LTCS–Report On the Complexity and Expressiveness of Description Logics with Counting

Simple counting quantiﬁers that can be used to compare the number of role successors of an individual or the cardinality of a concept with a ﬁxed natural number have been employed in Description Logics (DLs) for more than two decades under the respective names of number restrictions and cardinality restriction on concepts. Recently, we have considerably extended the expressivity of such quantiﬁers by allowing to impose set and cardinality constraints formulated in the quantiﬁer-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA) on sets of role successors and concepts, respectively. We were able to prove that this extension does not increase the complexity of reasoning. In the present paper, we investigate the expressive power of the DLs obtained this way, using appropriate bisimulation characterizations and 0–1 laws as tools for distinguishing the expressiveness of diﬀerent logics. In particular, we show that, in contrast to most classical DLs, these logics are no longer expressible in ﬁrst-order predicate logic (FOL), and we characterize their ﬁrst-order fragments. In most of our previous work on DLs with QFBAPA-based set and cardinality constraints we have employed ﬁniteness restrictions on interpretations to ensure that the obtained sets are ﬁnite. Here we dispense with these restrictions to make the comparison with classical DLs, where one usually considers arbitrary models rather than ﬁnite ones, easier. It turns out that doing so does not change the complexity of reasoning.


Introduction
Description Logics (DLs) [6] are a well-investigated family of logic-based knowledge representation languages, which are frequently used to formalize ontologies for application domains such as biology and medicine [17].To define the important notions of such an application domain as formal concepts, DLs state necessary and sufficient conditions for an individual to belong to a concept.These conditions can be Boolean combinations of atomic properties required for the individual (expressed by concept names) or properties that refer to relationships with other individuals and their properties (expressed as role restrictions).Adapting an example from [7,3]  can then be used to state that Computer Science authors need to belong to this concept description.
Numerical constraints on the number of role successors (so-called number restrictions) have been used in DLs for more than three decades [9,19,18].For example, using number restrictions, we can define prolific authors as those having published at least 100 papers: Prolific-author Person ( 100 published.Paper).
The exact complexity of reasoning in ALCQ, the DL that has all Boolean operations and number restrictions of the form ( n r.C) and ( n r.C) as concept constructors, was determined by Stephan Tobies [28,30]: it is PSpace-complete without CIs and ExpTime-complete w.r.t.CIs, independently of whether the numbers occurring in the number restrictions are encoded in unary or binary.Note that, using unary coding of numbers, the number n is assumed to contribute n to the size of the input, whereas with binary coding the size of the number n is log n.Thus, for large numbers, using binary coding (or coding w.r.t.any base larger than 1) is more realistic.Numerical constraints have also been used in DLs to formulate cardinality restrictions on concepts (CRs) [5,29].For example, the CRs1 |Conference ∃uses.Easychair| 75000 and |Person ∃uses.Easychair| 3000000 state that at least 75 thousand conferences and at most 3 million persons use the conference management system Easychair. 2Whereas number restrictions are local in the sense that they consider role successors of an individual under consideration (e.g. the papers published by a particular author), cardinality restrictions on concepts (CRs) [5,29] are global, i.e., they consider all individuals in an interpretation.Cardinality restrictions can express CIs since, clearly, C D is equivalent to |C ¬D| 0. They are, however, considerably more expressive.The higher expressivity of CRs over CIs can, for example, be seen from the fact that CIs in ALCQ are closed under disjoint union of models, but models of a CR like |A| 1 are clearly not (see Section 2.3 below for more details).
In addition, CRs increase the complexity of reasoning: for the DL ALCQ, consistency w.r.t.CIs is ExpTime-complete [30], but consistency w.r.t.CRs is NExpTime-complete if the numbers occurring in the CRs are assumed to be encoded in binary [29].With unary coding of numbers, consistency stays ExpTime-complete even w.r.t.CRs [29], but the above example considering 3 million conferences clearly shows that unary coding is not appropriate if numbers with large values are employed.It should be noted that both number restrictions and CRs can be expressed in C 2 , the two-variable fragment of first-order logic with counting quantifiers [14,25], whose satisfiability problem is known to be NExpTime-complete [26].
Whereas C 2 , and thus also number restrictions and CRs, are expressible in FOL, the counting extensions considered in the present paper actually leave the realm of FOL.The classical number restrictions available in ALCQ can only be used to compare the number of role successors of an individual with a fixed natural number.They cannot relate numbers of different kinds of role successors to each other.This would, e.g., be required to describe persons that have developed more theorem provers than conference management systems, without fixing what these numbers actually are.To overcome this deficit, we have extended ALCQ by allowing the statement of constraints on role successors that are more general than the number restrictions of ALCQ [1].To formulate these constraints, we have used the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA) [22], in which one can express Boolean combinations of set constraints and numerical constraints comparing the cardinalities of sets.In the resulting logic, called ALCSCC, the above constraint regarding theorem provers and conference managements systems can be expressed using a cardinality constraint on the role successors: Person succ(|developed In general, such a succ-expression considers the set of all role successors of a given individual, and requires certain subsets to satisfy the stated QFBAPA constraints.In our example, for a person Andrei to belong to this concept, the cardinality of the set of developed -successors of Andrei that belong to the concept TP (collecting all theorem provers) must be larger than the cardinality of the set of developed -successors of Andrei that belong to the concept CMS (collecting all conference management systems).
Adding such cardinality constraints strictly extends the expressive power of ALCQ.In [1] it was shown that the constraint succ(|r| = |s|), which describes individuals that have the same number of r-successors as s-successors, cannot be expressed in ALCQ.In [4], the constraint succ(|r ∩ A| = |r ∩ ¬A|) describing individuals such that the number of r-successors belonging to A is the same as the number of r-successors not belonging to A was shown to be not even expressible in first-order logic.Intuitively, both kinds of constraints can, e.g., be used to describe people that have the same number of sons and daughters, where in the first constraint one uses roles son and daughter , whereas in the second one uses the role child and the concept Male.In spite of this considerable increase of the expressive power, we were able to show in [1] that this does not increase the complexity of reasoning: like for ALCQ, the complexity of the satisfiability problem in ALCSCC is PSpace-complete without CIs and ExpTime-complete w.r.t.CIs.While the PSpace result also follows from previous work [11] on modal logics with Presburger constraints, the ExpTime result was new.
Just like classical number restrictions, CRs can only relate the cardinality of a concept to a fixed number.In [7], we have introduced and investigated more general constraints on the cardinalities of concepts, which we called extended cardinality constraints.The main idea was again to use QFBAPA to formulate and combine these constraints.An example of a constraint expressible this way, but not expressible using CRs is 2 • |Paper ∀topic.DL| |Paper ∀topic.AR| which states that papers with topic Automated Reasoning outnumber papers with topic Description Logic by a factor of at least two.In [7] it is shown that, in the DL ALC, the complexity of reasoning w.r.t.extended cardinality constraints (NExpTime for binary coding of numbers) is the same as for reasoning w.r.t.CRs.In addition, the paper introduces a restricted version of this formalism, which can express CIs, but not CRs, and shows that this way the complexity can be lowered to ExpTime.The NExpTime upper bound for the general case actually also follows from the NExpTime upper bound in [31] for a more expressive logic with n-ary relations and function symbols, but the ExpTime result for the restricted case was new.
In [2,3], we combined the work in [1] and [7] by considering extended cardinality constraints in ALCSCC.This turned out to be non-trivial since the local cardinality constraints of ALCSCC may interact with the global ones in the extended cardinality constraints.Nevertheless, we were able to show that the complexity results (NExpTime-complete in general, and ExpTimecomplete in the restricted case) hold not only for ALC, but also for ALCSCC.
The purpose of the present paper is twofold.On the one hand, we give a compact representation of the known complexity results for the DLs with extended counting facilities mentioned above, and transfer them to a setting where arbitrary rather than just finite models are considered (see below).On the other hand, we investigate the expressive power of these DLs in detail.A first step in this direction was already made in [4], where the expressive power of concept descriptions was examined using appropriate bisimulation relations.Basically, we show there that ALCSCC is not expressible in FOL, and determine a sub-logic of ALCSCC, called ALCCQU, that is the first-order fragment of ALCSCC.We also shows that ALCCQU is more expressive that ALCQ.Here, we recall these results, and then extend them to TBoxes, CRs, and extended cardinality constraints, by adapting methods and ideas from [24].As in [4], we consider variants of QFBAPA and ALCSCC that allow for possibly infinite sets and interpretations, respectively.This change has no influence on the complexity of reasoning, but it eases the comparison with classical DLs, for which one usually employs arbitrary models rather than finite ones when defining the semantics.It also adds flexibility since finiteness can be expressed in these logics, and thus one can actually switch between arbitrary model reasoning and finite model reasoning.

