Ontology consistency has been shown to be undecidable for a wide variety of fairly inexpressive fuzzy Description Logics (DLs). In particular, for any t-norm \starting with" the Lukasiewicz t-norm, consistency of crisp ontologies (w.r.t. witnessed models) is undecidable in any fuzzy DL with conjunction, existential restrictions, and (residual) negation. In this paper we show that for any t-norm with Godel negation, that is, any t-norm not starting with Lukasiewicz, ontology consistency for a variant of fuzzy SHOI is linearly reducible to crisp reasoning, and hence decidable in exponential time. Our results hold even if reasoning is not restricted to the class of witnessed models only.
Fuzzy Description Logics (DLs) were introduced over a decade ago to represent and reason with vague or imprecise knowledge. Since their introduction, several variants of these logics have been studied; in fact, in addition to the constructors and kinds of axioms used, fuzzy DLs have an additional degree of liberty in the choice of the t-norm that speci es the semantics. An extensive, although slightly outdated survey of the area can be found in [18].
Very recently, it was shown that some fuzzy DLs lose the nite model property in the presence of GCIs [3]. This eventually led to a series of undecidability results [1, 2, 12, 10]. Most notably, for t-norms that \start with" the Lukasiewicz t-norm, consistency of crisp ontologies becomes undecidable for the inexpressive fuzzy DL -NE L, which allows only the constructors conjunction, existential restrictions and (residual) negation [10].
So far, the only known decidability results for fuzzy DLs rely on a restriction of the expressivity: either by allowing only nitely-valued semantics [6, 8], by limiting the terminological knowledge to be acyclic or unfoldable [14, 11, 5], or by using the very simple Godel semantics [4, 21{23]. Moreover, with very few exceptions [6, 8], reasoning is usually restricted to the class of witnessed models.1 In fact, witnessed models were introduced in [14] to correct the previous algorithms for fuzzy DL reasoning. ? Partially supported by the DFG under grant BA 1122/17-1 and in the Collaborative
Research Center 912 \Highly Adaptive Energy-E cient Computing". 1 All fuzzy logics with nitely-valued semantics have the witnessed model property.
In this paper we show that for any t-norm with Godel negation, ontology consistency is decidable in the very expressive fuzzy DL -SHOI 8, which is the sub-logic of -SHOI where value restrictions 8 are not allowed. For these logics, ontology consistency w.r.t. general models is linearly reducible to consistency of a crisp SHOI ontology, and hence decidable in exponential time. In particular, this holds for the product t-norm. We emphasize that our proofs do not depend on the models being witnessed or not, hence decidability is shown for reasoning w.r.t. both, general models and witnessed models.
Since a t-norm has Godel negation i it does not start with the Lukasiewicz t-norm [17], this yields a full characterization of the decidability of ontology consistency (w.r.t. witnessed models) for all fuzzy DLs between -NE L and -SHOI 8. We also provide the rst decidability results w.r.t. general models for in nitely-valued, non-idempotent fuzzy DLs. 2
Mathematical fuzzy logic [13] generalizes classical logic by allowing all the real numbers from the interval [0; 1] as truth values. The interpretation of the di erent logical constructors depends on the choice of a triangular norm (t-norm for short). A t-norm is an associative, commutative, and monotone binary operator on [0; 1] that has 1 as its unit element. The dual operator of a t-norm is the t-conorm de ned as x y = 1 ((1 x) (1 y)). Notice that 0 is the unit of the t-conorm, and hence x
y = 0 i x = 0 and y = 0: i y
Every continuous t-norm de nes a unique residuum ) such that x
x ) z for all x; y; z 2 [0; 1]. It is easy to see that for all x; y 2 [0; 1]
y
(
z
yg, Based on the residuum, one can de ne the unary precomplement x = x ) 0. Three important continuous t-norms are the Godel, product and Lukasiewicz t-norms shown in Table 1, together with their t-conorms and residua. These are fundamental in the sense that every continuous t-norm can be described as an ordinal sum of copies of these t-norms [20].
In this paper, we are interested in t-norms that do not have zero divisors. An element x 2 (0; 1) is called a zero divisor for if there is a z 2 (0; 1) such that x z = 0. Of the three fundamental continuous t-norms, only the Lukasiewicz t-norm has zero divisors. In fact, every element of the interval (0; 1) is a zero divisor for this t-norm. The Godel and product t-norms are just two elements of the uncountable class of continuous t-norms without zero divisors. Proposition 1. For any t-norm
without zero divisors and every x 2 [0; 1], 1. x ) y = 0 i x > 0 and y = 0, and 2. x = 0 i x > 0.
