Temporal Query Answering in the Description Logic EL

1 Introduction

Context-aware systems use data collected at runtime to recognize certain predefined situations and trigger adaptations. For example, an operating system might be able to recognize that a video application is out of user focus (e.g., by corresponding sensors) and then adapt application parameters to optimize the energy consumption of the system. A straightforward approach is to encode the situations into queries over a database containing the sensor data. However, in general, sensors do not completely describe the environment (e.g., to date, sensors cannot capture the intentions of users), and usually additional knowledge about the behavior of the environment is available. For example, if the user has not been watching the video for a longer period of time because he is using another application on a second screen, then the video does not need to be displayed in the highest resolution.

Ontology-based data access [Poggi et al., 2008; Decker et al., 1998] remedies this situation by adopting the openworld assumption, where facts not present in the data are assumed to be unknown rather than false, and by employing an ontology to encode background knowledge. This is done using axioms of an appropriate ontology language, for example a description logic (DL) [Baader et al., 2003]. In this paper, we focus on ontologies in the lightweight DL EL, which allows for efficient reasoning [Baader et al., 2005; Lutz et al., 2009] and is successfully applied in practice (e.g., in large biomedical ontologies like SNOMEDCT 1 ). In this setting, the data is collected into a fact base (or ABox) containing assertions about individuals using unary and binary predicates, called concepts and roles, respectively. Thus, we can represent both static knowledge about active applications as well as dynamic knowledge about the current context: VideoApplication(app1),NotWatching(user1). Background knowledge is represented in the ontology (or TBox) using socalled general concept inclusions (GCIs) like VideoApplication ⊓ ∃hasUser.NotWatching

⊑ ∃hasState.OutOfFocus,

saying that a video application whose user is currently not watching the video is out of user focus. ABox and TBox together are called knowledge base. We can use a conjunctive query (CQ) like ψ(x) := ∃y.hasState(x, y)∧OutOfFocus(y) over this knowledge base to identify applications x that can potentially be assigned a lower priority. However, complex situations typically depend also on the behavior of the environment in the past. For example, the operating system should not switch between configurations every time the user is not watching for one second, but only after this has been the case for a longer period of time.

For that reason, we investigate temporal conjunctive queries (TCQs), originally proposed in [Baader et al., 2013; 2015]. They allow to combine conjunctive queries via the operators of the propositional linear temporal logic LTL [Pnueli, 1977; Lichtenstein et al., 1985]. Hence, we can use the TCQ

to obtain all applications that were out of user focus during the three previous (#-) moments of observation, were prioritized by the operating system at some point in time, and the priority has not (¬) changed since (S) then. 2

The semantics of TCQs is based on temporal knowledge bases (TKBs), which, in addition to the background knowledge (which is assumed to hold globally, i.e., at every point in time), contains a sequence of ABoxes A0,A1, . . . ,An, representing the data collected at specific points in time. We designate with n the most recent time of observation (the current time point), at which the situation recognition is performed.

1 http://www.ihtsdo.org/snomed-ct/

2 Although our formalism does not support it yet, priority values can be represented if the underlying DL allows for so-called concrete domains.

We also investigate the related temporalized formalism ELLTL, in which axioms, i.e., assertions or GCIs, are combined using LTL-operators. This approach was first suggested in [Baader et al., 2012].

In our setting, the axioms in the TKB do not explicitly refer to temporal information, but are written in a classical (atemporal) DL; only the query is temporalized. In contrast, [Lutz et al., 2008; Artale et al., 2007; 2014; Gutiérrez-Basulto et al., 2014] extend classical DLs by temporal operators, which then occur within the knowledge base. However, most of these logics yield high reasoning complexities, even if the underlying atemporal DL has tractable reasoning problems. Lower complexities are only obtained by either considerably restricting the set of temporal operators or the underlying DL.

