News on Temporal Conjunctive Queries

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1 News on Temporal Conjunctive Queries Veronika Thost Center for dvancing Electronics Dresden (cfaed), TU Dresden stract. Temporal query languages are important for stream processing, and ontologies for stream reasoning. Temporal conjunctive queries (TCQs) have therefore een investigated recently together with description logic ontologies, and the knowledge we have aout the comined complexities is rather complete. However, often the size of the queries and the ontology is negligile, and what costs is the data. We prove a new result on the data complexity of ontology-ased TCQ answering and close the gap etween co-np and ExpTime for many description logics. Keywords: temporal queries, description logics, data complexity 1 ntroduction The temporal nature of data is important in many applications, and the We offers more and more streaming sources and datasets. Ontologies play an important role in this context: y linking data from heterogeneous sources to the concepts and relations descried in an ontology, the integration and automated processing of the data can e consideraly enhanced. Queries formulated in the astract vocaulary of the ontology can then e answered over all the linked datasets. Medical domain ontologies written in description logics (DLs) may, for example, capture the facts that the varicella zoster virus (VZV) is a virus, that chickenpox is a VZV infection, and that a negative allergy test implies that no allergies are present, y concept inclusions: VZV Virus, Chickenpox VZVnfection, NegllergyTest llergyto. Here, Virus is a concept name that represents the set of all viruses, and llergyto is a role name, representing a inary relation connecting patients to allergies; llergyto refers to the domain of this relation. possile data source storing patient data (e.g., allergy test results and findings) could look as follows: PD Name 1 nn 2 Bo 3 Chris PD llergytest Date 1 neg pos neg PD Finding Date 1 Chickenpox VZV-nfection VZV-nfection The data is then connected to the ontology y mappings [10], which in our example may link the tuple (1, Chickenpox, ) to the facts HasFinding(1, x) and Chickenpox(x). Conceptually, we thus regard a sequence of fact ases, one for each time point we have data for.

2 Ontology-ased query answering (OBQ) over the aove knowledge can then, for example, assist in finding appropriate participants for a clinical study, y formulating the eligiility criteria as queries over the usually linked and heterogeneous patient data. The following are examples of in- and exclusion conditions for a recently proposed clinical trial: 1 (i) the patient should have een previously infected with VZV or previously vaccinated with VZV vaccine; (ii) the patient should not e allergic to VZV vaccine. We focus on temporal conjunctive queries (TCQs), which were originally proposed y [2, 4]. TCQs allow to comine conjunctive queries (CQs) via the Boolean operators and the temporal operators of propositional linear temporal logic LTL [9]. The aove criteria can e specified with the following TCQ Φ(x), to otain all eligile patients x: ( ( ) ( P y.hasfinding(x, y) VZVnfection(y) P y.vaccinatedwith(x, y) VZVVaccine(y) )) ( y.llergyto(x, y) VZVVaccine(y) ) We here use the temporal operator some time in the past ( P ) and consider the symols llergyto and VZVVaccine to e rigid, which means that their interpretation does not change over time; that is, we assume someone having an allergy to have this allergy for his whole life. The OBQ scenario outlined aove is similar to the classical one [5], ut we use temporal queries and consider a (finite) sequence of fact ases. The ontology is written in a classical DL (i.e., one can take one of the many existing ontologies) and assumed to always hold. n contrast, so-called temporal DLs extend classical DLs y temporal operators, which then occur within the ontology (see [11] for an overview). But most of these logics yield high reasoning complexities, even if the underlying atemporal DL allows for tractale reasoning. Lower complexities are only otained y either consideraly restricting the set of temporal operators or the DL. The comined and data complexity of TCQ entailment have een studied for various DLs in the past [4, 3, 7, 6]. n a nutshell, we have that the comined complexity strongly varies etween PSpace and 2-ExpTime depending on the DL considered and the rigid names allowed, which often increase complexity. 2 The data complexity for the lightweight DLs etween DL-Lite core and DL-Lite H horn is generally LogTime, and the one for EL without rigid symols is P, and co-np with rigid symols. For all other DLs investigated so far, containment in co-np has only een shown for the case without rigid roles. This includes expressive DLs such as SHQ and is interesting since already standard conjunctive query entailment is co-np-hard in these DLs, which means that we get the temporal features for free. However, rigid roles are considered as an important feature for modeling and often expressive DLs are needed; for instance, simple disjunctions of the form Male Female ( everyone is male or female ) cannot e expressed in DL-Lite H horn or EL. Yet, the proposed algorithms for such cominations are at least exponential in the data. n this paper, we first close the co-np/exptime gap for the DL-Lite H krom which allows for disjunctions as in the example and prove that TCQ entailment For some very expressive DLs, we have co-2-nexptime-hardness/decidaility.

