The Abstract Expressive Power of First-Order and Description Logics with Concrete Domains

Concrete domains have been introduced in description logic (DL) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. The primary research goal in this context was to find restrictions on the concrete domain such that its integration into certain DLs preserves decidability or tractability. In this paper, we investigate the abstract expressive power of both first-order and description logics extended with concrete domains, i.e., we analyze which classes of first-order interpretations can be expressed using these logics, compared to what first-order logic without concrete domains can express. We demonstrate that, under natural conditions on the concrete domain D (which also play a role for decidability), extensions of first-order logic (FOL) or the well-known DL ALC with D share important formal characteristics with FOL, such as the compactness and the Löwenheim-Skolem properties. Nevertheless, their abstract expressive power need not be contained in that of FOL, though in some cases it is. To be more precise, we show, on the one hand, that unary concrete domains leave the abstract expressive power within FOL if we are allowed to introduce auxiliary predicates. On the other hand, we exhibit a class of concrete domains that push the abstract expressive power beyond that of FOL. As a by-product of these investigations, we obtain (semi-)decidability results for some of the logics with concrete domains considered in this paper.


INTRODUCTION
Description Logics (DLs) [8] are a prominent family of logic-based knowledge representation formalisms, which offer a good compromise between expressiveness and the complexity of reasoning and are the formal basis for the Web ontology language OWL 2. 1 To accommodate diverse application domains, the DL community has developed logics whose expressive power is tailored towards what is needed in these domains while leaving reasoning decidable.In many cases, however, the added expressiveness turned out to be useful also in other applications.For example, concrete domains, which enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons), were introduced in [6] motivated by a mechanical engineering application [7].Due to their usefulness in many applications, they are included in the OWL 2 standard, albeit in the restricted form of unary concrete domains (called datatypes), where all predefined predicates have arity one [16].
Most DLs [8] are decidable fragments of first-order logic (FOL), i.e., their expressive power [4,17] is below that of FOL, but there are also decidable DLs whose knowledge bases (KBs) cannot always be expressed by an FOL sentence [5].A case in point are DLs with concrete domains [6,18,20], at least at first sight.In such DLs, the abstract interpretation domain is complemented by the concrete domain, and partial functions can be used to assign values in the concrete domain to abstract objects.These values can then be constrained using the predefined predicates of the concrete domain.For example, assume that we want to model physical objects, collected in a concept (i.e., unary predicate) PO, which can be decomposed into their proper parts using a role (i.e., binary predicate) hpp for "has proper part."If we want to take the weight of such objects into account, it makes sense to assign a number for its weight to every physical object using a feature (i.e., partial function) , and to state that this weight is positive and that proper parts are physical objects that have a smaller weight than the whole.Using the syntax employed in [10,20] and in the present paper, these conditions can be expressed with the help of value restrictions and concrete domain restrictions w.r.t. an appropriate concrete domain by the following concept inclusion (CI): PO ⊑ ∀hpp.PO ⊓ ∃ .( 1 > 0) ⊓ ∀, hpp  .>( 1 ,  2 ).(1) Depending on what kind of decomposition into proper parts we have in mind, we can use the rational numbers or the integers as concrete domain.The former would be more appropriate for settings like cutting a cake, where a given piece can always be cut into even smaller parts, whereas the latter is more appropriate for settings where physical objects are composed of finitely many atomic parts that cannot be divided any further.Interestingly, as we will show in this paper, this decision also has an impact on the formal properties that the logic (in the example, the well-known DL ALC [26]) extended with such a concrete domain satisfies.If we employ the integers, then for any element of PO there is a positive integer such that the length of all hpp-chains issuing from it are bounded by this number.Using this fact, it is easy to show that the logic at hand is not compact, i.e., there may be unsatisfiable infinite sets of sentences for which all finite subsets are satisfiable.In particular, this implies that the abstract expressive power of this logic, which considers only the abstract domain and the interpretation of concept and role names, but ignores the feature values, cannot be contained in FOL.For the rational numbers, the results obtained in this paper imply that the extension of ALC or FOL with this concrete domain shares the compactness and the Löwenheim-Skolem property with FOL.The reason is that the rational numbers with > are homomorphism -compact [9,10], which means that a countable set of constraints is solvable iff all its finite subsets are solvable.We can, however, prove that the abstract expressive power of these logics is nevertheless not contained in FOL, though we cannot use a compactness argument to show this.
