On the Undecidability of Fuzzy Description Logics with GCIs with Lukasiewicz t-norm
aa r X i v : . [ c s . L O ] J u l On the Undecidability of Fuzzy Description Logics with GCIswith Lukasiewicz t -norm Marco Cerami Umberto Straccia
IIIA - CSIC ISTI - CNRBellaterra, Catalunya Pisa, [email protected] [email protected]
June 16, 2018
Abstract
Recently there have been some unexpected results concerning Fuzzy DescriptionLogics (FDLs) with General Concept Inclusions (GCIs). They show that, unlike theclassical case, the DL
ALC with GCIs does not have the finite model property under Lukasiewicz Logic or Product Logic and, specifically, knowledge base satisfiability isan undecidable problem for Product Logic. We complete here the analysis by showingthat knowledge base satisfiability is also an undecidable problem for Lukasiewicz Logic.
Description Logics (DLs) [1] play a key role in the design of
Ontologies . Indeed, DLs areimportant as they are essentially the theoretical counterpart of the
Web Ontology LanguageOWL 2 [19], the standard language to represent ontologies.It is very natural to extend DLs to the fuzzy case and several fuzzy extensions of DLscan be found in the literature. For a recent survey on the advances in the field of fuzzyDLs, we refer the reader to [18]. Besides the enrichment of DLs with fuzzy features, one ofthe challenges of the research in this community is the fact that different families of fuzzyoperators (or fuzzy logics) lead to fuzzy DLs with different computational properties.Decidability of fuzzy DLs is often shown by adapting crisp DL tableau-based algorithmsto the fuzzy DL case [8, 21, 22, 23, 25, 26], or a reduction to classical DLs [5, 6, 7, 9, 24],or relying on some Mathematical Fuzzy Logic [13] based procedures [11, 12, 14, 15].However, recently there have been some unexpected surprises [2, 3, 4]. Indeed, unlikethe classical case, for the DL
ALC with GCIs (i) [4] shows that it does not have thefinite model property under Lukasiewicz Logic or Product Logic, illustrates that somealgorithms are neither complete not correct, and shows some interesting conditions underwhich decidability is still guaranteed; and (ii) [2, 3] show that knowledge base satisfiability1s an undecidable problem for it under Product Logic. Also worth mentioning is [10], whichillustrates the undecidability of knowledge base satisfiability if one replaces the truth set[0 ,
1] with complete De Morgan lattices equipped with a t-norm operator.In this paper, we complete the analysis by showing that knowledge base satisfiability isan undecidable problem for the DL
ALC with GCIs under [0 , ALC
In this section we are going to introduce the general definitions of L-
ALC based on Lukasiewicz t -norm. Syntax.
Let A be a set of concept names , R be a set of role names . Concept namesdenote unary predicates, while role names denote binary predicates. The set of L- ALC concepts are built from concept names A (also called atomic concepts) using connectivesand quantification constructs over roles R as described by the following syntactic rules: C → ⊤ | ⊥ | A | C ⊓ C | C ⊔ C | ¬ C | ∃ R.C | ∀
R.C . An assertion axiom is an expression of the form h a : C, n i ( concept assertion , a is an instanceof concept C to degree at least n ) or of the form h ( a , ): R, n i ( role assertion , ( a , a ) isan instance of role R to degree at least n ), where a, a , a are individual names, C is aconcept, R is a role name and n ∈ (0 ,
1] is a rational (a truth value). An
ABox A consistsof a finite set of assertion axioms.A General Concept Inclusion (GCI) axiom is of the form h C ⊑ C , n i ( C is a sub-concept of C to degree at least n ), where C i is a concept and n ∈ (0 ,
1] is a rational. A concept hierarchy T , also called TBox , is a finite set of GCIs. In what follows we will usethe following shorthands: • C ⊑ C for h C ⊑ C , i and a : C for h a : C, i ; • C ≡ C for the two axioms C ⊑ C and C ⊑ C ; • C → C for ¬ C ⊔ C ; • C ↔ C for ( C → C ) ⊓ ( C → C ); • min { C , C } for C ⊓ ( C → C ), and min { C , . . . , C n } for min { . . . min { C , C } , . . . } ; • max { C , C } for ( C → C ) → C and max { C , . . . , C n } for max { . . . max { C , C } , . . . } ; Each symbol may have super- and/or subscripts. I ( x ) = 0 ⊤ I ( x ) = 1( C ⊓ D ) I ( x ) = C I ( x ) ⊗ D I ( x )( C ⊔ D ) I ( x ) = C I ( x ) ⊕ D I ( x )( ¬ C ) I ( x ) = ⊖ C I ( x )( ∀ R.C ) I ( x ) = inf y ∈ ∆ I { R I ( x, y ) ⇒ C I ( y ) } ( ∃ R.C ) I ( x ) = sup y ∈ ∆ I { R I ( x, y ) ⊗ C I ( y ) } Table 1: Semantics for L-
ALC . • n · C for the n -ary disjunction C ⊔ . . . ⊔ C ;Finally, a knowledge base K = hT , Ai consists of a TBox T and an ABox A . Semantics.