3 is in co-np in data complexity, even with rigid roles. Then, we show that this also holds for much more expressive DLs, such as LCH. 2 Preliminaries Description logics focus on individual names, which are interpreted as constants; concepts, which are interpreted as sets; and roles, which are interpreted as inary relations. ccordingly, DL signatures are ased on three kinds of symols: individual names N, concept names N C, and role names N, all of which are non-empty, pairwise disjoint sets. We focus on the DL DL-Lite H krom [1]. DL-Lite H krom. Let a, N, N C, and P N. n DL-Lite H krom, the sets of roles, asic concepts, and concepts are defined as follows:, S ::= P P, B, C ::=, D ::= B B where denotes the inverse role operator. DL-Lite H krom axioms are the following kinds of expressions: concept inclusions (Cs) are of the form B C, B C, or B C; role inclusions (s) of the form S; and assertions of the form B(a), B(a), P (a, ), or P (a, ). DL-Lite H krom ontology is a finite set of concept and role inclusions, and an Box is a finite set of assertions. Together, an ontology O and an Box form a knowledge ase (KB) K := O, written K = O,. We sometimes also refer to the Box as fact ase or simply as the data. Without loss of generality, we assume that, if the S is contained in O, then we also have S O and S O; and that O contains the trivial axioms for all roles occurring in O. The set of roles is denoted y N. For a given KB K := O, we denote y N (K) and N () the set of individual names that occur in K and ; y N C (O) and N (O) the sets of concept and role names occurring in K; and y N (O) the set of roles occurring in K. B(O) and C(O) denote the sets of all asic concepts and, respectively, concepts that can e uilt from the symols in N C (O) and N (O). We may also use the areviation (P ) := P for P N. DL interpretation = (, ) consists of a non-empty set, the domain of, and an interpretation function, which assigns to every N C a set, to every P N a inary relation P, and to every a N an element a such that, for all a, N with a, we have a (unique name assumption). The function is extended to all roles and concepts: P = {(y, x) (x, y) P }, =, = {x y, (x, y) }, ( D) = \ D. n interpretation satisfies (or is a model of) an axiom α, written = α, if: α = X Y is a C or, and X Y ; α = ( )B(a) and a B (a B ); α = ( )P (a, ) and (a, ) P ((a, ) P ). satisfies (or is a model of) a KB K, written = K, if it satisfies all axioms contained in it. KB K is consistent (or satisfiale) if it has a model, and it is inconsistent (or unsatisfiale) otherwise. K entails an axiom α, written K = α, if all models of K also satisfy α. This terminology and notation is extended to

4 (single) axioms, ontologies, and Boxes y regarding each as a (singleton) KB. We denote non-entailment y K = α. n the temporal setting, we assume that some concept and role names are designated as eing rigid (vs. flexile) as outlined in Section 1. f a concept (axiom) contains only rigid symols, then we may call it a rigid concept (axiom). We denote y N C N C the rigid concept and y N N the rigid role names. Temporal Semantics. n infinite sequence = ( i ) i 0 of interpretations i = (, i ) is a DL-LTL structure if it respects rigid names, that is: X i = X j for all X N N C N and i, j 0. Oserve that the interpretations in a DL-LTL structure share one domain (constant domain assumption). We may use that terminology in other settings in that we consider interpretations 1,..., l to respect rigid names if they agree on the interpretation of all rigid symols. Temporal Knowledge Bases. temporal knowledge ase (TKB) is of the form K = O, ( i ) 0 i n with an ontology O and a non-empty, finite sequence of Boxes. We assume all concept and role names occurring in some Box of a TKB to also occur in its ontology. N (K) denotes the set of all individual names occurring in the TKB K. Note that every KB can e regarded as a TKB with an Box sequence of length one. DL-LTL structure = ( i ) i 0 over a domain is a model of a TKB K = O, ( i ) 0 i n, written = K, if i = O for all i 0 and i = i for all i [0, n]. TKB is consistent (or satisfiale) if it has a model, and it is inconsistent (or unsatisfiale) otherwise. Temporal Conjunctive Queries. Let N V e the set of variales, and N T := N N V e the set of terms. conjunctive query (CQ) is of the form y 1,..., y m.ψ, where y 1,..., y m N V and ψ is a (possily empty) finite conjunction of concept atoms of the form (t) and role atoms of the form (s, t), where N C, N and s, t N T. The set of temporal conjunctive queries (TCQs) is defined as follows, where ϕ is a CQ: Φ, Ψ ::= ϕ Φ Φ Ψ F Φ P Φ Φ U Ψ Φ S Ψ TCQ Φ is a CQ literal if it is of the form ( )ϕ with ϕ eing a CQ; it is positive if Φ = ϕ, and otherwise negative. We denote the set of individuals occurring in a TCQ Φ y N (Φ). s in propositional LTL, we may use areviations true 3 and false. The empty conjunction and disjunction are interpreted as true and false, respectively. s usual, the semantics is defined in a model-theoretic way, ased on the notion of homomorphisms. mapping π : N T (ϕ) is a homomorphism of a CQ ϕ into an interpretation = (, ) if π(a) = a for all a N (ϕ), π(t) for all concept atoms (t) in ϕ, and (π(s), π(t)) for all role atoms (s, t) in ϕ. satisfies (or is a model of) ϕ, written = ϕ, if there is such a homomorphism. For a given DL-LTL structure = ( i ) i 0, an i 0, and TCQ Φ, the satisfaction relation, i = Φ is defined y induction on the 3 For instance, true may denote a fix TCQ ϕ ϕ, where ϕ is an aritrary CQ.

