Temporalized EL Ontologies for Accessing Temporal Data: Complexity of Atomic Queries

1 Introduction

In recent years, the use of ontologies to enrich plain data with a semantic layer has become one of the outstanding applications of description logic (DLs) technologies in the Semantic Web. The ontology-based data access (OBDA) setting provides information systems with various advantages, e.g., a friendlier vocabulary for accessing heterogenous data is given by the ontology, and means of querying potentially incomplete data are provided by taking account of the implicit knowledge derived from the data and the ontology. Due to the increasing need to account for the temporal dimension of data available on the Web [Roth and Tan, 2013; Dong and Tan, 2015], the DL community has recently investigated extensions of the OBDA paradigm for temporal data. The initial efforts concentrated on temporal query languages with atemporal ontologies [Guti´errez-Basulto and Klarman, 2012; Klarman and Meyer, 2014; Baader, Borgwardt, and Lippmann, 2015; Borgwardt, Lippmann, and Thost, 2015; Borgwardt and Thost, 2015]. On the other hand, temporal ontology languages can enhance conceptual modelling with temporal aspects [Artale et al., 2015], which are required, e.g., in applications managing data from sensor networks. In this line, the research has focused on temporal extensions of DL-Lite that support rewritability of temporal queries into the monadic second-order logic with order or into twosorted first-order logic with < and + [Artale et al., 2013b;

2015]. Since standard relational database management systems have such built-in predicates, they can in principle evaluate the FO(<↪+)-rewritings. However, no temporal extensions of other classical DLs have been investigated yet in the context of OBDA, which is partly because of the intractability and often even undecidability of the standard reasoning tasks (e.g., subsumption) [Artale et al., 2007; Guti´errez-Basulto, Jung, and Lutz, 2012; Guti´errez-Basulto, Jung, and Schneider, 2014]. On the other hand, temporal data has also been studied in classical database theory [Chomicki and Toman, 2005]. In their seminal paper, Chomicki and Imielinski [1988] identified DATALOG1S as a decidable extension of DATALOG with one successor function. Here we make the first (to the best of our knowledge) attempt to link temporal OBDA with temporal deductive databases [Chomicki, 1990; Baudinet, Chomicki, and Wolper, 1993].

In this paper, we study TEL, a temporal extension of EL [Baader, Brandt, and Lutz, 2005]. The underlying DL component, EL, underpins the OWL 2 EL profile of OWL 2 and the medical ontology SNOMED CT, which provides the vocabulary for electronic health records (EHRs). Indeed, applications managing EHRs must be able to provide information, e.g., on when and for how long some drug has been prescribed to a patient, so that drugs that interact adversely are not prescribed at the same time. Clinical trials [Shankar et al., 2008; O'Connor et al., 2009] also require a unified conceptual model for specifying temporal constraints of protocol entities such as 'a viable participant should have had a vaccination with live virus 5 days ago' or 'blood tests of a patient should be run every 3 days'. These statements can be encoded in TEL:

Patient u ©3 P ReqBloodTest v ReqBloodTest▷ (2)

Patient u ©5 P 9vaccinated▷LiveVirus v ViableParticip↪ (1)

Our main objective is to establish the limits of decidability and tractability of the query answering problem over TEL ontologies, in terms of both data and combined complexity. In order to set the foundations, we focus on temporal atomic queries. On the one hand, an atomic query ViableParticip(x↪ t) with the temporal concept inclusion (1) effectively encodes a tree-shaped temporal conjunctive query. On the other hand, using (1) to extend the vocabulary with a concept ViableParticip is closer to the spirit of the OBDA paradigm than repeating the same conjunction in all similar user queries. Moreover, a recurrent pattern ReqBloodTest is expressible as an atomic query ReqBloodTest(x↪ t) with the temporal concept inclusion (2) but not expressible as a query without temporal concept inclusions like (2). As we shall see, even for atomic queries rather surprising (and challenging) results are obtained.