Regarding temporal properties formulated over atemporal DLs, ALC-LTL, a variant of EL-LTL over the more expressive DL ALC, was first considered in [Baader et al., 2012]. This was the basis for introducing TCQs over ALC-TKBs in [Baader et al., 2013], which was extended to SHQ in [Baader et al., 2015]. However, reasoning in ALC is not tractable, and context-aware systems often need to deal with large quantities of data and adapt fast. TCQs over several lightweight logics have been regarded in [Borgwardt et al., 2015], but only over a fragment of LTL without negation. In [Artale et al., 2007], the complexity of LTL over axioms of several members of the DL-Lite family of DLs has been investigated. However, nothing is known about TCQs over these logics.

In this paper, we want to answer TCQs over TKBs formulated in EL and in particular investigate both the combined and the data complexity of the temporal query entailment problem. Moreover, we determine the complexity of satisfiability of EL-LTL-formulae, and additionally consider the special case where only global GCIs are allowed [Baader et al., 2012]. As usual, we consider rigid concepts and roles, whose interpretation does not change over time. In this regard, we distinguish three different settings, depending on whether concepts or roles (or both) are allowed to be rigid. Since rigid concepts can be simulated by rigid roles [Baader et al., 2012], only three cases need to be considered.

Our results are summarized in Table 1. The complexity of EL-LTL is often lower than that of ALC-LTL, for which satisfiability is EXPTIME-, NEXPTIME-, and 2-EXPTIMEcomplete, respectively, in the three settings we consider [Baader et al., 2012]. This partially confirms and refutes the conjecture from [Baader et al., 2012] that EL-LTL is as hard as ALC-LTL. Using only global GCIs, the complexity matches that of (unrestricted) DL-Litekrom-LTL [Artale et al., 2007]. Regarding TCQs, the complexity is even more reduced compared to ALC (and SHQ), where TCQ entailment is in EXPTIME, CO-NEXPTIME, and 2-EXPTIME, respectively, w.r.t. combined complexity, and in CO-NP, CO-NP, and EXPTIME, respectively, w.r.t. data complexity [Baader et al., 2015]. The only lower bounds that directly apply to the problems considered here are PSPACE-hardness of LTL [Sistla and Clarke, 1985] and P-hardness of CQ entailment in EL w.r.t. data complexity [Calvanese et al., 2006].

Our results are based on known techniques for ALC-LTL and TCQs over ALC-TKBs [Baader et al., 2012; 2015], but we had to significantly adapt them and to combine them with new approaches, in particular for some of the hardness proofs and for the PSPACE upper bounds. Full proofs of all results can be found in the technical reports [Borgwardt and Thost, 2015a; 2015b].

2 EL and LTL

We introduce the formalisms underlying the temporal languages we consider in this paper.

The DL part focuses on the logic EL. Let NC, NR, NI be sets of concept-, role-, and individual names, respectively. Concepts are built from concept names using the constructors conjunction (C⊓D), existential restriction (∃r.C for r ∈ NR), and top concept (⊤). An axiom is either an assertion of the form A(a) or r(a, b), where A ∈ NC, r ∈ NR, and a, b ∈ NI, or a general concept inclusion (GCI) of the form C ⊑ D for concepts C,D. An ABox is a finite set of assertions, a TBox is a finite set of GCIs, and a knowledge base (KB) is a pair 〈T , A〉 consisting of a TBox T and an ABox A.

An interpretation I has a non-empty domain ∆I and an interpretation function ·I that assigns to every A ∈ NC a set AI ⊆ ∆I , to every r ∈ NR a relation rI ⊆ ∆I × ∆I , and to every a ∈ NI an element aI ∈ ∆I . This function is extended to concepts as follows: (C ⊓ D)I := C I ∩ DI ; (∃r.C)I := {x | ∃y : (x, y) ∈ rI ∧y ∈ C I }; ⊤I := ∆I . An interpretation I satisfies (or is a model of) A(a) if aI ∈ AI ; r(a, b) if (aI , bI ) ∈ rI ; C ⊑ D if C I ⊆ DI ; a set of axioms or KB if it satisfies all its axioms. A KB K entails an axiom α

(written K |= α) if every model of K is also a model of α.