5 TCQ Φ Condition for, i = Φ CQ ϕ i = ϕ Φ W, i = Φ Φ Ψ, i = Φ and, i = Ψ F Φ, i + 1 = Φ P Φ i > 0 and, i 1 = Φ Φ U Ψ there is a k i, such that, k = Ψ and, for all j, i j < k, we have, j = Φ Φ S Ψ there is a k, 0 k i, such that, k = Ψ and, for all j, k < j i, we have, j = Φ Fig. 1. Semantics of TCQs given a DL-LTL structure = ( i) i 0. structure of Φ as specified in Figure 1. is a model of Φ w.r.t. a TKB K if = K and, n = Φ. TCQ Φ is satisfiale w.r.t. a TKB K if it has a model w.r.t. K; and Φ is entailed y a TKB K, written K = Φ, if every model of K is also a model of Φ w.r.t. K. We denote the fact that, i = Φ and K = Φ do not hold y, i = Φ and K = Φ. Oserve that a model of a TCQ must satisfy the query at the current time point n, which is different for propositional LTL if n > 0. Without loss of generality, we assume that the CQs contained in a TCQ Φ use disjoint variales and denote y Q Φ the set of exactly those CQs. 4 We further assume that TCQs contain only individual names that occur in the Boxes, and only concept and role names that occur in the ontology, and that all CQs contained in TCQs are connected (i.e., the corresponding Gaifman graph is connected); it is easy to show that this is without loss of generality. Solving TCQ Satisfiaility. The TCQ satisfiaility prolem can e split into two separate ones: one in propositional LTL and one or several atemporal ones in DL [4, Lemma 4.7]. The former tests the satisfiaility of the propositional astraction of the given TCQ Φ at n, which is otained from Φ y replacing the CQs ϕ 1,..., ϕ m Q Φ y propositional variales p 1,..., p m, respectively. The idea is that the worlds w 0, w 1,... in the LTL model characterize the satisfaction of the CQs from Q Φ in the respective DL interpretations 0, 1,... such that, to otain i, we only have to check the satisfiaility of the conjunction of CQ literals induced y w i w.r.t. the atemporal KB O, i, where i = for i > n. From the latter it can e seen that, assuming k to e the numer of different worlds occurring in the LTL model, it is sufficient to look for n k corresponding DL interpretations. More precisely, the prolems are linked y a set W = {W 1,..., W k } 2 {p1,...,pm}, which collects all worlds occurring in the LTL model, and a mapping ι: [0, n] [1, k] that maps time points to indexes from W and points out the first n + 1 worlds, which have to reflect the knowledge given in the respective Boxes. The DL part is defined as r-satisfiaility; the set W is r-satisfiale w.r.t. ι and a TKB K iff there are interpretations 0,..., n, J 1,..., J k as follows: 4 f the variales were not disjoint, we could simply rename them.