Our main contributions are complexity bounds, algorithms and rewritability into DATALOG1S for atomic query answering in fragments of TEL. Since query answering over unrestricted TEL turns out to be undecidable (in data complexity), we investigate its fragments to attain decidability and tractability. First, for TEL◦, which allows only the 'next-' ©F and 'previoustime' ©P operators, we identify ultimate periodicity as a natural semantic condition ensuring decidability, more precisely, PSPACE data complexity (the question of decidability of the full TEL◦ is left open for future work). Then, we identify a number of fragments with better computational properties. (i) For the fragment of TEL◦ without rigid (not changing over time) roles on the right-hand side of concept inclusions, we construct a polynomial rewriting into DATALOG1S, and so, establish PSPACE-completeness for data complexity. This fragment contains all EL ontologies as well as both (1) and (2). (ii) Over temporally acyclic TEL◦-ontologies (with rigid roles and concepts), query answering is PTIME-complete in both data and combined complexity. This tractable fragment contains (1) and fully captures all atemporal EL ontologies and may prove particularly useful in applications; it, however, does not contain (2). (iii) Query answering over DL-acyclic TEL◦ ontologies is NC1 -complete for data complexity (in principle, highly parallelizable). This fragment contains many acyclic EL ontologies as well as both (1) and (2) (note that large parts of SNOMED CT are in fact acyclic). We remark that our two novel acyclicity conditions (each constraining only one dimension) are inspired by the 'traditional' notion of acyclicity in (temporal extensions of) DLs [Haase and Lutz, 2008; Guti´errez-Basulto, Jung, and Schneider, 2015]. Finally, (iv) we show that the language with only 3P and 3F (sometime in the past/future) on the left-hand side of concept inclusions also enjoys PTIME query answering.

Omitted proofs can be found at tinyurl.com/TempEL16.

2 Preliminaries

We begin by introducing TEL, a temporal extension of the classical DL EL. Let NC, NR, NI be countably infinite sets of concept, role and individual names, respectively. We assume that NR is partitioned into two infinite sets, N rig R and N loc R , of rigid and local role names, respectively. TEL concepts are defined by the following grammar:

where A 2 NC, r 2 NR, and ⇤ 2 ¶F↪ P♦. A TEL-TBox (ontology) T is a finite set of concept inclusions (CIs) C v D and concept definitions (CDs) C ⌘ D for TEL concepts C↪ D. Data is given in terms of temporal ABoxes A, which are finite sets of assertions of the form A(a↪ n) and r(a↪ b↪ n), where A 2 NC, r 2 NR, a↪ b 2 NI, and n 2 Z. We denote by ind(A) the sets of individual names occurring in A, and by tem(A) the set ¶n 2 Z ♣ minA  n  maxA♦, where minA and maxA are, respectively, the minimal and maximal time points in A. The size, ♣T ♣ and ♣A♣, of T and A is the number of symbols required to write T and A, respectively, with time points n 2 Z encoded in unary. A temporal knowledge base (KB) K is a pair (T ↪ A).

An interpretation I is a structure (∆I ↪ (In)n2Z), where each In is a classical DL interpretation with domain ∆I : we have AIn ✓ ∆I and rIn ✓ ∆I⇥∆I . Rigid roles r 2 N rig R do not change their interpretation in time: rIn = rI0 for all n 2 Z. We usually write AI↪n and rI↪n instead of AIn and rIn , respectively, and the mapping ·I↪n is extended to complex TEL-concepts as follows:

where op⇤ stands for - if ⇤ = P and for + if ⇤ = F . We use strict 3⇤ (k > 0) but our results do not depend on the choice.

For ABoxes A we adopt the standard name assumption: aI↪n = a for all a 2 ind(A), n 2 Z (and thus ind(A) ✓ ∆I ). The relation ♣= is extended to ABoxes by taking I ♣= A(a↪ n) iff a 2 AI↪n and I ♣= r(a↪ b↪ n) iff (a↪ b) 2 rI↪n ; I is a model of A iff I ♣= ↵ for all ↵ 2 A. An interpretation I is a model of a KB (T ↪A), written I ♣= (T ↪A), iff I ♣= T and I ♣= A. Finally, K ♣= A(a↪ n) if I ♣= A(a↪ n) for every model I of K.