DLs with counting quantifiers
In this section, we formally introduce the DLs with extended counting facilities mentioned in the introduction, and recall the known complexity results for reasoning in these logics.As mentioned above, we will not restrict the semantics to finite models.For this reason, the result originally obtained for the "finite model" case need to be adapted.We start with introducing the infinite variant of QFBAPA on which all our logics are based.

An infinite variant of QFBAPA
We recall the definition of QFBAPA ∞ as introduced in [4]. 3In this logic one can build set terms by applying Boolean operations (intersection ∩, union ∪, and complement • c ) to set variables as well as the constants ∅ and U. Set terms s, t can then be used to state inclusion and equality constraints (s = t, s ⊆ t) between sets.For example, if Vampire and Easychair are set variables, then the set constraint Vampire ∩ Easychair = ∅ says that vampires are not easychairs.
Presburger Arithmetic (PA) expressions are built from integer variables, integer constants, and set cardinalities |s| using addition as well as multiplication with an integer constant.They can be used to form numerical constraints of the form k = and k < , where k, are PA expressions.For example, the numerical constraint says that there are more than ten times as many vampires as there are easychairs and barstools together.A QFBAPA ∞ formula is a Boolean combination of set and numerical constraints.
The semantics of set terms and set constraints is defined using substitutions σ that assign a set σ(U) to U and subsets of σ(U) to set variables. 4The evaluation of set terms and set constraints by such a substitution is defined in the obvious way, using the standard notions of intersection, union, complement, 5 inclusion, and equality for sets.PA expressions are evaluated over N ∞ = N ∪ {∞}, i.e., the non-negative integers extended with a symbol for infinity.Thus, substitutions additionally assign elements of N ∞ to PA variables.The cardinality expression |s| is evaluated under σ as the cardinality of σ(s) if this set is finite, and as ∞ if σ(s) is not finite. 6When evaluating PA expressions w.r.t. a substitution σ, we employ the usual way of adding, multiplying, and comparing integers, extended by the following rules that deal with infinity: • N for all positive integers N , N < ∞ and ∞ < N for all non-negative integers N , and ∞ = ∞ as well as ∞ < ∞.
A solution σ of a QFBAPA ∞ formula φ is a substitution that evaluates φ to true, using the above rules for evaluating set and numerical constraints and the usual interpretation of the Boolean operators occurring in φ.The formula φ is satisfiable if it has a solution.
Note that, in QFBAPA ∞ , we can enforce infinity of a set although we do not allow the use of ∞ as a constant.For instance, |s| = ∞ is not an admissible numerical constraint, but it is easy to see that the constraint |s| + 1 = |s| can only be satisfied by a substitution that assigns an infinite set to the set term s.
The set constraints in QFBAPA ∞ are actually syntactic sugar since they can be expressed using numerical constraint.In fact, the set constraint s ⊆ t is equivalent to the numerical constraint |s ∩ t c | 0. Note that, for finite sets, this could equivalently be expressed as |s ∪ t| = |t|, but for infinite sets the latter constraint is not equivalent to s ⊆ t.Since set constraints are syntactic sugar and > and < can easily be simulated in N ∞ using and , we can assume without loss of generality that any QFBAPA ∞ formula is a Boolean combination of atomic QFBAPA ∞ formulae of the form where the s i , t j are set terms and the N i , M j are non-negative integers.
The logic CQU as introduced in [4] is obtained from QFBAPA ∞ by restricting numerical constraints to be of the form k N and k N , i.e., a CQU formula is a Boolean combination of set constraints and numerical constraints of this restricted form.By using the same arguments as above, we can show that any CQU formula is equivalent to a Boolean combinations of atomic formulae of the form (2) where k = 0 or = 0.In addition, in this setting sums can be expressed using disjunction.For example, saying that |s| + |t| 1 is equivalent to saying that |s| 0 and |t| 1, or |s| 1 and |t| 0. Thus, when it comes to expressive power, we can assume without loss of generality that formulae of CQU are Boolean combinations of numerical restrictions of the form |s| N or |s| N .
It is actually not hard to see that the logic CQU as defined here and in [4] has the same expressivity as C 1 , the one-variable fragment of first-order logic with counting (see, e.g., [27]).
The logic originally called CQU in [13] is the fragment where only conjunctions of atomic restrictions of the form |s| N or |s| N can be used.However, when using CQU within our DLs with counting quantifiers, this difference is irrelevant since the Boolean operations are available anyway on the DL level.