6 the interpretations share the same domain and respect rigid names, the interpretations are models of O, J i is a model of χ i := p j W i ϕ j p j W i ϕ j for all i [1, k], i is a model of i and χ ι(i) for all i [0, n]. Oserve that, regarding data complexity, W and ι can e guessed in constant and linear time, respectively. [4, Lem. 4.12] show that the LTL satisfiaility prolem w.r.t. a given W and ι can e decided in polynomial time. However, regarding r-satisfiaility, [4] only show memership in ExpTime. The critical point with r-satisfiaility is the requirement that the interpretations for the n + k + 1 relevant time points share a common domain, so the individual satisfiaility tests have to e done together. The trivial approach is to rename the flexile names for all i [0, n + k]. This requires however that the ontology is extended y corresponding axioms; that is, it grows with the data and impacts complexity. 3 Characterizing r-satisfialility We regard a TCQ Φ, a TKB K = O, ( i ) 0 i n in DL-Lite H krom, a set W 2 {p1,...,pm} such that W = {W 1,..., W k }, and a mapping ι: [0, n] [1, k], as descried in the previous section. The goal is to propose a characterization of r- satisfiaility of W w.r.t. ι and K which, in contrast to existing characterizations, is tailored to DL-Lite H krom and shows that the r-satisfiaility prolem is in NP. Oserve that the functions of the shared domain in the definition of r- satisfiaility are mainly two: (i) to synchronize the interpretation of rigid symols regarding the named individuals; (ii) to guarantee that the satisfiaility of the conjunctions χ i, i [1, k], which is represented y the respective interpretation J i, is not contradicted y the interpretation of the rigid names in the other interpretations, especially in no i with i [0, n]. The idea is to look for similar interpretations 0,..., n, J 1,..., J k, ut to not require a shared domain: ased on K, W, and ι, we specify a polynomial amount of additional data that hence can e guessed in polynomial time, which captures knowledge restricting the interpretation of the individual and rigid names and simulates the shared domain; the additional data then allows us to check the conditions for r-satisfiaility for each of the interpretations independently of the other interpretations nondeterministically, in polynomial time. Without loss of generality, we can restrict our focus to certain canonical interpretations, ased on the standard chase [8]; we introduce elements of the form u a1... l, a N, 1,..., l N. Note that we also apply this general approach for DL-LiteH horn and EL in [6, 7], ut we do not have to deal with nondeterminism there; that is, there is only one canonical interpretation for a KB. Definition 1 (Canonical nterpretation). Let K = O, e a consistent DL-Lite H krom knowledge ase. For all N C and P N, define: 0 := {a (a) }, P 0 := {(a, ) P (a, ) } {(a, u ap ) P (a) } {(u ap, a) P (a) }.

7 Then, iterate over all i 0: for all X N C N define X i+1 := X i ; apply one of the following rules for all N C, P N,, S N, and B, C B(O); and increment i; (d, e) (P ) i denotes the fact that (e, d) P i, and d ( ) i denotes the existence of an element e such that (d, e) i : f B O and e B i, then add e to i+1. f B O and e B i : if e N (), then add (e, u e ) to i+1 ; if e = u ϱ, then add (e, u ϱ ) to i+1. f O, (d, e) i, then add d to i+1. f S O and (d, e) i, then add (d, e) to S i+1. f B C O, e B i, e C i, and every other rule (i.e., for a C without negation) that applies to e or a tuple containing e has een applied in a step j < i, then add e to B i+1 or C i+1. The set K u collects the aove introduced new elements. canonical interpretation K for K is then defined as follows ased on such a sequence of rule applications, for all a N (), N C, and P N : K := N () K u, a K := a, K := i, P K := i=0 P i. i=0 Note that the assumptions in Section 2 aout the additional axioms in the ontology ensure that, whenever there is a named individual a ( ) i for some i 0, then a has an -successor of the form u a in the corresponding canonical interpretation, and similar for unnamed elements. We denote the restriction of a canonical interpretation to a named individual a and its unnamed successors y a. f K is consistent, then there is a canonical interpretation for K that is a model of K. We denote the set of all those canonical models y K. n what follows, we specify the additional data to overtake the Functions (i) and (ii). Specifically, we define a set of Boxes containing assertions that (i) (largely) fix the interpretations on the named individuals (i.e., relations to unnamed successors are not fully taken into account yet), and (ii) ensure oth that the positive CQ literals are satisfied as required and that the negative CQ literals are not satisfied if they must not. For simplicity, for i [1, k], we define n+i := and extend ι such that ι(n + i) := i. For the synchronization of the named individuals, we use name-boxes, which are similar to the Box types defined for DL-Lite H horn in [6], ut we include flexile symols. The idea is to then guess n + k + 1 name-boxes and require them to agree on the rigid assertions. Definition 2 (Name-Box). name-box for a set of individual names w.r.t. O is a set of assertions formulated over and all symols in B(O) and N (O) such that α iff α /. Second, for all i [0, n + k], define Q i := {ϕ j p j W ι(i) }. Let the set N aux N contain an individual name a i x for each i [0, n + k] and each variale x occurring in a CQ in Q i. Note that, ecause of our assumption that the

