: Infinitely Valued Gödel Semantics for Expressive Description Logics

http://lat.inf.tu-dresden.de/research/reports-abs.html#ZaTu-LTCS-13-06 Paper (PDF): http://lat.inf.tu-dresden.de/research/../data/reports/2013/ZaTu-LTCS-13-06.pdf Franz Baader, Oliver Fernández Gil, and Barbara Morawska: Hybrid Unification in the Description Logic EL. 13-07, Chair of Automata Theory, Institute of Theoretical Computer Science, Technische Universität Dresden, Dresden, Germany, 2013. See http://lat.inf.tu-dresden.de/research/reports.html. BibTeX entry: http://lat.inf.tu-dresden.de/research/reports-bib.html#BaFM-LTCS-13-07 Abstract: http://lat.inf.tu-dresden.de/research/reports-abs.html#BaFM-LTCS-13-07 Paper (PDF): http://lat.inf.tu-dresden.de/research/../data/reports/2013/BaFM-LTCS-13-7.pdf Franz Baader and Benjamin Zarries̈: Verification of Golog Programs over Description Logic Actions. 13-08, Chair of Automata Theory, TU Dresden, Dresden, Germany, 2013. See http://lat.inf.tu-dresden.de/research/reports.html. BibTeX entry: http://lat.inf.tu-dresden.de/research/reports-bib.html#BaZa-LTCS-13-08 Paper (PDF): http://lat.inf.tu-dresden.de/research/../data/reports/2013/BaZa-LTCS-13-08.pdf Stefan Borgwardt, Felix Distel, and Rafael Peñaloza: Goedel Description Logics: Decidability in the Absence of the Finitely-Valued Model Property. 13-09, Chair for Automata Theory, Institute for Theoretical Computer Science, Technische Universität Dresden, Dresden, Germany, 2013. See http://lat.inf.tu-dresden.de/research/reports.html. BibTeX entry: http://lat.inf.tu-dresden.de/research/reports-bib.html#BoDP-LTCS-13-09 Abstract: http://lat.inf.tu-dresden.de/research/reports-abs.html#BoDP-LTCS-13-09 Paper (PDF): http://lat.inf.tu-dresden.de/research/../data/reports/2013/BoDP-LTCS-13-09.pdf Benjamin Zarrieß and Jens Claßen: On the Decidability of Verifying LTL Properties of Golog Programs. 13-10, Chair of Automata Theory, TU Dresden, Dresden, Germany, 2013. Extended version. See http://lat.inf.tu-dresden.de/research/reports.html. BibTeX entry: http://lat.inf.tu-dresden.de/research/reports-bib.html#ZaCla-LTCS-13-10 Abstract: http://lat.inf.tu-dresden.de/research/reports-abs.html#ZaCla-LTCS-13-10 Paper (PDF): http://lat.inf.tu-dresden.de/research/../data/reports/2013/ZaCla-LTCS-13-10. pdf Daniel Borchmann: Model Exploration by Confidence with Completely Specified Counterexamples. 13-11, Chair of Automata Theory, Institute of Theoretical


Introduction
Description Logics (DLs) are a well-studied family of knowledge representation formalisms [1].They constitute the logical backbone of the standard Semantic Web ontology language OWL 2, 1 and its profiles, and have been successfully applied to represent the knowledge of many and diverse application domains, particularly in the bio-medical sciences.DLs describe the domain knowledge using concepts (such as Patient) that represent sets of individuals, and roles (hasChild) that represent connections between individuals.Ontologies are collections of axioms formulated over these concepts and roles, which restrict their possible interpretations.The typical axioms considered in DLs are assertions, like alice:Patient, providing knowledge about specific individuals; general concept inclusions (GCIs), such as Patient Human, which express general relations between concepts; and role inclusions hasChild hasChild hasGrandchild between (chains of) roles.Different DLs are characterized by the constructors allowed to formulate complex concepts, roles, and axioms.
Fuzzy Description Logics (FDLs) have been introduced as extensions of classical 1 http://www.w3.org/TR/owl2-overview/DLs to represent and reason with vague knowledge.The main idea is to use truth values from the interval [0, 1] instead of only true and false.In this way, one can give a more fine-grained semantics to inherently vague concepts like LowFrequency or HighConcentration, which can be found in biomedical ontologies like SNOMED CT, 2 and Galen. 3Different FDLs are characterized not only by the constructors they allow, but also by the way these constructors are interpreted.To interpret conjunction in complex concepts like ∃hasHeartRate.LowFrequency ∃hasBloodAlcohol.HighConcentration, a popular approach is to use so-called t-norms [27].The semantics of the other logical constructors can then be derived from these t-norms in a principled way, as suggested in [20].Following the principles of mathematical fuzzy logic, existential and value restrictions are interpreted as suprema and infima of truth values, respectively.However, to avoid problems with infinitely many truth values, reasoning in fuzzy DLs is often restricted to so-called witnessed models [21], in which these suprema (infima) are required to be maxima (minima); i.e. the degree is witnessed by at least one domain element.
Unfortunately, most FDLs become undecidable when the logic allows to use GCIs and negation under witnessed model semantics [2,13,18].One of the few exceptions are FDLs using the Gödel t-norm, which is defined as min{x, y}, to interpret conjunctions [12].In the absence of an involutive negation constructor and negated assertions, such FDLs are even trivially equivalent to classical DLs [13].However, in the presence of the involutive negation, reasoning becomes more complicated.Despite not being as well-behaved as finitely valued FDLs, which use a finite total order of truth values instead of the infinite interval [0, 1], it was shown using an automata-based approach that reasoning in Gödel extensions of ALC exhibits the same complexity as in the classical case, i.e. it is ExpTime-complete [12].A major drawback of this approach is that it always has an exponential runtime, even when the input ontology has a simple form.
In the present paper, we present a combination of the automata-based construction for ALC from [12] and automata-based algorithms and reduction techniques developed for more expressive finitely valued FDLs [6,10,11,14,15,31].We exploit the forest model property of classical DLs [17,25] to encode order relationships between concepts in a fuzzy interpretation in a manner similar to the Hintikka trees from [12].However, instead of using automata to determine the existence of such trees, we reduce the fuzzy ontology directly into a classical ALCOQ ontology, which enables us to use optimized reasoners for classical DLs.In addition to the cut-concepts of the form C p for a fuzzy concept C and a value p, which are used in the reductions for finitely valued DLs [6,10,31], we employ order concepts C D expressing relationships between fuzzy concepts.In contrast to the reductions for finitely valued Gödel FDLs [6,7], our reduction does not produce an exponential blowup in the nesting depth of concepts in the input ontology.
Although our reduction deals with the Gödel extension of SROIQ, it is not correct if all three constructors nominals (O), inverse roles (I), and number restrictions (Q) are present in the ontology, since then one cannot restrict reasoning to forest-shaped models [32].However, it is correct for SRIQ, SROQ, and SROI, and we obtain several complexity results that match the currently best known upper bounds for reasoning in (sublogics of) these DLs.In particular, we show that reasoning in Gödel extensions of SRIQ is 2-ExpTime-complete, and for SHOI and SHIQ it is ExpTime-complete.