Concept descriptions that count
We are now ready to define the DL ALCSCC ∞ and some of its sub-logics.Basically, ALCSCC ∞ provides us with Boolean operations on concepts and constraints on role successors, which are expressed in QFBAPA ∞ .In these constraints, role names and concept descriptions can be used as set variables, and there are no PA variables allowed.
Definition 1 (Syntax of ALCSCC ∞ ).Given finite, disjoint sets N C of concept names and N R of role names, the set of ALCSCC ∞ concept descriptions over the signature (N C , N R ) is inductively defined as follows: • , ⊥, and every concept name in N C is an ALCSCC ∞ concept description over (N C , N R ); • if C, D are ALCSCC ∞ concept descriptions over the signature (N C , N R ), then so are C D, C D, and ¬C; • if Con is a set or numerical constraint of QFBAPA ∞ using role names and already defined ALCSCC ∞ concept descriptions over the signature (N C , N R ) as set variables, then succ(Con) is an ALCSCC ∞ concept description over (N C , N R ).
For example, the description (1) in the introduction is an ALCSCC ∞ concept description that uses the QFBAPA ∞ numerical constraint |developed ∩ TP | > |developed ∩ CMS |, in which developed , TP , CMS are viewed as set variables.Of course, successor constraints can also be nested, as in the ALCSCC ∞ concept description which describes all individuals having as many friends that have developed at least one conference management system as they have friends that have developed at least one theorem prover.
For the sake of simplicity, we will sometimes use "concept" in place of "concept description," and often dispense with explicitly mentioning the signature.As usual in DL, the semantics of ALCSCC ∞ is defined using the notion of an interpretation.The function • I is inductively extended to ALCSCC ∞ concept descriptions over (N C , N R ) by interpreting , , and ¬ respectively as intersection, union and complement as well as as ∆ I and ⊥ as the empty set.Successor constraints are evaluated according to the semantics of QFBAPA ∞ : to determine whether d ∈ succ(Con) I or not, • U is evaluated as ars I (d) (i.e., the set of all role successors of d), • ∅ as the empty set, • roles r occurring in Con as r I (d) (i.e., the set of r-successors of d), • and concept descriptions D as D I ∩ars I (d) (i.e., the set of role successors of d that belong to D).Note that, by induction, the sets D I are well-defined.
Then d ∈ succ(Con) I iff the substitution obtained this way is a solution of the QFBAPA ∞ formula Con.
The ALCSCC ∞ concept description C is satisfiable if there is an interpretation I such that The sub-logics ALCQ, ALCQt, and ALCCQU of ALCSCC ∞ can be obtained from ALCSCC ∞ by restricting the successor constraint appropriately: • The DL ALCQ is the fragment of ALCSCC ∞ in which only successor constraints of the form succ(|C ∩ r| N ), succ(|C ∩ r| N ) are allowed, where N is a natural number, r is a role name, and C is an ALCQ concept description.These constraints are usually written as ( N r.C) and ( N r.C), and are called qualified number restrictions.
• The DL ALCQt is the fragment of ALCSCC ∞ in which only successor constraints of the form succ(|C ∩τ | N ), succ(|C ∩τ | N ) are allowed, where N is a natural number, τ is a safe role type, and C is an ALCQt concept description.A safe role type is an intersection of role names r (positive occurrence) and complements r c of role names (negative occurrence) such that every role name in N R occurs either positively or negatively, and at least one role name occurs positively.Using the syntax for qualified number restrictions, these constraints can be written as ( N τ.C) and ( N τ.C).
• The DL ALCCQU is the fragment of ALCSCC ∞ in whose successor constraints only constraints of CQU are allowed.
By definition, ALCQ is a sub-logic of ALCCQU, and ALCCQU is a sub-logic of ALCSCC ∞ .In addition, ALCQt is clearly a sub-logic of ALCCQU.In addition, it is shown in [4] that any ALCCQU concept description can be expressed by an equivalent ALCQt concept description, and thus that ALCCQU and ALCQt have the same expressive power.
We will see below that ALCSCC ∞ concept descriptions can in general not be expressed in FOL.
In contrast, the concept descriptions of the three fragments introduced above can be expressed by first-order formulae with one free variable.This is well-known for ALCQ [8], and can be shown for ALCQt by a simple adaptation of the first-order translation for ALCQ, where safe role types are translated into first-order formulae with two variables in the obvious way.For ALCCQU this follows from its equivalence with ALCQt.
Proposition 3. If L ∈ {ALCQ, ALCQt, ALCCQU}, then L is a fragment of FOL, i.e., for every L concept description C there exists a first-order formula with one free variable C (x) such that C and C (x) are equivalent in the sense that, for every interpretation I, we have The logic ALCSCC ∞ and its sub-logics are local in the sense that the decision on whether a certain individual belongs to a concept depends only on this individual and other individuals connected via roles to it.For this reason, evaluating a concept in the disjoint union of interpretations corresponds to evaluating it separately in the single interpretations.To be more precise, given a family (I ν ) ν∈N of interpretations, we define their disjoint union I = ν∈N I ν as The following is now easy to show, using the locality of ALCSCC ∞ concept descriptions mentioned above.
Lemma 4. Let C be an ALCSCC ∞ concept description.Then we have

TBoxes and cardinality boxes
In classical DLs, terminological knowledge is represented using so-called TBoxes, which are finite sets of CIs of the form C D for concepts C, D. Cardinality boxes extend TBoxes by allowing for the formulation of cardinality constraints also on this level.To simplify the comparison with cardinality boxes, in which Boolean combinations of numerical constraints are allowed, we also consider Boolean TBoxes.3.An L ECBox is a Boolean combination of inequations of the form where the C i , D j are L concept descriptions and the N i , M j are natural numbers.
4. An L RCBox is a conjunction of of inequations of the form where the C i , D j are L concept descriptions and the N i , M j are positive natural numbers.
We say that the interpretation I is a model of 4. an inequation of the form (4) if The notion of a model is extended to Boolean combinations of such constraints in the obvious way.
Obviously, the CI C D can be expressed by the CR |C ¬D| |⊥|, and this CR is also expressible by ECBoxes and RCBoxes.Thus, TBoxes can be expressed using CBoxes, ECBox, or RCBoxes.CBoxes and RCBoxes are clearly expressible by ECBoxes.However, the expressiveness of CBoxes and RCBoxes appears to be orthogonal.While the former only allow us to compare concept cardinalities with a fixed number, this is exactly what is prohibited in RCBoxes.On the other hand, RCBoxes enable us to compare the cardinalities of different concepts whereas this is not possible in CBoxes.
In case the underlying DL L is expressible in FOL, L TBoxes and L CBoxes are clearly also expressible in FOL.Together with Proposition 3 this observation yields the following: Corollary 6.If L ∈ {ALCQ, ALCQt, ALCCQU}, then L (Boolean) TBoxes and CBoxes can be expressed in FOL, i.e., for every (Boolean) L TBox or CBox T there exists a first-order sentence T such that T and T are equivalent in the sense that they have the same interpretations as models.
We can use disjoint unions to show inexpressibility results for some of our box formalisms.Definition 7. Let L ∈ {ALCQ, ALCQt, ALCCQU, ALCSCC ∞ } and T be a (Boolean) TBox, CBox, RCBox, or ECBox.We say that the models of T are closed under disjoint union if the following holds: if the interpretations I ν for ν ∈ N are models of T , then their disjoint union I = ν∈N I ν is also a model of T .The models of T are invariant under disjoint union if additionally the implication in the other direction holds, i.e., if the disjoint union I = ν∈N I ν is a model of T , then so are the interpretations I ν for ν ∈ N .
Using Lemma 4, the positive statements of the following proposition are easy to show.As an immediate consequence of the above lemma, we obtain the following inexpressibility results.

Reasoning in the introduced logics
For a DL L, the fundamental inference problems are satisfiability and subsumption of concepts: • Given an L concept C, the satisfiability problem asks whether C is satisfiable, i.e., whether there is an interpretation I such that C I = ∅.
• Given L concepts C, D, the subsumption problem asks whether C is a subconcept of D (written C D), i.e., whether C I ⊆ D I holds for all interpretations I.
If the DL L can express ⊥ as well as conjunction and negation of concepts, then subsumption and satisfiability can be reduced to each other in polynomial time since C D holds iff C ¬D is unsatisfiable, and C is unsatisfiable iff C ⊥ holds.
These two inference problems can also be considered w.r.t. the kinds of boxes introduced in the previous subsection.Let C, D be L concepts and T an L TBox, CBox, ECBox, or RCBox.Then we say that Since the prerequisites required for the reductions mentioned above are satisfied by the DLs ALCQ, ALCQt, ALCCQU, and ALCSCC ∞ , and all our box formalisms can express CIs, we can restrict the attention to the satisfiability problem in case there is no box, and to the consistency problem in case there is a box, when investigating the complexity of reasoning.

Reasoning without a box in ALCSCC ∞ and its sub-logics
The satisfiability problem in ALCQ was shown to be PSpace-complete in [28].In [1] it was proved that this result can be extended to ALCSCC, and in [4] it was demonstrated that the same is true for ALCSCC ∞ .Since we have a PSpace lower bound for ALCQ, which is the least expressive DL considered in this paper, even for unary coding of numbers, as well as a PSpace upper bound for ALCSCC ∞ , which is the most expressive one, even for binary coding, this determines the exact worst-case complexity of the satisfiability problem for all the DLs introduced above.