From a semantics point of view, an axiom h α, n i constrains the truth de-gree of the expression α to be at least n . In the following, we use ⊗ , ⊕ , ⊖ and ⇒ todenote Lukasiewicz t -norm, t -conorm, negation function, and implication function, respec-tively [17]. They are defined as operations in [0 ,
1] by means of the following functions: a ⊗ b = max { , a + b − } a ⊕ b = min { , a + b }⊖ a = 1 − aa ⇒ b = min { , − a + b } , where a and b are arbitrary elements in [0 , t -conorm) and negation in theusual way: a ⇒ b = ⊖ a ⊕ b . Note also that for any implication defined from a continuoust-norm ⊗ , it holds that: x ⇒ y = max { z | x ⊗ z ≤ y } , which is equivalent to the condition: y ≥ x ⊗ z iff ( x ⇒ y ) ≥ z .A fuzzy interpretation (or model) is a pair I = (∆ I , · I ) consisting of a nonempty (crisp)set ∆ I (the domain ) and of a fuzzy interpretation function · I that assigns:1. to each atomic concept A a function A I : ∆ I → [0 , R a function R I : ∆ I × ∆ I → [0 , a an element a I ∈ ∆ I such that a I = b I if a = b ( Unique NameAssumption , different individuals denote different objects of the domain).The fuzzy interpretation function is extended to complex concepts as specified in Table 1(where x, y ∈ ∆ I are elements of the domain). Hence, for every complex concept C we geta function C I : ∆ I → [0 , satisfiability of axioms is then defined by the followingconditions: 3. I satisfies an axiom h a : C, α i if C I ( a I ) ≥ α ,2. I satisfies an axiom h ( a, b ): R, α i if R I ( a I , b I ) ≥ α ,3. I satisfies an axiom h C ⊑ D, α i if ( C ⊑ D ) I ≥ α where( C ⊑ D ) I = inf x ∈ ∆ I { C I ( x ) ⇒ D I ( x ) } . It is interesting to point out that the satisfaction of a GCI of the form h C ⊑ D, i is exactlythe requirement that ∀ x ∈ ∆ I , C I ( x ) ≤ D I ( x ) (i.e., Zadeh’s set inclusion); hence, in thisparticular case for the satisfaction it only matters the partial order and not the exact valueof the implication ⇒ .As it is expected we will say that a fuzzy interpretation I satisfies a KB K in case thatit satisfies all axioms in K . And it is said that a fuzzy KB K is satisfiable iff there exist afuzzy interpretation I satisfying every axiom in K .In this paper, we mainly focus on witnessed models. This notion (see [14]) correspondsto the restriction to the DL language of the notion of witnessed model introduced, in thecontext of the first-order language, by H´ajek in [16]. Specifically, a fuzzy interpretation I is said to be witnessed iff it holds that for every complex concepts C, D , every role R , andevery x ∈ ∆ I there is some1. y ∈ ∆ I such that ( ∃ R.C ) I ( x ) = R I ( x, y ) ⊗ C I ( y )2. y ∈ ∆ I such that ( ∀ R.C ) I ( x ) = R I ( x, y ) ⇒ C I ( y )If I satisfies only condition 1. then I is said to be weakly witnessed . Note that for Lukasiewicz logic, condition 1. and 2. are equivalent, so I is weakly witnessed iff I iswitnessed. Thorough the paper we will rely on the notion of witnessed interpretation only,but keep in mind that the results apply, thus, to weakly witnessed interpretations as well.Note also that it is obvious that all finite fuzzy interpretations (this means that ∆ I is afinite set) are indeed strongly witnessed but the opposite is not true.Sometimes (see, e.g. , [3]), the notion of witnessed interpretatations is strengthened toso-called strongly witnessed interpretations by imposing that additionally that for everycomplex concepts C, D and every x ∈ ∆ I there is some • y ∈ ∆ I such that ( C ⊑ D ) I = C I ( y ) ⇒ D I ( y )has to hold. We do not deal with strongly witnessed interpretations here.A fuzzy KB K is said to be satisfiable iff there exist a fuzzy interpretation I satisfyingevery axiom in K . 4 Undecidability of L-