The temporal component of our formalisms is based on propositional LTL [Pnueli, 1977]. LTL-formulae are built from a set of propositional variables P using conjunction (φ ∧ ψ), negation (¬ψ), next (#φ), previous (#-φ), until (φUψ), and since (φSψ). An LTL-structure I = (wi)i≥0 is an infinite sequence of worlds wi ⊆ P. Validity of an LTLformula φ in I at time point i ≥ 0 (written I, i |= φ) is defined inductively:

I, i |= p iff p ∈ wi (for p ∈ P) I, i |= φ ∧ψ iff I, i |= φ and I, i |= ψ I, i |= ¬φ iff not I, i |= φ I, i |= #φ iff I, i +1 |= φ I, i |= #-φ iff i > 0 and I, i -1 |= φ I, i |= φUψ iff there is k ≥ i such that I, k |= ψ and I, j |= φ for all j with i ≤ j < k I, i |= φSψ iff there is 0 ≤ k ≤ i, such that I, k |= ψ and I, j |= φ, for all j with k < j ≤ i

An LTL-formula φ is satisfiable if there is an LTL-structure I with I, 0 |= φ. Note that this logic is usually called Past-LTL due to the operators #- and S. The presence of past operators does not affect the complexity of the satisfiability problem [Lichtenstein et al., 1985], but allows to write some formulae more succinctly [Laroussinie et al., 2002]. As usual, one can express other temporal operators such as eventually (3φ) and always (2φ) in this logic.

3 Temporal Query Entailment in EL

As described in the introduction, in our temporal formalism we can designate certain concept and role names as being rigid, which means that their interpretation is not allowed to change over time. For this purpose, we fix a set NRC ⊆ NC of rigid concept names and a set NRR ⊆ NR of rigid role names.

Temporal conjunctive queries (TCQs) [Baader et al., 2015] are constructed exactly as LTL-formulae, except that conjunctive queries (CQs) [Abiteboul et al., 1995] take the place of the propositional variables. A conjunctive query is of the form ∃x1, . . . , xm.ψ, where x1, . . . , xm are variables and ψ is a conjunction of atoms of the form A(t) or r(t, t′ ), where A ∈ NC, r ∈ NR, and t, t′ are individual names or variables. A Boolean TCQ does not contain free variables. A CQ-literal is either a CQ or a negated CQ; and a union of conjunctive queries (UCQ) is a disjunction of CQs.

The semantics of TCQs is also very similar to that of LTLformulae. However, instead of LTL-structures one has to consider infinite sequences I = (Ii)i≥0 of interpretations. Following [Baader et al., 2015], we make the constant domain assumption (i.e., the interpretations all have the same domain ∆). Furthermore, we have to ensure that the rigid names are respected; that is, we require that sIi = sIj holds for all symbols s ∈ NI ∪NRC ∪NRR and i, j ≥ 0. Validity of a TCQ φ in I at time point i ≥ 0 (again denoted by I, i |= φ) is now defined exactly as for LTL in Section 2, with the obvious exception of CQs. For these, we adopt the classical semantics based on homomorphisms [Chandra and Merlin, 1977]. More precisely, the fact that I, i |= ψ for a CQ ψ is equivalent to ψ being satisfied by Ii (written Ii |= ψ), which is the case if there is a homomorphism π mapping the variables and individual names of ψ into ∆ such that: π(a) = aIi for all a ∈ NI; π(t) ∈ AIi for all concept atoms A(t) in ψ; and (π(t), π(t′ )) ∈ rIi for all role atoms r(t, t′ ) in ψ.

We now consider temporal knowledge bases (TKBs) of the form K = 〈T , (Ai)0≤i≤n〉, where T is a TBox and the Ai are ABoxes. As described in the introduction, T represents the global knowledge about the application domain, whereas the Ai contain data about different time points. A sequence I = (Ii)i≥0 of interpretations as above satisfies (or is a model of) K (written I |= K) if we have Ii |= T for all i ≥ 0, and Ii |= Ai for all i, 0 ≤ i ≤ n. A Boolean TCQ φ is satisfiable w.r.t. K if there is a model I of K such that I, n |= φ, and it is entailed by K (written K |= φ) if for all models I of K it holds that I, n |= φ. Recall that we are interested in the current time point n, for which the most recent data (An) is available.