Note that it is quite natural to consider concrete domains that are homomorphism -compact.In fact, in the presence of CIs, integrating even rather simple concrete domains into the DL ALC may cause undecidability [9,19].To overcome this problem, the notion of -admissible concrete domains was introduced in [20], and it was proved that integrating such a concrete domain into ALC leaves reasoning decidable also in the presence of CIs.By definition, all -admissible concrete domains are homomorphism -compact.As examples of -admissible concrete domains, Allen's interval algebra [1] and the region connection calculus RCC8 [24] are provided in [20].Using well-known notions and results from model theory, additional -admissible concrete domains were exhibited in [9,10].A simpler, but considerably more restrictive way of achieving decidability is to use unary concrete domains.Decidability for the expressive DL SH OQ extended with such concrete domains is shown in [16].In [3,25], conjunctive query answering in extensions of the inexpressive DL DL-Lite with unary concrete domains is investigated.
In the next section, we will introduce FOL with concrete domains and then define DLs with concrete domains as fragments.Subsequently, we present two variants of the notion of abstract expressive power, one where one can use auxiliary predicates on the first-order side to express sentences of the logic with concrete domains, and one where this is not allowed.Section 3 is dedicate to proving that FOL and DLs with concrete domains share a number of interesting formal properties with FOL, provided that the employed concrete domain is homomorphism -compact and its set of predicates is closed under negation.In Section 4, we show, on the one hand, that FOL with a unary concrete domain can be expressed in FOL if we are allowed to use auxiliary predicates.In addition, if we restrict the logic with unary concrete domain to a decidable fragment like the guarded or the two-variable fragment with counting, then decidability on the concrete domain side yields decidability of the whole logic.On the other hand, we provide conditions on concrete domains such that ALC extended with such a concrete domain cannot be expressed in FOL.Basically, these are concrete domains whose predicates are closed under negation and in which equality is definable.

LOGICS WITH CONCRETE DOMAINS
We introduce first-order logic with concrete domains, from which we obtain DLs with concrete domains as fragments.Then, we define the notion of abstract expressive power of a logic with concrete domains.We assume that the reader is familiar with syntax, semantics, and basic results for first-order logic.

Concrete domains.
A concrete domain is a -structure  for a relational signature , i.e., it consists of a set , called its domain, together with relations   ⊆   for each -ary relation symbol  ∈ .A constraint system Γ for  is a set of atoms of the form  ( 1 , . . .,   ) where  ∈  has arity  and the   are variables.The constraint system Γ is satisfiable in  if there is an assignment ℎ (also called homomorphism) of elements of  to the variables in Γ such that (ℎ( 1 ), . . ., ℎ(  )) ∈   for all atoms  ( 1 , . . .,   ) in Γ.We call such a homomorphism a solution of Γ in .For example, consider the structure  := (Q, >) of rational numbers with the standard ordering relation, which we write infix.The set The concrete domain  is homomorphism -compact if the following holds for any countable constraint system Γ for : Γ is satisfiable in  iff all its finite subsets are satisfiable in .For example,  is homomorphism -compact.This follows from the results in [9,10], but is also a consequence of the fact that a constraint system is satisfiable in  iff it does not contain a cycle of the form  1 >  2 ,  2 >  3 , . . .,   >  1 .In contrast ℨ := (Z, >) is not homomorphism -compact: the constraint system Γ := {  >   | ,  ∈ Q,  >  } is countable and unsatisfiable in ℨ since it requires the existence of infinitely many integers between whatever integers are assigned to  0 and  1 ; however, any finite subset is clearly satisfiable.
First-order logic with concrete domains.Let  be a concrete domain over a relational signature ,  be a first-order signature (which may also contain function symbols), and F be a countable set of feature symbols.The formulae of first-order logic with the concrete domain , FOL F  () (or simply FOL() if  and F are irrelevant or clear from the context), are obtained by extending the usual inductive definition for FOL with the following two base cases: • definedness predicates Def ( )() with  ∈ F and  a -term, • concrete domain predicates  ( 1 , . . .,   )( 1 , . . .,   ) with  ∈  of arity ,   ∈ F , and   -terms.The semantics of FOL() formulae is defined inductively, using a first-order interpretation ℑ = (, • ℑ ) for  extended with a set  of partial functions   :  ⇀  for  ∈ F , and an assignment  mapping variables to elements of  .The semantics of terms, Boolean connectives and first-order quantifiers is defined as usual, where we denote the interpretation of a term  by ℑ and  as  ) must be defined for  = 1, . . ., .The tuple (ℑ, ) is a model of the FOL() sentence  (i.e., formula without free variables), in symbols (ℑ, ) |= , if (ℑ, ),  |=  for some (and thus all) assignments .
