Consistency in Fuzzy Description Logics over Residuated De Morgan Lattices

Fuzzy description logics can be used to model vague knowledge in application domains. This paper analyses the consistency and satisfiability problems in the description logic SHI with semantics based on a complete residuated De Morgan lattice. The problems are undecidable in the general case, but can be decided by a tableau algorithm when restricted to finite lattices. For some sublogics of SHI, we provide upper complexity bounds that match the complexity of crisp reasoning.


Introduction
Description Logics (DLs) [1] are a family of knowledge representation formalisms that are widely used to model application domains.In DLs, knowledge is represented with the help of concepts (unary predicates) and roles (binary predicates) that express the relationships between concepts.They have been successfully employed to formulate ontologies-especially in the medical domain-like Galen1 and serve as the underpinning for the current semantic web language OWL 2. 2Standard reasoning in these logics includes concept satisfiability (is a given concept non-contradictory?) and ontology consistency (does a given ontology have a model?).These and other reasoning problems have been studied for DLs, and several algorithms have been proposed and implemented.
One of the main challenges in knowledge representation is the correct modeling and use of imprecise or vague knowledge.For example, medical diagnoses from experts are rarely clear-cut and usually depend on concepts like HighBloodPressure that are necessarily vague.Fuzzy variants of description logics were introduced in the nineties as a means to tackle this challenge.Their applicability to the representation of medical knowledge was studied in [19].
Fuzzy DLs generalize (crisp) DLs by providing a membership degree semantics for their concepts.Thus, e.g.130/85 belongs to the concept HighBloodPressure with a lower degree than, say 140/80.In their original form, membership degrees are elements of the real-number interval [0, 1], but this was later generalized to lattices [18,23].The papers [18,23] consider only a limited kind of semantics over lattices, where conjunction and disjunction are interpreted through the lattice operators meet and join, respectively.
In this paper, we consider a more general lattice-based semantics that uses a triangular norm (t-norm) and its residuum as interpretation functions for the logical constructors.We study fuzzy variants of the standard reasoning problems like concept satisfiability and ontology consistency in this setting.
We show that concept satisfiability in ALC under this semantics is undecidable in general, even if we restrict ourselves to a very simple class of infinite lattices.However, we show with the help of a tableaux-based algorithm that decidability of reasoning can be regained-even for the more expressive DL SHI-if the underlying lattice is required to be finite.Moreover, we describe a black-box method that can be used to transform any decision algorithm for (a small generalization of) satisfiability into a decision procedure for consistency.

Preliminaries
We start with a short introduction to residuated lattices, which will be the base for the semantics of the fuzzy DL L-SHI.For a more comprehensive view on these lattices, we refer the reader to [13,15].

Lattices
A lattice is a triple (L, ∨, ∧), consisting of a carrier set L and two idempotent, associative, and commutative binary operators join ∨ and meet ∧ on L that satisfy the absorption laws 1 ∨ ( 1 ∧ 2 ) = 1 = 1 ∧ ( 1 ∨ 2 ) for all 1 , 2 ∈ L. These operations induce a partial order ≤ on L: 1 ≤ 2 iff 1 ∧ 2 = 1 .As usual, we write 1 < 2 if 1 ≤ 2 and 1 = 2 .A subset T ⊆ L is called an antichain (in L) if there are no two elements 1 , 2 ∈ T with 1 < 2 .Whenever it is clear from the context, we will use the carrier set L to represent the lattice (L, ∨, ∧).
The lattice L is distributive if ∨ and ∧ distribute over each other, finite if L is finite, and bounded if it has a minimum and a maximum element, denoted as 0 and 1, respectively.It is complete if joins and meets of arbitrary subsets T ⊆ L, t∈T t and t∈T t, respectively, exist.Clearly, every finite lattice is also complete, and every complete lattice is bounded.
Given a lattice L, a t-norm is an associative and commutative binary operator on L that is monotonic and has 1 as its unit.A residuated lattice is a lattice L with a t-norm ⊗ and a binary operator ⇒ (called residuum) such that for all If L is a distributive lattice, then the meet operator 1 ∧ 2 always defines a continuous t-norm, called the Gödel t-norm.In a residuated De Morgan lattice L, the tconorm ⊕ is defined as as 1 ⊕ 2 := ∼(∼ 1 ⊗ ∼ 2 ).The t-conorm of the Gödel t-norm is the join operator 1 ∨ 2 .
For example, consider the finite lattice L 4 , with the elements f, u, i, and t as shown in Figure 1.This lattice has been used for reasoning about incomplete and contradictory knowledge [5] and as a basis for a paraconsistent rough DL [25].
In our blood pressure scenario, the two degrees i and u may be used to express readings that are potentially and partially high blood pressures, respectively.The incomparability of these degrees reflects the fact that none of them can be stated to belong more to the concept HighBloodPressure than the other.
For the rest of this paper, L denotes a complete residuated De Morgan lattice with t-norm ⊗ and residuum ⇒, unless explicitely stated otherwise.

