Undecidability of Fuzzy Description Logics

Undecidability of Fuzzy Description Logics

Stefan Borgwardt

and Rafael Pe ˜naloza

Theoretical Computer Science, TU Dresden, Germany {stefborg,penaloza}@tcs.inf.tu-dresden.de

Abstract

Fuzzy description logics (DLs) have been investigated for over two decades, due to their capacity to formalize and rea- son with imprecise concepts. Very recently, it has been shown that for several fuzzy DLs, reasoning becomes undecidable.

Although the proofs of these results differ in the details of each specific logic considered, they are all based on the same basic idea.

In this paper, we formalize this idea and provide sufficient conditions for proving undecidability of a fuzzy DL. We demonstrate the effectiveness of our approach by strengthen- ing all previously-known undecidability results and provid- ing new ones. In particular, we show that undecidability may arise even if only crisp axioms are considered.

1 Introduction

Description logics (DLs) (Baader et al. 2003) are a family of logic-based knowledge representation formalisms, which can be used to represent the knowledge of an application domain in a formal way. They have been successfully used for the definition of medical ontologies, like SNOMEDCT¹ and GALEN,²but their main breakthrough arguably was the adoption of the DL-based language OWL (Horrocks, Patel- Schneider, and van Harmelen 2003) as the standard ontology language for the semantic web.

Fuzzy variants of description logics have been introduced to deal with applications where concepts cannot be speci- fied in a precise way. For example, in the medical domain a high body temperature is often a symptom for a disease.

When trying to represent this knowledge, it makes sense to seeHighas a fuzzy concept: there is no precise point where a temperature becomes high, but we know that 36^◦C belongs to this concept with a lower membership than 39^◦C. A more detailed description of the use of fuzzy semantics in medical applications can be found in (Molitor and Tresp 2000).

A great variety of fuzzy DLs can be found in the litera- ture (see (Lukasiewicz and Straccia 2008; Garc´ıa-Cerda˜na,

∗Partially supported by the German Research Foundation (DFG) in the Collaborative Research Center 912 “Highly Adaptive Energy-Efficient Computing”.

Copyright c2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.

1http://www.ihtsdo.org/snomed-ct/

2http://www.opengalen.org/

Armengol, and Esteva 2010) for a survey). In fact, fuzzy DLs have several degrees of freedom for defining their ex- pressiveness. In addition to the choice of concept construc- tors (such as conjunctionuor existential restriction∃), and the type of axioms allowed (like acyclic concept definitions or general concept inclusions), one must also decide how to interpret the different constructors, through a choice of functions over the domain of fuzzy values[0,1]. These func- tions are typically determined by a continuous t-norm (like G¨odel, Łukasiewicz, or product) that interprets conjunction;

there exist uncountably many such t-norms, each with dif- ferent properties. For example, under the product t-norm semantics, existential- (∃) and value-restrictions (∀) are not interdefinable, while under the Łukasiewicz t-norm they are.

Even after fixing the t-norm, one can choose whether to in- terpret negation by the involutive negation operator, or using the residual negation. An additional level of liberty comes from selecting the class of models over which reasoning is considered: either all models, or so-called witnessed models only (H´ajek 2005).

Most existing reasoning algorithms have been developed for the G¨odel semantics, either by a reduction to crisp rea- soning (Straccia 2001; Bobillo et al. 2009), or by a simple adaptation of the known algorithms for crisp DLs (Stoilos et al. 2005; 2006; Tresp and Molitor 1998). However, meth- ods based on other t-norms have also been explored (Bobillo and Straccia 2007; 2008; 2009; Straccia and Bobillo 2007;

Stoilos and Stamou 2009). Usually, these algorithms reason w.r.t. witnessed models.³

Very recently, it was shown that the tableaux-based algo- rithms for logics with semantics based on t-norms other than the G¨odel t-norm and allowing general concept inclusions were incorrect (Baader and Pe˜naloza 2011a; Bobillo, Bou, and Straccia 2011). This raised doubts about the decidabil- ity of these logics, and eventually led to a series of undecid- ability results for fuzzy DLs (Baader and Pe˜naloza 2011a;

2011b; 2011c; Cerami and Straccia 2011). All these pa- pers, except (Baader and Pe˜naloza 2011c), focus on one specific fuzzy DL; that is, undecidability is proven for a specific set of constructors, axioms, and underlying seman- tics. A small generalization is made in (Baader and Pe˜naloza

3In fact, witnessed models were introduced in (H´ajek 2005) to correct the algorithm from (Tresp and Molitor 1998).

2011c), where undecidability is shown for a whole family of t-norms—specifically, all t-norms “starting” with the prod- uct t-norm—and two variants of witnessed models.

Abstracting from the particularities of each logic, the proofs of undecidability appearing in (Baader and Pe˜naloza 2011a; 2011b; 2011c; Cerami and Straccia 2011) follow similar ideas. The goal of this paper is to formalize this idea and give a general description of a proof of undecidability, which can be instantiated to different fuzzy DLs. More pre- cisely, we describe a general proof method based on a re- duction from the Post Correspondence Problem and present sufficient conditions for the applicability of this method to a given fuzzy DL.

We demonstrate the effectiveness of our approach by pro- viding several new undecidability results for fuzzy DLs.

In particular, we improve the results from (Baader and Pe˜naloza 2011a; Cerami and Straccia 2011) by showing that a weaker DL suffices for obtaining undecidability, and the results from (Baader and Pe˜naloza 2011b; 2011c), by allow- ing a wider family of t-norms. We also prove the first unde- cidability results for reasoning w.r.t. general models. An in- teresting outcome of our study is that, for the product t-norm and any t-norm “starting” with the Łukasiewicz t-norm, un- decidability can arise even if only crisp axioms are allowed.

Due to a lack of space, some technical details have been left out of this paper. Full proofs and details can be found in the technical report (Borgwardt and Pe˜naloza 2011c).

2 T-norms and Fuzzy Logic

Fuzzy logics are formalisms introduced to express imprecise or vague information (H´ajek 2001). They extend classical logic by interpreting predicates as fuzzy sets over an inter- pretation domain. Given a non-empty domainD, afuzzy set is a functionF :D →[0,1]fromDinto the real unit inter- val[0,1], with the intuition that an elementδ∈ Dbelongs to F withdegreeF(δ). The interpretation of the logical con- structors is based on appropriate truth functions that gener- alize the properties of the connectives of classical logic to the interval[0,1]. The most prominent truth functions used in the fuzzy logic literature are based on t-norms (Klement, Mesiar, and Pap 2000).

At-normis an associative and commutative binary oper- ator⊗: [0,1]×[0,1]→[0,1]that has1as its unit element, and is monotonic, i.e., for everyx, y, z ∈ [0,1], ifx ≤ y, thenx⊗z≤y⊗z. If⊗is a continuous t-norm, then there exists a unique binary operator⇒, called theresiduum, that satisfiesz ≤x⇒yiffx⊗z ≤yfor everyx, y, z∈[0,1].