8 CQs in Φ have no variales in common, each a i x N aux can e unamiguously associated to a CQ containing x. Then, Qi denotes the Box otained from Q i y instantiating all variales with the corresponding names from N aux. We use these Boxes to ensure that the positive CQ literals are satisfied as required. While the former is similarly done in [6, 7], the nondeterminism allowed in DL-Lite H krom requires a more careful construction of the unnamed parts of 0,..., n, J 1,..., J k (i.e., since they have to satisfy all Cs of the form B C in O, we have to specify them correspondingly): we must ensure that the interactions of 0,..., n, J 1,..., J k in those parts, which are caused y the rigid names, do not lead to the satisfaction of some ϕ j Q Φ in some J i ( i ) although we have that p j W i (p j W ι(i) ). The idea for the construction of 0,..., n, J 1,..., J k is to not consider aritrary trees of unnamed successors for all the individuals in all the interpretations, ut to define prototypical ones whose size is constant in the data, that fix the interpretations, and which we then copy for all named individuals that are sufficiently similar. To this end, we define types, which are generally independent of the data; for every interpretation and individual name, there is however exactly one type characterizing the former on the latter. type captures the asic concepts satisfied on a name and, in particular, relevant homomorphisms of CQs from Q Φ w.r.t. the named individual and its unnamed successors; in particular, it does not explicitly refer to individual names. temporal type is a set of types. The idea is to consider prototypical trees of unnamed successors for each temporal type as additional data: we use a set of prototypical tree-boxes (one per type) over the same names, which agree on the interpretation of the rigid names, and are such that every Box represents some interpretation on the unnamed successors that fits to the respective type. For instance, if a type specifies a CQ ϕ Q Φ to e not satisfied (w.r.t. the unnamed successors), then ϕ is not satisfied in the Box. Our main contriution is that we show that such tree-boxes whose size is independent of the data do exist in the case of r-satisfiaility and that we can assemle the interpretations 0,..., n, J 1,..., J k from these Boxes: for every individual name a and all i [0, n + k], we guess a type T a,i a polynomial amount of information that represents i (or J i ) on a; the set of all these types for a is a temporal type and yields the prototypical successors to choose; that is, we copy the elements in the corresponding set of tree-boxes and then specify i (or J i ) on these elements according to the Box for T a,i. Oserve that we use finite Boxes, which means that every of the Boxes contains enough information to define the interpretations on other required successors (i.e., we may copy the elements several times). Since the Boxes capture all the rigid information from other time points, this allows us to test the satisfaction of the negative CQ literals for every of the n + k + 1 interpretations individually. Definition 3 (Type). asic type is a set B C(O) such that B B iff B B for all B B(O); given such a asic type, the corresponding set of assertions is defined as B (a) := {D(a) D B}. The asic type of an individual name a in an interpretation is the set BT(a, ) := {D C(O) a D }.

9 type is a triple (B, M, Q) with a asic type B, a set M ϕ Q Φ 2 N T(ϕ) of term sets, and a set Q Q Φ of CQs. The type of an individual name a in a canonical interpretation is the triple T(a, ) := (BT(a, ), M, Q) where Q Q Φ contains exactly the CQs that are satisfied in a, and M contains all sets S of terms for which there are a CQ ϕ Q Φ and a partial homomorphism π : N T (ϕ) a of ϕ into a with a range(π) and dom(π) = S. temporal type is a set of types. We assume every temporal type τ to e an ordered set and use τ i to refer to the i-th type in τ. We denote the set of all temporal types y T. t is left to specify the prototypical tree Boxes of unnamed successors for a given temporal type τ. We first construct Boxes for the types and, in a second step, ensure that the conditions specified y the types are satisfied in them, respectively. These Boxes ( τi ) 1 i τ, initially empty, are constructed iteratively, ased on canonical interpretations, amongst others for these Boxes. During the iteration, we therefore assume that these interpretations are (nondeterministically) extended correspondingly (i.e., to cover the new elements of the Boxes). Let s = τ. For all i [1, s], consider some τi O,τi Bi () ; is a fresh individual name and B i the asic type in τ i. Our procedure takes the sequence ( τi ) 1 i s as input. t then repeatedly iterates over the (extended) interpretations and extends the Boxes ( τi ) 1 i τ until nothing changes any more. Example 1. We consider an ontology containing the inclusions B, B C,, C S,, where only is rigid. Let further τ = 3 and the input canonical interpretations for τ 1, τ 2, τ 3 e as follows. u u u C S u S u u u u Note that all elements that are no instances of or C instantiate B, is dotted. fter one iteration over the interpretations, the Boxes τi for i [1, 3] are: a a a S 2 a a S 2 a 1 a 1 C S a 1 3 a a 3 a 1 a 3 3 a τ1 is otained from τ1 y introducing names representing the unnamed elements, flexile roles get a superscript. ll names and the rigid assertions from τ1 are then added to the other Boxes, and the interpretations τ2 and τ3 are extended correspondingly. The aove τ2 is then otained from this τ2,