For a (non-Boolean) TCQ φ, a mapping a of the free variables in φ to the individual names of K is a certain answer to φ w.r.t. K if K |= a(φ), where a(φ) is obtained from φ by replacing the free variables according to a. As usual, the problem of computing all certain answers can be reduced to exponentially many entailment tests. Therefore, we investigate in the following the complexity of the TCQ entailment problem in EL. We do this indirectly, via the satisfiability problem, which has the same complexity as non-entailment.

We consider both data complexity, where the TBox T and the TCQ φ are assumed to be fix and the complexity is measured only w.r.t. the size of the input ABoxes (Ai)0≤i≤n; and combined complexity, where also the influence of T and φ is taken into account. As described in the introduction, we further distinguish the three cases where (i) no rigid names are available (NRC = NRR = ∅); (ii) only rigid concept names are allowed (NRR = ∅, but NRC ̸= ∅); and (iii) also rigid role names can be used (NRR ̸= ∅).

We next state an auxiliary result about satisfiability of (atemporal) conjunctions of CQ-literals. Note that, for this case, it suffices to consider an ordinary KB instead of a TKB.

Lemma 1. W.r.t. combined complexity, deciding whether a Boolean conjunction of CQ-literals ψ is satisfiable w.r.t. a KB K = 〈T ,A〉 can be reduced to several P-tests, whose number is polynomial in the number of conjuncts of ψ and exponential in the size of the largest negated conjunct in ψ.

Proof Sketch. We can reduce this problem to the UCQ nonentailment problem 〈T ,A ∪ A′ 〉 ̸|= σ1 ∨ . . . ∨ σm, where A′ is obtained from instantiating the variables of the CQs occurring positively in ψ with fresh individual names, and σi are the CQs occurring negatively in ψ. Using the algorithm in [Rosati, 2007], this can be solved by a series of polynomial non-entailment tests, one for every possible CQ resulting from some σi by unifying some of the terms of σi. The claim follows from the fact that there are at most exponentially many such unifiers.

3.1 The Upper Bounds

In this section, we describe the general approach used in [Baader et al., 2012; 2015] to solve the satisfiability problem. In the following, let K = 〈T , (Ai)0≤i≤n〉 be a TKB and

φ be a Boolean TCQ. For ease of presentation, we assume that all concept and role names occurring in (Ai)0≤i≤n or φ also occur in T , and that all individual names occurring in φ also occur in (Ai)0≤i≤n. These assumptions do not affect the complexity results.

The main idea is to consider two separate satisfiability problems-one in LTL and the other in EL-that together imply satisfiability of φ w.r.t. K. The LTL part analyzes the propositional abstraction φp of φ, which contains the propositional variables p1, . . . , pm in place of the CQs α1, . . . , αm from φ (where each αi was replaced by pi). Furthermore, let S ⊆ 2{p1,...,pm} be a set that specifies the worlds that are allowed to occur in an LTL-structure satisfying φp . This condition is formalized by the following LTL-formula:

where X := {p1, . . . , pm} \ X is the complement of X ∈ S. However, for checking satisfiability of φ w.r.t. K, it is not sufficient to find such a set S and then test whether φpS is satisfiable (at time point n). We must also ensure that S can indeed be induced by a model of K in the following sense.

Definition 2. Let S = {X1, . . . ,Xk} ⊆ 2{p1,...,pm} and ι : {0, . . . , n} → {1, . . . , k}. S is r-satisfiable (w.r.t. ι and K) if there are interpretations J1, . . . , Jk, I0, . . . , In such that

• they share the same domain and respect rigid names; 3
• the interpretations are models of T ;
• each Ji is a model of χi := ∧ pj ∈Xi αj ∧ ∧ pj ∈Xi ¬αj; and
• each Ii is a model of Ai and χι(i).