For every continuous t-norm⊗andx, y ∈ [0,1], we have (i)x⇒y= 1iffx≤yand (ii)1⇒y=y(H´ajek 2001).

Three important continuous t-norms are the G¨odel, prod- uct and Łukasiewicz t-norms, shown in Table 1.

We say that a t-norm⊗(a, b)-containsthe t-norm⊗⁰, for 0≤a < b≤1, if for everyx, y∈[0,1]it holds that

(a+ (b−a)x)⊗(a+ (b−a)y) =a+ (b−a)(x⊗⁰y).

In this case, if⇒and⇒⁰ denote the residua of⊗and⊗⁰, respectively, then it also holds that for everyx > y,

(a+ (b−a)x)⇒(a+ (b−a)y) =a+ (b−a)(x⇒⁰ y).

Name t-norm (x⊗y) Residuum (x⇒y) G¨odel min{x, y}

1 ifx≤y y otherwise product x·y

1 ifx≤y y/x otherwise Łukasiewicz max{x+y−1,0} min{1−x+y,1}

Table 1: Three t-norms and their residua

Moreover, for everyx∈[a, b]andy /∈ [a, b], we have that x⊗y = min{x, y}. Intuitively,⊗behaves like a scaled- down version of⊗⁰ in the interval [a, b], and as the G¨odel t-norm if exactly one of the arguments belongs to[a, b].

We say that a t-normcontains⊗⁰if it(a, b)-contains⊗⁰ for some0 ≤ a < b≤ 1. A consequence of the Mostert- Shields Theorem (Mostert and Shields 1957) is that every continuous t-norm⊗that is not the G¨odel t-norm must con- tain the product or the Łukasiewicz t-norm. Notice that⊗ may contain both t-norms; in fact, it may even contain in- finitely many instances of these t-norms over disjoint inter- vals. For example, the t-norm defined by

x⊗y=







2xy ifx, y∈[0,0.5]

max{x+y−1,0.5} ifx, y∈[0.5,1]

min(x, y) otherwise,

(0,0.5)-contains the product t-norm, and(0.5,1)-contains the Łukasiewicz t-norm.

We denote the product and Łukasiewicz t-norms byΠand Ł, respectively. In general, a continuous t-norm that is not the G¨odel t-norm may contain several instances of the prod- uct and Łukasiewicz t-norms. In the following, we always choose and fix a representative, and use the notationΠ^(a,b) to express that the t-norm(a, b)-contains the product t-norm, and similarly for Ł^(a,b). Since our constructions differ ac- cording to the t-norm, it is important to emphasize that the representative is fixed throughout the whole construction.

Fuzzy logics are sometimes extended with the involutive negation ∼x := 1−x(Zadeh 1965; Esteva et al. 2000).

If ⊗is the Łukasiewicz t-norm, then this operator can be expressed through the equality∼x=x⇒0. However, for any other continuous t-norm∼is not expressible in terms of

⊗and its residuum⇒.

3 Fuzzy Description Logics

Just as classical description logics, fuzzy DLs are based on concepts, which are built from the mutually disjoint sets N_C,N_RandN_Iofconcept names,role names, andindividual names, respectively, using different constructors. A wide va- riety of constructors can be found in the literature. For this paper, we consider only the constructors >(top), ⊥(bot- tom),u(conjunction),→(implication),¬(involutive nega- tion),(residual negation),∃(existential restriction), and

∀(value restriction). When restricted to classical semantics, this set of constructors corresponds to the crisp DLALC.

Definition 1 (concepts). (Complex) concepts are built in- ductively fromNCandNRas follows:

Name > ⊥ u → ¬ ∃ ∀

EL √ √ √

ELC √ √ √ √

NEL √ √ √ √

AL √ √ √ √

ALC √ √ √ √ √

IAL √ √ √ √ √ √

Table 2: Some relevant DLs and the constructors they allow.

• every concept nameA∈NCis a concept

• if C, D are concepts and r ∈ NR, then>, ⊥, C uD, C→D,¬C,C,∃r.C, and∀r.Care also concepts.

We will use the expressionCⁿ to denote then-ary con- junction of a conceptCwith itself; formally,C⁰ :=>and Cⁿ⁺¹:=CuCⁿfor everyn≥0.

Different DLs are determined by the choice of construc- tors used. The DLELallows only for the constructors>,u, and∃. AL additionally allows value restrictions. Follow- ing the notation from (Cerami, Garc´ıa-Cerda˜na, and Esteva 2010), the lettersCandIexpress that the involutive negation or implication and bottom constructors are allowed, respec- tively.NELextendsELwith the residual negation construc- tor. Table 2 summarizes this nomenclature.

The knowledge of a domain is represented using a set of axioms that express the relationships between individuals, roles, and concepts.

Definition 2(axioms). Anaxiomis one of the following:

• Ageneral concept inclusion (GCI)is of the formCvD for conceptsCandD.⁴

• Anassertionis of the formhe:C . piorh(d, e) :r . pi for a conceptC,r∈N_R,d, e∈N_I, and.∈ {≥,=}. This axiom is called acrisp assertionifp= 1, aninequality assertionif.is≥and anequality assertionif.is=.

• Acrisp role axiomis of the formcrisp(r)forr∈NR. Anontologyis a finite set of axioms. It is called aclassical ontologyif it contains only GCIs and crisp assertions.

As with the choice of the constructors, the axioms influ- ence the expressivity of the logic. Our logics always al- low at least classical ontologies. Given a DL L, we will use the subscripts≥,=, andcto denote that also inequal- ity assertions, equality assertions, and crisp role axioms are allowed, respectively. For instance,EL_≥,cdenotes the logic ELwhere ontologies can additionally contain inequality as- sertions and crisp role axioms, but not equality assertions.

Compared to classical DLs, fuzzy DLs have an additional degree of freedom in the selection of their semantics since the interpretation of the constructors depends on the t-norm chosen. Given a DLLand a continuous t-norm⊗, we obtain the fuzzy DL⊗-Lwith the following semantics.

4One can also consider fuzzy GCIshC v D ≥ pi(see, e.g.

(Straccia 1998)). Since our proofs of undecidability do not require these more general axioms, we do not consider them here.

Definition 3(semantics). Aninterpretation I = (D^I,·^I) consists of a non-empty domainD^I and an interpretation function·^Ithat assigns to everye∈N_Ian elemente^I∈ D^I, to everyA∈NCa fuzzy setA^I:D^I→[0,1], and to every r∈N_Ra fuzzy binary relationr^I:D^I× D^I→[0,1].