Description Logics with concrete domains.For an arbitrary DL DL, a given concrete domain  can be integrated into DL with the help of concrete domain restrictions.Concrete domain restrictions for  are concept constructors of the form ∃ p. ( x) or ∀ p. ( x), with p =  1 , . . .,   a sequence of  feature paths,  a -ary predicate of , and x =  1 , . . .,   a -tuple of variables.In the context of this paper, a feature path is either a feature name  or an expression   with  a feature name and  a role name.We denote the DL obtained from DL by adding these restrictions as concept constructors with DL ().For example, ALC() has, in addition to the concrete domain restrictions introduced above, the concept constructors conjunction (⊓), disjunction (⊔), negation (¬), existential restriction (∃ .),and value restriction (∀ .).
To define the semantics of DL (), we assume that concepts of DL can inductively be translated into FOL formulae with one free variable  using a translation function   .For example,   (⊓) :=   () ∧   () and   (∃ .):= ∃.( (, ) ∧   ()).We extend this translation function to map concepts of DL () to formulae of FOL() by providing the translation of concrete domain restrictions.Taking x, p as defined above, let  ⊆ {1, . . .,  } be such that   =     if  ∈  and   =   otherwise.We define ȳ :=  1 , . . .,   by setting   =   if  ∈  and   =  otherwise, and z as the sequence of variables   with  ∈  .The translation of concrete domain restrictions is then defined as follows, where  (, ȳ) abbreviates The semantics of TBoxes (i.e., finite sets of CIs  ⊑ ) of the DL DL () is then defined in the usual way by translation into FOL() sentences:  ⊑  is translated into ∀ . () →   ().It is easy to see that the semantics of concrete domain restrictions given by the translation in (2) coincides with the direct modeltheoretic semantics in [10,20].In [20], extensions of the predicates of a concrete domain  by disjunctions of its base predicates are allowed to be used in concrete domain restrictions, whereas in [10] even predicates first-order definable from the base predicates are considered.These extensions can clearly also be translated into FOL().We denote them as DL ∨ + () and DL fo (), respectively.
Abstract expressive power.If we want to compare the expressive power of (a fragment of) FOL with that of (a fragment of) FOL(), we have the problem that the semantic structures they are based on differ in that, for the latter, one additionally has a collection of partial functions into the concrete domain.To overcome this difference, we say that the first-order interpretation ℑ is an abstract model of the FOL() sentence , in symbols ℑ |=  , if there is an interpretation of the feature symbols  such that (ℑ, ) |= .The FOL sentence  is an abstract definition of the FOL() sentence  if the abstract models of  are exactly the models of  .In this case we also say that  and  are abstractly equivalent.
Example 2.1.Consider the unary concrete domain  := (N, even, odd) where even, odd are unary relations with the standard meaning.The ALC() TBox T := { ⊑ ∃ .even(), ⊑ ∃ .odd()} is abstractly equivalent to the ALC TBox T ′ := { ⊑ ¬}.In fact,  and  must be interpreted as disjoint sets in any model of T .Conversely, any model of T ′ can be extended to a model of T by defining  to yield 0 for the elements of , 1 for the elements of , and no value for all other elements.We will show in the next section that such a definability result always holds for unary concrete domains.However, in general one may need to introduce auxiliary predicates to express the concrete domain restrictions.The following definition allows for such additional predicates.Let  be an FOL() sentence and  an FOL sentence that may contain auxiliary predicates not occurring in .Then  is an abstract projective definition of  if the abstract models of  are exactly the reducts of the models of  , where in a reduct we just forget about the interpretation of the auxiliary predicates.In this case we also say that  and  are abstractly projectively equivalent.The abstract expressive power of (a fragment of) FOL() is determined by which classes of abstract models can be defined by its sentences.Definition 2.2.Given a fragment  of FOL(), we say that its abstract expressive power is (projectively) contained in a fragment  of FOL if every sentence of  has an abstract (projective) definition in .
Example 2.3.In the introduction we have given an example showing that, for a concrete domain  over the integers with predicates  >  and  > 0, the abstract expressive power of ALC() is not contained in FOL.The argument we have used there (which is based on the fact that FOL is compact, but ALC() is not) also works in the projective setting.In fact, the CI (1) enforces that, for any element of PO, there is a positive integer such that the length of all hpp-chains issuing from it are bounded by this number.Assume that  is an FOL sentence that is an abstract projective definition of this CI.Clearly we can write, for all  ≥ 1, an FOL sentence   that says that the constant  is an element of PO and the starting point of an hpp-chain of length .Then any finite subset of { } ∪ {  |  ≥ 1} is satisfiable, but the whole set cannot be satisfiable since the CI (1) enforces a finite bound on the length of chains issuing from .Since FOL is compact, this shows that  cannot be a first-order sentence.