The existence of Ji ensures that the conjunction χi of CQliterals induced by Xi is satisfiable; a set S containing an Xi for which this does not hold cannot be induced by a model of K. The interpretations Ii represent the first n + 1 elements of such a model, which must additionally satisfy the ABoxes Ai. The mapping ι chooses a world for each ABox.

We now call φp t-satisfiable (w.r.t. S and ι as above) if there is an LTL-structure I = (wi)i≥0 with I, n |= φpS and wi = Xι(i) for all i, 0 ≤ i ≤ n. Intuitively, I is the propositional abstraction of the model of φ w.r.t. K we are looking for. The following was shown in [Baader et al., 2015] for SHQ, and remains valid in our setting.

Lemma 3. φ is satisfiable w.r.t. K iff there are S and ι as above such that S is r-satisfiable w.r.t. ι and K and φp is tsatisfiable w.r.t. S and ι.

Since t-satisfiability is independent of the DL part, we can also reuse the following result from [Baader et al., 2015].

Lemma 4. Checking t-satisfiability of φpS w.r.t. S and ι is

• in EXPTIME w.r.t. combined complexity, and
• in P w.r.t. data complexity.

Given this, we already obtain some of the upper bounds.

3 This is defined as for sequences of interpretations.

Theorem 5. TCQ entailment in EL is

• in CO-NP w.r.t. data complexity and in CO-NEXPTIME w.r.t. combined complexity even if NRR ̸= ∅,
• in P w.r.t. data complexity if NRC = NRR = ∅.

Proof Sketch. Recall that we show the complementary results by regarding TCQ satisfiability. Let K = 〈T , (Ai)0≤i≤n〉 be a TKB and φ be a Boolean TCQ. For the first two results, we can simply guess S and ι as required for Lemma 3; note that S is of constant size in the size of the input ABoxes. By Lemma 4, the required t-satisfiability test can be done within the claimed time bounds. For the r-satisfiability test, we use a technique from [Baader et al., 2015] that constructs an exponentially large conjunction χS,ι of CQ-literals and TBox TS,ι such that it remains to check satisfiability of χS,ι w.r.t. TS,ι. Since the CQ-literals in χS,ι are essentially of the same size as the CQs in φ, we can apply Lemma 1 to decide this problem via exponentially many EXPTIME-tests w.r.t. combined complexity. Moreover, the number of conjuncts of χS,ι and the size of TS,ι are linear in the size of the input ABoxes, and thus we obtain an upper bound of P w.r.t. data complexity.

For the last result, observe first that in the absence of rigid names the satisfiability tests of Definition 2 are largely independent of each other. Hence, it suffices to define S as the set of all sets Xj for which χj is satisfiable w.r.t. T . Likewise, we consider, for each ABox Ai, the set ι′ (i) of all indices j of worlds Xj for which χj is satisfiable w.r.t. 〈T ,Ai〉, and employ a modified t-satisfiability test w.r.t. these sets. This results in a deterministic polynomial-time procedure.

It remains to consider the case where NRR = ∅, but possibly NRC ̸= ∅, under combined complexity (see Table 1). Note that the satisfiability tests of Definition 2 are not independent in this case. Nevertheless, we can guess polynomially many additional data (see AR and Q¬R below) that allow us to separate these tests. We then combine these with the PSPACEprocedure for LTL-satisfiability of [Sistla and Clarke, 1985] in order to obtain the claimed upper bound.

We assume here that the sequence of input ABoxes consists only of one empty ABox; this is without loss of generality since the ABoxes can be encoded into the TCQ without affecting the (combined) complexity [Baader et al., 2015]. We thus consider a TKB K = 〈T , ∅〉 and a Boolean TCQ φ.

Before stating the main result, we first give some auxiliary definitions. Let ψ be a CQ that does not contain any individual names and is tree-shaped (i.e., the directed graph described by its atoms is a tree), and let x be the root of this tree. Then Con(ψ) abbreviates the concept Con(ψ, x), where

A subset B of the rigid concept names occurring in T is a witness of ψ w.r.t. T if there are r1, . . . , rℓ, ℓ ≥ 0, such that T |= (d B) ⊑ ∃r1. . . . ∃rℓ.Con(ψ). Intuitively, if a model of T contains an element satisfying d B, then ψ is satisfied.