An interpretation I is said to be a model of C v D, written I ♣= C v D, if C I↪n ✓ DI↪n , for all n 2 Z; and a model of C ⌘ D if C I↪n = DI↪n , for all n 2 Z. We call I a model of a TBox T , written I ♣= T , if I ♣= ↵ for all ↵ 2 T . Note that TBoxes are interpreted globally in the sense that all CIs and CDs must be satisfied at every time point. A concept D subsumes a concept C with respect to T , written T ♣= C v D, if I ♣= C v D for all models I of T .

As the query language, we consider temporal atomic queries (TAQs) of the form A(x↪ t) with A 2 NC, x an individual variable and t a temporal variable. Given K = (T ↪A), a certain answer to A(x↪ t) over K is a pair (a↪ n) 2 ind(A) ⇥ tem(A) with K ♣= A(a↪ n). We study the complexity of the query answering problem over temporal knowledge bases:

TAQ answering

Our results concern both the combined and data complexity of the problem: for data complexity, the TBox is fixed. As usual, for a complexity class C and a class X of TBoxes, we say that TAQ answering over X is C-hard in data complexity if there is some T 2 X such that answering TAQs over T is C-hard. Conversely, TAQ answering over X is in C in data complexity if, for all T 2 X, answering TAQs over T is in C.

As classes X, we will in particular look at full TEL and its fragments TEL3 and TEL◦ in which, respectively, only the temporal operators 3⇤ and ©⇤ are allowed. Note that 3⇤ on the left-hand side and 2⇤ (with the usual semantics) on the right-hand side of CIs can be expressed in TEL◦, e.g., instead of 3PA v X or, equivalently, A v 2FX, take A v A0 and ©PA0 v A0 uX, for a fresh A0 . Thus, rigid concepts, which do not change their interpretation in time, can be expressed in these two fragments using 3P3F on the left-hand side of CIs.

3 Query Answering in TEL: Undecidability

We first pinpoint different sources of complexity for the query answering problem in TEL in order to identify computationally well-behaved fragments in the sequel.

We begin by showing that TAQ answering over TEL3 is undecidable. The known undecidability of subsumption in TEL3 [Artale et al., 2007] translates only into the combined complexity of TAQ answering. We strengthen the result to obtain undecidability in data complexity by reducing the halting problem for the universal Turing machine. We exploit the crucial observation that disjunction, although not in the syntax, can be simulated with 3⇤ [Artale et al., 2007].

Theorem 1 TAQ answering over TEL3 is undecidable in data complexity.

The proof can also be adapted to the non-strict semantics of 3⇤ using the chessboard technique [Gabbay et al., 2003].

TEL◦, unlike TEL3 , is not capable of expressing disjunction. Still, its expressiveness makes answering TAQs hard:

Theorem 2 TAQ answering over TEL◦ is non-elementary in combined complexity and PSPACE-hard in data complexity.

The proof of PSPACE-hardness is close in spirit to that for DATALOG1S [Chomicki and Imielinski, 1988]; note that the lower bound holds even in the restriction of TEL◦ without 9r▷C on the right-hand side of CIs. For the non-elementary lower bound, we take inspiration in the construction for the product modal logic LTL⇥K [Gabbay et al., 2003, Theorem 6.34]. Our proof requires a careful implementation of the yardstick technique [Stockmeyer, 1974] with only Horn formulas.

Decidability of TAQ answering in full TEL◦ is left open as interesting and challenging future work; more insights on the difficulty of the problem are given in Sec. 4. Nevertheless, we show that extending TEL◦ with certain DL constructs that are harmless for data complexity of atemporal query answering [Krisnadhi and Lutz, 2007] immediately leads to undecidability. Let TELI ◦ and TELF ◦ be the extensions of TEL◦ with inverse roles r- and functionality axioms func(r), respectively. 1 For both languages, we reduce the halting problem for the universal Turing machine to prove:

Theorem 3 TAQ answering over TELI ◦ and TELF ◦ is undecidable in data complexity.

In the rest of the paper, we study decidability and complexity of TAQ answering in various fragments of TEL◦ and TEL3 .

