1 Introduction
The DLLite description logics allow for representing conceptual data models like UML class and ER diagrams formally [Calvanese et al.2007]. In DLLite, concept inclusions (CIs) can, for instance, capture the fact that every master’s student is a student and that every student is enrolled:
Here, and are concept names that represent the sets of all (master’s) students; is a role name representing a binary relation connecting students to degree programs; and refers to the domain of that relation. Since knowledge is often temporal, several temporal extensions of DLLite have been investigated [Artale et al.2007, Artale et al.2014, Artale et al.2015, Borgwardt et al.2015, Borgwardt and Thost2015a, Thost et al.2015, Thost2017], which allow some qualitative operators of linear temporal logic (LTL) to occur within the axioms and/or for combining the axioms with such operators. For example, we may actually want to say that every student enrolled at some time in the past () and, after enrollment, pays the fee in the next () month:  
After the initial research on those and other temporal description logics, recent studies have also considered metric operators and hence quantitative temporal logics (TLs) [Alur and Henzinger1993], yet mostly for more expressive logics [GutiérrezBasulto et al.2016, Baader et al.2017, Brandt et al.2017], such as for the description logic (DL) , which is propositionally complete [Baader et al.2007]. Metric operators refer to concrete intervals and allow for describing temporal information more precisely. They are clearly also interesting for DLLite, allowing, for instance, the statement that the payment must happen at some time in the future, maximally three months after enrolling:
In temporal DLs, subsets of the symbols may additionally be distinguished as socalled rigid symbols, to describe information that does not change over time. For instance, while a bachelor’s student may become a master’s student, he or she will always stay male or female. Declaring concepts and as rigid thus allows modeling the knowledge more faithfully, but often increases reasoning complexity. Similarly, we may consider intervalrigid symbols, to express that certain knowledge always must remain valid for a specific period of time [Baader et al.2017]. To describe that a master’s degree lasts at least two years, the concept may be declared as rigid.
Special symbols  none  rigid  intervalrigid  intervalrigid,  
only global CIs  
LTL  *  2ExpSpace  Thm.4  2ExpSpace  Thm.3  ExpSpace Thm.3 
LTL  *  ExpSpace  [1]  ExpSpace  Thm.7  ExpSpace 
LTL  *  ExpSpace  [2]  ExpSpace  Thm.3  PSpace Thm.9 
LTL  *  PSpace  [2]  ExpSpace  Thm.8  PSpace 
LTL  ExpSpace  ExpSpace  ExpSpace  Thm.3  ExpSpace  
LTL  PSpace Thm.11  NExpTime  Thm.12  ExpSpace  Thm.10  PSpace 
LTL  PSpace  PSpace  ExpSpace  Thm.10  PSpace 
In this paper, we study the combined complexity of satisfiability in various fragments of , the metric temporal extension of (interpreted over integer time) where TL operators may be used both within the DL axioms and for combining them; the ‘’ hints at the binary encoding of interval boundaries. In a second dimension, we consider rigid and intervalrigid symbols. Our complexity results are summarized in Table 1. Next to the Bool fragment, we consider the Krom and Horn fragments of DLLite, and thus extend the results of [Artale et al.2007] on fragments regarding the metric operators and intervalrigid symbols. LTL is generally as expressive as LTL but exponentially more succinct. Observe that satisfiability in LTL is ExpSpacecomplete, the same as in LTL [Artale et al.2007], but better than in LTL, where we have 2ExpSpace [GutiérrezBasulto et al.2016, Baader et al.2017]. Targeting better complexities, we also investigate the fragments where the TL operators must not occur within the DL axioms, denoted DLTL (e.g., LTL), and, on the other hand, the restriction to global CIs, which must not be combined arbitrarily but have to be always satisfied. Most importantly, we show that the intervalrigid symbols cannot only be used to simulate rigid symbols, but also that their expressive power may lead beyond the spirit of DLLite: if the DL axioms can be arbitrarily combined with LTL operators, then intervalrigid symbols can be used to express the operator, and conjunction and disjunction in global CIs, so that many fragments collapse. We can show containment in ExpSpace in most cases (note that LTL etc.). Moreover, we can extend the PSpace result of [Artale et al.2014] for LTL restricted to global CIs to the setting with intervalrigid names; note that this restriction does not seem to hurt in many applications, such as with conceptual modeling (see the introductory examples). Containment in PSpace can also be shown for LTL, but then rigid symbols yield a surprising jump in complexity. Nevertheless, this result contrasts the 2ExpTimecompleteness we have for LTL in this setting [Baader et al.2012]. Our results strongly depend on the fact that, in some dialects, interval (and hence metric) operators must not occur within concepts, and on a result we show: intervalrigid roles can be simulated through corresponding concepts. This simplifies reasoning and shows that the DLLite fragments represent rather special DLs also in the quantitative temporal setting. All the results also hold if we consider instead of as the temporal dimension.