However, compactness of ALC() for a given concrete domain  does not guarantee that its abstract expressive power is projectively contained in FOL.
Example 2.4.Consider the concrete domain  ′ := (Q, >, =).The results shown in the next section imply that the logic FOL( ′ ) is compact, and thus also its fragment ALC( ′ ).Nevertheless, the abstract expressive power of ALC( ′ ) is not projectively contained in FOL.To see this, consider the TBox T := {⊤ ⊑ ∃ ,  .=( 1 ,  2 ) ⊓ ∀ ,   .>( 1 ,  2 )} and assume that there is an FOL formula  that is an abstract projective definition of it.Then (Q, >), where > is the interpretation of  , is an abstract model of T .In fact, one can use the identity function to interpret the feature  .Thus, (Q, >) can be extended to a model of  (by appropriate interpretations of the auxiliary predicates contained in  , if any).Since (Q, >) satisfies the formula  := ∀, .( (, ) ∨  =  ∨  (, )), we can conclude that  ∧  is satisfiable.The upward Löwenheim-Skolem property of FOL yields an uncountable model of  ∧ .Since  is an abstract projective definition of T , the reduct ℜ of this uncountable model to the signature consisting of  must be extendable to a model of T .This means that there is an interpretation   of  such that (ℜ, ) is a model of T .The conjunct ∃ ,  .=( 1 ,  2 ) on the right-hand side of the CI forces   to be total.Let ,  be distinct elements of ℜ.Since ℜ satisfies , we know that  and  are related via  , in one direction or the other.Then the restriction ∀ ,   .>( 1 ,  2 ) yields   () ≠   (), and thus   is injective.However, since ℜ is uncountable and Q is countable, there cannot be an injective function from the domain of ℜ to Q.

FIRST-ORDER PROPERTIES OF LOGICS WITH CONCRETE DOMAINS
First-order logic satisfies a number of interesting formal characteristics, usually shown in any introductory textbook in logic [12,14].Assuming that Φ is an at most countable set of sentences in our target language, these properties can be specified as follows: (Downward) Löwenheim-Skolem: If Φ is satisfiable, then it has a model whose domain is at most countable; (Upward) Löwenheim-Skolem: If Φ has a model with an infinite domain, then it has a model with an uncountable domain; (Countable) Compactness: If every finite subset of Φ is satisfiable, then Φ is satisfiable; Recursive enumerability: The set of unsatisfiable sentences is recursively enumerable (r.e.).
We will show that, under natural conditions on the concrete domain , FOL() shares most and ALC() shares all of these properties with FOL.The first condition is that  is homomorphism -compact, i.e., that constraint solving in  is compact in the sense that a countable constraint system for  is satisfiable iff every of its finite subsets is satisfiable.As mentioned before, this property is part of the -admissibility condition, which guarantees decidability of ALC().The second condition is that the concrete domain  is closed under negation, i.e. for every predicate symbol  of  there is a predicate symbol   of  such that d ∈   iff d ∉    .This condition appears in the definition of admissibility for concrete domains [6], and is needed since our logics can express negation of concrete domain predicates.If it is not satisfied (as, e.g., for the concrete domain  ′ in Example 2.4), then one can extend the concrete domain by the missing predicates.However, then homomorphism -compactness needs to hold for the extended concrete domain (as is the case for the extension of  ′ by the complements of its predicates).
We assume in this section that the concrete domain  is homomorphism -compact and closed under negation.The main tool for showing our results is a satisfiability-preserving translation of sets of FOL() sentences into sets of FOL sentences.
Rewriting to first-order logic.Let Φ be an at most countable set of FOL() sentences.We translate Φ into a set of FOL sentences Φ FOL by replacing every atom of the form  ( 1 , . . .,   )( 1 , . . .,   ) occurring in Φ with   1 ,...,  ( 1 , . . .,   ), where for every -ary concrete domain predicate  and features  1 , . . .,   we assume that   1 ,...,  is a new -ary predicate symbol in the first-order signature.Similarly, every atom of the form Def ( )() is replaced with Def  () where Def  is a new predicate symbol for every feature  .
Every set Γ of atoms of the form   1 ,...,  ( 1 , . . .,   ) induces the constraint system Γ := { ( To capture the semantics of the concrete domain predicates and the definedness predicate, we additionally consider the set of FOL sentences Ψ  where: • assuming that x :=  1 , . . .,   , we add for each of the new predicate symbols   1 ,...,  the sentences • for every finite set Γ of atoms of the form   1 ,...,  ( 1 , . . .,   ) we add the sentence ∀x .Γ → ⊥ if the constraint system Γ is unsatisfiable in , where x collects all the variables occurring in Γ.