4 Foundations of Query Answering in TEL◦

In the sequel, we assume that TEL◦-TBoxes are in normal form, that is, they consist of CIs of the form

In this section, we lay the groundwork for the development of algorithms for query answering in fragments of TEL◦ by introducing canonical quasimodels, which are succinct abstract representations of the universal model of the KB, see also [Artale et al., 2013b; 2015]. They can also be viewed as a generalization of the canonical structures used for query answering in pure EL [Lutz, Toman, and Wolter, 2009].

1 with the usual semantics: (r-)I↪n = ¶(e↪ d) ♣ (d↪ e) 2 rI↪n ♦; I ♣= func(r) iff e1 = e2, for all (d↪ e1)↪ (d↪ e2) 2 rI↪n and n 2 Z.

where A, A0 and B are concept names and X is a basic concept of the form A, ©⇤A, or 9r▷A, for a concept name A. Observe that, without loss of generality, ©⇤ is restricted to the left-hand side of CIs: for instance, A v ©FB is equivalent to ©PA v B. It is routine to show that every TEL◦-TBox can be transformed into the normal form by introducing fresh concept names; see, e.g., [Baader, Brandt, and Lutz, 2005].

• (t1) if AuA0 v B 2 T and A↪A0 2 ⇡(n), then B 2 ⇡(n);

Fix now a KB (T ↪A) with a TEL◦-TBox T in normal form and let CN be the set of concept names in (T ↪A). A map ⇡: Z! 2CN is a trace for T if it satisfies the following:

Traces are the building blocks of quasimodels: they represent the temporal evolution of individual domain elements. For example, for T = ¶©PC v B↪©PB v C♦, the map ⇡ such that ⇡(i) is ¶B♦ for odd i and ¶C♦ for even i is a trace for T .

In order to describe interactions of domain elements, we require more notation. Let ⇡ be a trace for T . For a rigid role r 2 N rig R , the r-projection of ⇡ is a map projr(⇡): Z ! 2CN that sends each i 2 Z to ¶A ♣ 9r▷B v A 2 T ↪B 2 ⇡(i)♦; for a local role r 2 N loc R , projr(⇡) is defined in the same way on 0 but is ; for all other i 2 Z. Given a map %: Z ! 2CN and n 2 Z, we say that ⇡ contains the n-shift of % and write % ✓n ⇡ if %(i - n) ✓ ⇡(i), for all i 2 Z. For example, let T = ¶9r▷B v B0 ♦ with rigid role r. In the picture below, the trace ⇡a contains the 1-shift of the r-projection of ⇡B:

If r is local then ⇡a has to contain B0 only at 1 (but not at 3, etc.). We are now fully equipped to define quasimodels.

Let D = ind(A) [ CN henceforth. A quasimodel Q for (T ↪A) is a set of traces ⇡d, d 2 D, for T such that

• (q2) B 2 ⇡B(0), for all B 2 CN;
• (q1) A 2 ⇡a(n), for all A(a↪ n) 2 A;
• (q3) projr(⇡b) ✓0 ⇡a, for all r(a↪ b↪ n) 2 A;
• (q4) if A 2 ⇡d(n) then projr(⇡B) ✓n ⇡d, for all d 2 D, n 2 Z and A v 9r▷B in T .

Intuitively, quasimodels represent models of (T ↪ A): each ⇡a stands for the ABox individual a; each ⇡B, on the other hand, represents all individuals that witness B for CIs A v 9r▷B in T . The latter is, in fact, the crucial abstraction underlying quasimodels. Note that traces ⇡B are normalized: B occurs at time point 0, which is compensated by the shift operation in (q4). For example, in the picture above, if A v 9r▷B 2 T then, in any model, a has an r-successor that belongs to B at moment 1. Such a successor can be obtained as a 'copy' of trace ⇡B shifted by 1 so that its origin, 0, matches moment 1 for a. Then, by (q4), a belongs to B0 at all odd moments.

For the purposes of query answering we need to identify canonical (minimal) quasimodels. We define the canonical quasimodel as the limit of the following saturation (chase-like) procedure. Start with initially empty maps ⇡d, for d 2 D, and apply (t1)-(t2), (q1)-(q4) as rules: (q3), for example, says 'if B0

r(a↪ b↪ n) 2 A and A 2 projr(⇡b)(i), then add A to ⇡a(i).' Then we have the following characterization:

Theorem 4 Let Q = ¶⇡d ♣ d 2 D♦ be the canonical quasimodel of (T ↪ A) with T a TEL◦-TBox. Then, for any A 2 CN, (T ↪A) ♣= A(a↪ i) iff A 2 ⇡a(i), for a 2 ind(A), i 2 Z.