Preliminaries
We consider vague statements taking truth degrees from the infinite interval [0, 1], where the Gödel t-norm min{x, y} is used to interpret logical conjunction.The semantics of implications is given by the residuum of this t-norm; i.e., x ⇒ y := 1 if x y, y otherwise.
Note that min is monotone in both arguments, and hence preserves arbitrary infima in suprema, while ⇒ is monotone in the second argument and antitone in the first argument.We furthermore have the following useful property.
We recall some basic definitions from [12].An order structure S is a finite set containing at least the numbers 0, 0.5, and 1, and an involutive unary operation inv : S → S such that inv(x) = 1 − x for all x ∈ S ∩ [0, 1].A total preorder over S is a transitive and total binary relation ⊆ S × S. For x, y ∈ S, we write x ≡ y if x y and y x.Notice that ≡ is an equivalence relation on S. The total preorders considered in [12] have to satisfy additional properties, e.g. that 0 and 1 are always the least and greatest elements, respectively.These properties can be found in our reduction in the axioms of red(U) (see Section 4).
We now define the fuzzy description logic G-SROIQ.Let N I , N C , and N R be three mutually disjoint sets of individual names, concept names, and role names, respectively, where N R contains the universal role r u .The set of (complex) roles is the elements of the form r − are called inverse roles.Since there are several syntactic restrictions based on which roles appear in which role axioms, we start by defining role hierarchies.A role hierarchy R h is a finite set of (complex) role inclusions of the form w r p , where r = r u is a role name, w ∈ (N − R ) + is a non-empty role chain not including the universal role, and p ∈ (0, 1].Such a role inclusion is called simple if w ∈ N − R .We extend the notation • − to inverse roles r − and role chains w = r 1 . . .r n by setting (r − ) − := r and w − := r − n . . .r − 1 .We recall the regularity condition from [5,23].Let ≺ be a strict partial order on • w is of the form rr or r − , or • w is of the form r 1 . . .r n , rr 1 . . .r n , or r 1 . . .r n r, and for all 1 i n it holds that r i ≺ r.
An role hierarchy R h is regular if there is a strict partial order ≺ as above such that each role inclusion in R h is ≺-regular.A role name r is simple (w.r.t.R h ) if for each w r p ∈ R h we have that w is of the form s or s − for a simple role s.This notion is well-defined since the regularity condition prevents any cyclic dependencies between role names in R h .An inverse role r − is simple if r is simple.In the following, we always assume that we have a regular role hierarchy R h .
Concepts in G-SROIQ are built from concept names using the constructors listed in the upper part of Table 1, where C, D denote concepts, p ∈ [0, 1], n ∈ N, a ∈ N I , r ∈ N − R , and s ∈ N − R is a simple role.The restriction to simple roles in at-least restrictions is necessary to ensure decidability [24].We also use the common DL constructors := 1 (top concept), ⊥ := 0 (bottom concept), C D := ¬(¬C ¬D) (disjunction), and n s.C := ¬( (n + 1) s.C) (at-most restriction).
Notice that in [7], fuzzy at-most restrictions are defined using the residual negation: n s.C := ( (n + 1) s.C) → ⊥.This has the effect that the value of n r.C is always either 0 or 1 (see the semantics below).However, this discrepancy in definitions is not an issue since our reduction can handle both cases.The use of truth constants p for p ∈ [0, 1] is not standard in FDLs, but it allows us to simulate fuzzy nominals [4] of the form {p 1 /a 1 , . . ., p n /a n } with p i ∈ [0, 1] and The semantics of G-SROIQ is based on G-interpretations I = (∆ I , • I ) over a non-empty domain ∆ I , which assign to each individual name a ∈ N I an element a I ∈ ∆ I , to each concept name A ∈ N C a fuzzy set A I : ∆ I → [0, 1], and to each role name r ∈ N R a fuzzy binary relation r I : ∆ I × ∆ I → [0, 1].This interpretation is extended to complex concepts and roles as defined in the last column of Table 1, for all d, e ∈ ∆ I .We restrict all reasoning problems to witnessed G-interpretations [21], which intuitively require the suprema and infima in the semantics to be maxima and minima, respectively.Formally, a G-interpretation • a disjoint role axiom dis(r, s) p if min{r I (d, e), s I (d, e)} 1 − p holds for all d, e ∈ ∆ I ; • a reflexivity axiom ref(r) p if r I (d, d) p holds for all d ∈ ∆ I ; • an ontology if it satisfies all its axioms.
An ontology is consistent if it has a (witnessed) model.
We can simulate other common role axioms in G-SROIQ [7,22] by those we introduced above: • transitivity axioms tra(r) p by rr r p ; • symmetry axioms sym(r) p by r − r p ; • asymmetry axioms asy(s) p by dis(s, s − ) p ; • irreflexivity axioms irr(s) p by ∃s.Self ¬p 1 ; and We consider ¬¬C to be equal to C, and thus sub(O) is of quadratic size in the size of O.We denote by V O the closure under the involutive negation x → 1−x of the set of all truth degrees appearing in O (either in axioms or in truth constants), together with 0, 0.5, and 1.This set is of linear size.
Other common reasoning problems for FDLs, such as concept satisfiability and subsumption can be reduced to consistency [12]: the subsumption between C and D to degree q w.r.t. a TBox T and an RBox R is equivalent to the inconsistency of ({ a:C → D < q }, T , R), and the satisfiability of C to degree q w.r.t.T and R is equivalent to the consistency of ({ a:C q }, T , R).
The letter I in G-SROIQ denotes the presence of inverse roles and the universal role.If such roles are not allowed, the resulting logic is written as G-SROQ.
Likewise, G-SRIQ indicates the absence of nominals, and G-SROI that of atleast and at-most restrictions.Replacing the letter R with H indicates that RBoxes are restricted to simple role inclusions, ABoxes are restricted to order assertions, and local reflexivity is not allowed; however, the letter S indicates the presence of transitivity axioms.Hence, in G-SHOIQ we are allowed to use role inclusions of the forms r s p and rr r p .Disallowing axioms of the first type removes the letter H, while the absence of transitivity axioms is denoted by replacing S with ALC.
Classical DLs are obtained from the above definitions by restricting the set of truth values to 0 and 1.The semantics of a classical concept C is then viewed as a set C I ⊆ ∆ I instead of the characteristic function C I : ∆ I → {0, 1}, and likewise for roles.In this setting, all axioms (also order assertions) are restricted to be of the form α 1 , and usually this is simply written as α, e.g.C D instead of C D 1 .We also use C ≡ D to abbreviate C D and D C. Furthermore, the implication constructor C → D, although usually not included in classical DLs, can be expressed via ¬C D.
In this paper, we provide a reduction from a G-SROIQ ontology to a classical ALCOQ ontology.For all sublogics of G-SROIQ that do not contain the constructors O, I, and Q at the same time, the reduction preserves consistency.Before we describe the main reduction, however, we provide a characterization of role hierarchies using (weighted) finite automata.