Theorem 3.1.Let  be a homomorphism -compact concrete domain that is closed under negation.The set Proof."⇐" Assume that Φ FOL ∪ Ψ  is satisfiable.Since this is a countable set of first-order formulae, we apply the downward Löwenheim-Skolem property of FOL to get an at most countable model ℑ of Φ FOL ∪ Ψ  .We show that we can extend ℑ with an interpretation  of the features such that (ℑ, ) is a model of Φ.To this purpose, introduce a fresh variable   for every  ∈  and consider the set Γ ℑ consisting of all atoms   1 ,...,  (  1 , . . .,    ) such that   1 ,...,  ( 1 , . . .,   ) is satisfied in ℑ, where  1 , . . .,   ranges over all elements of ℑ and  1 , . . .,   over all feature names.Due to our construction of Ψ  and the fact that ℑ is a model of this set, we know that all finite subsets of Γ ℑ are satisfiable in .Since Γ ℑ is countable, homomorphism -compactness implies that there exists a solution ℎ of it in .For all feature names  and elements  ∈  for which the variable    occurs in Γ ℑ , we define   () := ℎ(   ).Otherwise, we choose an arbitrary value for   () if Def  () is true in ℑ, and leave   () undefined otherwise.The fact that, together with this interpretation of the features , the FOL interpretation ℑ is indeed a model of Φ, is an immediate consequence of the following two claims: is true in (ℑ, ).
To show the first claim, assume that Def  () is true in ℑ.Then   () is defined either by the solution ℎ of the constraint system Γ ℑ in  or it has received some arbitrary value.If Def  () is not true in ℑ, then   () cannot have been defined in terms of ℎ, since otherwise an expression   1 ,...,  ( 1 , . . .,   ) that is true in ℑ would have to exist such that  =   and  =   .But then Ψ  would have enforced Def  () to be true in ℑ, leading to a contradiction.In addition, since Def  () is not true in ℑ, no arbitrary value is assigned to   ().Thus   () is undefined.
"⇒" Assume that Φ is satisfiable in FOL() by the interpretation ℑ of the FOL part and the interpretation  of the features.We extend ℑ to an interpretation ℑ ′ that also takes the new predicates Def  and   1 ,...,  into account: •  ∈ Def ℑ ′  iff   () is defined, • ( 1 , . . .,   ) ∈ (  1 ,...,  ) ℑ ′ iff (  1 ( 1 ), . . .,    (  )) ∈   .Since (ℑ, ) makes Φ true, it is easy to see that ℑ ′ is a model of Φ FOL .In addition, it is a model of Ψ  due to the semantics of concrete domain restriction in FOL() and the fact that   is the complement of  in .□ Thanks to this theorem, we can transfer the properties of FOL introduced above to FOL().Corollary 3.2.FODcompactness If  is a homomorphism compact concrete domain that is closed under negation, then FOL() is countably compact and satisfies the downward Löwenheim-Skolem property.Homomorphism -compactness is also a necessary condition for countable compactness.In general, FOL() need not satisfy the upward Löwenheim-Skolem property.If the finite unsatisfiable constraint systems for  are r.e., then so are the unsatisfiable sentences of FOL().
Proof sketch.Compactness follows from Theorem 3.1.In fact, if Φ is unsatisfiable, then this theorem and compactness of FOL yield a finite subset Ψ of Φ FOL ∪ Ψ  that is unsatisfiable.Then translating Ψ ∩ Φ FOL back to FOL() yields an unsatisfiable finite subset of Φ.The downward Löwenheim-Skolem property follows from the construction of the abstract model ℑ in the if-direction of Theorem 3.1, which is at most countable.
Next, consider the concrete domain  = := (Q, =, ≠), which is closed under negation and easily seen to be homomorphism compact.The FOL ( = ) sentence states that  is an injective function from the domain of an abstract model of  up into Q.Thus, no abstract model of  up can have an uncountable domain, as Q is is countable.
Finally, assume that Φ = { } for an FOL() sentence .The assumption that the finite unsatisfiable constraint systems for  are r.e.entails that the set Φ FOL ∪ Ψ  is r.e. as well.We can now dovetail a partial decision procedure for unsatisfiability of finite sets of FOL sentences with the enumeration of Φ FOL ∪Ψ  to get a procedure that terminates iff Φ FOL ∪Ψ  is unsatisfiable.Together with Theorem 3.1 this shows that unsatisfiability of FOL() sentences is partially decidable, and thus r.e.□ For ALC with a concrete domain, we can strengthen the result of Corollary 3.2 as following.