This function is extended to concepts as follows:

• >^I(x) = 1, ⊥^I(x) = 0,

• (CuD)^I(x) =C^I(x)⊗D^I(x),

• (C→D)^I(x) =C^I(x)⇒D^I(x),

• (¬C)^I(x) = 1−C^I(x), (C)^I(x) =C^I(x)⇒0,

• (∃r.C)^I(x) = sup_y∈DI(r^I(x, y)⊗C^I(y)),

• (∀r.C)^I(x) = inf_y∈DI(r^I(x, y)⇒C^I(y)).

We say that an interpretationI⁰ is anextensionofI if it has the same domain asI, agrees withI on the interpreta- tion ofNC, NR, andNI and additionally defines values for some new concept names not appearing inNC.

The reasoning problem that we consider in this paper is ontology consistency; that is, deciding whether there is an interpretation satisfying all the axioms in an ontology.

Definition 4(consistency). An interpretationI = (D^I,·^I) satisfiesthe GCICvDifC^I(x)≤D^I(x)for allx∈ D^I. Itsatisfiesthe assertionhe:C . pi(resp.,h(d, e) :r . pi) if C^I(e^I). p(resp.,r^I(d^I, e^I). p). Itsatisfiesthe crisp role axiomcrisp(r)ifr^I(x, y)∈ {0,1}for allx, y∈ D^I. It is a modelof an ontologyOif it satisfies all the axioms inO.

An ontology isconsistentif it has a model.

Notice that the GCIsCvDandD vCare satisfied iff C^I(x) =D^I(x)for everyx∈ D^I. It thus makes sense to abbreviate them through the expressionC≡D.

In fuzzy DLs, reasoning is often restricted to a spe- cial kind of models, called witnessed models (H´ajek 2005;

Bobillo and Straccia 2009). An interpretation I is called witnessedif for every conceptC,r∈N_R, andx∈ D^Ithere existy, y⁰ ∈ D^Isuch that

• (∃r.C)^I(x) =r^I(x, y)⊗C^I(y), and

• (∀r.C)^I(x) =r^I(x, y⁰)⇒C^I(y⁰).

This means that the suprema and infima in the semantics of existential and value restrictions are actually maxima and minima, respectively. Restricting to this kind of models changes the reasoning problem since there exist consistent ontologies that have no witnessed models (H´ajek 2005).

We also consider a weaker notion of witnessing, where witnesses are required only for the existential restrictions

∃r.>evaluated to1. Formally,Iis called>-witnessedif for everyr∈NRandx∈ D^I such that(∃r.>)^I(x) = 1, there is ay∈ D^Iwithr^I(x, y) = 1. Obviously, every witnessed interpretation is also >-witnessed. We use the subscripts wand>to denote that reasoning is restricted to witnessed and>-witnessed models, respectively. Thus,⊗w-ELC rep- resents the logic⊗-ELCrestricted to witnessed models.

In general, a fuzzy DL is determined by three parameters:

the classLof constructors and axioms it allows, the t-norm

⊗that describes its semantics, and the class of modelsxover which reasoning is considered. In the following, we will use the expression⊗x-Lto denote an arbitrary fuzzy DL.

Before we present our general framework for proving un- decidability, it is worth to relate the fuzzy DLs introduced according to their expressive power. For every choice of constructorsLand t-norm⊗, the inequality concept asser- tion he : C ≥ qi can be expressed in ⊗-L₌ using the axioms he : A = qi, A v C, where A is a new con- cept name. For every t-norm⊗,⊗-NELis a sublogic of

⊗-IELsince(C)^I(x) = (C→ ⊥)^I(x). It also holds that Ł-ELC,Ł-NEL,Ł-IEL,Ł-ALC, andŁ-IALare all equiv- alent (H´ajek 2001): the residual and involutive negation are equivalent and can express implication together with conjunction (C → D)^I = ¬(Cu ¬D)^I, and the dual- ity between value and existential restrictions (∀r.C)^I =

¬(∃r.¬C)^I holds. However, in general these logics have different expressive power; if any t-norm different from Łukasiewicz is used, then(¬∃r.¬C)^I6= (∀r.C)^I.

4 Showing Undecidability

We now describe a general approach for proving that the consistency problem for a fuzzy DL⊗x-L is undecidable.

This approach is based on a reduction from the undecidable Post correspondence problem (Post 1946).

Definition 5(PCP). LetP ={(v1, w₁), . . . ,(v_n, w_n)}be a finite set of pairs of words over the alphabetΣ ={1, . . . , s}

withs > 1. ThePost correspondence problem (PCP)asks whether there is a finite sequencei1. . . ik ∈ {1, . . . , n}⁺ such thatv_i1. . . v_ik =w_i1. . . w_ik. If this sequence exists, it is called asolutionforP.

We defineN :={1, . . . , n}and forν =i₁. . . i_k ∈ N⁺, we use the notationvν=vi₁. . . vi_kandwν =wi₁. . . wi_k.

LetP ={(v1, w₁), . . . ,(v_n, w_n)}be an instance of the PCP. We can representP by itssearch tree, which has one node for everyν ∈ N^∗, whereεrepresents the root, andνi is thei-th successor ofν,i∈ N. Each nodeν in this tree is labelled with the wordsvν, wν ∈Σ^∗.

We reduce the PCP to the consistency problem of⊗x-Lin two steps. We first construct an ontologyO_P that describes the search tree of P using two designated concept names V, W. More precisely, we will enforce that for every model I of OP and everyν ∈ N^∗, there is an x_ν ∈ D^I such that V^I(xν) = enc(vν)andW^I(xν) = enc(wν), where enc: Σ^∗→[0,1]is an injective function that encodes words overΣinto the interval[0,1](see Section 4.1).

Once we have encoded the wordsv_νandw_νusingV and W, we add axioms that restrict every node to satisfy that V^I(x_ν)6=W^I(x_ν). This will ensure thatP has a solution iff the ontology is inconsistent (see Section 4.2).

Recall that the alphabet Σ consists of the first s posi- tive integers. We can thus view every word in Σ^∗ as a natural number represented in base s+ 1. On the other hand, every natural number nhas a unique representation in bases+ 1, which can be seen as a word over the alpha- betΣ₀ := Σ∪ {0} = {0, . . . , s}. This is not a bijection since, e.g. the words001202and1202represent the same number. However, it is a bijection between the setΣΣ^∗₀and the positive natural numbers. We will in the following inter- pret the empty wordεas0, thereby extending this bijection to{ε} ∪ΣΣ^∗₀and all non-negative integers.

In the following constructions and proofs, we will view elements ofΣ^∗₀both as words and as natural numbers in base s+ 1. To avoid confusion, we will use the notationuto express thatuis seen as a word. Thus, for instance, ifs= 3, then3·2²= 30(in base4), but3·2²= 322. Furthermore, 000is a word of length3, whereas000is simply the number 0. For a wordu=α₁· · ·α_mwithα_i ∈Σ₀,1≤i≤m, we denote as←−u the wordαm· · ·α1∈Σ^∗₀.