The Fuzzy DL L-SHI
The fuzzy DL L-SHI is a generalization of the crisp DL SHI that uses the elements of L as truth values, instead of just the Boolean true and false.The syntax of L-SHI is the same as in SHI with the addition of the constructor →.
Definition 1 (syntax of L-SHI).Let N C , N R , and N I be pairwise disjoint sets of concept-, role-, and individual names, respectively, and N + R ⊆ N R a set of transitive role names.The set of (complex) roles is N R ∪{r − | r ∈ N R }.The set of (complex) concepts C is obtained through the following syntactic rule, where A ∈ N C and s is a complex role: The semantics of this logic is based on functions specifying the membership degree of every domain element in a concept C.
Definition 2 (semantics of L-SHI).An interpretation is a pair I = (∆ I , • I ) where ∆ I is a non-empty domain, and • I is a function that assigns to every individual name a an element a I ∈ ∆ I , to every concept name A a function A I : ∆ I → L, and to every role name r a function r I : ∆ I × ∆ I → L, where r I (x, y) ⊗ r I (y, z) ≤ r I (x, z) holds for all r ∈ N + R and x, y, z ∈ ∆ I .The function • I is extended to L-SHI concepts as follows for every x ∈ ∆ I : where (r − ) I (x, y) = r I (y, x) for all x, y ∈ ∆ I and r ∈ N R .
Notice that, unlike in crisp SHI, existential and universal quantifiers are not dual to each other, i.e. in general, (¬∃s.C) I (x) = (∀s.¬C)I (x) does not hold.Likewise, the implication constructor → cannot be expressed in terms of the negation ¬ and conjunction .
The axioms of this logic are those of crisp SHI, but with associated lattice values, which express the degree to which the restrictions must be satisfied.Definition 3 (axioms).An assertion can be a concept assertion of the form a : C or a role assertion of the form (a, b) : s , where C is a concept, s is a complex role, a, b are individual names, ∈ L, and ∈ {=, ≥}.If is =, then it is called an equality assertion.A general concept inclusion (GCI) is of the form C D, , where C, D are concepts, and ∈ L. A role inclusion is of the form s s , where s and s are complex roles.
An ontology (A, T , R) consists of a finite set A of assertions (ABox ), a finite set T of GCIs (TBox ), and a finite set R of role inclusions (RBox ).The ABox A is called local if there is an individual a ∈ N I such that all assertions in A are of the form a : C = , for some concept C and ∈ L. I is a model of the ontology (A,T ,R) if it satisfies all axioms in A, T , R.
Given an RBox R, the role hierarchy R on the set of complex roles is the reflexive and transitive closure of the relation Using reachability algorithms, the role hierarchy can be computed in polynomial time in the size of R.An RBox R is called acyclic if it contains no cycles of the form s R s , s R s for two roles s = s .
The fuzzy DL L-ALC is the sublogic of L-SHI where no role inclusions, transitive roles, or inverse roles are allowed.SHI is the sublogic of L-SHI where the underlying lattice contains only the elements 0 and 1, which may be interpreted as false and true, respectively, and the t-norm and t-conorm are conjunction and disjunction, respectively.
Recall that the semantics of the quantifiers require the computation of a supremum or infimum of the membership degrees of a possibly infinite set of elements of the domain.To obtain effective decision procedures, reasoning is usually restricted to a special kind of models, called witnessed models [16].
Definition 4 (witnessed model).Let n ∈ N. A model I of an ontology O is n-witnessed if for every x ∈ ∆ I , every role s and every concept C there are x 1 , . . ., x n , y 1 , . . ., y n ∈ ∆ I such that In particular, if n = 1, the suprema and infima from the semantics of ∃s.C and ∀s.C are maxima and minima, respectively, and we say that I is witnessed.
The reasoning problems for SHI generalize to the fuzzy semantics of L-SHI.Example 6.It is known that coffee drinkers and salt consumers tend to have a higher blood pressure.On the other hand, bradycardia is highly correlated with a lower blood pressure.This knowledge can be expressed through the TBox { CoffeeDrinker HighBloodPressure, i , SaltConsumer HighBloodPressure, i , Bradycardia ¬HighBloodPressure, i }, over the lattice L 4 from Figure 1.The degree i in these axioms expresses that the relation between the causes and HighBloodPressure is not absolute.Consider the patients ana, who is a coffee drinker, and bob, a salt consumer with bradycardia, as expressed by the ABox We can deduce that both patients are an i-instance of HighBloodPressure, but only bob is an i-instance of ¬HighBloodPressure.Notice that if we changed all the degrees from the GCIs to the value t, the ontology would be inconsistent.
We will focus first on a version of the consistency problem where the ABox is required to be a local ABox; we call this problem local consistency.We show in Section 5 that local consistency can be used for solving other reasoning problems in L-SHI if L is finite.Before that, we show that satisfiability and (local) consistency are undecidable in L-ALC, and hence also in L-SHI, in general.