Corollary 3.3.ALCDcompactness Let  be a homomorphism -compact concrete domain that is closed under negation, and L be either ALC(), ALC ∨ + () or ALC fo ().Then L is countably compact and satisfies the upward and the downward Löwenheim-Skolem property.Homomorphism -compactness is also a necessary condition for countable compactness.
Proof sketch.The downward Löwenheim-Skolem property and countable compactness are an immediate consequence of the fact that L can be expressed in FOL().Regarding necessity of homomorphism -compactness, it is easy to see that a counterexample to this property for  can also be turned into a counterexample to countable compactness of L, similar to the construction for FOL().The upward Löwenheim-Skolem property is an immediate consequence of the fact that, like ALC [8], its extension L is closed under disjoint unions.□

FIRST-ORDER (NON-)DEFINABILITY AND DECIDABILITY
In Section 2, we have seen an example of a unary concrete domain  and an ALC() TBox T such that T is abstractly equivalent to an ALC TBox T ′ .Basically, the first part of this section generalizes this result to all unary concrete domains  that are closed under negation.To be more precise, we show that, in this setting, every FOL() sentence has an abstract projective definition in FOL, and likewise every ALC() TBox is abstractly projectively equivalent to an ALC TBox.As a byproduct of these results, we are able to show that, under the additional assumption that constraint satisfaction in  is decidable, extending the guarded or two-variable fragments with counting of FOL with such a concrete domain leaves the resulting logic decidable.Regarding non-definability, Section 2 presents an example of a homomorphism -compact concrete domain  ′ and an ALC( ′ ) TBox T that has no abstract projective definition in FOL.In the second part of this section, we will generalize this result from  ′ to countable concrete domains in which (in)equality is appropriately definable.We will also show that adding such concrete domains to FOL destroys the upward Löwenheim-Skolem property.

Unary concrete domains
We recall that a concrete domain is unary if it contains only unary relations.Assume that  is a unary concrete domain that is closed under negation.Let  be an FOL() sentence and Φ := { }.The rewriting approach described in Section 3 produces a singleton set Φ FOL consisting of an FOL sentence  FOL and a set Ψ  of FOL sentences consisting of • finitely many sentences ∀ . () → Def  (), • finitely many sentences ∀ .¬ () →    () ∨ ¬ Def  (), • finitely many sentences of the form ∀ .Γ → ⊥ where Γ is a set of atoms The first two points are justified by the fact that we can restrict our attention to the concrete domain predicates and feature symbols that occur in .Regarding the last point, we notice that, in a setting where all concrete domain predicates are unary, constraints of the form   () and   (), where  ≠  or  ≠ , cannot influence each other.Thus, one can restrict the attention to constraint systems built using a single feature name  and variable .In fact, any unsatisfiable constraint systems must contain an unsatisfiable one of this form.Since we can again restrict the attention to the concrete domain predicates and feature symbols occurring in , and the name of single variable is irrelevant, there are only finitely many sentences of this form.Overall, this rewriting approach yields an FOL sentence  :=  FOL ∧ Ψ  .We cannot directly apply Theorem 3.1 to conclude that  and  are equisatisfiable since we have not assumed that  is homomorphism -compact.The proof of the following results shows that, even without this assumption, we obtain the stronger result that  is a first-order abstract projective definition of .
Corollary 4.1.Let  be a unary concrete domain that is closed under negation.Then, every FOL() sentence has an abstract projective definition in FOL.
Proof.Let  be a FOL() sentence and  the FOL sentence obtained by the rewriting process described above.First, we show that every model of  is an abstract model of .Let ℑ be a model of  .Since  is unary, the constraint system Γ ℑ considered in the proof of Theorem 3.1 contains all expressions   (  ) such that   () holds in ℑ for  a feature name and  ∈  .For every feature name  and  ∈  , let Γ , be the subsystem of Γ ℑ containing all and only expressions of the form   (  ).We notice that each of these subsystems is finite, and that they partition Γ ℑ .In particular, Γ ℑ is satisfiable in  iff Γ , is satisfiable in  for every  and  ∈  .The satisfiability of each such Γ , in  is a consequence of the fact that ℑ is a model of  , and thus of Ψ  .Otherwise, Ψ  would contain the sentence ∀ .Γ , → ⊥ and this would lead to a contradiction.We conclude that Γ ℑ has a solution ℎ in , which we use as in the proof of Theorem 3.1 to define an interpretation  of feature names such that (ℑ, ) is a model of .