Automata for Complex Role Inclusions
Let O = (A, T , R) be a G-SROIQ ontology.We extend the idea from [23] of using finite automata to characterize all role chains that imply a given role w.r.t.R h .
In our setting, we need to use a certain kind of weighted automata [19], which use as input symbols the roles in rol(O), and compute a weight for any given input word.
Definition 3.1 (WFA).A weighted finite automaton (WFA) is a quadruple A = (Q, q ini , wt, q fin ), consisting of a non-empty set Q of states, an initial state q ini ∈ Q, a transition weight function wt : Q × (rol(O) ∪ {ε}) × Q → [0, 1], and a final state q fin ∈ Q.Given an input word w ∈ rol(O) * , a run of A on w is a non-empty sequence of pairs r = (w i , q i ) 0 i m such that (w 0 , q 0 ) = (w, q ini ), (w m , q m ) = (ε, q fin ), and for each i, 1 i m, it holds that w i−1 = x i w i for some x i ∈ rol(O) ∪ {ε}.The weight of such a run is wt(r) := min m i=1 wt(q i−1 , x i , q i ).The behavior of A on w is ( A , w) := sup r run of A on w wt(r).
We often denote by q x,p −→ q ∈ A the fact that wt(q, x, q ) = p.Further, for a state q of A, we denote by A q the automaton resulting from A by making q the initial state.The following connection is easy to see by the definition of the behavior of a WFA.Proposition 3.2.Let A be a WFA, q x,p −→ q ∈ A, and w ∈ rol(O) * .Then ( A q , xw) min{p, ( A q , w)}.
A mirrored copy A − is constructed from A by exchanging initial and final states, and replacing each transition q x,p −→ q by q x − ,p −−→ q, where ε − := ε.Proposition 3.3.Let A be a WFA, A be a mirrored copy of A, and w ∈ rol(O) * .Then ( A , w) = ( A , w − ).
Following [23], we now construct, for each role r, a WFA A r that recognizes all role chains that "imply" r w.r.t.R h (with associated degrees).This construction proceeds in several steps.The first automaton A 0 r contains the initial state i r , the final state f r , and the transition i r r,1 − → f r , as well as the following transitions for each w r p ∈ R: where all states q i w are distinct.Here and in the following, all transitions that are not explicitly mentioned have weight 0.
The WFA A 1 r is now defined as A 0 r if there is no role inclusion of the form r − r p ∈ R; otherwise, A 1 r is the disjoint union of A 0 r and a mirrored copy of A 0 r , where i r is the only initial state, f r is the only final state, and the following transitions are added for the copy f r of f r and the copy i r of i r : i r ε,p Finally, we define the WFA A r by induction on ≺ as follows: r with s = r. 4 For each such transition, we add ε-transitions with weight 1 from q to the initial state of A 1 s and from the final state of A 1 s to q .
• The automaton A r− is a mirrored copy of A r .
The difference to the construction in [23] is only the inclusion of the appropriate weights for each considered role inclusion.As shown in [23], the size of each A r is bounded exponentially in the length of the longest chain r 1 ≺ • • • ≺ r n for which there are role inclusions The following lemma describes the promised characterization of the role inclusions in R in terms of the behavior of the automata A r .Intuitively, the degree to which the interpretation of w must be included in the interpretation of r is determined by the behavior of A r on w.For the other direction, assume that I satisfies R, and let r ∈ rol(O), w ∈ rol(O) + , and d, e ∈ ∆ I .We prove the claim by well-founded induction on ≺.It suffices to show the claim for all role names r since A r − is a mirrored copy of A r .
If ( A r , w) = 0 or w I (d, e) = 0, then the claim is trivially satisfied.If both values are > 0, then due to the construction of A r there must be • a word w = r 1 . . .r n ∈ rol(O) + such that r i ≺ r or r i = r holds for all 1 i n, and where, if r i = r, then w i = r, and thus ( A r i , w i ) = 1.Since we have ( A r i , w i ) > 0, 1 i n, we know by the construction of A r i that all w i are non-empty.
Since w = w 1 . . .w n , we have where we set d 0 := d and d n := e.For any such choice of , by the induction hypothesis.But this also holds for r i = r since then w i = r.Hence, we obtain We proceed by a case distinction on the transitivity and symmetry properties of r in R.
1. Assume that no role inclusions of the form rr r p or r − r p occur in R. Since ( A 1 r , w ) > 0, by construction of A 1 r = A 0 r we know that w is of the form w = u 1 . . .u m tv 1 . . .v k such that • either t r p ∈ R or t = r (and then we set p := 1), • rv j r p j ∈ R for all 1 j k, and 2. Consider the case that rr r p t ∈ R, but there is no role inclusion r − r p ∈ R. Then w must be of the form • for each 1 o , either t (o) r p (o) ∈ R or t (o) = r (and then we set p (o) := 1), • rv and 1 j k o , and The claim can be obtained by the same arguments as in Case 1.Note that the axiom rr r p t is only needed if > 1.
3. If r − r p s ∈ R, but there is no role inclusion rr r p ∈ R, then w is of the form w = u 1 . . .u m tv 1 . . .v k , where in the latter two cases we set p := 1), • rv j r p j ∈ R or v − j r r p j ∈ R for all 1 j k, and • ( A 1 r , w ) = min{p s , p 0 } if one of the "inverse" cases applies, and ( A 1 r , w ) = p 0 otherwise, where p 0 := min{p 1 , . . ., p m , p, p 1 , . . ., p k }.The claim can be obtained as in Case 1.
4. If both rr r p t and r − r p s are present in R, then w is a non-empty sequence of words of the form described in Case 3, and the claim can be shown as before.
For the universal role r u , we define A ru as above based on the role inclusions r − u r u 1 , r u r u r u 1 , and r r u 1 for all r ∈ rol(O).Hence, A ru accepts any (non-empty) word w ∈ rol(O) + with degree 1, and it is easy to see that Lemma 3.4 also holds for r u .
We define the relation p as the "transitive closure" of the simple role inclusions in R: we set r p s iff p is the supremum of the values min{p 1 , . . ., p n } over all sequences r r 1 p 1 , . . ., r n−1 s p n in R. Note that r 1 r because of the empty sequence.Proposition 3.5.For a simple role r and w ∈ rol(O) * , we have and s p r, 0 otherwise.