The procedure for constructing the canonical quasimodel deals with infinite data structures (traces) and is generally not terminating. So, although Theorem 4 provides a criterion for certain answers, it does not immediately yield a decision algorithm for full TEL◦. We remark that known techniques for dealing with such infinite structures cannot be easily applied: for example, MSO (over Z), a standard tool for decidability proofs in temporal DLs [Gabbay et al., 2003], is not sufficient to encode the canonical quasimodel directly because (q4) requires +. In fact, the key to showing decidability for (fragments of) TEL◦ is finding a finite representation of traces.

The starting point of the rest of the paper is a semantic condition on the canonical quasimodel, ultimate periodicity, which ensures decidability in data complexity. Let T be a TEL◦-TBox and Q the canonical quasimodel for (T ↪ ;). We say that T is ultimately periodic, if there is p 2 N such that all ⇡B, B 2 CN, in Q are ultimately p-periodic, that is, for each B 2 CN, there are positive integers mP ↪ pP ↪ mF ↪ pF  p satisfying the following conditions:

Intuitively, an ultimately p-periodic trace has repeating sections on the left and on the right:

-mP -2pP

-mP

The condition of ultimate periodicity is rather natural. On the practical side, it is motivated by applications with recurrent patterns such as health care support [Shankar et al., 2008], see CIs (1) and (2) in Section 1. From the theoretical point of view, any satisfiable LTL formula has an ultimately periodic model [Manna andWolper, 1984].

We next show that ultimate periodicity is indeed sufficient for decidability in data complexity.

Theorem 5 TAQ answering over ultimately periodic TEL◦- TBoxes is PSPACE-complete in data complexity.

PSPACE-hardness follows from (the proof of) Theorem 2. We prove the matching upper bound by rewriting an ultimately periodic TEL◦-TBox T into DATALOG1S [Chomicki and Imielinski, 1988]. First, we take temporal rules

which reflect rigidity of roles and standard EL concept inclusions on ABox individuals. Second, we observe that, for any trace ⇡d in the canonical quasimodel Q of any (T ↪A), if A 2 ⇡d(n) and A v 9r▷B 2 T then, by (q4), ⇡d contains the n-shift not only of projr(⇡B) but also of ⇡A. Since T is ultimately periodic, for each trace ⇡B, we fix integers mP , pP , mF , pF and take the following rules with a fresh predicate FB:

and symmetric rules with mP , pP and fresh PB. Intuitively, the rules in the first line replicate the (irregular) part of ⇡B from 0 to mF . The two rules in the last line add recurring markers FB at the start of each period while the rules in the second line replicate the period of ⇡B starting from each marker FB.

The required DATALOG1S-program ⇧T contains all the rules above (note that CIs of the form ©⇤A v B are also covered by the rules for traces ⇡B). Using the canonical quasimodel and Theorem 4, it is readily seen that ⇧T is equivalent to T : for every temporal ABox A, the answers to ⇧T over A coincide with the certain answers to (T ↪ A). Theorem 5 follows from PSPACE data complexity in DATALOG1S [Chomicki and Imielinski, 1988] and independence of ⇧T fromA.

Observe that Theorem 5 does not imply decidability of full TEL◦ since it is open whether every TEL◦-TBox is ultimately periodic. We thus turn our attention to sufficient syntactic conditions for ultimate periodicity and obtain tight complexity bounds for both data and combined complexity for the resulting fragments. We consider two types of conditions: restricted use of rigid roles and acyclicity of concept inclusions.

5 Restricted Use of Rigid Roles

We first consider TEL◦loc , the restriction of TEL◦ in which only local roles are allowed. Due to the reduced interaction between the temporal and DL component, we obtain data tractability.

Theorem 6 TAQ answering over TEL◦loc is PSPACE-complete in combined and PTIME-complete in data complexity.