Undecidability
To show undecidability, we use a reduction from the Post Correspondence Problem [21] to strong satisfiability in L-ALC over a specific infinite lattice.The reduction uses ideas that have been successfully applied to showing undecidability of reasoning for several fuzzy description logics [2,3,12].
Although the basic idea of the proof is not new, it is interesting for several reasons.First, previous incarnations of the proof idea focused on decidability of ontology consistency [3,11,12], while we are concerned with strong -satisfiability.Second, most of the previous undecidability results only hold for reasoning w.r.t.
witnessed models, but the current proof works for both witnessed and general models.Finally, in contrast to an earlier version of this proof [10], the employed lattice has a quite simple structure in the sense that it is a total order that has only the two limit points −∞ and ∞ instead of infinitely many.Note that any distributive lattice without limit points is already finite and reasoning in finite residuated De Morgan lattices is decidable (see Sections 4 and 5).
Definition 7 (PCP).Let P = {(v 1 , w 1 ), . . ., (v n , w n )} be a finite set of pairs of words over the alphabet Σ = {1, . . ., s} with s > 1.The Post Correspondence Problem (PCP) asks for a finite non-empty sequence i If this sequence exists, it is called a solution for P.
We consider the lattice Z ∞ whose domain is Z ∪ {−∞, ∞} with the usual ordering over the integers and −∞ and ∞ as the minimal and maximal element, respectively.Its De Morgan negation is This is in fact a residuated lattice with the following residuum: Given an instance P of the PCP, we will construct a TBox T P such that the designated concept name S is strongly ∞-satisfiable iff P has no solution.Recall that the alphabet Σ consists of the first s positive integers.Thus, every word in Σ + can be seen as a positive integer written in base s + 1; we extend this intuition and denote the empty word by 0. We encode each word u ∈ Σ * with the number −u ≤ 0.
The idea is that the TBox T P describes the search tree of P with the nodes {1, . . ., n} * .At its root ε, it encodes the value v ε = w ε = ε, which is represented by 0, using the concept names V and W .These concept names are used throughout the tree to express the values v ν and w ν at every node ν ∈ {1, . . ., n} * .Additionally, we will use the auxiliary concept names V i and W i to encode the constant words v i and w i , respectively, for each i ∈ {1, . . ., n}.These will be used to compute the concatenation v νi = v ν v i at each node.
To simplify the reduction, we will use some abbreviations.• We now introduce the TBox T 0 := n i=0 T i P that encodes the search tree of the instance P of the PCP: The TBox T 0 P initializes the search tree by ensuring for every model I and every domain element x ∈ ∆ I that satisfies S I (x) = ∞ that the values of V and W are both 0, which is the encoding of the empty word.Each TBox T i P ensures the existence of an r i -successor for every domain element and describes the constant pair (v i , w i ) using the concepts V i and W i , i.e. it forces that V I i (x) = −v i and W I i (x) = −w i for every x ∈ ∆ I .Using the last two axioms, the search tree is then extended by concatenating the words v and w produced so far with v i and w i , respectively.In the following, we will describe this in more detail.
Consider the interpretation I P over the domain ∆ I P = {1, . . ., n} * , where for all ν, ν ∈ {1, . . ., n} * and i ∈ {1, . . ., n}, It is easy to see that I P is in fact a model of T 0 and it strongly satisfies S with degree ∞.Moreover, every model of this TBox that strongly ∞-satisfies S must "include" I P in the following sense.
Proof.We construct the function g by induction on ν and set g(ε) := x 0 .Since I is a model of T 0 P and S I (x 0 ) = ∞, we have V I (x 0 ) ≥ 0 and ∼ V I (x 0 ) ≥ 0, i.e.V I (x 0 ) = 0, and similarly W I (x 0 ) = 0.In the same way, for every i ∈ {1, . . ., n}, V I i (x 0 ) and W I i (x 0 ) are restricted by T i P to be −v i and −w i , respectively.Let now ν ∈ {1, . . ., n} * and assume that g(ν) already satisfies the condition.For each i ∈ {1, . . ., n}, the first axiom of T i P ensures that y∈∆ I r I i (g(ν), y) ≥ 1.Thus, there is y i ∈ ∆ I such that r I i (g(ν), y i ) ≥ 1.We define g(νi) := y i .By Proposition 8, we have and similarly for W I (y i ).The claim for V i and W i can be shown as above.
This proposition shows that every model of T 0 encodes a description of the search tree for a solution of P. Thus, to decide the PCP, it suffices to detect whether there is a node ν ∈ {1, . . ., n} + of I P where V I P (ν) = W I P (ν).We accomplish this using the TBox The interpretation I P is a model of T iff V I P (ν) = W I P (ν) holds for every ν ∈ {1, . . ., n} + .
Lemma 10.P has a solution iff S is not ∞-satisfiable w.r.t.T P := T 0 ∪ T .
For the other direction, let I be a model of T P and x 0 ∈ ∆ I such that S I (x 0 ) = ∞.
As mentioned before, since the interpretation I P is witnessed, undecidability holds even if we restrict reasoning to n-witnessed models, for any n ∈ N.
Theorem 11.Strong satisfiability is undecidable in L-ALC, even if L is a countable total order with at most two limit points and reasoning is restricted to nwitnessed models.
This theorem also shows that (local) consistency is undecidable in L-ALC since S is strongly ∞-satisfiable w.r.t.T P iff ({ a : S = ∞ }, T P ) is locally consistent, where a is an arbitrary individual name.Notice that these do not exclude the existence of classes of infinite lattices for which reasoning in L-SHI is decidable.
If we restrict to finite lattices, then a tableau algorithm can be used for reasoning.