The Reduction
We now describe the reduction from O to a classical ALCOQ ontology red(O).This reduction always uses nominals, even in the logic G-SRIQ.However, if number restrictions are not allowed (e.g. in G-SROI), then red(O) is an ALCO ontology.
As a first pre-processing step, we eliminate role assertions (a, b):r from the ABox by replacing them with the equivalent concept assertions a:∃r.{b}; this simplifies the following reduction.We now extend the set sub(O) by the following elements (and their negations): • We add all nominals {a} for a ∈ ind(O) to be able to distinguish all named domain elements.
• We further consider all concepts ∃r.Self with r ∈ rol(O) (also for nonsimple roles), in order to represent the degrees to which a domain element is connected to itself, e.g. for reflexivity axioms.
• We add all "concepts" of the form ∀A q r .C (∃A q r .C) for all ∀r.C (∃r.C) occurring in O and all states q of A r .These concepts help to transfer the constraints imposed by the existential and value restrictions along all role chains that imply the possibly non-simple role r.The semantics of ∀A.C is defined as follows: The idea is that in our reduction we do not need to explicitly represent all role connections, but only a "skeleton" of connections which are necessary to satisfy the witnessing conditions for role restrictions.The restrictions for all implied role connections are then handled by the concepts ∀A r .C and ∃A r .C by simulating the transitions of A r ; each transition corresponds to a role connection to a new domain element.Note that we do not need to introduce concepts of the form n A r .C since all roles in at-least restrictions must be simple, i.e. there can be no role chains of length > 1 that imply them (at least not with a degree > 0).
The main idea of the reduction is that instead of precisely defining the interpretation of all concepts at each domain element, it suffices to consider a total preorder on them.For example, if an axiom restricts the value of C → D at each domain element to be 0.5, then we do not have to find the exact values of C and D, but only to ensure that either C I (d) D I (d) or else D I (d) 0.5.This information is encoded by total preorders over the order structure U that is defined below.The other main insight for our reduction is that we consider only (quasi-)forest-shaped models of O [17].In such a model, the domain elements identified by individual names serve as the roots of several tree-shaped structures.The roots themselves may be arbitrarily interconnected by roles.Due to nominals, there may also be role connections from any domain element back to the roots.Note that complex role inclusions may actually imply role connections between arbitrary domain elements, but the underlying tree-shaped "skeleton" is what is important for reasoning purposes (for details, see [17] and our correctness proof in [16]).This dependence on forest-shaped models is the reason why our reduction works only for G-SROI, G-SROQ, and G-SRIQ-even classical ALCOIQ does not have the forest model property [32].
We then define the order structure U as follows: In order to describe such total preorders over U with a classical ALCOQ ontology, we use special concept names of the form α β for α, β ∈ U.This differs from previous reductions for finitely valued FDLs [7,9,31] in that we not only consider cut-concepts of the form q α with q ∈ V O , but also relationships between different concepts. 5We use the abbreviations α β := β α , α < β := ¬ α β , and similarly for = and >.Furthermore, we define the complex expressions and extend these notions to α β ⇒ γ etc., for ∈ {<, =, >}, analogously.
In our reduction, we additionally use the special concept name AN to identify the anonymous domain elements, i.e. those which are not of the form b I for any b ∈ ind(O).The reduction uses only one role name r.The reduced ontology red(O) consists of the parts red(U), red(A), red(AN), red(↑), red(R), red(T ), and red(C) for all C ∈ sub(O), which we describe in the following.We want to emphasize that red(O) is formulated in ALCOQ, whenever O is in G-SRIQ or G-SROQ, and in ALCO if O is a G-SROI ontology.This is due to the fact that we always use nominals to distinguish the named from the anonymous part of the forest-shaped model, and the inverse of r is not needed in the reduction.
The first part of red(O) is These axioms ensure that at each domain element the relation " " forms a total preorder that is compatible with the values in V O , and that inv is an antitone operator.
To describe the behavior of all named elements, we use the following axioms: {a} .
The following axioms ensure that the order of a node in a tree-shaped part of the model is known at each of its successors via the elements of sub ↑ (O): We now come to the reduction of the RBox: The concepts and axioms concerning the universal role, inverse roles, and (local) reflexivity statements are only included in the reduction if the logic under consideration supports them.
These axioms ensure that the various elements of U that represent the values of role connections, such as (a, b):r, ∃r.Self, and r, respect the axioms in R.
Although the simple role inclusions r s p are handled by the automata A r , we include them also in red(R).The reason is that the reduction of at-least restrictions below does not need to use these automata since only simple roles can occur in them.
The GCIs in T can be translated in a straightforward manner: We now come to the reductions of the concepts.Intuitively, each red(C) describes the semantics of C in terms of its order relationships to other elements of U. Note that the semantics of the involutive negation ¬C = inv(C) is already handled by the operator inv (see red(U) above): The reductions of role restrictions are more involved.In particular, in the case of value and existential restrictions we have to deal with non-simple roles, for which we employ the automata A r from the previous section.
∃r. AN Here and in the following definitions, we label with (I) those concepts or axioms that are contingent on the presence of inverse roles in the source logic.Likewise, terms labeled with (S) are only included if (local) reflexivity statements are allowed, and similarly for (N) and nominals.
The second axiom of red(∀r.C) ensures the existence of a witness for ∀r.C at each anonymous domain element.For example, assume that the preorder represented by the concepts α β at some domain element d satisfies 0.5 < ∀r.C < 1.
The first possibility is that the above axiom creates an anonymous element e that is connected to d via r, and hence by red(AN) we know that e satisfies 0.5 < ∀r.C ↑ < 1.The axiom further requires that ∀r.C ↑ r ⇒ C, which implies that ∀r.C ↑ C and r > C. We will see below that the reduction of ∀A r .C further ensures that ∀r.C ↑ r ⇒ C, and thus we get ∀r.C ↑ = C. Hence, e can be seen as an abstract representation of the witness of ∀r.C at d; the precise value of the r-connection between d and e (represented by the element r) is irrelevant, as long as it is strictly greater than the value of C at e.The other disjuncts of this axiom deal with the possibility that d itself, its predecessor, or a named domain element acts as the witness for the value restriction in a similar way.The assertions deal with the case of a named domain element, in which case the first two options above (self and inverse) are subsumed by the last option.
Together with the first axiom of red(∀r.C), the following axioms ensure that no other r-successor of d violates the lower bound on r ⇒ C given by ∀r.C at d: −→q ∈A red x,p,q (∀A q .C) red ε,p,q (∀A q .C) := { (∀A q .C) p ⇒ (∀A q .C) } red s,p,q (∀A q .C) := {AN (∀A q .C) min{p, s − } ⇒ ∀A q .C ↑ , (I) ∀r. AN → ∀A q .C ↑ min{p, s} ⇒ (∀A q .C) } ∪ {∃r.{a} (∀A q .C) min{p, ( * , a):s} ⇒ a:(∀A q .C) , ∃r.{a} a:(∀A q .C) min{p, ( * , a): Recall that A r in particular contains the transition i r Using arbitrary runs through the automaton A r , we can ensure that no other r-successor of d violates the value restriction.For example, if r I (d, e 1 ) = 0.3 and r I (e 1 , e 2 ) = 0.5 for two other (anonymous) domain elements e 1 , e 2 , and we further have the role inclusion rr r 0.7 , then we know that r I (d, e 2 ) must be at least 0.5.Although this r-connection is not explicitly represented in our forest-based encoding, concepts of the form ∀A q r .C are appropriately transferred from d via e 1 to e 2 in order to ensure that the value of C at e 2 satisfies 0.5 < (∀r.C) I (d) r I (d, e 2 ) ⇒ C I (e 2 ).In this example, since we know only that r I (d, e 2 ) 0.5, it must be ensured that C I (e 2 ) r I (d, e 2 ).
The reduction for existential restrictions can be defined similarly to that for value restrictions, but replacing with (and vice versa) and ⇒ with min.
We now come to the final component of red(O).If we do not have inverse roles, (local) reflexivity, or nominals, then we fix the numbers z i , z s , or m, respectively, to 0, 0, or n−z i −z s , which effectively eliminates the conjuncts using these constructors from the above axioms.
The reduction of at-least restrictions works similarly to the one of value restrictions: the first axiom ensures the existence of the n required witnesses, while the second one ensures that no n different elements can exceed the value of the at-least restriction.Unfortunately, the number of named successors cannot be counted using a classical at-least restriction in our encoding, since these named successors do not know about the degree of the role connection from an anonymous element; otherwise they would have to store a possibly infinite amount of information since they may have infinitely many anonymous role predecessors.For this reason, the above axioms first guess how many (m − n) and which (S) named elements are connected to the current domain element to the appropriate degrees (given by ( * , a):r).For named elements, however, this guessing is not necessary.
This reduction is correct in the sense that the resulting ontology red(O) has a classical model iff O has a G-model.As mentioned before, this holds only for the sublogics SRIQ, SROQ, and SROI that have the forest model property [17].However, the correctness is not affected by the presence or absence of (local) reflexivity statements.