We now consider all possible assertions over the individual names and the rigid concept names occurring in the input, together with their negations. An ABox type AR is a set of such assertions such that A(a) ∈ AR iff ¬A(a) /∈ AR. Given S = {X1, . . . , Xk} ⊆ 2{p1,...,pm} , we define the KBs KiR := 〈T ,AR ∪ AQi 〉, 1 ≤ i ≤ k, where the ABox AQi contains the CQs occurring positively in χi with the variables replaced by fresh individual names. A tuple (AR, Q¬R ), where AR is an ABox type and Q¬R is a subset of {α1, . . . , αm}, is r-complete (w.r.t. S) if the following hold:

• (R1) For all i ∈ {1, . . . , k}, KiR has a model.
• (R2) For all i ∈ {1, . . . , k} and pj ∈ Xi, we have KiR ̸|= αj .
• (R3) For all i ∈ {1, . . . , k}, all tree-shaped α ∈ Q¬R , and all witnesses B of α w.r.t. T , we have KiR ̸|= ∃x.B(x).
• (R4) For all αj ∈ Qφ \ Q¬R , we have pj ∈ ⋂ S.

The idea is to fix the interpretation of the rigid names on all named individuals and specify the CQs that are allowed to occur negatively in S via the guessed data (AR,Q¬R ). (R1) and (R2) ensure that exactly the queries specified by Xi, together with the assertions from AR, can be satisfied w.r.t. T . (R3) ensures that there is a model of KiR that does not satisfy any of the witnesses of the tree-shaped queries in Q¬R (the canonical model [Lutz et al., 2009]). Finally, (R4) makes sure that only the queries from Q¬R can occur negatively in any X ∈ S.

We can show that r-satisfiability of S is characterized by the existence of such an r-complete tuple. To actually obtain a PSPACE decision procedure from this result, we adapt the PSPACE-Turing machine from [Sistla and Clarke, 1985] that successively guesses propositional worlds and checks whether these can be assembled into an LTL-structure satisfying φp . We use a modified version of this Turing machine that first guesses a tuple (AR,Q¬R ) as described above, and then proceeds as before, but, for each guessed world Xi, additionally checks whether the KB KiR satisfies Conditions (R1)-(R4). For (R1), note that the negated assertions in AR do not pose a problem, as they can be simulated using nominals and the bottom constructor [Baader et al., 2005]. Moreover, the non-entailment tests in (R2) and (R3) can be done using only the positive assertions in AR. Finally, to check whether a given set B is actually a witness of a tree-shaped CQ α ∈ Q¬R , it suffices to do a reachability test in the completion graph of T [Baader et al., 2005].

Theorem 6. If NRR = ∅, but possibly NRC ̸= ∅, then TCQ entailment in EL is in PSPACE w.r.t. combined complexity.

3.2 What Makes It Hard

If NRR ̸= ∅, we can show CO-NEXPTIME-hardness w.r.t. combined complexity by adapting the proof of NEXPTIMEhardness of satisfiability in ALC-LTL from [Baader et al., 2012]. The latter reduces the 2n+1 -bounded domino problem [Lewis, 1978; Börger et al., 1997] and the result already holds if only concept names are allowed to be rigid. However, ALC-LTL-formulae are built by replacing the propositional variables in LTL-formulae by axioms of the more expressive DL ALC, which may contain concept negation (¬) and disjunction (⊔). In a nutshell, the original proof represents the positions in the 2n+1 × 2n+1 domino grid in two different ways: for each position, there is a specific time point representing it, as well as a domain element xi. This dual representation facilitates the encoding of the domino conditions.

We describe some interesting adaptations necessary to apply the proof for the case of conjunctive queries and a global ELTBox; the detailed proof can be found in the technical report.