A Tableaux Algorithm for Local Consistency
Before presenting a tableau algorithm [4] that decides local consistency by constructing a model of a given L-SHI ontology, we discuss previous approaches to deciding consistency of fuzzy DLs over finite residuated De Morgan lattices in the presence of GCIs.
A popular method is the reduction of fuzzy ontologies into crisp ones, which has so far only been done for finite total orders [7,8,23].Reasoning can then be performed through existing optimized reasoners for crisp DLs.The main idea is to translate every concept name A into finitely many crisp concept names A ≥ , one for each truth value , where A ≥ collects all those individuals that belong to A with a truth degree ≥ .The lattice structure is expressed through GCIs of the form A ≥ 2 A ≥ 1 , where 2 is a minimal element above 1 , and analogously for the role names.All axioms are then recursively translated into crisp axioms that use only the introduced crisp concept and role names.The resulting crisp ontology is consistent iff the original fuzzy ontology is consistent.
In general such a translation is exponential in the size of the concepts that occur in the fuzzy ontology.The reason is that, depending on the t-norm used, there may be many possible combinations of values 1 , 2 for C, D, respectively, that lead to C D having the value = 1 ⊗ 2 , and similarly for the other constructors.All these possibilities have to be expressed in the translation.If after the translation one uses a crisp DL reasoner, which usually implement tableaux algorithms with a worst-case complexity above NExpTime, one gets a 2-NExpTime reasoning procedure.In contrast, our tableau algorithm has a worst-case complexity of NExpTime, matching the complexity of crisp SHI.
To the best of our knowledge, at the moment there exists only one (correct) tableaux algorithm that can deal with a finite total order of truth values and GCIs [22], 3 but it is restricted to the Gödel t-norm.The main difference between this algorithm and ours is that we non-deterministically guess the degree of membership of each individual to every relevant concept, while the approach from [22] sets only lower and upper bounds for these degrees; this greatly reduces the amount of non-determinism, but introduces several complications when a t-norm different from the Gödel t-norm is used.
We present a straightforward tableaux algorithm with a larger amount of nondeterminism that nevertheless matches the theoretical worst-case complexity of tableaux algorithms for crisp SHI.It is loosely based on the crisp tableaux algorithm in [17].A first observation that simplifies the algorithm is that since L is finite, we can w.l.o.g.restrict reasoning to n-witnessed models.
Proposition 12.If the maximal cardinality of an antichain of L is n, then every interpretation in L-SHI is n-witnessed.
For simplicity, we consider only the case n = 1.For n > 1, the construction is similar, but several witnesses have to be produced for satisfying each existential and value restriction.The necessary changes in the algorithm are described at the end of this section.We can also assume w.l.o.g. that the RBox is acyclic.The proof of this follows similar arguments as for crisp SHI [24].
Proposition 13.Deciding local consistency in L-SHI is polynomially equivalent to deciding local consistency in L-SHI w.r.t.acyclic RBoxes.
In the following, let O = (A, T , R) be an ontology where A is a local ABox that contains only the individual name a and R is an acyclic RBox.We first show that O has a model if we can find a tableau; intuitively, a possibly infinite "completed version" of A. Later we describe an algorithm for constructing a finite representation of such a tableau.Complete: For every row of Table 1, the following condition holds: "If trigger is in T, there are values such that assertions are in T." These conditions help to abstract from the interplay between transitive roles and existential and value restrictions.It suffices to satisfy the above conditions to make certain that O has a model.