Proof. For the rst claim, we prove only the if direction, since the other direction is known to hold for every t-norm [17]. Assume that x > 0 and y = 0. Then x ) y = x ) 0 = supfz j z x = 0g. Since has no zero divisors, z x > 0 for all z > 0. Therefore fz j z x = 0g = f0g and thus x ) y = 0. The second statement follows from the rst one since x = x ) 0. tu In particular, this implies that if the t-norm does not have zero divisors, then its precomplement is the so-called Godel negation, i.e. for every x 2 [0; 1], x = (0 if x > 0
1 otherwise. 1(x) = (1 if x > 0
0 if x = 0: It can be shown that the converse also holds: if the precomplement is the Godel negation, then the t-norm has no zero divisors.
We now de ne the function 1 that maps fuzzy truth values to crisp truth values by setting, for all x 2 [0; 1], For a t-norm without zero divisors it follows from Proposition 1 that 1(x) = x for all x 2 [0; 1]. This function is compatible with negation, the t-norm, the corresponding t-conorm, implication and suprema.
Lemma 2. Let non-empty sets X be a t-norm without zero divisors. For all x; y 2 [0; 1] and all
[0; 1] it holds that 1. 1( x) = 1(x), 2. 1(x y) = 1(x) 1(y), 3. 1(x y) = 1(x) 1(y), 4. 1(x ) y) = 1(x) ) 1(y), and 5. 1 (supfx j x 2 Xg) = supf1(x) j x 2 Xg.
Proof. It holds that 1( x) = x = 1(x) which proves 1. Since
have zero divisors, x y = 0 i x = 0 or y = 0. This shows that
Both 1(x y) and 1(x) 1(y) can only have the values 0 or 1. Hence, (2)
proves the second statement. Following similar arguments we obtain from (
-SHOI
8 A fuzzy description logic usually inherits its syntax from the underlying crisp description logic. We consider the constructors of SHOI with the addition of !, which in the crisp case can be expressed by t and :.
De nition 3 (syntax). Let NC, NR, and NI, be disjoint sets of concept, role, and individual names, respectively, and NR+ NR be a set of transitive role names. The set of (complex) roles is NR [ fr j r 2 NRg. -SHOI (complex) concepts are constructed by the following syntax rule:
C ::= A j > j ? j fag j :C j C u C j C t C j C ! C j 9s:C j 8s:C; where A is a concept name, a is an individual name, and s is a complex role. The inverse of a complex role s (denoted by s) is s if s 2 NR and r if s = r . A role s is transitive if either s or s belongs to N+.
R
Given a continuous t-norm , concepts in the fuzzy DL -SHOI are interpreted by functions specifying the membership degree of each domain element to the concept. The interpretation of the constructors is based on the t-norm and the induced operators , ), and .
De nition 4 (semantics). An interpretation is a pair I = ( I ; I ), where the domain I is a non-empty set and I is a function that assigns to every concept name A a function AI : I ! [0; 1], to every individual name a an element aI 2 I , and to every role name r a function rI : I I ! [0; 1] such that rI (x; y) rI (y; z) rI (x; z) holds for all x; y; z 2 I if r 2 NR+. The function I is extended to complex roles and concepts as follows for every x; y 2 I , { (r )I (x; y) = rI (y; x), { >I (x) = 1, ?I (x) = 0, { fagI (x) = 1 if aI = x and 0 otherwise, { (:C)I (x) = CI (x), { (C1 u C2)I (x) = C1I (x) C2I (x), { (C1 ! C2)I (x) = C1I (x) ) C2I (x), { (9s:C)I (x) = supz2 I sI (x; z) CI (z), and { (8s:C)I (x) = infz2 I sI (x; z) ) CI (z).
(C1 t C2)I (x) = C1I (x)
C2I (x), I is nite if its domain I is nite, and crisp if AI (x); rI (x; y) 2 f0; 1g for all A 2 NC, r 2 NR, and x; y 2 I .
Recall from the previous section that : is interpreted by the Godel negation i the t-norm does not have zero divisors. In particular, (:C)I (x) 2 f0; 1g holds for every concept C, interpretation I, and x 2 I , i.e. the value of :C is always crisp.