• Instead of the formula 2¬(⊤ ⊑ ¬N), we use the TCQ 2(∃x.r(x, a) ∧ N(x)) to create the elements xi. This connects them to a fixed individual a via the new rigid role r and allows us to refer back to them later.
• Formulae of the form (⊤ ⊑ A) ∨ (⊤ ⊑ ¬A) are used to express that A is either satisfied by all domain elements or by none. The second axiom can be expressed by the negated CQ ¬∃x.A(x). The first axiom cannot easily be expressed by a TCQ; however, for the hardness proof it suffices to ensure that A is satisfied by all elements xi, which can be identified via their rconnection to a. Thus, we can replace the first axiom by the CQ A(a) and the (global) GCI ∃r.A ⊑ A.

Theorem 7. If NRR ̸= ∅, then TCQ entailment in EL is CO-NEXPTIME-hard w.r.t. combined complexity.

The last remaining result concerns the data complexity of TCQ entailment if rigid concept names are allowed. We show NP-hardness of satisfiability by a reduction of the 3-SAT problem [Karp, 1972], considering a propositional 3-CNF formula ψ = ∧0≤i<ℓ li,1 ∨ li,2 ∨ li,3. We construct a TCQ φ and a TKB Kψ = 〈T , (Aψt )0≤t<4ℓ〉 such that ψ is satisfiable iff φ is satisfiable w.r.t. Kψ. We use four ABoxes to represent each clause: one to identify the start of a new clause (via C(c)), and the following three to encode the literals of this clause via the individual names al (see also Figure 1):

Then, we enforce through

that one of the clause's literals is satisfied (indicated by T (c)). Using the rigid concept A, we express that a literal al and its complement a¬l cannot both be true at the same time. Finally, we use the TBox T := {∃s.T ⊑ A} to connect the satisfaction of the literal of a clause (T(c)) with the truth of the corresponding literal (A(al)).

Note that both φ and T are of constant size, and the size of (Aψt )0≤t<4ℓ is linear in the size of ψ.

Theorem 8. If NRC ̸= ∅, then TCQ entailment in EL is CO-NP-hard w.r.t. data complexity.

4 Temporal Subsumption in EL

We now consider a related temporal formalism based on EL, where the atoms of the temporal formulae are not CQs, but axioms [Baader et al., 2012]. More formally, EL-LTLformulae are defined exactly as LTL-formulae, except that instead of propositional variables they contain assertions and GCIs. As in Section 3, the semantics are given by infinite sequences of interpretations. Validity of an EL-LTL-formula φ in I = (Ii)i≥0 at time point i ≥ 0 (written I, i |= φ) is defined as in Section 2, with the exception of axioms α, where we define I, i |= α iff Ii satisfies α. As in [Baader et al., 2012], we investigate the satisfiability of EL-LTLformulae, i.e., deciding whether there is a sequence I such that I, 0 |= φ. A corresponding entailment problem would be the question whether I, 0 |= ψ always implies that I, 0 |= φ, but this can easily be reduced to the unsatisfiability of ψ∧¬φ.

ALC-LTL-formulae [Baader et al., 2012] can be reformulated as TCQs over ALC-TKBs [Baader et al., 2015]. However, this is not the case for EL: GCIs of the form ⊤ ⊑ A cannot directly be simulated by TCQs; and conversely, cyclic CQs like ∃x, y.r(x, y) ∧ r(y, x) cannot be expressed by ELLTL-formulae. Hence, these two satisfiability problems are not directly comparable. Nevertheless, satisfiability of ELLTL-formulae turns out to be always harder than that of TCQs (see Table 1). It does not make sense to consider data complexity here because the assertions are part of the formula.

The proof techniques employed for EL-LTL-formulae are similar to those we have presented in Section 3. For instance, we can show NEXPTIME-hardness using a similar construction as in the proof of Theorem 7, which is even closer to that of [Baader et al., 2012] and does not use rigid role names.

Theorem 9. If NRC ̸= ∅, then satisfiability in EL-LTL is NEXPTIME-hard.