2 Preliminaries
We first introduce LTL and for and establish the relation to LTL.
Syntax. Let , and be countably infinite sets of concept names, role names, and individual names, respectively. In , roles and (basic) concepts are defined as follows, where :
where is the inverse role constructor.
LTL concepts are defined based on concepts:
where is an interval of the form or with and , given in binary. Concepts in LTL and LTL restrict LTL concepts in that they must not contain interval operators ().
axioms are the following kinds of expressions: concept inclusions of the form
(1) 
where if , if , and if ; and assertions of the form or with being concepts and .
LTL axioms are defined accordingly but based on LTL concepts .
Define LTL formulas as follows:
where is an LTL axiom, and is as above.
MTL formulas are defined accordingly, but based on axioms .
LTL restricts LTL in that formulas must not contain quantitative temporal operators (), but they may contain the qualitative versions ; and correspondingly for LTL and MTL.
As usual, we denote the empty conjunction () by and the empty disjunction () by . For a given MTL formula , the sets of individual names, role names, and roles occurring in are denoted, respectively, by , , and ; and the closure under single negation of all concepts (formulas) and subconcepts (subformulas) occurring in by (). Note that contains negated concepts, but we only consider these sets on a syntactic level so that it is not problematic that the semantics of such expressions is not defined in some dialects. is the size of . We may also mention the Core fragment of DLLite, the intersection of the Horn and the Krom fragment. In the following, we may use the notion DLLite to address the fragments in general.
Semantics. A DL interpretation over a nonempty set , called the domain, defines an interpretation function that maps each concept name to a subset of , each role name to a binary relation on and each individual name to an element of , such that if , for all (unique name assumption). The mapping is extended to roles by defining and to DLLite concepts:
A (temporal DL) interpretation is a structure , where each , , is a DL interpretation over (constant domain assumption) and for all and (i.e., the interpretation of individual names is fixed). The mappings are extended to LTL concepts as follows:
where denotes the set for all and intervals as above; is defined analogously. The concept requires to be satisfied at some point in the interval , and to hold at all time points before that; and similar for . The validity of an LTL formula in at time point (written ) is inductively defined. For CIs, we have ; and further:
The Boolean operators and are defined as abbreviations in the usual way. We further define , , , , , and , where are either concepts or formulas [Baader et al.2007, Kurucz et al.2003]. In accordance with the notation, the empty conjunction is interpreted as , and the empty disjunction as . We may use negated concept names in , interpreted as , which can be simulated by a fresh name via CIs and . We further assume that neither nor occur in assertions, in the following; this is w.l.o.g., since we always allow the operators in front of assertions on axiom level.
Relation to LTL. LTL restricts LTL in that it only allows the qualitative temporal operators, but that does actually not decrease the expressivity: every formula can be transformed into an equisatisfiable formula and similarly for concepts and for ; denotes a sequence of operators. Likewise, is equivalent to . However, if this transformation is recursively applied to subformulas, then the size of the resulting formula is exponential: ignoring the nested operators, its syntax tree has polynomial depth and an exponential branching factor; and the formulas have exponential depth, but introduce no branching. This blowup cannot be avoided in general [Alur and Henzinger1993, GutiérrezBasulto et al.2016]. Yet, an interesting result for LTL, the restriction of LTL to intervals of the form and , is given in [Baader et al.2017, Thm. 2], and that reduction works similarly in our setting with past operators: each LTL formula can be translated in polynomial time into an equisatisfiable LTL formula. The reduction is particularly modular in that, if the formula contains only global CIs (which are formally introduced in the next paragraph), then this is still the case after the reduction.