Reasoning w.r.t. CBoxes and ECBoxes
Consistency of ALCQ CBoxes was shown to be NExpTime-complete in [29] if binary coding of numbers is used, whereas for unary coding it stays in ExpTime.In [2,3] we were able to prove a NExpTime upper bound for consistency of ALCSCC ECBoxes with numbers encoded in binary.Basically, the proof of this result takes the ALCSCC ECBoxes E and translates it into an exponentially larger QFBAPA formula δ E that is satisfiable iff E is consistent.Since satisfiability in QFBAPA is NP-complete for binary coding of numbers, this yields the NExpTime upper bound for ALCSCC.This results can easily be transferred to ALCSCC ∞ by using the same translation, but then testing satisfiability of δ E in QFBAPA ∞ rather than in QFBAPA.In [4] it is shown that the satisfiability problem in QFBAPA ∞ is also in NP.
The reason why the coding of numbers is irrelevant in the presence of ECBoxes is that one can use iterated multiplication to create large numbers from small ones (see [7] for a more detailed argument).

Reasoning w.r.t. TBoxes and RCBoxes
It is well-known that consistency of ALCQ TBoxes is an ExpTime-complete problem [30].This result was extended in [1] to ALCSCC TBoxes, and in [4] it was argued that it also holds for ALCSCC ∞ .RCBoxes were introduced in [7] to obtain a restriction of ECBoxes that lowers the complexity of the consistency problem from NExpTime to ExpTime.For ALCSCC (i.e., the case of finite models), it is shown in [2,3] that the consistency problem for RCBoxes is ExpTime-complete.It would not be hard to demonstrate that the approach employed there to prove the ExpTime upper bound for the "finite model" case can be adapted to the infinite case as well.However, below we give a simpler proof of this result for ALCSCC ∞ , which uses that fact that it is sufficient to consider solutions of inequations of the form (5) where the concepts C i are either empty or have infinite cardinality.
Recall that an ALCSCC ∞ RCBox R is a system of inequations of the form where the C i are ALCSCC ∞ concept descriptions and the N i are positive integers.Our algorithm reduces consistency of ALCSCC ∞ RCBoxes to consistency of ALCSCC ∞ TBoxes.It receives an ALCSCC ∞ RCBox R as input and initializes the ALCSCC ∞ TBox T as T := ∅.It then proceeds with the following steps: 1. Check whether the ALCSCC ∞ TBox T is consistent.If this is not the case, then terminate with failure.Otherwise, for all concepts C occurring in an inequation of R, check whether T implies C ⊥.If this is the case, then add C ⊥ to T .Then proceed with the next step.
2. For all inequations of the form (5) such that C j ⊥ belongs to T for all k + 1 j k + , add C i ⊥ to T for all i, 1 i k.If no new CI has been added to T , then terminate with success.Otherwise, continue with the previous step.
Lemma 12.The algorithm terminates after a polynomial number of iterations and it succeeds iff the RCBox R is consistent.
Proof.Termination after a polynomial number of iterations is an immediate consequence of the fact that only polynomially many CIs of the form C ⊥ can be added to T since the concepts C for which such a CI can be added must occur in an inequation in R.
Now, assume that R is consistent, and let I be a model of R. By an induction on the number of iterations, it is easy to show that we must have C I = ∅ for all CIs added to T during the run of the algorithm.Consequently, in Step 1 the algorithm can never fail since I is a model of T .Since the algorithm always terminates, it must thus succeed.
Next, assume that the algorithm succeeds with the final TBox T .Then T is consistent, and for every concept C occurring in an inequation of R such that C ⊥ does not belong to T , there is a model I C of T such that C I C = ∅.By using closure under disjoint union of models of ALCSCC ∞ TBoxes, this implies that there is an interpretation I ∞ such that the following holds for all concepts C occurring in an inequation of R:  It remains to shows that I ∞ is a model of R. Thus, consider an inequation of the form (5) in R. If there is a j with k + 1 j k + such that C I∞ j is infinite, then clearly this inequation is satisfied by I ∞ .Otherwise, C j ⊥ belongs to T for all k + 1 j k + , and thus also C i ⊥ belongs to T for all i with 1 i k.This shows that, again, the inequation is solved.
Since consistency of ALCSCC ∞ TBoxes can be tested in exponential time [4], the overall complexity of our algorithm is ExpTime.
Combining this result with the known lower bounds for TBox consistency, we thus obtain the following: Theorem 14.If L ∈ {ALCQ, ALCQt, ALCCQU, ALCSCC ∞ }, then consistency of L (Boolean) TBoxes and RCBoxes is ExpTime-complete independently of whether numbers are encoded in unary or binary.
To explain the ExpTime upper bound for Boolean TBoxes, note that one can reduce consistency of a Boolean TBox to exponentially many consistency tests for TBoxes.In fact, one can bring the Boolean TBox into disjunctive normal form and then test every disjunct for consistency.At first sight, such a disjunct is not a TBox since it may contain negated CIs, but one can replace negated CIs ¬(C D) with CIs ( 1 r.(C ¬D)) for new roles r.We show in Proposition A.1 that this replacement works as intended.
The complexity results for "box consistency" in ALCSCC ∞ and its sub-logics are summarized in Table 1.

Expressivity of concept descriptions
The purpose of this section is to compare the expressive power of the concept description languages of the DLs ALCQ, ALCQt, and ALCSCC ∞ .Since we already know that ALCQt and ALCCQU have the same expressiveness, we will not consider ALCCQU explicitly here.Our results, which have been presented first in [4], make use of appropriate bisimulation relations for the first-order expressible logics ALCQ and ALCQt.

Bisimulation relations for ALCQ and ALCQt
Let τ be a safe role type, r 1 , . . ., r k the role names occurring positively in τ , and s 1 , . . ., s the role names occurring negatively, i.e., τ = r 1 ∩ . . .∩ r k ∩ s c 1 ∩ . . .∩ s c .For a given interpretation I and an element d ∈ ∆ I , we define Since τ is safe, we must have k 1, and thus τ I (d) ⊆ r I 1 (d) ⊆ ars I (d).
Definition 15 (ALCQt bisimulation).Let I 1 and I 2 be interpretations of N C and N R .The relation ρ ⊆ ∆ I1 × ∆ I2 is an ALCQt bisimulation between I 1 and I 2 if for all A ∈ N C and all safe role types τ over N R the following three properties are satisfied: such that ρ contains a bijection between D 1 and D 2 ; The notion of an ALCQ bisimulation (called counting bisimulation in [24]) is obtained from the above definition by replacing safe role types τ over N R with role names r ∈ N R .ALCQ bisimilarity (written (I 1 , d 1 ) ∼ ALCQ (I 2 , d 2 )) and ALCQ equivalence (written (I 1 , d 1 ) ≡ ALCQ (I 2 , d 2 )) are obtained by replacing ALCQt in the above definition with ALCQ.The next proposition states that concepts of ALCQ and ALCQt are invariant under the respective notion of bisimulation.For ALCQ, this was first shown in [24] and for ALCQt in [4].A detailed proof for L = ALCQt can be found in Appendix A.
This result is already sufficient for showing that ALCQt is not expressible in ALCQ.