Recall that for everyp, q∈[0,1],p=qiffp⇒q= 1and q ⇒p= 1. Thus,P has no solution iff for everyν ∈ N⁺ eitherenc(vν)⇒enc(wν)<1orenc(wν)⇒enc(vν)<1 holds. Instead of performing this test directly, we will con- struct a word whose encoding bounds these residua. Clearly, the precise word and encoding must depend on the t-norm used. The needed properties are formalized by the follow- ing definition.

Definition 6(valid encoding function). enc: Σ^∗₀→[0,1]is avalid encoding functionfor⊗if it is injective on{ε}∪ΣΣ^∗₀ and there exist two wordsuε, u+ ∈Σ^∗₀ such that for every ν ∈ N⁺it holds thatv_ν6=w_νiff either

enc(vν)⇒enc(wν)≤enc(uε·u+|ν|) or enc(w_ν)⇒enc(v_ν)≤enc(u_ε·u₊^|ν|).

For every continuous t-norm⊗except the G¨odel t-norm, we give a valid encoding function, which depends on whether⊗contains the product or the Łukasiewicz t-norm.

If ⊗ (a, b)-contains the product t-norm, then we define enc(u) =a+(b−a)2^−u∈(a, b]for everyu∈Σ^∗₀. If⊗is of the formŁ^(a,b), thenenc(u) =a+(b−a)(1−0.←−u)∈(a, b].

Lemma 7. The functionsencdescribed above are valid en- coding functions.

Proof. [Π^(a,b)] Letv 6=wand assume w.l.o.g. thatv < w.

Thenv+ 1≤wand hence2^−w≤2^−(v+1)≤2^−v/2. This implies that

enc(v)⇒enc(w) = a+ (b−a)2^−w/2^−v

≤ a+ (b−a)/2 =enc(1)<1.

Conversely, ifv = w, then(enc(v) ⇒ enc(w)) = 1and (enc(w)⇒enc(v)) = 1.Thus,uε= 1andu+ =εsatisfy the condition of Definition 6.

[Ł^(a,b)] Letk = max{|v_i|,|w_i| | i ∈ N }be the maximal length of a word inP. Then, for everyν∈ N⁺,|vν| ≤ |ν|k and|wν| ≤ |ν|k. Ifvν6=wν, these words must differ in one of the first|ν|kdigits. Thus, either

enc(vν)⇒enc(wν)

=a+ (b−a) min{1,1 + 0.←v−ν−0.←w−ν}

= min{b, a+ (b−a)(1 + 0.←v−ν−0.←w−ν)}

≤a+ (b−a)(1−(s+ 1)^−|ν|k)

=enc((s+ 1)^|ν|k)<1

orenc(wν) ⇒ enc(vν)≤ enc((s+ 1)^|ν|k).⁵ Ifvν = wν, then both residua are1. Thus,uε= 1andu+= 0^kgive the desired result.

5We have(s+ 1)^|ν|k= 1·0^|ν|kand(s+ 1)^−|ν|k= 0.0^|ν|k·1.

Variants of the above encoding functions and words uε, u+ have been used before to show undecidability of fuzzy description logics based on the product (Baader and Pe˜naloza 2011c) and Łukasiewicz (Cerami and Straccia 2011) t-norms. For the rest of this paper, encrepresents a valid encoding function for⊗.

4.1 Encoding the Search Tree

As a first step for our reduction to the consistency problem in fuzzy DLs, we simulate the search tree for the instanceP using the concept namesV, W. Since we will later use this construction to decide whether a solution exists, we desig- nate the concept nameM to represent the bounduε·u+|ν|

from Definition 6. We useVi, Wi, M+to encode the words v_i, w_i, u₊, and the role namesr_ito distinguish the succes- sors in the search tree. We start by constructing the inter- pretationIP = (N^∗,·^IP), wheree^I₀^P = εand for every ν∈ N^∗andi∈ N,

• V^IP(ν) =enc(vν), W^IP(ν) =enc(wν),

• V_i^IP(ν) =enc(v_i), W_i^IP(ν) =enc(w_i),

• M^IP(ν) =enc(uε·u+|ν|), M₊^IP(ν) =enc(u+),

• r^I_i^P(ν, νi) = 1andr^I_i^P(ν, ν⁰) = 0ifν⁰ 6=νi.

Since every element ofN^∗has exactly oneri-successor with degree greater than0,I_P is a (>-)witnessed interpretation.

Our aim is to produce an ontology that can only be satis- fied by interpretations that “include” the interpretationI_P, as described by the following property.

Canonical model property (P₄):

⊗x-Lhas thecanonical model propertyif there is an on- tologyO_P such that for every modelI ofO_P there is a mappingg:D^IP → D^Iwith

A^IP(ν) =A^I(g(ν))andr_i^I(g(ν), g(νi)) = 1 for everyA∈ {V, W, M, M+} ∪Sn

j=1{Vj, Wj}, ν ∈ N^∗ andi∈ N.

Rather than trying to prove this property directly, we provide several simpler properties that together imply the canonical model property. We will often motivate the con- structions using onlyV andv_ν; however, all the arguments apply analogously toW, w_νandM, u_ε·u₊^|ν|.

To ensure that the canonical model property holds, we en- force the encoding of the search tree in an inductive way.

First, every model I must satisfy that A^IP(ε) = A^I(e^I₀) for every relevant concept name. This makes sure that the root εof the search tree is properly represented at the in- dividualg(ε) := e^I₀. Let now g(ν) be a node where all relevant concept names are interpreted as inI_P, andi∈ N. We need to ensure that there is a nodeg(νi)that also sat- isfies the property, andr^I_i(g(ν), g(νi)) = 1. We do this in three steps: first, we force the existence of an individ- ual y withr^I_i(g(ν), y) = 1 and set g(νi) := y. Then, we compute the valueenc(vνvi)fromV^I(g(ν)) =enc(vν) andV_i^I(g(ν)) = enc(v_i). Finally, we “transfer” this value to the previously created successor; that is, we ensure that

V^I(g(νi)) = enc(vνvi). The value V_j^I(g(ν)) for every j∈ N is also transferred toV_j^I(g(νi)).

Since the values ofVi,Wi, andM+are constant through- out the search tree, we additionally present an alternative approach that simply fixes these values for allx∈ D^I. This has the advantage that the initialization only has to consider the valuesenc(vε) =enc(wε) =enc(ε)andenc(uε).

Each step of the construction described above will be en- sured by a property of the underlying logic. These prop- erties, which will be used to produce the ontologyO_P, are described next. For each of the properties, we give examples of fuzzy DLs satisfying it. It is important to notice that the interpretationIP can be extended to a witnessed model of each of the ontologies that we introduce in the following.

The first property ensures the existence of anr-successor of degree1for every element of the domain.