Reasoning. We study the complexity of the satisfiability problem in LTL (and in its fragments): given an LTL formula , decide if there exists an interpretation such that . Additionally, we consider a syntactic restriction proposed in [Baader et al.2012]. An LTL formula is a formula with global CIs if it is of the form , where is a conjunction of CIs and is an LTL formula that does not contain CIs. Satisfiability w.r.t. global CIs represents the satisfiability problem w.r.t. such formulas. The problems and notions are correspondingly defined for the fragments of LTL.
Rigid Names. We especially consider a finite set of rigid symbols, whose interpretation must not change over time. That is, interpretations must respect these names, meaning: for all and . In addition, we consider a finite set of intervalrigid names, each of which must remain rigid for a specific period of time, determined by a function whose values are given in binary. Interpretations must also respect these names, meaning: for all with and : for every , there is a time point such that and for all ; and similarly for role names. In the following, let . Intuitively, any element (or pair of elements) in the interpretation of an intervalrigid name must be in that interpretation for at least consecutive time points; the name is rigid. The names in are flexible. We investigate the complexity of satisfiability w.r.t. different settings, in dependence of which kinds of (interval)rigid names may occur in the formula.
[Artale et al.2007] mention that rigid roles can be simulated using temporal constraints on unary predicates. In fact, in an LTL formula , a rigid role name can be simulated by considering to be flexible, introducing fresh rigid concept names and , and extending with the conjunct , and with a conjunct for each role assertion occurring in . Note that this reduction even works in the Core fragment, does not require temporal operators on the DL level, and only uses global CIs. We can extend the reduction to interval rigid symbols: for every rigid role , we introduce fresh rigid concept names and and CIs corresponding to the above ones, and we extend with the conjunct for each role assertion occurring in . An LTL formula referring only to intervals of the form is in the LTL fragment and can be translated in polynomial time into an equisatisfiable formula, as mentioned above; note that the interval operator does not occur on the concept level.
Lemma 1.
Satisfiability in LTL w.r.t. (interval)rigid names can be polynomially reduced to the setting where only (interval)rigid concepts are given. ∎
We hence restrict our attention to (interval) rigid concepts. Since LTL allows to express rigid concepts axiomatically using CIs and , rigid symbols are actually syntactic sugar in that language. This does not seem to be the case for fragments that do not allow for temporal operators within CIs, which we consider later, but, at least for the case with intervalrigid names, we will prove the contrary. Moreover, rigid concept names can be simulated using , and hence in LTL.
Relation to LTL. is more expressive than although it does not allow for inverse roles. Also, every LTL formula can be transformed into an equisatisfiable LTL formula as follows. First, we extend the set of role names to include all inverse roles for which we have that the role or occurs in . Then, is obtained from by replacing all occurrences of concepts of the form with by ^{1}^{1}1In an interpretation , a concept is interpreted as the set . ; by adding a CI for each as a conjunct; and by adding a conjunct for each role assertion or occurring in . Note that we use no temporal operators within the CIs, no metric ones, and that the CIs are global.
Lemma 2.
The LTL formula is satisfiable iff the LTL formula is satisfiable.
Proof.
() We consider an arbitrary model of and construct a model of . Specifically, interprets all symbols occurring in as does, and, for all , the interpretation of role names in , where , is equal to . Given this definition of , we obviously have that, for all , satisfies an axiom occurring in iff . Moreover, it is easy to see that the new axioms are satisfied, too. We thus have .
() Let now be a model of . We show this direction similarly, by constructing a model of . In particular, we assume to have the same domain as , to interpret all concept names as does, and to interpret all role names such that for all . The latter definition yields that if , and if for all . Together with (i) and the definition of , we thus obtain that iff for all . Given the latter and the fact that satisfies our extension of regarding the role assertions, it is easy to see that we get iff for all assertions occurring in . We get the same for CIs . Hence, we have . ∎
The reduction yields the below membership results.
Theorem 3.
Satisfiability w.r.t. both rigid and intervalrigid names is in ExpSpace in the following logics:

LTL restricted to global CIs,

LTL,

LTL.
In LTL, the problem is in 2ExpSpace.
Proof.
By Lemma 1, we can refer to the complexities for w.r.t. rigid and intervalrigid concepts only to obtain the results. The corresponding results all have been shown in [Baader et al.2017], for LTL restricted to global CIs, LTL, LTL, and for LTL, but only w.r.t. the fragments of the logics with only future operators and the natural numbers.
All the proofs apply the standard approach. In what follows we sketch it regarding the ExpSpace cases; the other one is similar. It is based on the fact we can restrict the focus to a kind of models—socalled quasimodels—of a special, regular shape and, specifically, to a part of doubleexponential size (one version of such a proof is presented in the next section). Such a quasimodel is a sequence of quasiworlds. A quasiworld describes the interpretation of all domain elements at a single time point (through closed subsets of ) and contains a closed set of subformulas of , those that are satisfied at the corresponding time point. Additionally, specific conditions hold for consecutive quasiworlds in a quasimodel, to guarantee that it describes the interpretations of all domain elements that have to be present to satisfy the given formula at w.r.t. all time points. In such a setting, we can restrict our focus to quasimodels of the form , where , and are sequences of quasiworlds of doubleexponential length, and do not contain a quasiworld twice, and does not contain a quasiworld more than twice. In a nutshell, this is shown by merging models of the form , containing a quasiworld twice, to models with . is obtained from an arbitrary given sequence of quasiworlds by considering all subconcepts containing the operator—w.r.t. all domain elements described in the model—, and also the subformulas containing . It can be shown that there is always a sequence as (i.e., one of the length of ), in which all these concepts are finally satisfied by the respective elements, and similar for the formulas and in the past direction; and that describes a model of . We regard the ExpSpace cases. In the original proofs, and also in our setting, the number of different quasiworlds is bounded by , meaning double exponentially in the input. Hence this also holds for the lengths of and with , and for . Therefore, the existence of a model of can be checked while using only exponential space: first, guess the starts of the periods and their lengths with ; second, guess the two sequences of quasiworlds (written in the order of the guessing) and by guessing one world after the other, respectively. Thereby only three quasiworlds have to be kept in memory at a time—the “current” quasiworld, the previous (next) one, and the first repeating one ()—and their sizes are exponentially bounded in the size of the input.
The proof for the 2ExpSpace case is similar, but the models are more complex and the bound there is triple exponential. ∎
Compared to other description logics, the DLLite logics thus present rather special cases also in the temporal setting. This is mainly due to the facts that rigid roles can be disregarded, and that, in some dialects, the interval operators may not occur within concepts. It is not directly clear how this affects the complexity results. In the remainder of the paper, we therefore look for fragments of LTL where satisfiability is not as complex as in the corresponding LTL fragment.
3 Ltl and IntervalRigid Names
We begin focusing on LTL and the fragments where temporal operators may occur on both concept and axiom level. Recall that we trivially have rigid symbols in these logics (Lem. 1
). Alas, the reduction of the word problem of doubleexponentially spacebounded deterministic Turing machines from the
2ExpSpacehardness proof for LTL over the natural numbers [GutiérrezBasulto et al.2016, Thm. 5] can be similarly done in LTL.Theorem 4.
Satisfiability in LTL without intervalrigid names is 2ExpSpacehard.
Proof.
The proof proposed for LTL is based on a reduction of the word problem of doubleexponentially spacebounded deterministic Turing machines. The LTL formula in that proof contains qualified existential restrictions on the right of CIs, sometimes prefixed by , but not nested and, apart from that, only constructs that are allowed in LTL. In particular, all the qualified existential restrictions are of the form , meaning that they all use the same role . Moreover, it can readily be checked that this feature is not critical since the role is otherwise not used in the formula. That is, the hardness result depends on the element the existential restriction forces to exist but not on the kind of the relation to its predecessor. Consequently, for each such restriction , we can introduce a fresh role name , and then create a similar LTL formula by adding the conjuncts for the introduced role names to the LTL formula and by replacing all concepts by . ∎
Given the 2ExpSpacehardness, the results in Thm. 3 for restrictions of LTL, or LTL, are really interesting.
Alternatively, we can regard other DLLite fragments. In fact, we show that satisfiability w.r.t. intervalrigid names is in ExpSpace in both LTL and LTL. This is particularly the case because and do not occur in concepts there. Our proof is an extension of the one for LTL [Wolter and Zakharyaschev1999] regarding intervalrigid names. Assume to be the given formula, and to be the exponentially larger formula obtained from it by simulating all  and operators (see Sec. 2). Note that since and here do not occur in concepts, but is exponentially larger than .
A concept type for is a set as follows:

[label=T0,leftmargin=*]

;

iff , for all ;

iff for exactly one , for all ;

for some implies , for all ;

for some implies , for all ;
the names are used to capture how long has been satisfied already. A named concept type for is a pair with and a concept type for . We denote such a tuple by and write instead of . Formula types are defined by the following conditions:

[label=T0’,leftmargin=*]

for all , iff ;

for all , iff .
Intuitively, a concept type describes the interpretation w.r.t. one domain element at a single time point; a formula type specifies constraints on the whole domain.
A quasiworld for is a triple , where is a set of unnamed types, is a set of named types containing exactly one named type for each , is a formula type, and:

[label=W0,leftmargin=*]

for all , iff implies for all types ;

for all and , iff there is a with .

for all , iff ;

for all , implies and .
Note that , , , and that the number of distinct quasiworlds for is double exponential and does not exceed
A quasiworld describes an interpretation at one time point. Regarding several time points, we first consider single sequences of types, which describe an interpretation on one element w.r.t. all time points.
A pair of concept types is suitable if we have:

for all , iff ;

for all , iff ;

for all and , iff either and , , or .
Let () be a sequence of quasiworlds for . We denote concept types in by for and, w.l.o.g., assume that every concept type in also occurs in . A run in is a sequence of concept types such that, for all :

[label=R0,leftmargin=*,series=run]

;

the pair is suitable;
denotes the element at index in a sequence .
Finally, a sequence of the form () is a quasimodel for if the following hold for all :

[label=M0,leftmargin=*]

for all , is a run in ;

for all , there is a run in with ;

for all , iff ;

for all , iff ;

for all , iff there is a ;

for all , iff there is a ;

.
Lemma 5.
An LTL formula is satisfiable w.r.t. intervalrigid names iff there is a quasimodel for .
Proof.
() Suppose that is satisfied in an interpretation that respects the intervalrigid names. For every , we define the quasiworld as follows:
where . Clearly, every represents a quasiworld, and the sequence is a quasimodel for .
() Let a quasimodel for of the form () be given. We define the interpretation as follows, based on the set of runs in :
Below, we sometimes denote by , where is the unique run for in . Given 2, we directly have that respects intervalrigid concepts.
To show that is a model of , we prove the following claim. Note that, by our assumption that for all and by 1, it also covers the named elements.
Claim.
For all runs , concepts and , we have iff
Proof of the claim. We argue by structural induction. Clearly, the claim holds for all concept names. It thus remains to consider the operators , , and .
Observe that we extend the original proof from [Wolter and Zakharyaschev1999] only in that we consider instead of , intervals with the operators and , and intervalrigid concepts; especially the former extensions are irrelevant. It is hence possible to consider only quasimodels of the form ,^{2}^{2}2For brevity, we may drop the brackets around sequences. where and are sequences of quasiworlds of doubleexponential length not containing a quasiworld twice, as outlined above. Recall that the number of different quasiworlds is bounded double exponentially in the input . This yields the following result.
Lemma 6.
If has a quasimodel, then it has a quasimodel of the form
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