Corollary 17 ([4]
).There is no ALCQ concept description C such that C is equivalent to the ALCQt concept description succ(|r ∩ s| 1).
In fact, if succ(|r ∩ s| 1) was equivalent to an ALCQ concept description, then it would need to be invariant under ALCQ bisimulation as stated in the above proposition.However, Fig. 1 shows two interpretations in which the individuals d 1 and d 2 are ALCQ bisimilar, but whereas d 1 belongs to succ(|r ∩ s| 1), the individual d 2 does not.
The following theorem states that ALCQ and ALCQt are exactly the fragments of first-order logic that are invariant under the respective notion of bisimulation.We say that a firstorder formula φ(x) with one free variable x is invariant under . For ALCQ this was first shown in [24] and the proof for ALCQt can be obtained by adapting this proof.A detailed proof that closes some small gaps of the one in [24] can be found in Appendix A.
Theorem 18 ( [24,4]).Let L ∈ {ALCQ, ALCQt} and φ(x) be a first-order formula with one free variable x.Then the following are equivalent: 1. there is an L concept description C such that C is equivalent to φ(x);

Comparison with ALCSCC ∞
One might think that invariance of ALCQt concept descriptions under ALCQt bisimulation could be used to show that ALCSCC ∞ concepts cannot be expressed in ALCQt.This is, however, not the case since ALCSCC ∞ concepts are also invariant under ALCQt bisimulation.
Here ALCSCC ∞ equivalence is defined in the obvious way, by considering all ALCSCC ∞ concept descriptions over the given signature.The main idea underlying the proof of this proposition is that all the PA expressions occurring in successor constraints can be transformed into the form where the N i are natural numbers, the τ i are safe role types, and the C i are ALCSCC ∞ concept descriptions.Then, one can show that, for individuals that are ALCQt bisimilar, expressions of the form |τ i ∩ C i | evaluate to the same number or to ∞ on their role successors.
Combining Proposition 19 and Theorem 18 for L = ALCQt, we can now conclude that ALCQt is exactly the first order fragment of ALCSCC ∞ .
Theorem 20 ([4]).For an ALCSCC ∞ concept description C, the following are equivalent: 1. C is equivalent to an FOL formula with one free variable; 2. C is equivalent to an ALCQt concept description.
The direction (2 ⇒ 1) is an immediate consequence of Proposition 3.For the other direction, assume that C is equivalent to the FOL formula φ(x).Then φ(x) is invariant under ALCQt bisimulation by Proposition 19, and thus equivalent to an ALCQt concept description by Theorem 18.
It remains to show that ALCSCC ∞ is more expressive than ALCQt.Note that, by Theorem 20, any ALCSCC ∞ concept that is not expressible in ALCQt is also not expressible in FOL.The following proposition, which was first stated in [4], is an easy consequence of Proposition 30 in Section 4.1.
Fig. 2 summarizes the results obtained in this section. DLs The relative expressivity of the DLs ALCQ, ALCQt, ALCCQU, and ALCSCC ∞ .

Expressivity of boxes
Here we extend the bisimulation characterizations of the previous section to the box formalisms introduced in Section 2.3.For (Boolean) TBoxes and the DL ALCQ, this was already done in [24].First, we recall these results and extend them to ALCQt.As a consequence, we also obtain characterizations of the first-order fragments of ALCSCC ∞ TBoxes and Boolean TBoxes.Second, we show similar results for CBoxes and ECBoxes.

TBoxes and Boolean TBoxes in ALCQ, ALCQt, and ALCSCC ∞
In order to deal with CIs, which make global statements about all individuals of an interpretation, we need to "globalize" the notion of a bisimulation.
• The L bisimulation ρ between I 1 and I 2 is global if for every d ∈ ∆ I1 there exists e ∈ ∆ I2 such that (d, e) ∈ ρ (and vice versa).
• The interpretations I 1 and I 2 are globally L bisimilar (written ) if there is a global L bisimulation ρ between I 1 and I 2 .
• The interpretations I 1 and I 2 are globally L equivalent (written I 1 ≡ g L I 2 ) if for every CI C D with C and D L concept descriptions we have that • The first-order sentence φ is invariant under global L bisimulation if I 1 |= φ and The following proposition is an easy consequence of the above definition and Proposition 16.
As an immediate consequence, we obtain invariance of (Boolean) L TBoxes (viewed as firstorder sentences) under global L bisimulation.This result can be used to show that Boolean ALCQ TBoxes cannot express ALCQt TBoxes.
Corollary 25.There is no Boolean ALCQ TBox that is equivalent to the ALCQt TBox T = {B succ(|r ∩ s| 1).
To prove this corollary, we can basically reuse the interpretations I 1 , I 2 and the ALCQ bisimulation ρ shown in Fig. 1, but where now additionally d 1 , d 2 belong to the concept B, whereas the other elements do not belong to B. Then ρ is a global ALCQ bisimulation between I 1 and I 2 .However, I 1 is a model of T , whereas I 2 is not, which shows that T cannot be equivalent to a Boolean ALCQ TBox by Corollary 24.
Global L bisimulations can also be used to characterize the first-order sentences that are equivalent to Boolean L TBoxes.For L = ALCQ, this was already shown in [24].A detailed proof for L = ALCQt can be found in Appendix A.
Theorem 26.Let L ∈ {ALCQ, ALCQt} and φ be a first-order sentence.Then the following are equivalent: 1.There exists a Boolean L TBox T such that T is equivalent to φ.
2. The sentence φ is invariant under global L bisimulation.
To distinguish TBoxes from Boolean TBoxes, one needs to use the fact that TBoxes are invariant under disjoint union, whereas Boolean TBoxes are not (see Proposition 8).
Theorem 27.Let L ∈ {ALCQ, ALCQt} and φ be a first-order sentence.Then the following are equivalent: 1.There exists an L TBox T such that T ≡ φ.
2. The sentence φ is invariant under global L bisimulation and under disjoint unions.
For L = ALCQ, this theorem was shown in [24] (see proof of Theorem 7 in [24]), and the adaptation of this proof to the case L = ALCQt is simple.
Using the fact that ALCSCC ∞ concept descriptions are invariant under ALCQt bisimulation (see Proposition 19 above), it is easy to see that Proposition 23 can be extended to ALCSCC ∞ as follows.
Combining this result with Theorems 26 and 27 for the case L = ALCQt, we thus obtain the following characterizations of the first order fragments of (Boolean) ALCSCC ∞ TBoxes.
Theorem 29.Let T be a (Boolean) ALCSCC ∞ TBox.Then the following are equivalent: 1. T is equivalent to a first-order sentence.
It remains to show that there are indeed ALCSCC ∞ TBoxes that cannot be expressed by a Boolean ALCQt TBox, and thus are not expressible in FOL.
Proof.It is sufficient to show that T cannot be expressed as an equivalent Boolean ALCQt TBox T .Together with Theorem 29, this yields our statement.We fix (N C , N R ) := ({A}, {r}) and assume by contradiction that such T exists over this signature.Note that, in this restricted signature, the only safe role type is the role r itself, and thus successor constraints are in fact qualified number restrictions for the role r.
Due to the semantic equivalence ( K r.D) ≡ ¬( (K + 1) r.D), we can assume that every qualified number restriction occurring in T is of the form ( K r.D) with K a natural number and D an ALCQt concept description.Let N be the largest natural number appearing in a qualified number restriction in T .Then, we define N := N + 1 and the sets S 1 := {1, . . ., N } and S 2 := {N + 1, . . ., 2N }.
The interpretation I over ({A}, {r}) of domain ∆ I = {0, 1, . . ., 2N } is defined by setting all elements of S 1 and S 2 as r-successors of 0 and A I := S 1 .Then I is clearly a model of T , and hence of T .We extend I to I by adding 2N + 1 to the domain and to the interpretation of A, and connecting 0 with 2N + 1 via the role r.We show that I is a model of T , by proving the following facts: 1.For all i ∈ {1, . . ., N }, j ∈ {1, . . ., N, 2N + 1}, and ALCQt concepts D we have i ∈ D I iff j ∈ D I .
3. For all ALCQt concepts D containing only numbers < N we have 0 Before showing these facts, first note that they indeed imply that I is a model of T .In fact, assume that this is not the case.Then there is a CI C 1 C 2 in T such that D := C 1 ¬C 2 is non-empty in I .Assume that j ∈ D I .If j ∈ {1, . . ., N, 2N + 1}, then (1) implies that 1 ∈ D I , contradicting the fact that I is a model of T .Similarly, we can show that the case where j ∈ {N + 1, . . ., 2N } leads to a contradiction.Finally, if j = 0, then we have 0 ∈ D I since D satisfies the restriction stated in (3).Again, this leads to a contradiction.
We show (3) by induction on the structure of D. The only interesting case is the one where D is of the form D = ( K r.E) for an ALCQt concept E and a number K < N .We observe that all the elements of S 1 are pairwise ALCQt bisimilar in I, and the same is true for the elements of S 2 .Combining this observation with Proposition 16, we obtain that, for the ALCQt concept E, at least one of the following holds: (a) If (a) holds, then |S 1 | = N > K yields 0 ∈ ( K r.E) I .Due to the ALCQt bisimilarity relations between elements of I and I mentioned above, Proposition 16 yields S 1 ⊆ E I , and thus 0 ∈ ( K r.D) I holds as well.The case (b) can be treated similarly.Finally, assume that (c) holds.The case K = 0 is trivial since then ( 0 r.D) ≡ .If K 1, then 0 / ∈ ( K r.D) I since none of the r-successors of 0 in I belong to E. It is straightforward to see that 1 and 2N + 1 are bisimilar in I .Thus, by Proposition 16 we obtain that {1, . . ., 2N + 1} ⊆ (¬E) I , and can conclude that 0 / ∈ ( K r.D) I .This concludes the proof of (3).
Summing up, we have seen that both I and I are models of T .However, this contradicts our assumption that T is equivalent to T since actually I is a model of T , but I is not.
Note that this proposition also implies Proposition 21.In fact, if succ(|r ∩ A| = |r ∩ ¬A|) was expressible in FOL, then there would exist an ALCQt concept C such that C ≡ succ(|r ∩ A| = |r ∩ ¬A|).But then the TBox T would be equivalent to the ALCQt TBox { C}, which is expressible in FOL by Corollary 6.