10 assumed to e as elow. Then, again, all names and rigid assertions from τ2 are added to the other Boxes, the interpretations are extended, and the aove τ3 is otained from the extended τ3, depicted elow, where c = a 1. a a a S 2 u C S u S u u u u a 1 u c u c a 1 Note that we do not depict all -successors; according to Definition 1, all elements instantiating must have such successors. We lastly show τ1 and τ3 extended for the aove τ3, where d = a 3, e = a 1 3, and f = a 3 3. Oserve that we assume that τ3 interprets u d in the same way as u, oth u d and u f according to u, and u e according to u c. a a S 2 a a S 2 u a 1 3 a a 3 a 1 u a 3 3 a u c u e u u c a 1 3 a a 3 a 1 u a 3 3 u u d a u u d n order to ensure that the size of the tree-boxes is finite, we specify a termination criterion ased on the maximal size m := max{ ϕ ϕ Q Φ } of a single of the CQs: we stop the introduction of new elements a ϱϱ ω with ϱ, ω > m if those would, thereafter, occur in a sutree (of depth > m) with root a ϱϱ that would e a copy of an already existing sutree of depth m with root a ϱ. This approach is correct if we extend the Boxes in a readth-first fashion and, especially, regard all of the canonical interpretations efore extending the trees one level deeper. We now specify the procedure TreeBox, which takes τ, the interpretation sequence ( τi ) 1 i s, and as input: For each domain element u ϱ of τi, introduce an individual name a ϱ if N, and otherwise a ϱ i; we assume that such individual names and role names containing superscripts do not occur in K. Similarly, for each domain element u cϱ of τi such that c = a σ N tree, introduce a new individual name a σϱ if N and otherwise a σϱ i. Let := N { i N \ N, 1 i s}. The set Ntree collects the new individual names, ut (*) a name a υ is only added if there are no names u f

11 a ϱ, and a σ in τ such that σ = ϱϱ ; υ = σσ with σ > m; and, for all ω m, a ϱω N tree iff a σω N tree, and a ϱω and a σω have the same asic type in any of the interpretations τl, l [1, s]. For the next step, we capture this relation using the function ν : τ i u N tree : for the aove elements u ϱ, define ν(u ϱ ) := a ϱ (i) and, for the others, define ν(u cϱ ) := a σϱ (i). Oserve that this function may map several elements to the same name; as it is the case for u and u d in Example 1, where we have ν(u ) = a 3 3 and ν(u d ) = a 3 3. n these cases, we assume that the unnamed successors of elements from N tree, such as u d, are interpreted in the same way as the original unnamed elements, such as u, for which ν was defined first (i.e., this must have happened when a was introduced, hence u d did not yet exist); and that the successors of the former are interpreted in the same way as the corresponding successors of the latter, and so on. For every a N tree introduced in the previous step, let u σ τ i u e one of the elements for which a was created. dd the following assertions to τi : for every B B(O) such that u σ B τ i, the assertion B(a); for every S N such that (σ, u σ) S τ i, the assertion S(σ, a); for every S N such that (u σ, u σ ) S τ i, the assertion S(ν(u σ ), a). dd all of the individual names and the rigid assertions added to τi also to all other τj, j [1, s]. 5 For every a N tree and B B(O) such that a B τ i, further add the assertion B(a) to τi. gain, add the rigid assertions to all τj, j [1, s]. The procedure outputs the sequence ( τi ) 1 i s. egarding the last item, note that it covers those names that were introduced for unnamed elements in other canonical interpretations. For them, we only have to explicitly capture the nondeterminism, w.r.t. asic concepts. ll relations on the corresponding named individuals are completely determined y the assertions added in the item efore (i.e., including those added in other iteration steps, maye for some τj with j i). s mentioned aove, we second have to ensure that the Box created for a type also satisfies the conditions specified y it. Definition 4 (Tree-Box). n Box produced y TreeBox given a name N as input is a tree-box for a type τ i = (B, M, Q) if: O, B () is consistent; for all ϕ Q Φ, ϕ Q iff O, B () = ϕ; for all S ϕ Q Φ 2 N T(ϕ), S M iff there are a CQ ϕ Q Φ and a partial homomorphism π : N T (ϕ) N () of ϕ into with range(π), S = dom(π). Because of (*), the sizes of the tree-boxes are finite, and they are independent of the data: given v := max{ N T (ϕ) ϕ Q Φ }, the maximal numer of terms occurring in one of the CQs, and t 2 B(O) + Q Φ +v Q Φ, the numer of possile types, the depth of how far we specify the prototypical trees is ounded y d := B(O) t N (O) t m + m; this follows from the facts that the names in N tree 5 ll names can e added, for example, y assuming to e rigid.