Soundness
We first show that, in SRIQ, SROQ, or SROI, if red(O) has a classical model, then O has a G-model.
Since red(O) contains only the role name r and no inverses, and hence is in ALCOQ, we can assume that it has a quasi-forest model I with the following properties [17]: • for each a ∈ ind(O), the set {u ∈ N * | (a, u) ∈ ∆ I } is prefix-closed; • for each a ∈ ind(O), we have a I = (a, ε); • for all a ∈ ind(O), u ∈ N * , and i ∈ N with (a, ui) ∈ ∆ I , we have ((a, u), (a, ui)) ∈ r I ; and • whenever ((a, u), (b, u )) ∈ r I , then a) a = b and u = ui for some i ∈ N or b) u = ε.
We assume here that all named individuals in ind(O) are interpreted by distinct elements in I.In general, we would have to consider sets of names from ind(O) as the roots of I. Since this is relevant only for number restrictions and (in)equality assertions, we ignore this in the following and only mention it at the appropriate places.
For any u = n 1 . . .n k ∈ N * with k 1, we denote by u ↑ := n 1 . . .n k−1 its predecessor.We denote by A the binary relation on U A defined by α A β iff c I ∈ α β I for an arbitrary c ∈ ind(O).This is a total preorder due to the axioms in red(U).We similarly define α a u β iff (a, u) ∈ α β I , for all α, β ∈ U. Since I satisfies red(A) and all domain elements are connected via r, we have A ⊆ a u for all (a, u) ∈ ∆ I .We further denote by ≡ A (≡ a u ) the equivalence relation induced by A ( a u ).As a first step in the construction of a G-model of O, we now construct a function v : We start defining v on U A .Let U A / ≡ A be the set of all equivalence classes of ≡ A .Then A yields a total order A on U are the elements of V O , then since I satisfies red(U), we have which is well-defined because of the axioms in red(U).On all α ∈ [p] A for p ∈ V O , we now define v(α) := p, which ensures that (P1) holds.For the equivalence classes that do not contain a value from V O , note that by red(U), every such class must be strictly between [p i ] A and [p i+1 ] A for some p i , p i+1 ∈ V O .We denote the n i equivalence classes between [p i ] A and [p i+1 ] A as follows: For every α ∈ E i j , 1 j n i , we set v(α) := p i + j n i +1 (p i+1 − p i ), which ensures that (P2) is also satisfied.Furthermore, 1 − p i+1 and 1 − p i are also adjacent in V O and we have A by the axioms in red(U).Hence, it follows from the definition of v(α) that (P3) holds.
We now construct the values of v(α, a, ε) using a similar technique.However, we now start by setting v(α, a, ε) A , and hence v(β) = v(β ) by (P2).We can now arrange all other values between those already fixed as shown above, thereby satisfying (P5)a)-c).Since a I = (a, ε) and I satisfies red(A), this construction also ensures that (P4) is satisfied.
We now proceed to define v(α, a, u) by induction on the structure of the tree {u | (a, u) ∈ ∆ I }.Assume that v(α, a, u ↑ ) has already been defined for all α ∈ U, and satisfies (P5)a)-c).By assumption, we have ((a, u ↑ ), (a, u)) ∈ r I , and by red(AN) we know that (a, u) ∈ AN I .We again use the same construction as before, but this time fixing all values v(α, a, u) := v(α, a, u ↑ ) for all α ∈ U A and v(α, a, u) := v(C, u ↑ ) for all C ∈ sub(O) and all α ∈ [ C ↑ ] u .This is well-defined by the same arguments as above and the fact that I satisfies red(↑).We then fix the remaining values as before.This construction ensures that (P5)a)-d) are satisfied.
Based on v, we now define the G-interpretation I f over the domain ∆ I f := ∆ I , where a I f := a I = (a, ε) for all a ∈ ind(O) 6 and A I f (d) := v(A, d) for all A ∈ N C ∩ sub(O) and d ∈ ∆ I f .The values for all other concept names are irrelevant and can be fixed arbitrarily.For all r ∈ N R ∩ rol(O), we first define a "simple" interpretation I 0 f as follows.and ((a, u), (b, ε)) ∈ r I , 0 otherwise In the absence of inverse roles, we set the second and fifth lines to 0; and if (local) reflexivity is not allowed, then the third line is 0; finally, if there are no nominals in our source logic, then the fifth and sixth lines are 0. Due to red(A) and red(∃r.Self), the same expressions as for role names can be used to evaluate inverse roles.In the case that r is simple, this already suffices.Otherwise, we use the automaton A r to complete I 0 f by additional links as follows: we set for all d, e ∈ ∆ I f .Note that this expression is also valid for simple roles r: by Proposition 3.5, we have ( A r , s) = p whenever s p r, ( A r , r) = 1, and ( A r , w) = 0 for all other words w, and moreover red(R) yields The expression (3) can also be used to evaluate inverse roles due to the semantics of role chains and Proposition 3.3.Finally, for the universal role r u , we have r due to the axioms in red(R).By the construction of A ur , we also have ( A ru , r u ) = 1, and hence the expression (3) also holds for the universal role and we have r I f u (d, e) = 1 for all d, e ∈ ∆ I .We now show by induction on the structure of C that where we exclude the auxiliary concepts of the form ∀A.C and ∃A.C.
For nominals {a}, we have {a} I f (d) = 1 if d = (a, ε), and {a} I f (d) = 0 otherwise.By red({a}) and (P5)a)-b), in the former case we have v({a}, d) = 1, while in the latter case it holds that v({a}, d) = 0.For local reflexivity concepts ∃r.Self, we have (∃r.Self) ) by the definition of r I f .For ¬C, we have by the induction hypothesis and (P5)c).The claim for , q, , and → can also be shown using the induction hypothesis, the semantics of these constructors, and the properties (P5)a)-b) of v.
For value restrictions ∀r.C, consider first the case that d = (a, u) with u = ε, and hence d ∈ AN I .By the second axiom in red(∀r.C), one of the following must hold: • We have d ∈ (∀r.C) r − ⇒ C ↑ , and thus (P5)b) and d) and the induction hypothesis yield Hence, (a, u ↑ ) can take the role of the witness for ∀r.C at d if we can show that the latter implication is v(∀r.C, d) for all successors.
• We have by similar arguments as above.In this case, we can choose d itself as the witness.
• There is an anonymous r-successor of d that satisfies ∀r.C ↑ r ⇒ C , which must be of the form (a, ui) for i ∈ N due to our assumption on the structure of I. We get • There is a b ∈ ind(O) such that (d, (b, ε)) ∈ r I , and again we have due to (P5)b) and d), red(A), and the induction hypothesis.
The assertions in red(∀r.C) similarly ensure the existence of witnesses for ∀r.C at all named domain elements.For the remainder of the claim, consider any e ∈ ∆ I f .
Due to the first axiom in red(∀r.C), we have as required, if we can show ( * ), i.e. it remains to show that holds for all d 0 , d n ∈ ∆ I f and w = r 1 . . .r n ∈ rol(O) + .Since ( A r , w) and w I 0 f (d 0 , d n ) are defined as suprema, it suffices to consider any run r = (w i , q i ) 0 i m with (w 0 , q 0 ) = (w, i r ), (w m , q m ) = (ε, f r ), and transitions q i−1 x i ,p i −−→ q i in A r , and any sequence d 1 , . . ., d m ∈ ∆ I f .To synchronize these two sequences, we define the mapping γ : {0, . . ., m} → {0, . . ., n}, where γ(0) := 0, and For all i, 0 i m.By the axiom (∀A qm r .C) C in red(∀A qm r .C) and the induction hypothesis, the claim for m implies ( * ).
For i = 0, (5) trivially holds.Assume now that it holds for i − 1.To show the claim for i, by Proposition 2.1 it suffices to show that v(∀A whenever x i = ε (and hence γ(i) = γ(i − 1)), and v(∀A for all x i = ε (for which we have γ(i) = γ(i − 1) + 1).
For the former case, the axioms in red ε,p i ,q i (∀A qm r .C) and (P5) directly yield the claim (6).If x i / ∈ ε, we make a case distinction on d γ(i) .If r , d γ(i) ) = 0, then the claim is trivially satisfied; otherwise, we must have one of the following cases: • If d γ(i) = (a, u) and d γ(i−1) = (a, u ↑ ), then we have (d γ(i−1) , d γ(i) ) ∈ r I and d γ(i) ∈ AN I .Hence, the third axiom in red r γ(i) ,p i ,q i (∀A as claimed in (7).
∈ AN I and inverse roles are allowed.Hence, the first axiom in red r γ(i) ,p i ,q i (∀A , then (local) reflexivity is allowed, and we similarly get by the second axiom in red r γ(i) ,p i ,q i (∀A .C).
This concludes the proof of (7), and hence that of (5) and of ( * ), which shows that (4) holds for all value restrictions.The proof for existential restrictions can be done using dual arguments.
For at-least restrictions n r.C, note first that r must be simple, and hence we have r I f (d, e) = r I 0 f (d, e) for all d, e ∈ ∆ I f .We first consider the case that d ∈ AN I , i.e. it is of the form (a, u) with u = ε.By the first axiom in red( n r.C), there are z i , z s ∈ {0, 1}, m ∈ {0, . . ., n − z i − z s } and S ⊆ ind(O) with |S| = n − m − z i − z s such that red z i ,zs,m,S, ( n r.C) is satisfied by d.
Further, we obtain by the induction hypothesis.
• If z s = 1, then local reflexivity is allowed and we have • Consider now any b ∈ ind(O).We know that nominals are allowed and (d, (b, ε)) ∈ r I , and thus Furthermore, all elements of S must be interpreted by different elements in I, and hence also in I f , even if we consider sets of individual names in the roots of I.
• Additionally, there are m different elements e 1 , . . ., e m ∈ ∆ I such that (d, e j ) ∈ r I and e j ∈ AN I for each e j , and hence they must be of the form (a, ui j ).We obtain, for every j, Note that all r-successors of d considered above, i.e. (a, u ↑ ), d, (b, ε), and e j , 1 j m, must be different; in particular, we do not consider nominals and inverse roles at the same time (since obviously O contains at-least restrictions), and thus even if u ↑ = ε, we do not have a ∈ S. Hence, these elements can serve as the witnesses for n r.C at d (assuming that its value is exactly v( n r.C), which is shown below).For named domain elements, the witnesses are created by the first kind of assertions in red( n r.C), where only two cases need to be considered (named and unnamed successors); note that all unnamed r-successors of (a, ε) must be of the form (a, i) due to our assumptions on the structure of I.
Assume now again that d = (a, u) ∈ AN I and that the elements found above are not witnesses for ( n r.C) I f (d) = v( n r.C, d).Then there must be n different elements e 1 , . . ., e n ∈ ∆ I f such that v( n r.C, d) < min{r I f (d, e j ), C I f (e j )} for all j, 1 j n.We show that we can find suitable z i , z s , m, and S such that red z i ,zs,m,S,< ( n r.C) is satisfied by d, which contradicts our assumption that I is a model of red(O).
• If inverse roles are allowed and there is an index j, 1 j n, such that e j = (a, u ↑ ), then we set z i := 1.By the induction hypothesis and our assumption above, we have • If (local) reflexivity is allowed and there is an index j, 1 j n, such that e j = d, then we set z s := 1, and get v( n r.C, d) < min{v(∃r.Self, d), v(C, d)}.
• If nominals are allowed, then we collect from the remaining elements those e j that are equal to (b, ε) for some b ∈ ind(O).Let S be the set all of all those individual names.• There are exactly m := n − |S| − z i − z s remaining elements e j .If nominals are not allowed, then no e j can be of the form (b, ε) for b ∈ ind(O) since r I f (d, e j ) > 0 and d is anonymous.If inverse roles are not allowed, then e j = (a, u ↑ ) due to the same reason.Similarly, if local reflexivity is not allowed, it cannot be the case that e j = d.Thus, we know for each of the remaining e j that e j = (a, ui j ) for some i j ∈ N and v( n r.C ↑ , e j ) = v( n r.C, d) < min{v(r, e j ), v(C, e j )}.
This shows that also the final part of red z i ,zs,m,S,< ( n r.C), namely the restriction m r.AN n r.C ↑ < min{r, C} is satisfied.
For named elements d = (a, ε), we can use a similar argument to contradict the second kind of assertions in red( n r.C).Note that there can be no anonymous element e j satisfying r I f (d, e j ) > 0 that is not of the form e j = (a, i j ) for some i j ∈ N, since otherwise we know from the definition of r I f that both inverse roles and nominals must be allowed, which cannot be the case since obviously number restrictions are allowed.
This concludes the proof of (4).It remains to show that I f is a model of O.
For the complex role inclusions in R, by Lemma 3.4 it suffices to show that w I f (d, e) ⇒ r I f (d, e) ( A r , w) holds for all r ∈ rol(O), w ∈ rol(O) + , and d, e ∈ ∆ I f .We can assume that w I f (d, e) > 0 and ( A r , w) > 0 since otherwise this inequation is trivially satisfied.Let now w = r 1 . . .r n , n 1.Then we have where we set d 0 := d and d n := e.Furthermore, for any choice of elements 7 Recall that we have eliminated all crisp role assertions from the ABox.
as required.• all d i are elements of ∆ I ;