Boolean CBoxes and ECBoxes in ALCQ, ALCQt, and ALCSCC ∞
In order to deal with CRs rather than CIs, we need to extend our notion of a global bisimulation to one that can also compare cardinalities of sets on the global level.The following definition is inspired by the first-order counting games used in [15] to analyze extensions of first-order logic by certain counting quantifiers.Definition 31.Let L ∈ {ALCQ, ALCQt} and I 1 , I 2 be interpretations.
ρ satisfies the following two properties: 1. if D 1 ⊆ ∆ I1 is finite, then there is a set D 2 ⊆ ∆ I2 such that ρ contains a bijection between D 1 and D 2 ; 2. if D 2 ⊆ ∆ I2 is finite, then there is a set D 1 ⊆ ∆ I1 such that ρ contains a bijection between D 1 and D 2 .
• The interpretations I 1 and I 2 are comparatively L bisimilar (written I 1 ∼ L I 2 ) if there is a comparative L bisimulation ρ between I 1 and I 2 .
• The interpretations I 1 and I 2 are comparatively L equivalent (written I 1 ≡ L I 2 ) if for all CRs |C| N (with C an L concept, N a natural number, and ∈ { , }) we have • The first-order sentence φ is invariant under comparative L bisimulation if I 1 |= φ and The following proposition states that CRs are indeed invariant under comparative bisimulation.
Proof.Assume that ρ is a comparative L bisimulation between the interpretations I This proposition obviously implies that (Boolean) CBoxes are invariant under comparative bisimulation.We show next that this is true even for ECBoxes (which subsumes the case of (Boolean) CBoxes).Every extended cardinality constraint occurring in E is of the form hold for 1 i k and 1 j (as just shown), every cardinality constraint occurring in E is evaluated in the same way in I 1 and I 2 .
Next, we want to show that Boolean L CBoxes are exactly the first-order sentences that are invariant under comparative L bisimulation.In contrast to the Sections 3 and 4.1, where we have stated the corresponding results (see Theorems 18 and 27) without proofs, here we will give a detailed proof.In fact, while the results for concept descriptions and TBoxes have been published before (in [24] and [4]), the results for CBoxes are published for the first time in the present paper.Note that the proofs of Theorems 18 and 27 have a structure that is very similar to the proof given below.
The first step is to show that the converse of Proposition 32 holds as well if we restrict the statement to so-called ω-saturated interpretations [24,10].When defining ω-saturated interpretations, one assumes that every domain element of an interpretation I can be used as a constant symbol in formulae, where d ∈ ∆ I interprets itself, i.e., d I := d.Let I be an interpretation of N C and N R .A (possibly infinite) set of first-order formulae Γ with free variables from a finite set {x 1 , . . ., x n }, predicate symbols from N C ∪ N R , and constant symbols from a finite subset of ∆ I is called • realizable in I if there is a variable assignment a : {x 1 , . . ., x n } → ∆ I such that I |= φ(a(x i1 ), . . ., a(x i k )) for every formula φ(x i1 , . . ., x i k ) ∈ Γ; • finitely realizable in I if every finite subset Γ of Γ is realizable in I.
The interpretation I is ω-saturated if, for every such set Γ, finite realizability in I implies realizability in I.
The following result from [10] implies that, though not every interpretations I is ω-saturated, one may without loss of generality assume that one has such an interpretation if one is only interested in the FOL sentences that the interpretation satisfies.
Theorem 34.For every interpretation I there exists an ω-saturated interpretation I that satisfies the same first-order sentences as I.
A further result that we will need in our proof of the converse of Proposition 32 is Hall's theorem [16].Given a finite family F = (S 1 , . . ., S N ) of sets, we say that F has a system of distinct representatives (SDR) if there are N distinct elements s 1 , . . ., s N such that s i ∈ S i for i = 1, . . ., N .
Theorem 35 (Hall).The family F = (S 1 , . . ., S N ) has a system of distinct representatives iff for all index sets I ⊆ {1, . . ., N } we have i∈I S i |I|.
The following lemma is an immediate consequence of Hall's theorem.It shows that the existence of an SDR can be characterized using a CBox.
Proposition 37. Let L ∈ {ALCQ, ALCQt} and I 1 , I 2 be ω-saturated interpretations.Then Proof.Let I 1 , I 2 be ω-saturated interpretations such that I 1 ≡ L I 2 .To demonstrate that these two interpretations are also comparatively L bisimilar, it is sufficient to prove that the binary relation Now, let G denote the set of L CRs that are satisfied by I − .We claim that G ∪{φ} has a model.In fact, otherwise first-order compactness would yield a finite subset G of G such that G ∪ {φ} also does not have a model.However, this would imply that φ → ¬ G is a tautology, which would yield ¬ G ∈ Cons(φ).This lead to a contradiction since now both G and ¬ G would need to be satisfied by I − .Thus, we have shown that G ∪ {φ} has a model I + , of which can again assume that it is ω-saturated.
We observe that I − and I + both satisfy exactly the CRs occurring in G, which implies that they are comparatively L equivalent.Since these two interpretations are also ω-saturated, Proposition 37 yields I − ∼ L I + .This contradicts our assumption that (2) holds since we have I + |= φ, but I − |= φ Thus, we have shown that (2) implies Cons(φ) |= φ, which concludes our proof.
Since ECBoxes are invariant under comparative L bisimulation by Corollary 33, Theorem 38 yields the following characterization of the first-order fragment of ECBoxes for the DLs ALCQ and ALCQt.
Theorem 39.Let L ∈ {ALCQ, ALCQt} and E be an L ECBox.Then the following are equivalent: 1.There exists a first-order sentence φ such that E ≡ φ.
2. E is equivalent to a Boolean L CBox C.
It remains to show that there are ALCQ ECBoxes that are not equivalent to a first-order sentence.Since it uses a technique different from the ones employed until now in this paper, we defer the proof of this result to the next section.
We close the current section by giving a characterization of the first-order fragment of ALCSCC ∞ ECBoxes.
Theorem 40.Let E be an ALCSCC ∞ ECBox.Then the following are equivalent: 1.There exists a first-order sentence φ such that E ≡ φ.
2. E is equivalent to a Boolean ALCQt CBox C.
Proof.To prove (1 =⇒ 2), assume that φ is a first-order sentence equivalent to E. It is easy to show that ALCSCC ∞ ECBoxes are invariant under comparative ALCQt bisimulation.Therefore, φ is also invariant under comparative ALCQt bisimulation.By Theorem 38, this implies that φ, and hence E, is equivalent to a Boolean ALCQt CBox C.
(2 =⇒ 1) is an immediate consequence of the fact that ALCQt CBoxes have a first-order translation (see Corollary 6).