Second, we must show that every abstract model  can be extended to a model of  by interpreting the new predicates of the form   and Def  appropriately.This can be done exactly as in the proof of Theorem 3.1.□ Recall that, in the proof of Theorem 3.1, we used the downward Löwenheim-Skolem property of first-order logic to ensure that the constraint system Γ ℑ is countable, a necessary requirement to be able to apply homomorphism -compactness.In the proof of Corollary 4.1, this was not possible since we had to show that the given model of  is an abstract model of .Fortunately, the fact that we can reduce satisfiability of Γ ℑ to that of the finite systems Γ , allowed us to dispense with this step and the requirement that  is homomorphism -compact.
For ALC() TBoxes T we can strengthen Corollary 4.1 by introducing a TBox T  that takes on the role of Ψ  in the FOL() setting.First, we introduce fresh concept names   and Def  for every feature name  and unary predicate  of  that occur in a concrete domain restriction of T .We denote with T FOL the ALC Decidability results.Note that, in the setting introduced in this subsection, the FOL() sentence Ψ  belongs both to the guarded and the two-variable fragment with counting of first-order logic, which are known to be decidable [2,15,22,23].Therefore, if the sentence  falls into one of these fragments, defined analogously to their first-order counterparts, it follows that the abstract projective definition  of  used in Corollary 4.1 also falls into this fragment.To ensure that satisfiability of FOL() sentences falling into one of these fragments is decidable, it is necessary to guarantee that Ψ  can effectively be computed.This is the case if checking satisfiability of a finite constraint system for  is decidable.Corollary 4.3.Let  be a unary concrete domain that is closed under negation.If constraint satisfiability for  is decidable, then satisfiability of sentences in the guarded or the two-variable fragment with counting of FOL() is decidable.
The first-order translations of many DLs considered in the literature actually belong to the guarded or the two-variable fragment with counting.Since, in the unary case, the translations of concrete domain restrictions into FOL() given in (2) also belong to these fragments, the above corollary yields decidability results for a great number of DLs extended with unary and decidable concrete domains.Note, however, that this does not cover the decidability result for SH OQ extended with unary concrete domains in [16] since the transitivity of roles specifiable in that DL cannot be expressed in the guarded or the two-variable fragment with counting.

In(equality) causes non-definability
We have seen above that the restriction to a unary concrete domain  ensures that every FOL() sentence has an abstract projective definition in FOL.This also implies that FOL() satisfies the upward Löwenheim-Skolem property.Without the restriction to predicates of arity 1, this need no longer be the case.In fact, Example 2.4 demonstrates that there exists a concrete domain  and an ALC() TBox T such that T does not have an abstract projective definition in FOL.In addition, the proof of Corollary 3.2 shows that, for the concrete domain  = , the logic FOL( = ) does not satisfy the upward Löwenheim-Skolem property.In the following, we extend these negative results from single examples to a large class of concrete domains.
Analyzing the two concrete examples, we see that they crucially depend on the fact that (in)equality can be expressed in the concrete domain under consideration.Following [10], we say that  is jointly diagonal (JD) if equality between elements of  can be expressed using a quantifier-free formula  = (, ) over the predicates contained in the signature of . 2 In [10], JD is part of the definition of -admissibility, and thus all -admissible concrete domains exhibited there satisfy this property.In Corollary 3.3 we use closure under disjoint union of models of ALC() TBoxes to show that ALC() has the upward Löwenheim-Skolem property.However, the fact that such a TBox then always has an uncountable model is not sufficient to apply the argument used in Example 2.4 to show that there exists an ALC() TBoxes that have no abstract projective first-order definition.In fact, such an uncountable model could be the uncountable disjoint union of countable models, and injectivity of the feature name  can possible only be enforced on the countable sub-models.This is why we needed the formula  in the proof given in that example, which states that any two distinct elements of the interpretation domain are linked by the role  .To adapt the idea underlying this proof to our more general setting, we make an additional assumption on the formula  = (, ) defining equality.Theorem 4.5.Let  be an at most countable concrete domain that is closed under negation and is JD, and assume that there is a quantifier-free definition of equality over  that uses only binary relations.Then, there is an ALC() TBox that has no abstract projective definition in first-order logic.
Proof.Let  = (, ) be a quantifier-free definition of equality over  that uses only binary relations.Let  be a binary relation that occurs in  = (, ) and   its complement, whose existence is guaranteed by the assumption of closure under negation.We can force a feature name  to act as a total function (in the spirit of the CI ⊤ ⊑ ∃ ,  .=( 1 ,  2 ) used in Example 2.4) with the ALC() TBox Equivalence to requiring totality of  follows from the fact that, for every  ∈ , the concrete domain  satisfies either  (, ) or   (, ).