Completeness
• if number restrictions are allowed, then we have to put some restrictions on this sequence of domain elements: d 1 is not equal to b I for any b ∈ N I ; if reflexivity is allowed, there is no index i such that d i = d i+1 ; if nominals are allowed, then no d i is equal to b I for any b ∈ N I ; and if inverse roles are allowed, then d 2 = a I , and there is no index i such that For ease of presentation, we assume that all individual names are interpreted by distinct elements of ∆ I .In general, however, we would have to consider equivalence classes of individual names as roots for I c , where a, b ∈ N I are equivalent iff a I = b I .Since this is only relevant for number restrictions, we will ignore this for most of the proof and only mention it at the appropriate places.For red(R), consider first a role inclusion of the form r s p ∈ R. Then r I (prev( ), tail( )) ⇒ s I (prev( ), tail( )) p for any ∈ ∆ Ic \ N I ; furthermore, every a ∈ N I also satisfies r ⇒ s p since −1 −1.A similar argument can be made for (a, b):r ⇒ (a, b):s p and ∃r.Self ⇒ ∃s.Self p , and for the reduction of disjoint role axioms.For every ref(r) p ∈ R and every d ∈ ∆ I , we have (∃r.Self) I (d) = r I (d, d) p. Finally, the three concepts for the universal role are obviously also satisfied at every domain element.
It remains to consider the axioms in red(C) for C ∈ sub(O).The reductions for , {a}, q, ∃r.Self, , and → obviously reflect the semantics of these constructors and are easy to verify.
We now consider the axioms in red(∀r.C).By Lemma 3.For red(∀A q .C), we first consider the axiom (∀A q .C) C for a final state q of A. We have for all d ∈ ∆ I , and hence I c satisfies this axiom.For any transition q ε,p − → q in A q , we have to satisfy the axiom (∀A q .C) p ⇒ (∀A q .C) .By Proposition 3.2, we get Consider now the axiom AN (∀A q .C) min{p, s − } ⇒ ∀A q .C ↑ for a transition q r,p − → q in A, and any ∈ AN Ic , which must be of the form ad 1 . . .d k with k 1.We have by Propositions 2.1 and 3.2.For (∀A q .C) min{p, ∃s.Self} ⇒ (∀A q .C) and any d ∈ ∆ I , we get by similar arguments.For ∀r. AN → ∀A q .C ↑ min{p, s} ⇒ (∀A q .C) , consider any , ∈ ∆ Ic with ( , ) ∈ r Ic and ∈ AN Ic .Thus, we must have = d for some d ∈ ∆ I , and we know that prev( d) = tail( ).We obtain Consider now the axiom ∃r.{a} (∀A q .C) min{p, ( * , a):s} ⇒ a:(∀A q .C) for any a ∈ ind(O), and ∈ ∆ Ic with ( , a) ∈ r Ic .We get = min{p, (( * , a):s) I ( )} ⇒ (a:(∀A q .C)) I ( ).
Finally, for ∃r.{a} a:(∀A q .C) min{p, ( * , a):s − } ⇒ (∀A q .C) and any a ∈ ind(O) and ∈ ∆ Ic with ( , a) ∈ r Ic , we obtain If reflexivity is allowed and we have e j = d k for one of these elements, we set z s := 1.If inverse roles are allowed and we have (i) k > 1 and e j = d k−1 or (ii) k = 1 and e j = a I , we set z i := 1.Note that the previous two elements identified for z s and z i must be different since otherwise we would have d k = d k−1 or d 1 = a I .If nominals are allowed, we define S to be the set of all individual names b ∈ ind(O) for which b I is among the remaining elements from e 1 , . . ., e n ; otherwise we set S := ∅.We thus have m := n−z i −z s −|S| remaining elements e j , and have uniquely identified one of the disjuncts of the axiom.We now show that for each of these elements e j the corresponding conjunct in this disjunct is satisfied, thus showing that the whole axiom is satisfied.
• If z s = 1, let e j be the element equal to d k .We have and thus the conjunct ( n r.C) min{∃r.Self, C} is satisfied by .• For any e j not corresponding to any of the previous cases, we know by the construction of ∆ Ic that e j ∈ ∆ Ic , and hence ( , e j ) ∈ r Ic and n r.C I ↑ ( e j ) min{r I (d k , e j ), C I (e j )} = min{r I ( e j ), C I ( e j )}.
Since there are m different such elements e j , the corresponding elements e j are also different, and m r.AN n r.C ↑ min{r, C} is satisfied by .
For the second axiom in red( n r.C), assume to the contrary that there is a = ad 1 . . .d k ∈ ∆ Ic , and numbers z i (if there are inverse roles), z s (if reflexivity is allowed), 0 m n−z i −z s , and a set S ⊆ ind(O) of cardinality n−m−z i −z s (which is 0 unless there are nominals) such that the corresponding conjunction is satisfied by in I c .Due to the construction of ∆ Ic , the above elements of ∆ I (prev( ), tail( ), a I for a ∈ S, and e j , 1 j m) must be different.But this contradicts the semantics of ( n r.C) I (tail( )).
For the first kind of assertions in red( n r.C), consider any a ∈ ind(O).Since I is witnessed, there must be n different elements e 1 , . . ., e n ∈ ∆ I such that For each e j , 1 j n, we make a case distinction on whether it is named or not.Since different elements of ∆ I induce different elements of ∆ Ic , this shows that the required at-least restriction is satisfied by a.
For the second kind of assertions in red( n r.C), assume to the contrary that there are n different r-successors 1 , . . ., n of a that are either anonymous and satisfy n r.C ↑ < min{r, C} , or named and satisfy (a: n r.C) < min{(a, * ):r, C} .
• If j satisfies AN, then it must be of the form ae j for some e j ∈ ∆ I and we have This concludes the proof of the following result.