PSPACE-hardness follows from the proof of PSPACE-hardness for entailment in Horn-LTL [Chen and Lin, 1993] and PTIMEhardness from atomic query answering in EL. For the upper bounds, let (T ↪A) be a KB with T a TEL◦loc -TBox and Q its canonical quasimodel. We take a propositional variable PA↪d for each A 2 CN and d 2 D and construct a HornLTL formula 'T ↪A whose minimal model is isomorphic to Q: variable PA↪d is true in the model at moment n iff A 2 ⇡d(n). We take the conjunction of the following formulas, for d 2 D:

where ©n is ©n F if n ≥ 0 and ©-n P if n < 0 and 2 is the 'globally' operator. It is readily verified that 'T ↪A is as required. Crucially, (q4) for local roles boils down to the last formula above. Since entailment in LTL is in PSPACE [Sistla and Clarke, 1985] and 'T ↪A is polynomial in the size of (T ↪A), we obtain membership in PSPACE for combined complexity.

To obtain the PTIME data complexity, observe that traces ⇡B, B 2 CN, are ultimately 2♣T ♣ -periodic because they are traces of the canonical quasimodel for (T ↪ ;); so, they can be maintained in constant space. Next, traces ⇡a, a 2 ind(A), are ultimately 2♣T ♣+♣A♣ -periodic, but a closer inspection reveals that the middle irregular section, mP +mF , is bounded by ♣A♣ + 2♣T ♣ , while both periods, pP and pF , by 2♣T ♣ ; see, e.g., Lemma 3 [Artale et al., 2013a]. Thus, Q can be stored in space bounded by a polynomial in ♣A♣. Since each rule application extends the traces, the saturation procedure for constructing Q terminates in polynomial time in the size of A.

Since ontologies without rigid roles at all may be too restrictive for applications, we consider TEL◦l-rig -TBoxes where rigid roles are allowed only in CIs of the form 9r▷B v A.

Theorem 7 TAQ answering over TEL◦l-rig is PSPACE-complete in data complexity and in EXPTIME in combined complexity.

PSPACE-hardness in data complexity follows from the proof of Theorem 2. For the upper bounds, we construct rewritings into DATALOG1S, similarly to ⇧T in Section 4 (Theorem 5).

6 Acyclicity Conditions

It is known that acyclicity conditions may lead to better complexity. In particular, acyclic TBoxes are a way of obtaining CTL-based temporal extensions of EL that have rigid roles and enjoy PTIME subsumption [Guti´errez-Basulto, Jung, and Schneider, 2015]. In DATALOG1S, a restriction on recursion has also been used to attain tractability [Chomicki, 1990]. From the application point of view, large parts of SNOMED CT and GO [Gene Ontology Cons., 2000] are indeed acyclic. So, we believe that the fragments we consider below are wellsuited for temporal extensions of such ontologies.

Acyclic TBoxes are finite sets of CDs A ⌘ C, A 2 NC, such that no two CDs have the same left-hand side, and there are no CDs A1 ⌘ C1↪ ▷ ▷ ▷ ↪ Ak ⌘ Ck in T such that Ai+1 occurs in Ci, for all 1  i  k (where Ak+1 := A1). We say A is defined in T if A ⌘ C 2 T and primitive otherwise.

Theorem 8 TAQ answering over acyclic TEL◦ is in LOGTIME-uniform AC0 in data complexity and in PTIME in combined complexity.

The LOGTIME-uniform AC0 upper bound is established by rewriting into FO(+): for a given TAQ A(x↪ t) and TBox T , we construct a two-sorted first-order formula 'T ↪A(x↪ t) with functions +1 and -1 on temporal terms such that (T ↪A) ♣= A(a↪ i) iff A (viewed as an interpretation) is a model of 'T ↪A(a↪ i), for all ABoxes A, a 2 ind(A), i 2 Z. We adapt the technique developed for atemporal EL [Bienvenu, Lutz, andWolter, 2012]:

where SA(x↪ t) is a disjunction of all B(x↪ t) for concept names B with T ♣= B v A, and Rr(x↪ y↪ t) is r(x↪ y↪ t) for r 2 N loc R and 9t0 r(x↪ y↪ t0 ) for r 2 N rig R .