Successor property (P_→):

⊗x-Lhas thesuccessor propertyif for everyr∈NRthere is an ontologyO_∃rsuch that for everyx-modelI ofO_∃r andx∈ D^Ithere is ay∈ D^Iwithr^I(x, y) = 1.

Lemma 8. For every t-norm⊗,⊗_>-ELand⊗-ELcsatisfy P_→.

Proof. [⊗_>-EL] LetO_∃r := {> v ∃r.>}. Any modelI of this ontology satisfies(∃r.>)^I(x) = 1for everyx∈ D^I. Since reasoning is restricted to >-witnessed models, there must be ay∈ D^Iwithr^I(x, y) = 1.

[⊗-ELc] We defineO_∃r := {> v ∃r.>,crisp(r)}. For any model I of this ontology and x ∈ D^I, we have (∃r.>)^I(x) = 1. If r^I(x, y) = 0 for all y ∈ D^I, then (∃r.>)^I(x) = sup_y∈DIr^I(x, y)⊗ >^I(y) = 06= 1. Since ris crisp, there must be ay∈ D^Iwithr^I(x, y) = 1.

Given this property, we create ri-successors for every nodeν ∈ N^∗with the ontology

OP,→:= [

i∈N

O∃r_i.

The concatenation property is satisfied if it is possible to compute the encoding of the concatenationu⁰ufrom the en- codings of two wordsuandu⁰, whereuis constant.

Concatenation property (P_◦):

⊗x-Lhas theconcatenation propertyif for allu∈Σ^∗₀and conceptsC,Cu, there is an ontologyO_C◦uand a concept nameD_C◦usuch that for everyx-modelI ofO_C◦u and x ∈ D^I, ifC_u^I(x) = enc(u)andC^I(x) = enc(u⁰)for u⁰∈ {ε} ∪ΣΣ^∗₀, thenD_C◦u^I (x) =enc(u⁰u).

Lemma 9. For any continuous t-norm⊗different from the G¨odel t-norm,⊗-ELsatisfiesP_◦.

Proof. The t-norm⊗must contain either the product or the Łukasiewicz t-norm. We divide the proof depending on the representative chosen for the encoding function.

[Π^(a,b)-EL] Since every word inΣ^∗₀is seen as a natural num- ber in bases+ 1, for everyu∈Σ^∗₀andu⁰ ∈ {ε} ∪ΣΣ^∗₀, we

haveu⁰(s+ 1)^|u|+u=u⁰u. We define the ontology O_C◦u:={D_C◦u≡C^(s+1)|u|uCu}.

Recall that for every interpretation I and x ∈ D^I, if C^I(x) =a+(b−a)p, then(C^m)^I(x) =a+(b−a)p^m. Let nowIbe a model ofO_C◦u,u⁰ ∈ {ε} ∪ΣΣ^∗₀, andx∈ D^I withC_u^I(x) =a+(b−a)2^−uandC^I(x) =a+(b−a)2^−u0. SinceIsatisfiesO_C◦u, we have

D^I_C◦u(x) =a+ (b−a)2⁻(^u0(s+1)|u|+u) =enc(u⁰u).

[Ł^(a,b)-EL] We define the ontology

O_C◦u:={C^0(s+1)|u| ≡C, D_C◦u≡C⁰uC_u}.

LetI be a model ofOC◦u,x∈ D^I, andC_u^I(x) = enc(u) andC^I(x) =enc(u⁰)for someu⁰ ∈ {ε} ∪ΣΣ^∗₀. From the first axiom it follows that

(C^0(s+1)|u|)^I(x) =C^I(x) =a+ (b−a)(1−0.←−

u⁰)∈(a, b].

Since⊗is monotone and(a, b)-contains the Łukasiewicz t- norm, it follows that (i)C^0I(x)> aand (ii)C^0I(x)≥biff C^I(x) = b, i.e. ifu⁰is the empty word. Recall that, when- everC^0I(x)∈[a, b]for some interpretationIandx∈ D^I, then((C⁰)^m)^I(x) = max{a, m C^0I(x)−b

+b} holds.

IfC^I(x)< b, thenC^0I(x)∈(a, b)and a+(b−a)(1−0.←−

u⁰) = max{a,(s+1)^|u| C^0I(x)−b +b}, and thusC^0I(x) =a+ (b−a)(1−(s+ 1)^−|u|0.←−

u⁰)and D^I_C◦u(x) =a+ (b−a)(1−0.←−u −(s+ 1)^−|u|0.←−

u⁰)

=enc(u⁰u).

Otherwise, u⁰ is the empty word and C^0I(x) ≥ b. Since C_u^I(x) ≤b, we know thatC^0I(x)⊗C_u^I(x) = C_u^I(x)and thusD^I_C◦u(x) =C_u^I(x) =enc(u) =enc(εu).

The goal of this property is to ensure that at every node with V^I(x) = enc(u) for some u ∈ {ε} ∪ ΣΣ^∗₀ and C_v^I_i(x) =vi, we haveD_V^I_◦v

i(x) =enc(uvi), and similarly forW, w_iandM, u₊. Thus, we define the ontology

O_P,◦:=

n

[

i=1

OV◦vi∪ OW◦wi∪ O_M◦u+ . By construction, the values of V^I(x) andW^I(x) should always be encodings of wordsvν, wν ∈ Σ^∗, whileM^I(x) might encode words inΣ^∗₀. To simplify the notation, we use the concept namesV_i, W_i, M₊instead ofC_vi, C_wi, C_u+in this ontology.

Once we have computed the concatenation of two words, we need to transfer it to the successors of the node, as en- sured by the following property.

Transfer property (P ):

⊗x-L has thetransfer property if for all concepts C, D and role namesrthere is an ontology O_C^r_D such that for everyx-model I ofO_Cr

D and everyx, y ∈ D^I, if r^I(x, y) = 1 andC^I(x) = enc(u)for some u ∈ Σ^∗₀, thenC^I(x) =D^I(y).

Lemma 10. For every t-norm⊗,⊗-ALand⊗-ELCsatisfy P .

Proof. Notice first that for any modelIof the⊗-ELaxiom

∃r.DvCand allx, y∈ D^Iwithr^I(x, y) = 1it holds that D^I(y) =r^I(x, y)⊗D^I(y)≤(∃r.D)^I(x)≤C^I(x).

We now add a restriction ensuring that alsoD^I(y)≥C^I(x) holds, depending on the expressivity of the logic used.

[⊗-AL] The axiomC v ∀r.Drestricts every modelI to satisfy that ifr^I(x, y) = 1, then

C^I(x)≤(∀r.D)^I(x)≤r^I(x, y)⇒D^I(y) =D^I(y).

Thus, the ontology O_Cr

D := {C v ∀r.D,∃r.D v C}

satisfies the condition.