ALC with GCIs
Our proof consists in a reduction of the reverse of the
Post Correspondence Problem (PCP)and follows conceptually the one in [2, 3, 10]. PCP is well-known to be undecidable [20],so is the reverse PCP, as shown next.
Definition 1 (PCP) . Let v , . . . , v p and w , . . . , w p be two finite lists of words over analphabet Σ = { , . . . , s } . The Post Correspondence Problem (PCP) asks whether there isa non-empty sequence i , i , . . . , i k , with ≤ i j ≤ p such that v i v i . . . v i k = w i w i . . . w i k .Such a sequence, if it exists, is called a solution of the problem instance. For the sake of our purpose, we will rely on a variant of the PCP, which we call
Reverse
PCP (RPCP). Essentially, words are concatenated from right to left rather than from leftto right.
Definition 2 (RPCP) . Let v , . . . , v p and w , . . . , w p be two finite lists of words over analphabet Σ = { , . . . , s } . The Reverse Post Correspondence Problem (RPCP) asks whetherthere is a non-empty sequence i , i , . . . , i k , with ≤ i j ≤ p such that v i k v i k − . . . v i = w i k w i k − . . . w i . Such a sequence, if it exists, is called a solution of the problem instance. For a word µ = i i . . . i k ∈ { , . . . , p } ∗ we will use v µ , w µ to denote the words v i k v i k − . . . v i and w i k w i k − . . . w i . We denote the empty string as ǫ and define v ǫ is ǫ .The alphabet Σ consists of the first s positive integers. We can thus view every word in Σ ∗ as a natural number represented in base s + 1 in which 0 never occurs. Using this intuition,we will use the number 0 to encode the empty word.Now we show that the reduction from PCP to RPCP is a very simple matter and it canbe done through the transformation of the instance lists to the lists of their palindromesdefined as follows: let Σ = { , . . . , s } be an alphabet and v = t t . . . t | v | a word over Σ,with t i ∈ Σ, for 1 ≤ j ≤ | v | , then the palindrome of v is defined as pal ( v ) = t | v | t | v |− . . . t . Lemma 3.
Let v , . . . , v p and w , . . . , w p be two finite lists of words over an alphabet Σ = { , . . . , s } . For every non-empty sequence i , i , . . . , i k , with ≤ i j ≤ p it holds that v i v i . . . v i k = w i w i . . . w i k iff pal ( v i k ) pal ( v i k − ) . . . pal ( v i ) = pal ( w i k ) pal ( w i k − ) . . . pal ( w i ) . ( P roof ) First we prove by induction on k , that, for every sequence v = v i v i . . . v i k ofwords over Σ, it holds that pal ( v ) = pal ( v i k ) pal ( v i k − ) . . . pal ( v i ). • The case k = 1 is straightforward. • Let v = v i v i . . . v i k and suppose, by inductive hypothesis, that pal ( v i v i . . . v i k − )= pal ( v i k − ) pal ( v i k − ) . . . pal ( v i ). It follows that pal ( v ) = pal ( v i v i . . . v i k − , v i k ) = pal ( v i k ) pal ( v i k − ) . . . pal ( v i ). 5ince the palindrome of a word is unique, we have that, if v i v i . . . v i k = w i w i . . . w i k ,then pal ( v i v i . . . v i k ) = pal ( w i w i . . . w i k ) and, thus, pal ( v i k ) pal ( v i k − ) . . . pal ( v i ) = pal ( w i k ) pal ( w i k − ) . . . pal ( w i ). Corollary 4.