Note that 'T ↪A is an FOZ -rewriting in the terminology of Artale et al. [2013b; 2015] because the temporal terms range over Z. However, the infinite interpretation of A is empty after at most ♣T ♣ steps from the ABox and so, 'T ↪A can be converted into an FO-rewriting whose temporal terms range over tem(A) only; see [Artale et al., 2015].

We next introduce novel notions of acyclicity that restrict only one dimension, DL or temporal.

DL Acyclicity

First, we introduce DL-acyclic TEL◦-TBoxes, which are wellsuited as temporal extensions of, say, biomedical ontologies that may require recurrent patterns but have an acyclic DL component. A TEL◦-TBox T with concept names CN is called DL-acyclic if there is a mapping `DL : CN ! N such that:

• (ii) ©⇤A v B implies `DL(A) = `DL(B);
• (i) Av9r▷B or 9r▷Bv A2 T implies `DL(A) > `DL(B);

A DL-acyclic TBox is of depth k if k is minimal such that a witnessing mapping `DL satisfies `DL(B)  k for all B 2 CN.

Theorem 9 TAQanswering over DL-acyclic TEL◦-TBoxes of depth k ≥ 1 is k-EXPSPACE-complete in combined complexity and NC1 -complete in data complexity.

A closer inspection of the non-elementary lower bound proof in Theorem 2 reveals that the TBox used is DL-acyclic and TAQ answering over TBoxes of depth k is k-EXPSPACE-hard. NC1 -hardness in data complexity follows [Artale et al., 2015] by reduction of the word problem of NFAs to TAQ answering, even without the DL dimension.

For the matching upper bounds, fix (T ↪A) with T of depth k. We devise a completion procedure, which is based on special LTL-formulas and implies ultimate periodicity of all traces in the canonical quasimodel of (T ↪A); cf. Section 5. Given any A, let the slice Ai consist of all A(a↪ i) 2 A, all r(a↪ b↪ i) 2 A and all r(a↪ b↪ i) with r(a↪ b↪ j) 2 A, for some j, and r 2 N rig R . The algorithm separates consequences of the role structure of A and local temporal consequences of T . In particular, it exhaustively extends A by all A(a↪ i) with either

It turns out that Ai in (6) can be replaced by its suitably defined quotient Bi. Intuitively, the logic can only distinguish distinct trees of depth k, whose number depends on ♣T ♣ only; so, the size of Bi is independent of ♣A♣. By induction on depth k, we define LTL-formulas 'a↪i of k-fold-exponential size characterizing all A 2 CN with (T ↪ Bi) ♣= A(a↪ i): we begin from formulas as in Theorem 6; the induction step takes account of the structure of Bi and incurs an exponential blowup.

Now, for combined complexity, observe that each of the polynomially many 'a↪i can be analyzed in k-EXPSPACE. For data complexity, observe that checking (T ↪ Bi) ♣= A(a↪ i) can be done in constant time. The second option in (6), however, cannot be implemented directly as the number of steps depends on ♣A♣. Instead, we construct a B¨uchi automaton that accepts precisely the traces for T and cast the second option in (6) as the question of whether all traces extending A have A at position i, which is a regular property and so, is in NC1 .

Temporal Acyclicity

We next relax acyclicity by admitting recursion in the DL dimension (but not in temporal); thus, temporally acyclic TBoxes include general EL-TBoxes. A TEL◦-TBox T with concept names CN is temporally acyclic if there is `◦ : CN ! N such that:

• (ii) 9r▷B v A or A v 9r▷B 2 T implies `◦(A) = `◦(B);
• (i) ©PAvB or ©FBvA2T implies `◦(B)=`◦(A)+1;
• (iii) AuA0 v B 2 T implies `◦(A) = `◦(A0 ) = `◦(B).

Temporally acyclic TBoxes cannot, unlike DL acyclic ones, express rigid concepts. Still, we can partition concept names NC into local N loc C and rigid N rig C and obtain the following:

Theorem 10 TAQ answering over temporally acyclic TEL◦ (with rigid concepts) is PTIME-complete in data and combined complexity.