Lemma 15.
O is locally consistent iff it has a tableau.
Table 1: The tableau conditions for L-SHI.
Proof.Let T be a tableau for O over the set Ind of individuals.We define Note that these values are either unique or undefined since T is clash-free.In this way, T immediately defines a rudimentary interpretation.However, transitive roles are not yet interpreted by transitive fuzzy relations.In the following, we denote by r T (z 1 , . . ., z n ) the value r T (z 1 , z 2 ) ⊗ . . .⊗ r T (z n−1 , z n ) for any sequence z 1 , . . ., z n ∈ Ind.This value is 1 if n = 1 since 1 is the unit element for ⊗.
We now define a proper model I of O by setting ∆ I := Ind, A I (x) = A T (x) for all concept names A and x ∈ Ind, r I (x, y) = n≥0 z 1 ,...,zn∈Ind r T (x, z 1 , . . ., z n , y) if the role r is transitive, and Thus, I correctly interprets transitive roles by transitive relations.This construction was inspired by a similar one used for crisp SHI in [17].It is well-defined if R is acyclic (see Lemma 13).By the invand R -conditions, I satisfies R and inverse roles are interpreted correctly.Furthermore, one can show by induction on the role depth that for every concept C we have C I (x) = C T (x) whenever the latter is defined.Together with the T -condition and the fact that A ⊆ T, this shows that I also satisfies A and T , and thus it satisfies O.
Let now I be a model of O.We can easily construct a tableau T over the set ∆ I of individuals as follows.For every concept C and x ∈ ∆ I , we add x : C = to T if C I (x) = .Similarly, for every role r and x, y ∈ ∆ I , we add the assertion (x, y) : r = r I (x, y) to T. We have A ⊆ T since I satisfies A. T is clash-free since the values are uniquely defined by I.
Furthermore, the semantics of L-SHI concepts and axioms yield completeness: consider the ∃ + -condition and assume that (∃s.C) I (x) = , r I (x, y) = 1 with r transitive, and r R s.Since the value 2 = (∃r.C) I (y) is defined, by monotonicity of ⊗ this value satisfies Similar arguments show that T satisfies the other completeness conditions.
We now present a tableaux algorithm that nondeterministically expands A to a tree-like ABox A that represents a model of O.It uses the conditions from Table 1 and reformulates them into expansion rules of the form: "If there is trigger in A and there are no values such that assertions are in A, then introduce values and add assertions to A." The rules ∃ and ∀ always introduce new individuals y that do not appear in A. Initially, the ABox A contains the single individual a.It is expanded by the rules in a tree-like way: role connections are only created by adding new successors to existing individuals.If an individual y was created by a rule ∃ or ∀ that was applied to an assertion involving an individual x, then we say that y is a successor of x, and x is the predecessor of y; ancestor is the transitive closure of predecessor.Note that the presence of an assertion (x, y) : r = in A does not imply that y is a successor of x-it could also be the case that this assertion was introduced by the inv-rule.We further denote by A x the set of all concept assertions from A that involve the individual x, i.e. are of the form x : C = for some concept C and ∈ L. To ensure that the application of the rules terminates, we need to add a blocking condition.We use anywhere blocking [20], which is based on the idea that it suffices to examine each set A x only once in the whole ABox A.
Let be a total order on the individuals of A that includes the ancestor relationship, i.e. whenever y is a successor of x, then y x.An individual y is directly blocked if for some other individual x in A with y x, A x is equal to A y modulo the individual names; in this case, we write A x ≡ A y and also say that x blocks y.It is indirectly blocked if its predecessor is either directly or indirectly blocked.A node is blocked if it is either directly or indirectly blocked.The rules ∃ and ∀ are applied to A only if the node x that triggers their execution is not blocked.All other rules are applied only if x is not indirectly blocked.