[⊗-ELC] For a modelIof∃r.¬Dv ¬Candr^I(x, y) = 1, 1−D^I(y) =r^I(x, y)⊗(1−D^I(y))

≤(∃r.¬D)^I(x)≤1−C^I(x).

Thus,O_C^r_D := {∃r.¬D v ¬C,∃r.D vC}satisfies the required condition.

To ensure that the values of enc(u_ε ·u₊^|ν|), enc(u₊), enc(v_νi), andenc(v_j)for everyj ∈ N are transfered from xto the successoryifor everyi∈ N, we use the ontology

O_P, := [

i∈N

OD_M◦u+^riM∪ O

M₊^riM₊

∪ [

i∈N

OD_V◦viriV ∪ O

DW◦wiriW

∪ [

i,j∈N

OV_j^riV_j ∪ O

W_j^riW_j.

The initialization property ensures that the root of the search tree can be encoded.

Initialization property (Pini):

⊗_x-L has theinitialization propertyif for every concept C, individual namee, andu ∈ Σ^∗₀ there is an ontology O_C(e)=u such thatC^I(e^I) = enc(u)for everyx-model IofOC(e)=u.

Lemma 11. For every t-norm⊗,⊗-EL=and⊗-ELC≥sat- isfyPini.

Proof. [⊗-EL=] If the equality assertionhe:C =enc(u)i is satisfied byI, thenC^I(e^I) =enc(u).

[⊗-ELC≥] We use the two axioms he : C ≥ enc(u)i and he : ¬C ≥ 1−enc(u)i. The first axiom expresses that C^I(e^I) ≥ enc(u), while the second requires that 1−C^I(e^I)≥1−enc(u), i.e.C^I(e^I)≤enc(u), holds.

To initialize the search tree, we need to fix an individ- ual namee0 at whichV andW are both interpreted as the encoding of the empty word andM as the encoding ofu_ε. Moreover, we need thatM+encodesu+and everyViand

Wiencodes the wordvi, wi, respectively. We thus define the ontology

OP,ini:=O_M(e0)=uε∪ O_M+(e0)=u+∪ O_V(e0)=ε

∪ O_W(e0)=ε∪

n

[

i=1

O_Vi(e0)=vi∪ O_Wi(e0)=wi .

In some cases, it suffices to consider a weaker version of Pini, where only the two wordsεanduεneed to be initial- ized.

Weak initialization property (P^w_ini):

⊗x-Lhas theweak initialization propertyif for every con- ceptC, individual namee, andu ∈ {ε, uε} there is an ontologyO_C(e)=usuch thatC^I(e^I) =enc(u)holds for everyx-modelIofO_C(e)=u.

Lemma 12. The logicΠ-ELCsatisfiesP^w_ini.

Proof. We haveenc(ε) = 1and hence the crisp assertion he:C≥1iyields the desired condition forε. Foruε= 1, we use the axiom C ≡ ¬C, which in particular restricts C^I(e^I) = 1−C^I(e^I)to be0.5 =enc(1).

For any logic satisfyingP^w_ini, any model of the ontology O_P^w_,ini:=O_V(e0)=ε∪ O_W(e0)=ε∪ O_M(e0)=uε, must contain an individual encoding the values ofV,Wand M at the root of the search tree ofP.

Note that the construction for Π-ELC works since we know thatu+ =ε, i.e. the value ofM is constant. In gen- eral, a constant interpretation of a concept name can be en- forced through the following property.

Constant property (P₌):

⊗x-Lhas theconstant propertyif for every concept name Cand wordu∈Σ^∗₀there is an ontologyOC=usuch that for everyx-model ofOC=uand everyx∈ D^I we have C^I(x) =enc(u).

Lemma 13. The logicΠ-ELCsatisfiesP=.

Proof. ConsiderOC=u := {H ≡ ¬H, C ≡ H^u}. From the first axiom it follows that for every modelI of this on- tology andx ∈ D^I, we have H^I(x) = 1−H^I(x), and thusH^I(x) = 0.5 = 2⁻¹. Thus, from the second axiom, C^I(x) = (2⁻¹)^u= 2^−u=enc(u).

The constant values ofVi,Wi, andM+are ensured by the ontology

O_P,=:=OM₊=u₊∪

n

[

i=1

OV_i=v_i∪ OW_i=w_i.

As described before, different combinations of these properties yield the canonical model property.

Theorem 14. If a logic ⊗x-L satisfies the properties P_◦, P_ini,P_→, andP , then it also satisfiesP₄.

Proof. We show thatOP :=OP,ini∪ OP,◦∪ OP,→∪ OP,

satisfies the conditions from the definition of P₄. For a modelI ofOP, we construct the functiong : N^∗ → D^I inductively as follows.

We first setg(ε) :=e^I₀. SinceIis a model ofO_P,ini, we have that V^I(g(ε)) = V^I(e^I₀) = enc(ε) = V^IP(ε), and likewise forW,M,M+,Vi, andWifor alli∈ N.

Let now ν be such that g(ν) has already been defined andV^I(g(ν)) = enc(vν),V_i^I(g(ν)) = enc(vi). I being a model ofO_P,◦ensures thatD^I_V◦v

i = enc(vνi). SinceI satisfiesOP,→, for eachi∈ Nthere must be ayi∈ D^Iwith r^I_i(g(ν), yi) = 1. Define nowg(νi) :=yi. O_P, ensures that V^I(g(νi)) = D_V^I_◦v

i(g(ν)) = enc(v_νi) = V^IP(νi) andV_i^I(g(νi)) = enc(v_i) = V_i^IP(νi)for alli ∈ N, and analogously forW,WiandM,M+.

From this theorem and Lemmata 8 to 11, we obtain the following result.

Corollary 15. If⊗is a continuous t-norm, but not the G¨odel t-norm, then the logics ⊗>-AL=, ⊗-AL=,c, ⊗>-ELC≥, and⊗-ELC_≥,csatisfyP₄.

Alternatively, we can substitute P_ini with the properties P^w_ini andP= and still obtain the canonical model property.

The proof of this is analogous to that of Theorem 14, using the ontologyOP :=O_P,ini^w ∪ OP,=∪ OP,◦∪ OP,→∪ OP, . Theorem 16. If⊗x-Lsatisfies the propertiesP_◦,P^w_ini,P=, P_→, andP , then it also satisfiesP₄.

With the help of Lemmata 8 to 13, we now obtain the following result.

Corollary 17. The logicsΠ>-ELCandΠ-ELCcsatisfyP4. It is a simple task to verify that the interpretationI_P can be extended to a model of the ontologyOP in all the cases described. We only need to use a unique new concept name for every auxiliary concept name appearing in the different ontologies. In fact, the values of these auxiliary concept names at each node ν are uniquely determined by the val- ues of the concept namesV, W, Vi, Wi, M, M+inν. More- over, since everyνhas exactly oner_i-successor with degree greater than 0 for everyi ∈ N, it follows that IP can be extended to a witnessed model ofO_P.