The RPCP is undecidable. ( P roof ) The proof is based on the reduction of PCPs to RCPs. For every instance ϕ =( v , w ) , . . . , ( v p , w p ) of PCP, let f be the function f ( ϕ ) = ( pal ( v ) , pal ( w )) , . . . , ( pal ( v p ) , pal ( w p )) . Clearly f is a computable function. Moreover, ϕ ∈ P CP if and only if there ex-ists a non-empty sequence i , i , . . . , i k , with 1 ≤ i j ≤ p such that v i v i . . . v i k = w i w i . . . w i k , that is, by Lemma 3, pal ( v i k ) pal ( v i k − ) . . . pal ( v i ) = pal ( w i k ) pal ( w i k − ) . . . pal ( w i ) i.e. , f ( ϕ ) ∈ RP CP . Therefore, ϕ ∈ P CP if and only if f ( ϕ ) ∈ RP CP . Undecidability of general KB satisfiability.
We show the undecidability by a reduc-tion of RPCPs to KB satisfiability problems. Specifically, given an instance ϕ of RPCP,we will construct a Knowledge Base O ϕ that is satisfiable iff ϕ has no solution.In order to do this we will encode words v from the alphabet Σ as rational numbers 0 .v in [0 ,
1] in base s + 1; the empty word will be encoded by the number 0.So, let us define the TBox T := { V ≡ V ⊔ V , W ≡ W ⊔ W } and for 1 ≤ i ≤ p the TBoxes 6 iϕ := { ⊤ ⊑ ∃ R i . ⊤ ,V ⊑ ( s + 1) | v i | · ∀ R i .V , ( s + 1) | v i | · ∃ R i .V ⊑ V,W ⊑ ( s + 1) | w i | · ∀ R i .W , ( s + 1) | w i | · ∃ R i .W ⊑ W h⊤ ⊑ ∀ R i .V , .v i i , h⊤ ⊑ ∀ R i . ¬ V , − .v i i , h⊤ ⊑ ∀ R i .W , .w i i , h⊤ ⊑ ∀ R i . ¬ W , − .w i i ,A ⊑ ( s + 1) max {| v i | , | w i |} · ∀ R i .A ( s + 1) max {| v i | , | w i |} · ∃ R i .A ⊑ A } . Now, let T ϕ = T ∪ p [ i =1 T iϕ . Further we define the ABox A as follows: A := { a : ¬ V, a : ¬ W, h a : A, . i , h a : ¬ A, . i} . Finally, we define O ϕ := hT ϕ , Ai . We now define the interpretation I ϕ := (∆ I ϕ , · I ϕ )as follows: • ∆ I ϕ = { , . . . , p } ∗ • a I ϕ = ǫ • V I ϕ ( ǫ ) = W I ϕ ( ǫ ) = 0, A I ϕ ( ǫ ) = 0 .