Knowledge is encoded using axioms, which restrict the class of interpretations that are considered and specify a degree to which the restrictions should hold. De nition 5 (axioms). A -SHOI-axiom is either an assertion of the form ha : C; `i or h(a; b) : s; `i, a GCI of the form hC v D; `i, or a role inclusion of the form hs v t; `i, where C and D are -SHOI-concepts, a; b 2 NI, s; t are complex roles, and ` 2 (0; 1]. An axiom is called crisp if ` = 1.
An interpretation I satis es an assertion ha : C; `i if CI (aI ) ` and an assertion h(a; b) : s; `i if sI (aI ; bI ) `. It satis es the GCI hC v D; `i if CI (x) ) DI (x) ` holds for all x 2 I . It satis es a role inclusion hs v t; `i if sI (x; y) ) tI (x; y) ` holds for all x; y 2 I .
A -SHOI-ontology (A; T ; R) is de ned by a nite set A of assertions (ABox), a nite set T of GCIs (TBox), and a nite set R of role inclusions (RBox). It is crisp if every axiom in A, T , and R is crisp. An interpretation I is a model of this ontology if it satis es all its axioms.
We consider also the logic -SHOI 8, which restricts -SHOI by disallowing the constructor 8. -SHOI 8-concepts, axioms and ontologies are de ned in the obvious way. Notice that, contrary to the crisp case, value- and existentialrestrictions are not dual. In fact, we will show in Section 4 that for every t-norm without zero divisors -SHOI is strictly more expressive than -SHOI 8.
Several reasoning problems are of interest in the area of fuzzy DLs. Here we focus only on deciding whether a -SHOI (or -SHOI 8) ontology is consistent ; that is, whether it has a model. We will show that, if the t-norm has no zero divisors, then consistency in -SHOI 8 is e ectively the same problem as consistency in crisp SHOI. Moreover, the precise values appearing in the axioms in the ontology are then irrelevant. The same is not true, however, for consistency in -SHOI.
Recall that the semantics of the quanti ers require the computation of a supremum or in mum of the membership degrees of a possibly in nite set of elements of the domain. In the fuzzy DL community it is customary to restrict reasoning to a special kind of models, called witnessed models [14]. De nition 6 (witnessed). An interpretation I is called witnessed if for every x 2 I , every role s and every concept C there are y1; y2 2 I such that (9s:C)I (x) = sI (x; y1)
CI (y1);
(8s:C)I (x) = sI (x; y2) ) CI (y2):
In particular, if an interpretation I is crisp or nite, then it is also witnessed. Witnessed models were introduced to simplify the reasoning tasks. In fact, although this concept was only formalized in [14], the earlier reasoning algorithms for fuzzy DLs semantics based on the Godel t-norm (e.g. [23]) implicitly used only witnessed models. We show that consistency of -SHOI 8-ontologies can be decided in exponential time, without restricting to witnessed models. 4
The existing undecidability results for fuzzy DLs all rely heavily on the fact that one can design ontologies that allow only models with in nitely many truth values. We shall see that for t-norms without zero divisors one cannot construct such an ontology in -SHOI 8. It is even true that all consistent -SHOI 8ontologies have a crisp model; that is, using at most two truth values. De nition 7. A fuzzy DL L has the crisp model property if every consistent L-ontology has a crisp model.
For the rest of this paper we assume that is a continuous t-norm that does not have zero divisors, and hence has the properties described in Section 2. In particular, Lemma 2 allows us to construct a crisp interpretation from a fuzzy interpretation by simply applying the function 1.
Let I be a fuzzy interpretation for the concept names NC and role names NR. We construct the interpretation J = ( J ; J ), where J := I and for all concept names A 2 NC, all role names r 2 NR, and all x; y 2 I ,
AJ (x) := 1 AI (x) and rJ (x; y) := 1 rI (x; y) : To show that J is a valid interpretation, we rst verify the transitivity condition for all r 2 NR+ and all x; y; z 2 J . From Lemma 2, we obtain rJ (x; y) rJ (y; z) = 1 rI (x; y) 1 rI (y; z) = 1 rI (x; y) rI (y; z) : Since I satis es the transitivity condition and 1 is monotonic, we have 1 rI (x; y) rI (y; z) 1 rI (x; z) = rJ (x; z); and thus rJ (x; y) rJ (y; z)
rJ (x; z).
Lemma 8. For all complex roles s and x; y 2 I , sJ (x; y) = 1(sI (x; y)).
Proof. If s is a role name, this follows directly from the de nition of J . If s = r for some r 2 NR, then sJ (x; y) = rJ (y; x) = 1(rI (y; x)) = 1(sI (x; y)). tu
The interpretation J preserves the compatibility of 1 with all the constructors of -SHOI 8.