We now use the property P4to prove undecidability of a fuzzy DL. The idea is to extendO_P so that every model I must satisfyV^I(g(ν))6=W^I(g(ν))for everyν ∈ N⁺, thus obtaining an ontology that is consistent if and only ifP has no solution.

4.2 Finding a Solution

For the rest of this section, we assume that ⊗x-L satisfies P4and for any given modelIofOP,gdenotes the function mapping the nodes ofI_Pto nodes inIgiven by the property.

Furthermore, we assume thatIP can be extended to anx- model ofO_P. These assumptions have been shown to hold for a variety of fuzzy DLs in the previous section.

The key to showing undecidability of⊗x-Lis to be able to express the restriction thatV andW encode different words at every non-root node ν ∈ N⁺ of the search tree. Since

encis a valid encoding function, andM encodes the word uε·u+|ν| at everyν ∈ N^∗, it suffices to check whether, for all ν ∈ N⁺, either(V → W)^IP(ν) ≤ M^IP(ν)or (W → V)^IP(ν) ≤ M^IP(ν)(recall Definition 6). This can easily be done in every logic that has the implication constructor→. However, this constructor is not necessary in general to show undecidability.

Solution property (P₆₌):

A logic⊗x-LsatisfyingP₄has thesolution propertyif there is an ontologyOV6=W such that

1. For everyx-model I of O_P ∪ OV6=W andν ∈ N⁺, either

V^I(g(ν))⇒W^I(g(ν))≤M^I(g(ν)) or W^I(g(ν))⇒V^I(g(ν))≤M^I(g(ν)).

2. If for everyν∈ N⁺we have either

V^IP(ν)⇒W^IP(ν)≤M^IP(ν) or W^IP(ν)⇒V^IP(ν)≤M^IP(ν),

thenI_P can be extended to a model ofO_P∪ OV6=W. Lemma 18. Let⊗be a continuous t-norm⊗different from the G¨odel t-norm and L contain either IAL or ELC. If

⊗x-LsatisfiesP₄andI_P can be extended to anx-model of OP, then⊗x-LsatisfiesP₆₌.

Proof. We divide the proof according to the underlying DL.

[IAL] We defineOV6=W as

{> v ∀r_i.(((V →W)u(W →V))→M)|i∈ N }.

This ontology is satisfied byIiff for everyx, y∈ D^Iand everyi∈ N we have

r^I_i(x, y)≤((V →W)u(W →V))^I(y)⇒M^I(y).

Let nowI be anx-model ofOP ∪ OV6=W. Since at least one of(V → W)^I(g(νi)),(W → V)^I(g(νi))must be1 andr_i^I(g(ν), g(νi)) = 1for everyν∈ N^∗andi∈ N, then it holds that eitherV^I(g(ν))⇒W^I(g(ν))≤M^I(g(ν))or W^I(g(ν))⇒V^I(g(ν))≤M^I(g(ν)).

For the second condition, consider an extensionI ofI_P that satisfiesOP and assume that it violatesOV6=W. Thus, there areν∈ N^∗,i∈ N such that

1>(∀ri.(((V →W)u(W →V))→M))^IP(ν).

Sinceνiis the onlyri-successor ofν, this implies that M^IP(νi)

<(V^IP(νi)⇒W^IP(νi))⊗(W^IP(νi)⇒V^IP(νi))

≤min{V^IP(νi)⇒W^IP(νi), W^IP(νi)⇒V^IP(νi)}.

[ELC] Consider the ontology

OV6=W:={∃ri.¬Y v ⊥ |1≤i≤n} ∪ (1) {X vXuX,> v ¬(Xu ¬X),he0:¬Y ≥1i, Y uXuV vY uXuWuM, (2) Y u ¬XuW vY u ¬XuV uM}. (3)

Every model of the axioms in (1) has to satisfy that ev- ery ri-successor with degree1 must belong toY with de- gree1, for everyi ∈ N. In particular, this means that for every model I of O_P ∪ OV6=W and every ν ∈ N⁺, we haveY^I(g(ν)) = 1. The next axiom ensures that for every x ∈ D^I, X^I(x) ≤ X^I(x)⊗X^I(x), and hence, X^I(x) must be an idempotent element w.r.t.⊗. In particular, this means that(Xu¬X)^I(x) = min{X^I(x),1−X^I(x)}(Kle- ment, Mesiar, and Pap 2000), and from the second axiom it follows thatX^I(x)∈ {0,1}.

Let nowI be a model of O_P ∪ OV6=W andν ∈ N⁺. If X^I(g(ν)) = 1, then from axiom (2) it follows that V^I(g(ν)) ≤ W^I(g(ν))⊗M^I(g(ν)). We consider two cases, according to the representative chosen in⊗.

[Π^(a,b)] We know that W^I(g(ν)) = enc(wν) > aand M^I(g(ν)) =enc(1)< b. Thus, for allm⁰> M^I(g(ν)), W^I(g(ν))⊗m⁰> W^I(g(ν))⊗M^I(g(ν))≥V^I(g(ν)).

[Ł^(a,b)] Since the length ofw_νis bounded by|ν|k, we have W^I(g(ν))⊗M^I(g(ν))

=a+ (b−a) max{0,1−0.←w−ν−(0.0^|ν|k·1)}

=a+ (b−a)(1−0.←w−ν−(0.0^|ν|k·1))∈(a, b).

Thus, for everym⁰ > M^I(g(ν)),

W^I(g(ν))⊗m⁰> W^I(g(ν))⊗M^I(g(ν))≥V^I(g(ν)).

In both cases, sinceW^I(g(ν))⇒V^I(g(ν))equals sup{z∈[0,1]|W^I(g(ν))⊗z≤V^I(g(ν))}, we have W^I(g(ν)) ⇒ V^I(g(ν)) ≤ M^I(g(ν)). Using an analogous argument, if X^I(g(ν)) = 0, then axiom (3) yieldsV^I(g(ν))⇒W^I(g(ν))≤M^I(g(ν)).

To show the second point ofP₆₌, consider an extension I ofIP that satisfiesOP, which exists by assumption. We show thatIcan be extended to a model ofOV6=W. We first setY^I(ν) = 1for everyν ∈ N⁺andX^I(ε) =Y^I(ε) = 0.

It remains to find values forX^I(ν)forν∈ N⁺.

By assumption, we know that one of the two residua V^IP(ν) ⇒ W^IP(ν)andW^IP(ν) ⇒ V^IP(ν) is smaller than or equal to M^IP(ν) < 1. However, one of them must be equal to 1. If V^IP(ν) ⇒ W^IP(ν) = 1 and W^IP(ν)⇒V^IP(ν)≤M^IP(ν), then we setX^I(ν) = 1, which trivially satisfies axiom (3) atν. By definition of the residuum, this implies thatW^IP(ν)⊗m⁰> V^IP(ν)for all m⁰ > M^IP(ν). Since⊗is continuous and monotone, this means thatV^IP(ν)≤W^IP(ν)⊗M^IP(ν), i.e. axiom (2) is also satisfied atν.