Complexity
We now analyze the complexity of the reduction.As in [23], the construction of the automata A r causes an exponential blowup in the size of R, which cannot be avoided [26].Independent of this, our reduction also involves an exponential blowup in the (binary encoding of) the largest number n involved in a number restriction in O, and in the number of individual names occurring in O, since the number of disjuncts in each GCI from red( n r.C) is linear in n • 2 |ind(O)| .Note, however, that this blowup only occurs if we consider both nominals and number restrictions.Hence, we obtain the following complexity results.Proof.The consistency of the ALCOQ ontology red(O) is decidable in exponential time in the size of red(O) [17].The first upper bounds thus follow from the fact that the size of red(O) is exponential in the size of O. 2-ExpTime-hardness holds already for G-SRIQ without involutive negation and only assertions of the form α p since in this case reasoning in G-SRIQ is equivalent to reasoning in classical SRIQ [13,26].Without complex role inclusions, i.e. restricting to simple role inclusions and transitivity axioms, the size of the automata A r is polynomial in the size of R [23].The other exponential blowup can be avoided by disallowing nominals or number restrictions.Hence, for G-SHOI and G-SHIQ, the size of red(O) is polynomial in the size of O, and the lower bound follows again from the reduction in [13] and ExpTime-hardness of consistency in classical ALC [29].
To the best of our knowledge, it is still open whether consistency in SROI and SROQ is actually 2-ExpTime-hard, even in the classical case [17,28]; the best known lower bound is given by the ExpTime-hardness for ALC [29].We also leave open the precise complexity of G-SHOQ, which is ExpTime-complete in the classical case [17,29].

Conclusions
Using a combination of techniques developed for infinitely valued Gödel extensions of ALC [12] and for finitely valued Gödel extensions of SROIQ [6,7], we derived several tight complexity bounds for consistency in sublogics of G-SROIQ.
Our reduction is more practical than the automata-based approach in [12] and does not exhibit the exponential blowup of the reduction from [6,7].However, it introduces a new kind of exponential blowup in the size of the binary encoding of numbers in number restrictions and the number of individual names occurring in the ontology.Beyond the complexity results, an important benefit of our approach is that it does not need the development of a specialized fuzzy DL reasoner, but can use any state-of-the-art reasoner for classical ALCOQ without modifications.For that reason, this new reduction aids in closing the gap between efficient classical and fuzzy DL reasoners.
A promising direction for future research is to integrate our reduction directly into a classical tableaux procedure.Observe that the axioms in red(C) are already closely related to the rules employed in (classical and fuzzy) tableaux algorithms (see, e.g.[3,8,23]).Such a tableaux procedure would need to deal with total preorders in each node, possibly using an external solver.
On the theoretical side, we want to extend our result to prove 2-NExpTimecompleteness of reasoning in G-SROIQ.As a prerequisite, we would have to eliminate the dependency on the forest-shaped structure of interpretations.It may be possible to adapt the tableaux rules from [22] for this purpose.It also remains open whether consistency in G-SHOQ is ExpTime-complete, as for its classical counterpart.
As done previously in [7], we can also combine our reduction with the one for infinitely valued Zadeh semantics.Although Zadeh semantics is not based on t-norms, it nevertheless is one of the most widely used semantics for fuzzy applications.It also has some properties that make it closer to the classical semantics, and hence is a natural choice for simple applications.

Table 1 :
Syntax and semantics of G-SROIQ Name Syntax Semantics (C I (d) / r I (d, e)) concept name A A I (d) ∈ [0, 1] truth constant p p conjunction C D min{C I (d), D I (d)} implication C → D C I (d) ⇒ D I (d) negation ¬C 1 − C I (d) existential restriction ∃r.C sup e∈∆ I min{r I (d, e), C I (e)} value restriction ∀r.C inf e∈∆ I r I (d, e) ⇒ C I (e) nominal {a} 1 if d = a I 0 otherwise at-least restriction n s.C sup e 1 ,...,en∈∆ I pairwise different n min i=1 min{s I (d, e i ), C I (e i )} local reflexivity ∃s.Self r I (d, d) role name r r I (d, e) ∈ [0, 1] inverse role r − r I (e, d) universal role r u 1 and concept C, there are e, e , e 1 , . . ., e n ∈ ∆ I such that e 1 , . . ., e n are pairwise different, (∃r.C) I (d) = min{r I (d, e), C I (e)}, (∀r.C) I (d) = r I (d, e ) ⇒ C I (e ), and ( n s.C) I (d) = n min i=1 min{s I (d, e i ), C I (e i )}.As we have seen already in the role inclusions, the axioms of G-SROIQ extend classical axioms by allowing to state a degree in (0, 1] to which the axioms hold.Moreover, we allow to compare the degrees of arbitrary classical assertions of the form a:C or (a, b):r for a, b ∈ N I , r ∈ N R , and a concept C.An order assertion [12] is of the form α p or α β for classical assertions α, β, ∈ {<, , =, , >}, and p ∈ [0, 1].An ordered ABox is a finite set of order assertions and individual (in)equality assertions of the form a ≈ b (a ≈ b) for a, b ∈ N I .A general concept inclusion (GCI) is of the form C D p for concepts C, D and p ∈ (0, 1].A TBox is a finite set of GCIs.A disjoint role axiom is of the form dis(r, s) p for two simple roles r, s ∈ N − R and p ∈ (0, 1].A reflexivity axiom is of the form ref(r) p for a role r ∈ N − R and p ∈ (0, 1].An RBox R = R h ∪ R a consists of a role hierarchy R h and a finite set R a of disjoint role and reflexivity axioms.An ontology O = (A, T , R) consists of an ABox A, a TBox T , and an RBox R. A G-interpretation I satisfies (or is a model of) • an order assertion α β if α I β I (where p I := p, (a:C) I := C I (a I ), and ((a, b):r) I := r I (a I , b I ));

•
negated role assertions (a, b):¬r p by (a, b):r 1 − p .For an ontology O, we denote by rol(O) the set of all roles occurring in O, together with their inverses; by ind(O) the set of all individual names occurring in O, and by sub(O) the closure under negation of the set of all subconcepts occurring in O.

Lemma 3 . 4 .
A G-interpretation I satisfies all role inclusions in R iff for every r ∈ rol(O), every w ∈ rol(O) + , and all x, y ∈ ∆ I , we have w I (x, y) ⇒ r I (x, y) ( A r , w).Proof.If I violates any w r p ∈ R, then there are d, e ∈ ∆ I such that w I (d, e) ⇒ r I (d, e) < p.Since ( A r , w) p by construction of A r , we get w I (d, e) ⇒ r I (d, e) < ( A r , w).

