . For every with , we have since satisfies . By the induction hypothesis, the fact that , 2, and 4, this implies that , and thus
Furthermore, by the existential restriction introduced in , we know that there exists a such that and . By the same arguments as above, we get
which concludes the proof of (1) for . As a byproduct, we have found in the element the witness required for satisfying the concept at .
Consider now . For any tuple of different domain elements with , by there must be an index , , such that . Using arguments similar to those introduced above, we obtain that
On the other hand, we know that there are different elements such that and for all , . As in the case of above, we conclude that
as required. Furthermore, are the required witnesses for at . This concludes the proof of (1).
It remains to be shown that is a model of . For every , we have , and thus by (1), 1, and 2. A similar argument works for handling order assertions of the form . To conclude, consider an arbitrary GCI and . By the definition of and 1, we have . Thus, (1) and 2 yield . Thus, satisfies all the axioms in , which concludes the proof. ∎
For the converse direction, we now show that it is possible to unravel every model of into a classical tree model of .
Lemma 2.
If has a model, then has a classical model.
Proof.
Given a model of , we define a classical interpretation over the domain of all paths of the form with , , . We set and
for all . We denote by the element if , and if . Similarly, we set to if , and to if . Finally, denotes whenever . For any and , we define as
if ;  
if ;  
if ;  
if ;  
if . 
Note that for this expression is only defined for . We fix the value of for all other arbitrarily, in such a way that for all we have iff . We can now define the interpretation of all concept names with as
It is easy to see that we have iff also for all other order expressions , and that satisfies . We now show that satisfies the remaining parts of .
For any order assertion we have . This implies that , and thus , as required. A similar argument works for assertions of the form . Consider now a GCI and any . We know that , and thus .
For , consider any , , and . Thus, it holds that . Every successor of in must be of the form . Since , we know that all successors of satisfy .
It remains to be shown that satisfies for all concepts . For
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