If the other residuum is equal to1, we setX^I(ν) = 0 and use dual arguments to show that axioms (2) and (3) are satisfied atν. We have thus constructed an extension ofIP

that satisfiesOV6=W.

If a fuzzy DL satisfies the propertyP6=, then consistency of ontologies is undecidable.

Theorem 19. Let⊗_x-LsatisfyP₆₌. ThenP has a solution iffOP∪ OV6=W is inconsistent.

Proof. If OP ∪ OV6=W is inconsistent, then in particular no extension ofI_P can satisfy this ontology. ByP₆₌, there must be aν ∈ N⁺such that bothV^IP(ν)⇒W^IP(ν)and W^IP(ν)⇒ V^IP(ν)are greater thanM^IP(ν). By Defini- tion 6 and byP₄, we haveenc(v_ν) =enc(w_ν), i.e.Phas a solution.

Assume now thatO_P∪OV6=W has a modelI. ByP₆₌, for everyν ∈ N⁺eitherV^I(g(ν))⇒W^I(g(ν))≤M^I(g(ν)) orW^I(g(ν))⇒V^I(g(ν))≤M^I(g(ν)). ByP₄, it follows that enc(v_ν) = V^I(g(ν)) 6= W^I(g(ν)) = enc(w_ν), and thusvν6=wνfor allν∈ N⁺, i.e.P has no solution.

Together with Corollaries 15 and 17, we obtain the fol- lowing undecidability results.

Corollary 20. For every continuous t-norm different from the G¨odel t-norm, ontology consistency is undecidable in the logics⊗_>-IAL=,⊗-IAL=,c,⊗_>-ELC_≥,⊗-ELC_≥,c, Π_>-ELC, andΠ-ELCc.

Since every extension ofIP is witnessed, from these re- sults it also follows that ontology consistency in the logics

⊗w-IAL=,⊗w-ELC≥, andΠ_w-ELCis undecidable.

4.3 Undecidability ofŁ^(0,b)-NEL

For the logicΠ-ELC, we were able to exploit the involutive negation and obtain undecidability of consistency ofclassi- calontologies; that is, no membership degrees other than1 are required to appear in the axioms. The same idea can be applied to show that ontology consistency is also undecid- able inŁ-ELC, which is equivalent toŁ-NEL. We show a stronger result: consistency inŁ^(0,b)_> -NELandŁ^(0,b)-NELc

forb >0is undecidable. The t-norms(0, b)-containing the Łukasiewicz t-norm cover an important family of t-norms, known as the Mayor-Torrens t-norms that have been studied in the literature (Klement, Mesiar, and Pap 2000).

If⊗(0, b)-contains the Łukasiewicz t-norm, then for ev- eryx ∈ (0, b] we have thatx ⇒ 0 = b−x; that is, the residual negation yields a “local involutive negation” over the interval [0, b]. Thus, the concept C will be inter- preted as the local involutive negation of the interpretation of C, whenever the latter is in this interval. Moreover, if 0≤D^I(x)< C^I(x)≤b, then

((CuD))^I(x)=b−(C^I(x) + (b−D^I(x))−b)

=b−C^I(x) +D^I(x) = (C→D)^I(x).

Thus, we abbreviate(CuD)asC * D. Additionally,

⊥can be expressed by>.

We encode a wordu ∈ Σ^∗₀ byenc(u) = b(0.←−u). The proof that this is indeed a valid encoding function uses sim- ilar arguments to the case forŁ^(a,b)of Lemma 7.

LetPbe an instance of the PCP as before and assume that v_ν 6=w_νfor someν ∈ N⁺. Then these words must differ in one of the first|ν|kdigits, and thus either

enc(vν)⇒enc(wν) =bmin{1,1−0.←v−ν+ 0.←w−ν}

≤b(1−(s+ 1)^−|ν|k)

=enc(ε·s^|ν|k)

orenc(wν)⇒enc(vν)≤enc(ε·s^|ν|k)<1. Conversely, if vν =wν, then both residua are1. Thus, the wordsuε =ε andu₊=s^ksatisfy the condition of Definition 6.

We will employ Theorem 16 to show that the logics Ł^(0,b)_> -NEL and Ł^(0,b)-NEL_c satisfy the canonical model property. Thus, we need to prove that they satisfyP→,P◦, P ,P^w_ini, andP=. By Lemma 8, they satisfy the successor property. We now show thatŁ^(0,b)-NELsatisfies the rest of the properties.

Concatenation property Analogous to Lemma 9, the ax- ioms(C⁰)^(s+1)|u| ≡CandD_C◦u ≡(C⁰)u(C_u) yield the concatenation of words represented byCwith the constant wordu.

Transfer property If C^I(x) = enc(w),w ∈ Σ^∗, then C^I(x)< b, and thus for every modelIof∃r.(D)vC ifr^I(x, y) = 1then

b−C^I(x) = (C)^I(x)≥(∃r.(D))^I(x)≥(D)^I(y).

IfD^I(y)< C^I(x)< b, then

(D)^I(y) =b−D^I(y)> b−C^I(x),

which yields a contradiction; henceC^I(x)≤ D^I(y)must hold. Together with the first part of the proof of Lemma 10, we have that the ontology

O_Cr

D:={∃r.(D)vC,∃r.DvC}

yields the transfer property.

Weak initialization property The assertionhe:C≥1i initializes the valueenc(u_ε) =enc(ε) = 0.

Constant property We have to restrict the value of a con- ceptC toenc(u)for some wordu ∈ Σ^∗₀. Foru = ε, the axiomCv ⊥suffices. Ifu∈Σ⁺₀, we employ the ontology

OC=u:={H^(s+1)|u| ≡H^(s+1)|u|, C≡H^2←−u}.

If an interpretationIsatisfies the first axiom, then for every x ∈ D^I we have−b = 2(s+ 1)^|u|(H^I(x)−b); that is H^I(x) =b−_2(s+1)^b |u|. From the second axiom it follows that

(C)^I(x) = maxn 0,2←−u

−_2(s+1)^b _|u|

+bo .

Since _(s+1)^←u−|u| = 0.←−u <1, we obtain

(C)^I(x) =b−b(0.←−u) =b−enc(u).

Sinceenc(u) < b, we have0 <(C)^I(x) < b, and thus 0< C^I(x)< band(C)^I(x) =b−C^I(x). From this, we obtain thatC^I(x) =enc(u).

One can easily extendIP to a model of the ontologyOP

that results from the above definitions. By Theorem 16, Ł^(0,b)_> -NEL and Ł^(0,b)-NEL_c satisfy the canonical model property. It remains to show that the solution property holds.