The total order avoids cycles in the blocking relation.One possibility is to simply use the order in which the individuals were created by the expansion rules.Note that the only individual a that occurs in A, which is the root of the tree-like structure represented by A, cannot be blocked since it is an ancestor of all other individuals in A. With this blocking condition, we can show that the size of A is bounded exponentially in the size of A, as in the crisp case [20].
Lemma 16.Every application of expansion rules to A terminates after at most exponentially many rule applications.
Proof.Let sub(O) denote the set of all subconcepts of concepts appearing in O and recall that every rule application expands A in a tree-like manner.Note that there are at most |L||sub(O)| possible concept assertions for one individual x.Thus, every node in this tree has at most |L||sub(O)| successors: one for each possible assertion with a quantified concept.Moreover, there can be at most 2 |L||sub(O)| non-blocked nodes in A at any time, and thus, when a node becomes blocked, at most exponentially many nodes become indirectly blocked.This shows that we obtain a tree of at most exponential size before every rule application is disallowed by the blocking condition.The claim now follows from the fact that every rule application adds at least one assertion to A and cannot remove assertions from A.
We say that A contains a clash if it contains two assertions that are equal except for their lattice value (see Definition 14).A is complete if it contains a clash or none of the expansion rules are applicable.The algorithm is correct in the sense that it produces a clash iff O is not locally consistent.
Lemma 17. O is locally consistent iff some application of the expansion rules to A yields a complete and clash-free ABox.
Proof.By Lemma 15, O is locally consistent iff it has a tableau.Assume first that T is a tableau for O over the set Ind of individuals.We show how to guide the application of the expansion rules in such a way that no clash is produced.
Observe that the initial ABox A is included in T by definition.We will ensure that the expansion rules add only assertions to A that are also in T. Assume that, for some row of Table 1, an expansion rule is applicable, i.e. trigger is in A and there are no values such that assertions are in A and the blocking condition does not apply.Since trigger is also in the tableau T, there must be values such that assertions are in T, and thus we can add assertions to A.
Since T is clash-free, this process cannot create any clashes in A. Lemma 16 shows that at some point A must also be complete.
Assume now that the expansion rules have produced a complete and clash-free ABox A. We define a tableau T for O over the set Ind := {x ∈ N I | x occurs in A and is not blocked} of individuals as follows: Thus, whenever y blocks z and z is not indirectly blocked, then all incoming role connections of z are "re-routed" back to y.Since the root a of the tree-like structure A has no predecessors, it cannot be blocked, and thus the initial ABox A is still contained in T. Furthermore, since A is clash-free, T is also clash-free.
Assume now that T violates the condition specified by some row of Table 1, i.e. there is trigger in T, but no values such that assertions are in T. a) If trigger involves only assertions from A, then the corresponding expansion rule was applied at some point and introduced values and assertions .If no new individual was introduced, all assertions must also be in T. We consider now the case of the ∃-rule; the ∀-rule can be handled similarly.As explained before, L-SHI has the n-witnessed model property for some n ≥ 1.
We have so far restricted our description to the case where n = 1.If n > 1, it does not suffice to generate only one successor for every existential and universal restriction, but one must produce n different successors to ensure that the degrees guessed for these complex concepts are indeed witnessed by the model.The only required change to the algorithm is in the rows ∃ and ∀ of Table 1, where we have to introduce n individuals y 1 , . . ., y n , and