We combine our assumptions about  = (, ) and closure under negation to obtain a quantifier-free and positive formula  ≠ (, ) that defines inequality over  and uses only binary relations of .To ensure that  ≠ (, ) does not contain negated predicates, we take the negation-normal form of ¬ = (, ) and replace every negated occurrence of  with its complement   .We introduce for every binary relation  that occurs in  ≠ (, ) a fresh role name   and a CI ⊤ ⊑ ∀ ,    .( 1 ,  2 ) and call T ≠ the resulting TBox.
Let T := T tot ∪ T ≠ and assume, by contradiction, that T is abstractly projectively equivalent to a first-order sentence .The interpretation ℑ with countable domain  :=  and  ℑ  :=   is an abstract model of T , where we interpret the feature name  using the identity over .Then, ℑ can be extended to a model ℑ ′ of .Using the upward Löwenheim-Skolem property of first-order logic, we find an uncountable interpretation  that is elementary equivalent to ℑ ′ in first-order logic (apply the property to the first-order theory of ℑ ′ , which is trivially satisfied by ℑ ′ ).This implies that  satisfies  and thus, by assumption, we can find an interpretation   of  such that (, ) is a model of T .
Since  is elementary equivalent to ℑ ′ , it also satisfies the above sentence and thus (, ) ∈ (  ≠ )  .Since (, ) is a model of T tot , both   () and   () must be defined.The fact that (, ) is a model of T ≠ ensures that (, ) ∈ (  )  implies (  (),   ()) ∈   for every predicate  occurring in  ≠ (, ).Therefore, (  (),   ()) ∈   ≠ holds, and consequently   () ≠   (), which implies that   is an injective function.This leads to a contradiction since we know that the domain  of  is at most countable, but the domain  of  is uncountable, and   is supposed to be an injective function from  to .We conclude that T is not abstractly projectively equivalent to any first-order sentence.□ To conclude this section, let us point out that the assumptions made in this theorem are not very restrictive.As already mentioned above, JD is part of the definition of -admissibility in [10].
Whereas [10] does not assume closure under negation of the set of concrete domain predicates, it requires that -admissible concrete domains are jointly exhaustive and pairwise disjoint (JEPD).It is easy to see that JEPD can replace closure under negation in the proof of the above theorem since the complement of any relation of  can then be expressed as the union of the other relations.Finally, note that the original work introducing -admissibility [20] assumed that all relations are binary, and many of the -admissible concrete domains exhibited in [10] also satisfy this restriction.

CONCLUSION
We have introduced the notion of abstract expressive power of a logic (FOL or a DL) with concrete domain, which is determined by which classes of abstract models (where one abstracts away the interpretation of feature names) can be defined by sentences of this logic.Our first main result is that such classes of abstract models share compactness and the downward Löwenheim-Skolem property with the ones definable by FOL if the employed concrete domain satisfies some reasonable model-theoretic assumptions.Under an additional computability assumption, the construction used to show these results also provides us with an effective procedure for enumerating all unsatisfiable sentences of the logic with concrete domain.An interesting topic for future research is to check which other properties of FOL (e.g., Craig interpolation [21] or the 0-1 law [13]) hold for (fragments of) FOL(), depending on certain properties satisfied by .It is well-known that ALC is the fragment of FOL that is closed under bisimulation [11].It would be interesting to see whether a similar result holds for ALC() and FOL(), based on an appropriate notion of bisimulation.
Our second main result is that, although sharing interesting properties with FOL, sentences of logics with concrete domain are often not (projectively) definable in FOL.An exception are logics with unary concrete domains, where we obtain FOL definability under weak additional assumptions.Given a logic with concrete domain, inexpressibility in FOL does not mean that none of its sentences are definable in FOL.Thus, one can ask if the existence of a (projective) definition in FOL for a given sentence is decidable.
TBox from T obtained by replacing ∃ .() with   , ∃  .() with ∃ . , ∀ .() with ¬ Def  ⊔   and ∀  .() with ∀ .(¬Def  ⊔   ).The ALC TBox T  consists of the following CIs: •   ⊑ Def  and ¬  ⊑    ⊔ ¬ Def  for every feature name  and unary relation  over  occurring in T , • Γ ⊑ ⊥ for every feature name  and every finite set Γ of concept names   s.t. the constraint system { () |   ∈ Γ} is unsatisfiable in .Then, T ′ := T FOL ∪ T  acts as the sentence  did in the proof of Corollary 4.1.Corollary 4.2.Let  be a unary concrete domain that is closed under negation.Then, every ALC() TBox has an abstract projective definition in ALC.