01, and for 1 ≤ i ≤ V I ϕ i ( ǫ ) = W I ϕ i ( ǫ ) = 07 for all µ, µ ′ ∈ ∆ I ϕ and 1 ≤ i ≤ pR I ϕ i ( µ, µ ′ ) = ( , if µ ′ = µi , otherwise • for every µ ∈ ∆ I ϕ , where µ = i i . . . i k = ǫ – V I ϕ ( µ ) = 0 .v µ , W I ϕ ( µ ) = 0 .w µ – A I ϕ ( µ ) = 0 . · ( s + 1) − P j ∈{ i ,i ,...,ik } max {| v j | , | w j |} – V I ϕ ( µ ) = 0 .v ¯ µ · ( s + 1) −| v ik | , W I ϕ ( µ ) = 0 .w ¯ µ · ( s + 1) −| w ik | , where ¯ µ = i i . . . i k − (last index i k is dropped from µ , and we assume that 0 .ǫ is 0), – V I ϕ ( µ ) = 0 .v i k , W I ϕ ( µ ) = 0 .w i k .It is easy to see that I ϕ is a witnessed model of O ϕ (note that e.g. , ( ∀ R i .V ) I ϕ ( µ ) = V I ϕ ( µi ). Moreover, as in [2] it is possible to prove that, for every witnessed model I of O ϕ , thereis a mapping g from I ϕ to I . Lemma 5.
Let I be a witnessed model of O ϕ . Then there exists a function g : ∆ I ϕ → ∆ I such that, for every µ ∈ ∆ I ϕ , C I ϕ ( µ ) = C I ( g ( µ )) holds for every concept name C and R I ϕ i ( µ, µi ) = R I i ( g ( µ ) , g ( µi )) holds for every i , with ≤ i ≤ p . ( P roof ) Let I be a witnessed model of O ϕ . We will build the function g inductively onthe length of µ .( ǫ ) Since I is a model of O ϕ , then there is an element δ ∈ ∆ I such that a I = δ . Since I is a model of A ϕ , setting g ( ǫ ) = δ , we have that V I ϕ ( ǫ ) = 0 = V I ( g ( ǫ )) andthe same holds for concept W . Moreover, since I is a model of T ϕ , we have that V I ( δ ) = ( V ⊔ V ) I ( δ ) and, therefore V I ϕ ( ǫ ) = 0 = V I ( g ( ǫ )) and the same holdsfor V , W and W . On the other hand, we have that A I ϕ ( ǫ ) = 0 .
01 = A I ( g ( ǫ )),as well. So, g ( ǫ ) = δ satisfies the condition of the lemma.( µi ) Let now µ be such that g ( µ ) has already been defined. Now, since I is awitnessed model and satisfies axiom ⊤ ⊑ ∃ R i . ⊤ , then for all i , with 1 ≤ i ≤ p ,there exists a γ ∈ ∆ I such that R I i ( g ( µ ) , γ ) = 1. So, setting g ( µi ) = γ we get1 = R I ϕ i ( µ, µi ) = R I i ( g ( µ ) , g ( µi )). Furthermore, by inductive hypothesis, wecan assume that V I ( g ( µ )) = 0 .v µ and W I ( g ( µ )) = 0 .w µ .Since I satisfies axiom V ⊑ ( s + 1) | v i | · ∀ R i .V , then 0 .v µ = V I ( g ( µ )) ≤ ( s +1) | v i | · ( ∀ R i .V ) I ( g ( µ )) = ( s + 1) | v i | · inf γ ∈ ∆ I { R I i ( g ( µ ) , γ ) ⇒ V I ( γ ) } ≤ ( s + 1) | v i | · R I i ( g ( µ ) , µi ) ⇒ V I ( µi ) = ( s + 1) | v i | · V I ( g ( µi )). However, I ϕ is not a strongly witnessed model of O ϕ . I satisfies axiom ( s + 1) | v i | · ∃ R i .V ⊑ V , then 0 .v µ = V I ( g ( µ )) ≥ ( s +1) | v i | · ( ∃ R i .V ) I ( g ( µ )) = ( s + 1) | v i | · sup γ ∈ ∆ I { R I i ( g ( µ ) , γ ) ⊗ V I ( γ ) } ≥ ( s + 1) | v i | · R I i ( g ( µ ) , µi ) ⊗ V I ( µi ) = ( s + 1) | v i | · V I ( g ( µi )). Therefore, ( s + 1) | v i | · V I ( g ( µi )) =0 .v µ and V I ( g ( µi )) = 0 .v µ · ( s + 1) −| v i | = V I ϕ ( µi ).Similarly, it can be shown that W I ( g ( µi )) = 0 .w µ · ( s + 1) −| w i | = W I ϕ ( µi ).Since I satisfies axioms h⊤ ⊑ ∀ R i .V , .v i i and h⊤ ⊑ ∀ R i . ¬ V , − .v i i , it fol-lows that ( ∀ R i .V ) I ( g ( µ )) ≥ .v i and ( ∀ R i . ¬ V ) I ( g ( µ )) ≥ − .v i . Therefore,for R I i ( g ( µ ) , g ( µi )) = 1 we have V I ( g ( µi )) = 0 .v i = V I ϕ ( µi ). Similarly, it canbe shown that W I ϕ ( µi ) = 0 .w i = W I ( g ( µi )).Now, since I satisfies axiom V ≡ V ⊔ V , then, V I ( g ( µi )) = V I ( g ( µi )) + V I ( g ( µi )) = 0 .v µ · ( s + 1) −| v i | + 0 .v i = 0 .v i v µ = V I ϕ ( µi ).Finally, by inductive hypothesis, assume that A I ( g ( µ )) = A I ϕ ( µ ) = 0 . · ( s +1) − P j ∈{ i ,i ,...,ik } max {| v j | , | w j |} , where µ = i i . . . i k .Since I satisfies axioms A ⊑ ( s + 1) max {| v i | , | w i |} · ∀ R i .A , we have that A I ( g ( µ )) ≤ ( s + 1) max {| v i | , | w i |} · ( ∀ R i .A ) I ( g ( µ )) ≤ ( s + 1) max {| v i | , | w i |} · A I ( g ( µi )) . Likewise, since I satisfies axioms ( s + 1) max {| v i | , | w i |} · ∃ R i .A ⊑ A , we have that A I ( g ( µ )) ≥ ( s + 1) max {| v i | , | w i |} · ( ∃ R i .A ) I ( g ( µ )) ≥ ( s + 1) max {| v i | , | w i |} · A I ( g ( µi ))and, thus, A I ( g ( µ )) = ( s + 1) max {| v i | , | w i |} · A I ( g ( µi )) . Therefore, A I ( g ( µi )) = ( s + 1) − max {| v i | , | w i |} · A I ( g ( µ ))= ( s + 1) − max {| v i | , | w i |} · A I ϕ ( µ )= ( s + 1) − max {| v i | , | w i |} · . · ( s + 1) − P j ∈{ i ,i ,...,ik } max {| v j | , | w j |} = 0 . · ( s + 1) − (max {| v i | , | w i |} + P j ∈{ i ,i ,...,ik } max {| v j | , | w j |} ) = 0 . · ( s + 1) − P j ∈{ i ,i ,...,ik,i } max {| v j | , | w j |} = A I ϕ ( µi ) , which completes the proof.From the last Lemma it follows that if the RPCP instance ϕ has a solution µ , for some µ ∈ { , . . . , p } + , then v µ = w µ and, thus, 0 .v µ = 0 .w µ . Therefore, every witnessed model I of O ϕ contains an element δ = g ( µ ) such that V I ( δ ) = V I φ ( µ ) = 0 .v µ = 0 .w µ = W I φ ( µ ) = W I ( δ ). Conversely, from the definition of I ϕ , if ϕ has no solution, then there is no µ suchthat 0 .v µ = 0 .w µ , i.e. , there is no µ such that V I φ ( µ ) = W I φ ( µ ).9owever, as O ϕ is always satisfiable, it does not yet help us to decide the RPCP. Wenext extend O ϕ to O ′ ϕ in such a way that an instance ϕ of the RPCP has a solution iffthe ontology O ′ ϕ is not witnessed satisfiable and, thus, establish that the KB satisfiabilityproblem is undecidable. To this end, consider O ′ ϕ := hT ′ ϕ , Ai , where T ′ ϕ := T ϕ ∪ [ ≤ i ≤ p {⊤ ⊑ ∀ R i . ( ¬ ( V ↔ W ) ⊔ ¬ A ) } . The intuition here is the following. If there is a solution for RPCP then, by theobservation before, there is a point δ in which the value of V and W coincide under I .That is, the value of ¬ ( V ↔ W ) is 0 and, thus, the one of ¬ ( V ↔ W ) ⊔ ¬ A ) is less than1. So, I cannot satisfy the new GCI in T ′ ϕ and, thus, O ′ ϕ is not satisfiable. On the otherhand, if there is no solution to the RPCP then in I ϕ there is no point in which V and W coincide and, thus, ¬ ( V ↔ W ) >
0. However, we will show that the value of ¬ ( V ↔ W )in all points is strictly greater than A and, as A ⊔ ¬ A is 1, so also ¬ ( V ↔ W ) ⊔ ¬ A will be1 in any point. Hence, I φ is a model of the aditional axiom in T ′ ϕ , i.e. , O ′ ϕ is satisfiable. Theorem 6.