Local Completion and Other Black-Box Reductions
In the following, we assume that we have a black-box procedure that decides local consistency in a sublogic of L-SHI.This procedure can be, e.g. the tableau-based algorithm from the previous section, or any other method for solving this decision problem.We show how to employ such a procedure to solve other reasoning problems for this sublogic.

Consistency
To reduce consistency of an arbitrary ontology O = (A, T , R) to local consistency, we first make sure that the information contained in A is consistent "in itself", i.e. if we only consider the individuals occurring in A. It then suffices to check a local consistency condition for each of the individuals.
Let Ind A denote the set of individual names occurring in A and sub(A, T ) the set of all subconcepts of concepts occurring in A or T .We first guess a set A of equality assertions of the forms a : C = and (a, b) : r = with a, b ∈ Ind A , C ∈ sub(A, T ), r ∈ N R , and ∈ L. We then check whether A is clash-free and satisfies the tableau conditions listed in Table 1, except the witnessing conditions ∃ and ∀.Additionally, we impose the following condition on A: "If there is an assertion α in A, then there is ∈ L such that and α = is in A." We call A locally complete iff it is of the above form and satisfies all of the above conditions.Guessing this set and checking whether it is locally complete can be done in polynomial time in the size of O. Let now A be a locally complete set for O and O x be locally consistent for every x ∈ Ind A .By Lemma 15, for each x ∈ Ind A there is a tableau T x for O x over the set Ind x of individuals.We can assume that the sets Ind x are mutually disjoint.Note that x ∈ Ind x for every x ∈ Ind A .
We now define C T (y) = whenever y : C = ∈ T x for some x ∈ Ind A .Similarly, we set r T (y, z) = if (y, z) : r = ∈ T x for some x ∈ Ind A .Note that, since T is clash-free and the sets Ind x are disjoint, these values are uniquely defined.To reconnect the individuals of Ind A , we additionally define r T (x, y) = whenever (x, y) : r = ∈ A.
As in the proof of Lemma 15, we can now define an interpretation I from these values by constructing the transitive closure of r T if r is transitive.It then holds that C I (x) = C T (x) whenever the latter is defined.Since the assertions in A satisfy A, I also satisfies A and by the T -and R -conditions, I satisfies T and R.This means that consistency in L-SHI is decidable in NExpTime.In [9], an automata-based algorithm was presented that can decide satisfiability and subsumption in L-ALCI in ExpTime.Moreover, if the TBox is acyclic, then this bound can be improved to PSpace.The algorithm can easily be adapted to decide local consistency.With the above reduction, this shows that consistency in L-ALCI w.r.t.general and acyclic TBoxes can be decided in ExpTime and PSpace, respectively.The same argument applies to any sublogic of L-SHI for which local consistency can be decided in ExpTime or PSpace.

Satisfiability, Instance Checking, and Subsumption
To decide whether a concept C is strongly -satisfiable w.r.t.O = (A, T , R), we can simply check whether (A ∪ {a : C ≥ }, T , R) is consistent for an arbitrary individual name a. Thus, strong -satisfiability is in the same complexity class as consistency.Moreover, we can easily compute the set of all values ∈ L such that the ontology (A ∪ {a : C ≥ }, T , R) is consistent by calling the decision procedure for consistency a constant number of times, i.e. once for each ∈ L. We can use this set to compute the best bound for the satisfiability of C. Formally, the best satisfiability degree of a concept C is the supremum of all ∈ L such that C is -satisfiable w.r.t.O. Since we can compute the set of all elements of L satisfying this property, obtaining the best satisfiability degree requires only a supremum computation.As the lattice L is fixed, this adds a constant factor to the complexity of checking consistency.
To check -instances, we can exploit the fact that a is not an -instance of C w.r.t.O iff there is a model I of O and a domain element x ∈ ∆ I such that C I (a I ) . This is the case iff there is a value such that the ontology (A ∪ {a : C = }, T , R) is consistent.Thus, -instances can be decided by calling the decision procedure for consistency a constant number of times, namely at most once for each ∈ L with .We can also compute the best instance degree for a and C, which is the supremum of all ∈ L such that a is an -instance of C w.r.Finally, note that C is -subsumed by D iff a is an -instance of C → D, where a is a new individual name.Thus, deciding -subsumption and computing the best subsumption degree can be done using the same approach as above.This shows that also strong satisfiability, instance checking, and subsumption in L-SHI are in NExpTime.This bound reduces to ExpTime or PSpace if we consider L-ALCI w.r.t.general or acyclic TBoxes, respectively [9].

Conclusions
We have studied fuzzy description logics with semantics based on complete residuated De Morgan lattices.We showed that even for the fairly inexpressive DL L-ALC, strong satisfiability w.r.t.general TBoxes is undecidable when the underlying lattice is infinite.For finite lattices, decidability is regained.In fact, local consistency can be decided with a nondeterministic tableaux-based procedure in exponential time.We conjecture that this upper bound can be improved to ExpTime either by an automata-based algorithm or with the help of advanced caching techniques [14].Other decision and computation problems can also be solved using a local consistency reasoner as a black box.In particular, this yields tight complexity bounds for deciding consistency in L-ALCI w.r.t.acyclic and general TBoxes-PSpace and ExpTime, respectively.
The presented tableaux algorithm has highly nondeterministic rules, and as such is unsuitable for an implementation.Most of the optimizations developed for tableaux algorithms for crisp DLs, like the use of an optimized rule-application ordering, can be transfered to our setting.However, the most important task is to reduce the search space created by the choice of lattice values in most of the rules.We plan to study these optimizations in the future.