Lemma 9. For every -SHOI 8-concept C and x 2 I , CJ (x) = 1 CI (x) .
Proof. We use induction over the structure of C. The claim holds trivially for C = ? and C = >. For C = A 2 NC it follows immediately from the de nition of J . It also holds for C = fag, a 2 NI, because fagI (x) can only take the values 0 or 1 for all x 2 I .
Assume now that the concepts D and E satisfy DJ (x) = 1(DI (x)) and EJ (x) = 1(EI (x)) for all x 2 I . In the case where C = D u E, Lemma 2 yields that for all x 2 I
CJ (x) = DJ (x)
EJ (x) = 1 DI (x)
1 EI (x) = 1 DI (x)
EI (x) = 1 CI (x) : Likewise, the compatibility of 1 with the t-conorm, the residuum, and the negation entails the result for the cases C = D t E, C = D ! E, and C = :D.
For C = 9s:D, where s is a complex role and D is a concept description satisfying DJ (x) = 1(DI (x)) for all x 2 I , we obtain 1 CI (x) = 1 (9s:D)I (x) = 1 sup sI (x; y)
y2 I = sup 1 sI (x; y) y2 I
DI (y) 1 DI (y) because 1 is compatible with the supremum and the t-norm. Lemma 8 yields sup 1(rI (x; y)) y2 I 1(DI (y)) = sup rJ (x; y) y2 I
DJ (y) = (9r:D)J (x): (4) Equations (3) and (4) prove 1(CI (x)) = CJ (x) for C = 9r:D.
We can use this lemma to show that the crisp interpretation J satis es all the axioms that are satis ed by I.
Lemma 10. Let O = (A; T ; R) be a -SHOI 8-ontology. If I is a model of O, then J is also a model of O. (3) tu Proof. We prove that J satis es all assertions, GCIs, and role inclusions from O. Let ha : C; `i, ` 2 (0; 1], be a concept assertion from A. Since the assertion is satis ed by I, CI(aI) ` > 0 holds. Lemma 9 yields CJ (aJ ) = 1 `. The same argument can be used for role assertions.
Let now hC v D; `i be a GCI in T and x 2 I. Since I satis es the GCI, we get CI(x) ) DI(x) ` > 0. By Lemmata 2 and 9, we obtain CJ (x) ) DJ (x) = 1(CI(x)) ) 1(DI(x)) = 1(CI(x) ) DI(x)) = 1 `; and thus J satis es the GCI hC v D; `i. A similar argument, using Lemma 8 instead of Lemma 9, shows that J satis es all role inclusions in R. tu
The previous results show that by applying 1 to the truth degrees we obtain a crisp model J from any fuzzy model I of a -SHOI 8-ontology O. Theorem 11. If is a t-norm without zero divisors, then -SHOI 8 has the crisp model property.
A trivial consequence of this theorem is that every consistent -SHOI 8ontology has also a witnessed model, since every crisp model is also crisp. Corollary 12. If is a t-norm without zero divisors, then -SHOI 8 has the witnessed model property.
In the next section we will use this result from Theorem 11 to show that -SHOI 8 ontology consistency can be decided in exponential time, by testing consistency of a (crisp) SHOI ontology. But rst, we show that value restrictions destroy the crisp model property, even if only crisp axioms are used. Example 13. Consider the -SHOI-ontology
O = fh> v ::A; 1i; ha : :8r:A; 1ig: The interpretation I = ( I; I) with I = N (the set of all natural numbers), aI = 1, AI(n) = 1=(n + 1), rI(1; n) = 1, and rI(n0; n) = 0 for all n; n0 2 N with n0 > 1 is a model of O, and hence O is consistent.
Let now J be a crisp interpretation satisfying the rst axiom in O. Then, AJ (x) = 1 for all x 2 J . This implies that (8r:A)J (aJ ) = inf rJ (aJ ; y) ) AJ (y)
y2 J = inf rJ (aJ ; y) ) 1
y2 J = 1: And thus, (:8r:A)J (aJ ) = 0, violating the second axiom. This means that O has no crisp model.
The example shows that no fuzzy DL with the constructor 8 and Godel negation2 has the crisp model property. A similar example in [14] demonstrates that no fuzzy DL with the constructors 9 and 8 and Godel negation has the witnessed model property.
Theorem 14. For any continuous t-norm and any fuzzy DL -L having the constructors >, :, and 8, -L does not have the crisp model property.