12 are uilt from elements of N (O), sometimes with suscripts from [1, t], and that we consider these names in t different interpretations. Finally, the additional data is a tuple as follows, polynomial in the data: (( i) 0 i n+k, (T a,i ) a N (K) N aux, ( τ,i ) τ T, ), where, 0 i n+k 0 i τ i is an name-box for N (K) N aux for all i [0, n + k], and all i contain the same rigid assertions; T a,i is a type for all a N (K) N aux and i [0, n+k], and T (1) a,i = BT(a, i ). ( τj ) 1 j τ is the result of applying TreeBox to τ, some sequence of corresponding canonical interpretations, and a fresh name for all τ T; and τi is a tree-box for τ i for all i [1, τ ] if there is a name a N (K) N aux such that τ = 0 j n+k {T a,j}. For names a, N and a type τ i, let τi [/a] e the Box otained from a tree-box τi y replacing every y a, also within individual names. For a tuple t as aove, we define the Box t tree,i for i [0, n+k] as the set that contains, for all a N (K) N aux, all assertions from τj [/a], where τ = {T a,i 0 i n+k} and τ j = T a,i. t tree denotes the Box that contains only the rigid assertions from t tree,i, for all i [0, n + k], and all names occurring in these Boxes6 ; y construction, all of the latter Boxes agree on those. Lemma 1. W is r-satisfiale w.r.t. ι and K iff there is a tuple t = (( i) 0 i n+k, (T a,i ) a N (K) N aux, ( τ ) τ T ), 0 i n+k as specified aove such that, for all i [0, n + k]: (C1) K i := O, i i Qι(i) t tree,i is consistent and, (C2) for all p j W ι(i), we have K i = ϕ j. Proof. ( ) s outlined in Section 2, we can consider the n+k+1 interpretations from the definition of r-satisfiaility integrated within a single interpretation if we rename the flexile symols accordingly. The advantage of this approach is that we can, w.l.o.g., assume that this interpretation is a canonical interpretation, which is not possile with the single interpretations from the definition of r- satisfiaility ecause of the shared domain. Hence, for every i [0, n + k + 1] and every flexile name X in N C (O) N (O), we introduce a fresh name X (i) called the i-th copy of X. f X is a more complex expressions (an axiom, CQ, or conjunction of CQ literals), X (i) is otained y replacing every occurrence of a flexile name y its i-th copy. By [4, Lem. 4.14], W is r-satisfiale w.r.t. ι and K iff the conjunction χ W,ι of CQ literals has a model w.r.t. O W,ι,, where: χ W,ι := 1 i k χ (n+1+i) i 0 i n χ (i) ι(i), χ(i) := O W,ι := {α (i) α O, 0 i n + k}, := ϕ (i) j p j W i n 1 {α (i) }. 0 i n, α i 6 s aove, this can e ensured, for example, y assuming to e rigid. ϕ (i) j, p j W i n 1

13 For simplicity, we often focus on some i [0, n + k] and consider the i-th copies of concept and role names as the (original) flexile names and disregard all other copies (ut not the rigid names); in the following, we refer to those parts of, in which the signature is smaller and renamed, as i. Similarly, we consider T(a, i ) to e the type of a in for i [0, n + k], and may refer to the set of all these types as the temporal type of a in. We then can define the components of the required tuple t for i [0, n + k] and c N (K) N aux easily: T c,i := T(c, i ) and i := {( )B(a) a N (K) N aux, B B(O), a B i (a B i )} {( )(a, ) a, N (K), N (O), (a, ) i ((a, ) i )}. For all τ T, we choose an aritrary a N (K) N aux with temporal type τ in ; for each i [1, τ ], define τi to e some j a with τ i = T(a, j ); and then define ( τi ) 1 i τ := TreeBox(τ, ( τi ) 1 i τ, a); if there is no such element a, the Boxes are empty. We can assume the latter algorithm to iterate only once over the types ecause the interpretations of the new names in all the interpretations are given already, y the interpretations of the elements for which they were introduced, respectively (i.e., we do not have to extend interpretations). t is easy to see that the tuple is as required that is, i represents the model of K if i ( ) we assume to e such that (1) the trees of unnamed successors for two names a, N (K) N aux that have the same temporal type in are isomorphic w.r.t. the rigid symols and that (2) it interprets those successors the same in i and j if T(a, i ) = T(a, j ). This is the case ecause t tree,i then is trivially satisfied for all i [0, n + k]: the