Fig. 3 summarizes the results obtained in this section and the next section.
5 ECBoxes and the 0-1 law for FOL Let φ be a first-order sentence over a relational signature δ.We denote by L n (δ) the set of interpretations over the signature δ with domain {1, . . ., n}, and with L n (φ) the number of these interpretations that are models of φ.We then set Theorem 41 (0-1 law of FOL [12]).For every first-order sentence φ, the limit (φ) always exists and is equal to 0 or 1.
One can use this theorem to prove that a sentence of a certain logic cannot be equivalent to a first-order sentence by showing that the corresponding limit either does not exist or is a number different from 0 or 1.An example for the former case would be a formula whose models are exactly the interpretations whose domain has even cardinality.We show now that ECBoxes can yield examples for the latter case.
Proposition 42.The ECBox E := |A| |¬A| is not expressible as a first-order sentence.
Proof.By contradiction, assume that E is equivalent to some first-order sentence φ.We restrict our attention to the function-free signature δ := {A} since the only relation symbol contained in E is the concept name A. If we consider interpretations I with domain ∆ I = {1, . . ., n}, then there are 2 n possible ways of interpreting A I , which shows that L n (δ) = 2 n .Among these interpretations, the ones where |A I | = j for 0 j n are exactly n j .Therefore, the number of interpretations with domain {1, . . ., n} over δ satisfying E, and hence φ, is Let n (φ) := L n (φ)/L n (δ).We show that the sequence L := ( n (φ)) n 1 is convergent and (φ) := lim n→∞ n (φ) = 1/2. 7This yields a contradiction: by Theorem 41, it should hold that (φ) = 0 or (φ) = 1.
To show that L converges to 1/2, it is sufficient to prove that both L 1 and L 2 have this limit.First, note that for n 1 the following identities are valid: n j=0 By (8), our claim clearly holds for L 2 .Indeed, for n 1 we have the existence of distinct τ -successors y 1 ,. . .,y N of y such that x i and y i are L bisimilar for 1 i N .Using our inductive hypothesis, this implies that y 1 , . . ., y N ∈ D I2 , thus y ∈ C I2 .We can prove that y ∈ C I2 implies x ∈ C I1 in a similar way.The proof that (11) holds for C = (( N τ.D)) follows from the fact that (( N τ.D)) ≡ ¬(( N + 1 τ.D)).
In order to prove that Theorem 18 holds, we first show that the converse of Proposition 16 holds when we restrict our statement to the class of ω-saturated interpretations mentioned in Section 4. As explained there, this restriction is justified by the fact that for every interpretation I one can find an ω-saturated interpretation I satisfying the same first-order sentences (Theorem 34).
Furthermore, we make use of Hall's theorem to show that the existence of an SDR for the set of τ -successors of a certain individual can be characterized using an ALCQt concept description.We extend the result proved in [24] for ALCQ and show that ALCQt equivalence implies ALCQt bisimilarity between individuals in ω-saturated interpretations.
is an L bisimulation between I 1 and I 2 relating d 1 and d 2 .The proof for L = ALCQ can be found in [24].We are left with proving the case L = ALCQt.
Condition 1 in Definition 15 is trivially satisfied by the definition of (13).To show that (13) fulfills condition 2 in Definition 15, let (e 1 , e 2 ) ∈ Eq and consider a finite set of τ -successors d 1 ,. . ., d N of e 1 , with τ a safe role type.In order to find distinct τ -successors d 1 ,. . ., d N of e 2 such that (d i , d i ) ∈ Eq for every 1 i N , we resort to the fact that I 2 is ω-saturated.In particular, if Γ j := Γ j,τ ∪ N i=1 Θ i is the set of first-order formulae defined by Γ j,τ := { N i=1 τ (e j , x i ) ∧ N j=i+1 x i = x j } Θ i := {C (x i ) | C is an ALCQt concept and d i ∈ C I1 } we want to show that if Γ 1 is realizable in I 1 then so is Γ 2 in I 2 .Then, the variable assignments realizing the two sets can be used to define a bijection f such that f (d i ) is a τ -successor of e 2 and (d i , f (d i )) ∈ Eq.
Under the variable assignment a(x i ) := d i we clearly obtain realizability of Γ 1 and each of its finite subsets in I 1 .Since I 2 is ω-saturated we only need to show that each finite subset Γ 2 of Γ 2 are realizable in I 2 .Without loss of generality, Γ 2 always includes Γ 2,τ since it is finite.We introduce the well-defined concept descriptions and notice that the first-order formula N i=1 (τ (e 1 , x i ) ∧ C i (x i )) ∧ is satisfied in I 1 under the variable assignment a(x i ) := d i .By definition of (15) it follows that d 1 ,. . ., d N form an SDR for the sets τ I1 (e 1 ) ∩ C I1 i with 1 i N .Then, by Proposition A.2 we obtain e 1 ∈ C I1 , where C is defined in (12).We assumed that e 1 and e 2 are ALCQt equivalent, thus e 2 ∈ C I2 .By Proposition A.2, the sets τ I2 (e 2 ) ∩ C I2 i with 1 i N have an SDR d 1 ,. . .,d N .Finally, we obtain that Γ 2 is realized in I 2 under the variable assignment b(x i ) := d i .Thanks to the fact that I 2 is ω-saturated, we deduce that Γ 2 is realizable in I 2 and conclude.
We can prove with a similar argument that condition 3 in Definition 15 holds for (13) and obtain that Eq fulfills all the conditions stated in Definition 15, hence it is an ALCQt bisimulation.Since we assumed that (I 1 , d 1 ) ≡ ALCQt (I 2 , d 2 ), this is sufficient to conclude.
Using Theorem A.1 we are finally able to show that Theorem 18 holds.
Theorem 18 ( [24,4]).Let L ∈ {ALCQ, ALCQt} and φ(x) be a first-order formula with one free variable x.Then the following are equivalent: 1. there is an L concept description C such that C is equivalent to φ(x); 2. φ(x) is invariant under ∼ L .
We prove that (2 =⇒ 1), showing that if we assume (2) and no L concept C is equivalent to φ(x) we are able to derive a contradiction.By using first-order compactness, we show that the set Cons(φ(x)) ∪ {¬φ(x)} where It is clear that Γ is closed under conjunction, due to the semantics of L.Moreover, we deduce that Γ ∪ {φ(x)} is satisfiable in an interpretation I + under the variable assignment b(x) := e.Otherwise, first-order compactness would yield a finite subset Γ of Γ such that {φ(x)} ∪ Γ was also unsatisfiable.This would imply that φ(x) → ¬ Γ (x) is a tautology.Accordingly, to a different domain, the concept of a Computer Science author can be formalized by the concept description Person ∃published.(Paper∀topic.CS), which uses the concept names Person and Paper and the role names published and topic as well as the concept constructors conjunction ( ), existential restriction (∃r.C), and value restriction (∀r.C).It describes the set of all persons that have published a paper all of whose topics lie in the area of Computer Science.The concept inclusion (CI) CS-author Person ∃published.(Paper∀topic.CS)