In particular, this means that -SHOI does not have the crisp model property and is strictly more expressive than -SHOI 8.
Corollary 15. If more expressive than is a t-norm without zero divisors, then -SHOI 8.
-SHOI is strictly 5
For a given -SHOI 8-ontology O, we de ne crisp(O) to be the crisp SHOIontology that is obtained from O by replacing all the truth values appearing in the axioms by 1. For example, for the ontology
O =
ha : C; 0:2i; h(a; b) : r; 0:8i; hC v D; 0:5i; hr v s; 0:1i we obtain crisp(O) =
ha : C; 1i; h(a; b) : r; 1i; hC v D; 1i; hr v s; 1i : Lemma 16. Let O be a -SHOI 8-ontology and I be a crisp interpretation. Then I is a model of O i it is a model of crisp(O).
Proof. Assume that crisp(O) has a model I. Let hC v D; `i, ` > 0, be an axiom from O. Since I is a model of crisp(O), it must satisfy hC v D; 1i; that is, CI (x) ) DI (x) 1 ` holds for all x 2 I . Thus I satis es hC v D; `i. The proof that I satis es assertions and role inclusions is analogous. Hence I is also a model of O.
For the other direction, assume that I satis es hC v D; `i with ` > 0. As I is a crisp interpretation it holds that CI (x) ) DI (x) 2 f0; 1g for all x 2 I . Together with CI (x) ) DI (x) ` > 0 we obtain CI (x) ) DI (x) = 1. Thus, I satis es the GCI hC v D; 1i. The same argument can be used for role inclusions and assertions. Thus, I is also a model of crisp(O). tu
In particular, a -SHOI 8-ontology O has a crisp model i crisp(O) has a crisp model. Together with Theorem 11, this shows that a -SHOI 8-ontology O is consistent i crisp(O) has a crisp model. Therefore, one can use any reasoning procedure for crisp SHOI to decide consistency of -SHOI 8-ontologies. Reasoning in crisp SHOI is known to be ExpTime-complete [15]. Recall that under crisp semantics, value restrictions can be expressed by negation and existential restrictions, and hence, crisp SHOI is equivalent to crisp SHOI 8. 2 Recall that a fuzzy DL has Godel negation i its semantics is based on a t-norm without zero divisors.
Theorem 18. The logics
nite model property. 6
-SHO 8 and -SI 8 and their sublogics have the Corollary 17. Deciding consistency in -SHOI 8 is ExpTime-complete.
Lemma 16 and Theorem 11 still hold when we restrict the semantics to the slightly less expressive logics -SHO 8, which does not allow for inverse roles, or -SI 8 which does not allow for nominals and role hierarchies. The crisp DLs SHO and SI are known to have the nite model property [16, 19], and -SI 8 and -SHO 8 inherit the nite model property from their crisp counterparts. In this paper we have described a family of expressive fuzzy DLs for which ontology consistency is decidable. More precisely, we have shown that if is a t-norm without zero divisors, consistency of -SHOI 8 ontologies is ExpTimecomplete, and hence as hard as consistency of (crisp) SHOI ontologies. Our construction shows that the fuzzy values appearing in -SHOI 8 ontologies are irrelevant for consistency: a -SHOI 8 ontology O has a (fuzzy) model i its crisp variant crisp(O), where the degrees of all the axioms in O are changed to 1, has a crisp model. This implies that -SHOI 8 has the crisp model property, and hence also the witnessed model property. If the constructor 8 is also allowed, hence obtaining the logic -SHOI, then these properties do not hold anymore.
For other reasoning problems such as entailment and subsumption it is unknown whether they are decidable in -SHOI 8. In [7] it is shown for -SHOI with witnessed models that subsumption and entailment, as well as computing the best subsumption and entailment degrees, cannot be reduced to crisp reasoning by simply mapping all nonzero truth degrees to 1. We conjecture that this is also the case in -SHOI 8.
It has recently been shown that if the t-norm has zero divisors, then consistency of crisp ontologies in the very inexpressive fuzzy DL -NE L w.r.t. witnessed models [10] and w.r.t. general models [9] is undecidable.3 Combining these results, we obtain a characterization of the decidability of consistency w.r.t. witnessed and general models for all fuzzy DLs between -NE L and -SHOI 8: it is decidable (in ExpTime) i has no zero divisors.
In future work, we plan to study reasoning problems in fuzzy DLs allowing for value restrictions without the restriction to witnessed models. In this direction it is worth looking at the decidability results for -ALC with product t-norm w.r.t. quasi-witnessed models [11].
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