The instance ϕ of the RPCP has a solution iff the ontology O ′ ϕ is not wit-nessed satisfiable. ( P roof ) Assume first that ϕ has a solution µ = i . . . i k and let I be a witnessed modelof O ϕ . Let ¯ µ = i i . . . i k − (last index i k is dropped from µ ). Then by Lemma5 it follows that there are nodes δ, δ ′ ∈ ∆ I such that δ = g ( µ ), δ ′ = g (¯ µ ), with V I ( δ ) = V I ϕ ( µ ) = W I ϕ ( µ ) = W I ( δ ) and R I i k ( δ ′ , δ ) = 1. Then ( V ↔ W ) I ( δ ) = 1.Since ( ¬ A ) I ( δ ) <
1, then ( ¬ ( V ↔ W ) ⊔¬ A ) I ( δ ) <
1. Hence there is i , with 1 ≤ i ≤ p ,such that ( ∀ R i . ( ¬ ( V ↔ W ) ⊔ ¬ A )) I ( δ ′ ) <
1. So, axiom ⊤ ⊑ ∀ R i . ( ¬ ( V ↔ W ) ⊔ ¬ A )is not satisfied and, therefore, O ϕ is not satisfiable.For the converse, assume that ϕ has no solution. On the one hand we know that I ϕ is a model of O ϕ . On the other hand, since ϕ has no solution, then there is no µ = i . . . i k such that v µ = w µ ( i.e. , 0 .v µ = 0 .w µ ) and, therefore, there is no µ ∈ ∆ I ϕ such that V I ϕ ( µ ) = W I ϕ ( µ ). Consider µ ∈ ∆ I ϕ and i , with 1 ≤ i ≤ p and assume,10ithout loss of generality, that V I ϕ ( µi ) < W I ϕ ( µi ). Then( V ↔ W ) I ϕ ( µi ) = ( V I ϕ ( µi ) ⇒ W I ϕ ( µi )) ⊗ ( W I ϕ ( µi ) ⇒ V I ϕ ( µi ))= 1 ⊗ ( W I ϕ ( µi ) ⇒ V I ϕ ( µi ))= W I ϕ ( µi ) ⇒ V I ϕ ( µi )= 1 − W I ϕ ( µi ) + V I ϕ ( µi )= 1 − ( W I ϕ ( µi ) − V I ϕ ( µi ))= 1 − (0 .w µi − .v µi ) ≤ − . · ( s + 1) − max {| v µi | , | w µi |} ≤ − . · ( s + 1) − P j ∈{ i ,i ,...,ik,i } max {| v j | , | w j |} = ( ¬ A ) I ϕ ( µi ) . Therefore, ( ¬ ( V ↔ W )) I ϕ ( µi ) ≥ A I ϕ ( µi ). As A I ϕ ( µi ) ⊕ ( ¬ A ) I ϕ ( µi ) = 1, it followsthat for every µ ∈ ∆ I ϕ and i , with 1 ≤ i ≤ p , it holds that ( ∀ R i . ( ¬ ( V ↔ W ) ⊔¬ A )) I ϕ ( µ ) = 1 and, therefore, I ϕ is a witnessed model of O ′ ϕ . References [1] Franz Baader, Diego Calvanese, Deborah McGuinness, Daniele Nardi, and Peter F.Patel-Schneider, editors.
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