(
∀A.C) I (d) := inf w∈rol(O) * inf e∈∆ I min{( A , w), w I (d, e)} ⇒ C I (e), where ε I (d, e) := 1 if d = e, and ε I (d, e) := 0 otherwise.Intuitively, it behaves like a value restriction, but instead of considering only the role r, we consider any role chain w, weighted by the behavior of A on w.Recall that for A r , this behavior represents the degree to which w implies r w.r.t.R h (see Lemma 3.4).
a, * ):s, ( * , a):s | a ∈ ind(O), r ∈ rol(O), s ∈ {r, ¬r}}, where sub ↑ (O) := { C ↑ | C ∈ sub(O)} and the function inv is defined by inv(C) := ¬C, inv(a:C) := a:¬C, inv(a, * ):r := (a, * ):¬r, etc.Total preorders on assertions in U A are used to describe the behavior of the named root elements in the forest-shaped model.For example, if the order is such that a:C > (a, b):r, the intention is that in the corresponding G-model I of O the value of C at a is strictly greater that the value of the r-connection from a to b, i.e. we have C I (a I ) > r I (a I , b I ).For each domain element of I, total preorders on the elements of sub(O) describe the degrees of all relevant concepts in a similar way.The elements of sub ↑ (O) are used to refer back to degrees of concepts at the unique predecessor element in the tree-shaped parts of the interpretation.For convenience, we also define p ↑ := p for all p ∈ V O .The elements r ∈ rol(O) represent the values of the role connections from the predecessor.The special elements ( * , a):r and (a, * ):r are used to describe role connections between arbitrary domain elements (represented by * ) and the named elements in the roots.
a):r = a:∃r.Self | a ∈ ind(O), r ∈ rol(O)} ∪ { (a, b):r = (b, a):r − | a, b ∈ ind(O) ∪ { * }, r ∈ rol(O)}, where c is an arbitrary individual name.The first two lines are responsible for enforcing that the ABox is satisfied and that information about the behavior of the named individuals is available throughout the whole model.The remaining axioms describe various equivalences for named individuals, e.g. that (a, b):r and ( * , b):r should have the same value when evaluated at a.The next axiom defines the concept AN of all anonymous elements, i.e. those that are not designated by an individual name: red(AN) := ¬AN ≡ a∈ind(O) ∀r.C ↑ r ⇒ C a∈ind(O) ∃r.{a} (∀r.C) ( * , a):r ⇒ a:C } ∪ (N) {a:∃r.AN ∀r.C ↑ r ⇒ C ¬AN (a:∀r.C) (a, * ):r ⇒ C | a ∈ ind(O)}

r, 1 −
→ f r from the initial state i r to the final state f r .By the first axiom in red(∀r.C) and the third axiom in red r,1,fr (∀A r .C), the witness e satisfies ∀r.C ↑ ∀A r .C ↑ r ⇒ (∀A fr r .C) Since f r is final, we further have (∀A fr r .C) C by red(∀A fr r .C), and hence ∀r.C ↑ r ⇒ C, as claimed above.The other axioms in red r,1,fr (∀A r .C) deal with the other kinds of possible successors (see above).
All (in)equality assertions a ≈ b (a ≈ b) in A are satisfied due to red(A) and the construction of I f .Consider any GCI C D p ∈ T and d ∈ ∆ I .By red(T ) and (P5)b), we have v(p, d) v(C, d) ⇒ v(D, d).Thus, (4) and (P5)a) yield C I f (d) ⇒ D I f (d) p.For ref(r) p ∈ R, by red(R) we have v(∃r.Self, d) p for all d ∈ ∆ I f , and hence r I f (d, d) p, as required.For any dis(r, s) p ∈ R, r and s are simple, and thus we can restrict our analysis to r I 0 f and s I 0 f .We have min{v((a, b):r, c, u), v((a, b):s, c, u)} 1 − p for all (c, u) ∈ ∆ I f and a, b ∈ ind(O)∪{ * }.Hence, min{r I

Conversely, we now
show that, in SRIQ, SROQ, or SROI, if O has a G-model, then red(O) has a classical model.Given a G-model I of O, we construct the classical interpretation I c , whose domain consists of all sequences of the form ad 1 . . .d k , where • a ∈ N I and k 0;

•••
4, we have (∀A r .C) I (d) = inf w∈rol(O) * inf e∈∆ I min{( A r , w), w I (d, e)} ⇒ C I (e) inf e∈∆ I r I (d, e) ⇒ C I (e) = (∀r.C) I (d) for all d ∈ ∆ I .Lemma 3.4 talks only about w ∈ rol(O) + , but it holds also for w = ε since then ( A r , w) = 0 due to the construction of A r .Hence, the axiom (∀r.C) (∀A r .C) is satisfied by I c .We consider the axiom AN (∀r.C) r − ⇒ C ↑ (∀r.C) (∃r.Self) ⇒ C ∃r. AN ∀r.C ↑ r ⇒ C a∈ind(O) ∃r.{a} (∀r.C) ( * , a):r ⇒ a:C , where the first disjunct is only present if inverse roles are considered, likewise for the second disjunct and (local) reflexivity, and the last disjunct is contingent on the presence of nominals.Let further ∈ AN Ic , i.e. = ad 1 . . .d k with k 1.Since I is witnessed, there is an e ∈ ∆ I with (∀r.C) I (d k ) = r I (d k , e) ⇒ C I (e).If e ∈ ∆ Ic , then ( , e) ∈ r I and e ∈ AN Ic .Furthermore, ∀r.C I ↑ ( e) = r I (d k , e) ⇒ C I (e) = r I ( e) ⇒ C I ( e), and hence e ∈ ∀r.C ↑ r ⇒ C Ic .Otherwise, i.e. in the case that e / ∈ ∆ Ic , there are three cases to consider: • Reflexivity is allowed and e = d k .Then (∀r.C) I ( ) = r I (d k , d k ) ⇒ C I (d k ) = (∃r.Self) I ( ) ⇒ C I ( ). • Nominals are allowed and e = b I for some b ∈ N I .Then we have ( , b) ∈ r Ic and (∀r.C) I ( ) = r I (d k , b I ) ⇒ C I (b I ) = (( * , b):r) I ( ) ⇒ (b:C) I ( ).Inverse roles are allowed and either (i) k > 1 and d k−1 = b I or (ii) k = 1 and a = b.In both cases, we have prev( ) = b I , and hence(∀r.C)I ( ) = r I (k m , prev( )) ⇒ C I (prev( )) = (r − ) I ( ) ⇒ C I ↑ ( ).Consider now the axiom a:∃r.AN ∀r.C ↑ r ⇒ C ¬AN (a:∀r.C) (a, * ):r ⇒ C for some a ∈ ind(O).Since I is witnessed, there is an element e ∈ ∆ I such that (∀r.C) I (a I ) = r I (a I , ⇒ C I (e).If e = b I for some b ∈ N I , then we have (a, b) ∈ r Ic and b / ∈ AN Ic .Furthermore, (a:∀r.C) I (b) = (∀r.C) I (a I ) = r I (a I , b I ) ⇒ C I (b I ), which is equal to ((a, * ):r) I (b) ⇒ C I (b), and hence we have b ∈ (a:∀r.C) (a, * ):r ⇒ C Ic .If e = b I for all b ∈ N I , then we have ae ∈ ∆ Ic , and thus (a, ae) ∈ r Ic .Moreover, ae ∈ AN Ic and ∀r.C I ↑ (ae) = (∀r.C) I (a I ) = r I (a I , e) ⇒ C I (e), which is equal to r I (ae) ⇒ C I (ae).

(
n r.C) I (a I ) = n min j=1 min{r I (a I , e j ), C I (e j )}.

•
If e j = b I for some b ∈ ind(O), then we have (a, b) ∈ r Ic , b ∈ (¬AN) Ic , and (a: n r.C) I (b) = ( n r.C) I (a I ) min{r I (a I , b I ), C I (b I )} = min{((a, * ):r) I (b), C I (b)}.• If e j = b I for all b ∈ ind(O), then we have ae j ∈ ∆ Ic , and thus (a, ae j ) ∈ r Ic .Furthermore, n r.C I ↑ (ae j ) = ( n r.C) I (a I ) min{r I (a I , e j ), C I (e j )} = min{r I (ae j ), C I (ae j )}.

(
n r.C) I (a I ) = n r.C I ↑ (ae j ) < min{r I (ae j ), C I (ae j )} = min{r I (a I , e j ), C I (e j )}.• If j does not satisfy AN, then it is of the form b and we obtain ( n r.C) I (a I ) = (a: n r.C) I (b) < min{((a, * ):r) I (b), C I (b)} = min{r I (a I , b I ), C I (b I )}.If we consider equivalence classes of individual names as roots for I c , all such b I are different.This again contradicts the semantics of ( n r.C) I (a I ).