For the upper bounds, we use the ideas from Theorem 5. The r-satisfiability condition is simpler since we do not have to consider ABoxes, and the χi are now conjunctions of ELliterals, which are axioms or negated axioms. As in Lemma 1, we first determine the complexity of satisfiability of such conjunctions. The main idea is to instantiate negated GCIs and to simulate negated assertions using nominals and the bottom constructor to construct an EL++-KB that has a model iff the original conjunction has a model. The former problem can be decided in polynomial time [Baader et al., 2005].

Lemma 10. Satisfiability of conjunctions of EL-literals can be decided in P.

This helps us to prove the following upper bounds.

Theorem 11. Satisfiability in EL-LTL is

• in NEXPTIME even if NRR ̸= ∅,
• in PSPACE if NRC = NRR = ∅.

Proof Sketch. The first result is obtained exactly as in the proof of Theorem 5, using the renaming technique from [Baader et al., 2012] and Lemma 10.

For the second upper bound, observe once more that the satisfiability tests of Definition 2 are independent in the absence of rigid concept and role names. Thus, we can again use the PSPACE-Turing machine from [Sistla and Clarke, 1985], where, in each step, we additionally execute a P-test according to Lemma 10.

Given the rather negative results for EL-LTL in the presence of rigid symbols, we now consider EL-LTL with global GCIs, as introduced in [Baader et al., 2012]. In this case, ELLTL-formulae are restricted to the form (2 ∧ T ) ∧ ψ, where T is a TBox and ψ is an EL-LTL-formula using only assertions. This is also a special case of a Boolean TCQ ψ over the TBox T , where the CQs in ψ do not contain any variables.

By an adaptation of the approach used in the proof of Theorem 6, we can extend the complexity of PSPACE even to the case where rigid roles are allowed. It suffices to guess an ABox type, which must now contain also (negated) role assertions for all rigid role names, together with a set of assertions of the form ∃r.A(a) for a rigid role name r. Intuitively, they specify the kinds of r-successors a must have at every time point. Again, the existence of such an ABox type and assertions characterizes the r-satisfiability of S, and we obtain the following result by an adaptation of the PSPACE-Turing machine from [Sistla and Clarke, 1985].

Theorem 12. Even if NRR ̸= ∅, then satisfiability in EL-LTL with global GCIs is in PSPACE.

5 Conclusions

We have characterized the computational complexity of two recently proposed temporal query languages over ontologies in EL. The data complexity of TCQ entailment implies that it may be possible to apply the approach of [Lutz et al., 2009] if no rigid names are allowed. But this is not a very interesting case since one cannot formulate temporal dependencies.

On the positive side, we show that the combined complexity of PSPACE inherited from LTL does not increase if rigid role names are disallowed, and also in the case that the query contains no variables and rigid role names are allowed. Furthermore, if we make the reasonable assumption that all relevant information about the rigid names (e.g., which applications belong to the concept VideoApplication) is available before the start of our context-aware system, then we do not need to guess the ABox type AR. It remains to be seen whether one can efficiently combine existing algorithms for LTL [Gastin and Oddoux, 2001] and EL [Lutz et al., 2009].

Regarding the conjecture about EL-LTL from [Baader et al., 2012], we have verified that EL-LTL has the same complexity as ALC-LTL if only rigid concept names are allowed. However, if rigid role names are considered, then the complexity decreases from 2-EXPTIME to NEXPTIME.

In future work, we want to investigate what happens if we replace EL by DL-Lite. While satisfiability in DL-LiteLTL is PSPACE-complete in all cases, the complexity of TCQ entailment over DL-Lite-TKBs remains open. Our hope is that TCQs can be rewritten into a first-order query over the database resulting from viewing the ABox sequence under the closed world assumption [Calvanese et al., 2006]. If the size of the rewriting is not too large, this may yield efficient algorithms for answering temporal queries. For a practical application, an implementation should also be based on suitable windows of the data rather than the whole history. We are also currently evaluating the utility of temporal query languages for situation recognition in operating systems.

Acknowledgments

This work was partially supported by the DFG in CRC 912. We also want to thank Franz Baader, Marcel Lippmann, and Carsten Lutz for fruitful discussions on the topic of this paper.