Definition 2 (
Semantics of ALCSCC ∞ ).Given finite, disjoint sets N C and N R of concept and role names, respectively, an interpretation of N C and N R consists of a non-empty set ∆ I and a mapping • I that maps every concept name A ∈ N C to a subset A I of ∆ I and every role name r ∈ N R to a binary relation r I over ∆ I .Given an individual d ∈ ∆ I and a role name r ∈ N R , we define r I (d) := {e ∈ ∆ I | (d, e) ∈ r I } (r-successors) and ars I (d) := r∈N R r I (d) (all role successors).

Definition 5 .
Let L be one of the DLs ALCQ, ALCQt, ALCCQU, or ALCSCC ∞ .1.A Boolean L TBox is a Boolean combination of CIs C D, where C, D are L concept descriptions.An L TBox is a conjunction of such CIs.2. A Boolean L CBox is a Boolean combination of CRs of the form |C| N and |C| N , where C is an L concept description and N is a natural number.An L CBox is a conjunction of such CRs.

1 .
the CI C D if C I ⊆ D I holds, 2. the CR |C| N if |C I | N , and of the CR |C| N if |C I | N , 3. an inequation of the form (3) if

Proposition 8 .
If L ∈ {ALCQ, ALCQt, ALCCQU, ALCSCC ∞ }, then 1. the models of L TBoxes are invariant under disjoint union; 2. the models of L RCBoxes are closed under disjoint union, but in general not invariant under disjoint union; 3. the models of L ECBoxes or CBoxes are in general not closed under disjoint union; 4. the models of Boolean L TBoxes are not closed under disjoint union.Regarding the negative statement in 2., consider the RCBox |A| + |B| |C| for concept names A, B, C. If we consider interpretation I 1 and I 2 in which A I1 contains one element, B I1 one element, C I1 one element, A I2 one element, B I2 one element, and C I2 three elements, then the disjoint union of I 1 and I 2 is a model of the RCBox, but I 1 is not.Regarding 3., it should be clear that the models of |A| 1 cannot be closed under disjoint union.Finally, it is also easy to see that the models of the Boolean TBox (A ⊥) ∨ (B ⊥) are not closed under disjoint union.

Figure 1 :
Figure 1: Two interpretations I 1 and I 2 and an ALCQ bisimulation ρ, which is not an ALCQt bisimulation.

1 and I 2 .
Let |C| N be a CR with C an L concept and N a natural number such that |C I1 | N .Then C I1 contains distinct elements d 1 , . . ., d N , and the fact that ρ is a comparative bisimulation implies that there exist distinct elements e 1 , . . ., e N ∈ ∆ I2 such that (d i , e i ) ∈ ρ for 1 i N .Thanks to Proposition 16, it follows that e 1 , . . ., e N ∈ C I2 , and thus |C I2 | N .We can prove that |C I2 | N implies |C I1 | N using an analogous argument.The case for CRs of the form |C| N follows from the semantic equivalence of |C| N and ¬(|C| N + 1).

Corollary 33 .
If E is an L ECBox with L ∈ {ALCQ, ALCQt} andI 1 ∼ L I 2 then I 1 |= E iff I 2 |= E. Proof.First, we show that I 1 ∼ L I 2 implies that |C I1 | = |C I2 | holds for all L concepts C. In fact, if |C I1 | = N isfinite, then I 1 satisfies the CRs |C| N and |C| N .By Proposition 32, I 2 must then satisfy these CRs as well, which shows that |C I1 | = N = |C I2 |.If |C I1 | is infinite, then I 1 satisfies the CRs |C|N for all N 0, and by Proposition 32, I 2 must satisfy all these CRs as well, which shows that |C I2 | is also infinite.

Proposition A. 2 .
Given d ∈ ∆ I , a safe role type τ and ALCQt concepts C 1 , . . ., C N , the set F := (τ I (d) ∩ C I 1 , . . ., τ I (d) ∩ C I N ) has an SDR iff d ∈ C I where C is the ALCQt concept defined by C := {(( k τ. i∈S C i )) | S ⊆ {1, . . ., N } and |S| = k}.(12) Proof.It is clear that if d 1 , . . ., d N is an SDR for F then d ∈ C I .The fact that d ∈ C I implies that d 1 , . . ., d N is an SDR for F is a consequence of Theorem 35.

Theorem A. 1 .
For L ∈ {ALCQ, ALCQt} and ω-saturated interpretations I 1 and I 2 , if (I 1 , d 1 ) ≡ L (I 2 , d 2 ) then (I 1 , d 1 ) ∼ L (I 2 , d 2 ).Proof.If (I 1 , d 1 ) and (I 2 , d 2 ) are L equivalent, we are able to show that the relation Eq L := {(d, e) ∈ ∆ I1 × ∆ I2 | (I 1 , d) ≡ L (I 2 , e)} Cons(φ(x)) := {C (x) | C is an L concept and φ(x) |= C (x)} (16) is satisfiable in an interpretation I − (w.l.o.g.ω-saturated, thanks to Theorem 34) under a variable assignment a(x) := d.Since φ(x) is not equivalent to any L concept C, every finite subset of Cons(φ) has a model satisfying ¬φ(x).Otherwise, Cons(φ(x)) would contain a finite subset S entailing φ(x) and S ≡ φ(x) would follow.The set Cons(φ(x)) is closed under Boolean combinations (consequence of the closure of L under Boolean constructors), hence S ∈ Cons(φ(x)) would hold and S = C (x) for some L concept C.This would lead to a contradiction because φ(x) would be equivalent to C, against our initial assumption.We notice that Cons(φ(x)) is a subset of Γ := {C (x) | C is an L concept and d ∈ C I − }.
for all models I of T .Again, it is well-known that these problems can be reduced to each other in polynomial time if ⊥, , ¬, , and qualified number restrictions of the form ( 1 r.C) are available.In fact, T is inconsistent iff T ⊥ holds, and C is satisfiable w.r.t.T iff T ∪ { ( 1 r.C)} is consistent, where r is a new role name occurring neither in C nor in T .

Table 1 :
Complexity results for consistency assuming binary coding of numbers• if C ⊥ does not belong to T , then the cardinality of C I∞ is infinite.