Figure 1 :
Figure 1: The De Morgan residuated lattice L 4 with ∼ u = u and ∼ i = i.
An interpretation I satisfies the assertion a : C if C I (a I ) and the assertion (a, b) : s if s I (a I , b I ) .It satisfies the GCI C D, if C I (x) ⇒ D I (x) ≥ holds for every x ∈ ∆ I .It satisfies the role inclusion s s if for all x, y ∈ ∆ I we have s I (x, y) ≤ s I (x, y).
Given two L-ALC concepts C and D and r ∈ N R , C ≡ D abbreviates the axioms C D, ∞ , D C, ∞ ; and C r D stands for the axioms C ∀r.D, ∞ , ∃r.D C, ∞ .For n ≥ 1, the concept C n is inductively defined by C 1 := C and C n+1 := C n C. Proposition 8. Let I be an interpretation and x ∈ ∆ I .• If I satisfies C ≡ D , then C I (x) = D I (x).• If I satisfies C r D and C I (x) ≤ 0, then C I (x) = D I (y) holds for all y ∈ ∆ I with r I (x, y) ≥ 1.

Definition 14 .
A tableau for O is a set T of equality assertions over a set Ind of individuals such that a ∈ Ind, A ⊆ T, and the following conditions are satisfied for all C, C 1 , C 2 ∈ sub(O), x, y ∈ Ind, r, s ∈ N R , and ∈ L: Clash-free: If x : C = ∈ T or (x, y) : r = ∈ T, then there is no ∈ L such that = and x : C = ∈ T or (x, y) : r = ∈ T, respectively.

Theorem 18 .
Assume that x : ∃r.C = ∈ A and x is not blocked.Then a new individual y was introduced, together with the assertions (x, y) : r = 1 and y : C = 2 , where 1 ⊗ 2 = .If y is not blocked, these assertions are also in T. If y is blocked by an individual z, then the assertion (x, z) : r = 2 is in T. Additionally, we have A y ≡ A z , and thus also z : C = 2 is in T. b) If trigger involves a role assertion (x, y) : r = 1 where (x, z) : r = 1 ∈ A and y blocks z, then x is not blocked and the corresponding expansion rule was applied to A with z instead of y.Consider the case of the ∃ ≤ -rule.Then the assertions x : ∃r.C = and z : C = 2 must be in A with 1 ⊗ 2 ≤ .Since A z ≡ A y , we have y : C = 2 in A and also in T. The rules ∃ + , ∀ ≥ , and ∀ + behave similarly.If the inv-rule was applied, then we have (z, x) : r = 1 ∈ A, and thus (y, x) : r = 1 is in T. If the R -rule was applied with r R s, then (x, z) : s = 2 ∈ A with some 2 ∈ L such that 1 ≤ 2 .Thus, we have (x, y) : s = 2 in T. c) If trigger involves a role assertion (x, y) : r = 1 where (z, y) : r = 1 ∈ A and x blocks z, then consider the concrete condition concerned.If it is the ∃ ≤ -condition, then we have x : ∃r.C = in T and also in A. Since A x ≡ A z , this implies that z : ∃r.C = is in A. Since z must be a successor of y, z is not indirectly blocked, and thus by the ∃ ≤ -rule there is y : C = 2 in A with Since the tableau rules are nondeterministic, Lemmata 16 and 17 together imply that the tableaux algorithm decides local consistency in NExpTime.Local consistency in L-SHI w.r.t.witnessed models can be decided in NExpTime.

Lemma 19 .
An ontology O = (A, T , R) is consistent iff there is a locally complete set A such that O x = ( A x , T , R) is locally consistent for every x ∈ Ind A .Proof.Let I be a model of O and A be the set of all assertions a : C = C I (a I ) and (a, b) : r = r I (a I , b I ) for a, b ∈ Ind A , r ∈ N R , and C ∈ sub(A, T ).Using the same arguments as in the proof of Lemma 15, we can show that A is locally complete.Furthermore, by construction I satisfies O x for any x ∈ Ind A .

Theorem 20 .
If local consistency in L-SHI can be decided in a complexity class C, then consistency in L-SHI can be decided in any complexity class that contains both NP and C.
t. O. Let L denote the set of all such that ({a : C = }, T , R) is consistent.The best instance degree for a and C is the infimum of all ∈ L since{ ∈ L | a is an -instance of C} = { ∈ L | ∀ : / ∈ L} = { ∈ L | ∀ ∈ L : ≤ } = L.

Lemma 21 .
If local consistency in L-SHI can be decided in a complexity class C, then strong satisfiability, instance checking, and subsumption in L-SHI can be decided in any complexity class that contains both NP and C.
Definition 5 (decision problems).Let O be an ontology, C, D be two concepts, a ∈ N I , and ∈ L. O is consistent if it has a (witnessed) model.C is strongly -satisfiable if there is a (witnessed) model I of O and x ∈ ∆ I with C I (x) ≥ .