The lower bounds are from EL. For the upper bounds, we show a small quasimodel property: traces of the canonical quasimodel of any (T ↪ A) with such a TBox T satisfy

where u = maxA and l = minA. Intuitively, the canonical quasimodel has a restricted temporal extension that stretches only ♣T ♣ time points beyond A. By the small quasimodel property, the procedure for constructing the canonical quasimodel can be implemented in polynomial time: traces ⇡d require only polynomial space, and rules (q1)-(q4) extend the traces.

Inflationary TEL3

Theorem 11 TAQ answering over TEL3 infl is PTIME-complete in both data and combined complexity.

Next, we follow an approach suggested by Artale et al. [2013b] (in the context of temporal DL-Lite) and restrict TEL3 by allowing 3⇤ only on the left-hand side of CIs. This fragment is denoted by TEL3 infl , for inflationary TEL (which is related to inflationary DATALOG1S [Chomicki, 1990]). Note that TEL3 infl extends general EL-TBoxes. Yet, the complexity is the same:

We need to show only the upper bounds. Observe that TEL3 infl can still be viewed as a fragment of TEL◦; see Section 2. In fact, one can show an analogue of Theorem 4 with the following replacement of (t2):

We establish a special shape of the traces in the canonical model of any (T ↪A). Let %: Z ! 2CN be a map and let l↪ u 2 Z with l  u. We say that % is an [l↪ u]-bow tie if

• for all i > u, we have %(i + 1) ◆ %(i), and if %(i + 1) = %(i) then all %(i0 ), for n0 ≥ i, coincide;
• symmetrically, for all i < l, we have %(i - 1) ◆ %(i), and if %(i - 1) = %(i) then all %(i0 ), for i0  i, coincide.

These properties mean that % grows monotonically to the right of u and to the left of l; in other words, % has inflationary behaviour. We prove that the traces ⇡d in the canonical quasimodel Q of (T ↪ A), for any A, enjoy the following properties:

• ⇡a is a [minA↪maxA]-bow tie, for each a 2 ind(A);

Thus, the traces in Q can be represented in polynomial space because only the middle section and at most ♣CN♣ steps at both ends need to be stored. Since the traces are extended with every rule application, the procedure terminates after polynomially many steps; Theorem 11 follows.

• ⇡B is a [0↪ 0]-bow tie, for each B 2 CN.

7 Discussion and FutureWork

We summarize the fragments of TEL, their relationships and the obtained complexity results in the following diagram:

where the solid lines are inclusions between DLs, the dashed line is a reduction that preserves answers to all queries (model conservative extension). The data complexity is indicated by shading and the combined complexity is below the language.

Our data-tractability results show theoretical adequacy of the identified fragments of TEL for data-intensive applications. Our two novel forms of acyclicity, DL- and temporal, are somewhat close in spirit to multi-separability [Chomicki, 1990]: the latter, however, puts a weaker restriction on recursion but a stricter one on the interaction between the temporal and data component. DL-acyclic TEL◦ is the first (to the best of our knowledge) DL shown to have NC1 -complete query answering (the large gap between data and combined complexity is also remarkable). On the practical side, there is evidence that such data-tractable fragments should be sufficient for many biomedical applications. Following the principles of OBDA, our framework provides a means of defining temporal concepts in the ontology for these applications: temporal concepts capture both (restricted) tree-shaped temporal conjunctive queries (CQs) and recurring temporal patterns.

As our immediate future work, we will address decidability of (full) TEL◦ and then consider CQs with the + operation on temporal terms. We expect that our positive results can be lifted to CQs using the combined approach [Lutz, Toman, and Wolter, 2009], which utilizes a structure similar to our canonical quasimodel. We will also study succinct and expressive representations of temporal data. For example, the only known algorithm for DATALOG1S with binary encoding of timestamps in the data runs in EXPTIME in the size of the data [Chomicki and Imielinski, 1988]. We, however, presume that careful materialization should be sufficient to deal with the issue. We will also consider interval encoding of temporal ABoxes, e.g., A(a↪ [n1↪ n2]), and settings capturing infinite temporal periodic data as introduced by Kabanza, St´evenne, and Wolper [1990] and Chomicki and Imielinski [1993].