interpretations that can e selected for the construction of a prototypical Box τ then all are isomorphic; that is, the construction neither depends on the name chosen as prototype for the temporal type nor on the indexes i whose interpretation i is chosen as prototype for a type. We lastly show that ( ) is a valid assumption. The proof is y contradiction. We hence assume that such a given model cannot e simplified in the descried way without loosing the property that it satisfies oth the KB and the conjunction of CQ literals. Let J e a corresponding adaptation of such a model, constructed as follows: For every temporal type τ for which there is an individual name a such that τ is the temporal type of a in, and for every j [1, τ ], select one index i j [0, n + k] such that τ j = T(a, ij ), and let τ,j e the (possily infinite) set of assertions representing ij on all unnamed successors of a. That is, τ,j covers all the successors contained in, ut it only descries w.r.t. the rigid names and the i j -th copies of flexile names. dapt as follows. For every a N (K) N aux with temporal type τ in, replace all unnamed successors y copies of the elements occurring in τ,j (for an aritrary j [1, τ ]). For every i [0, n + k], the interpretation of the i-th copies of names on these elements in J is then given y the corresponding flexile assertions in the one set τ,j for which we have τ j = T(a, i ). The interpretation of the rigid names is also given y these Boxes since they all agree on those

14 names. s with, we use interpretations J i to refer to J on the rigid names and on the i-th copies; that is, we consider the i-th copies as the (original) flexile names and disregard all other copies. The construction of J maintains the types, meaning that T(a, J i ) = T(a, i ) for all a N (K) N aux, i [0, n + k]. Let (B, M, Q) := T(a, i ), (B, M, Q ) := T(a, J i ). Note that, to replace the unnamed successors of a in, we chose unnamed successors of some N (K) N aux of the same temporal type as a in ; and that the interpretation in J i on a and its unnamed successors is given y the one on and its unnamed successors in some j with T(, j ) = T(a, i ). Clearly, for every homomorphism of some ϕ Q into J i a, there is a corresponding one into j if a does not occur in ϕ, y the definition of J i. This especially holds ecause the interpretation of the rigid symols on a and its unnamed successors in the whole interpretation J is fully determined y the interpretation of the rigid symols in j. f a occurs in ϕ, then the definition of T(a, i ) yields {a} M, and T(, j ) = T(a, i ) hence implies a = y the definition of T(, j ). The construction of J i a then, as efore, yields Q Q. For the other direction, we consider a homomorphism of some ϕ Q into i. But then we also have one of ϕ into j which, in turn, yields a corresponding one into J i. M = M follows y analogous arguments. We show B = B. Since we do not adapt the interpretation of the concept and role names w.r.t. only named individuals, it is left to focus on asic concepts of the form. The only critical case is thus the one where there is an element u a in ut not in J and a has no named -successor (in oth and J ) either; that is, there is no element u in, ut has a named -successor (again in oth and J ) y BT(a, i ) = BT(, j ). However, note that we assume the ontology to contain all Cs of the form. Hence, the element u must exist in y Definition 1. By construction, we thus get that u a exists in J, which contradicts the assumption. The case where an element u a exists in J ut not in is not critical w.r.t. possile changes of the asic type. Since we do not adapt the interpretation of the concept and role names w.r.t. only named elements, J =. egarding the named elements, J also satisfies O W,ι since the adaptation retains the asic types. egarding an unnamed element e, oserve that it is valid to argument ased on the single interpretations J i, i [0, n + k], instead of on J as a whole ecause the ontology contains no axioms where different kinds of copies occur in and J i represents the interpretation of rigid names in J. The interpretation of e in such a J i corresponds to the interpretation of an isomorphic element in some j, j [0, n + k]. But we have = O W,ι, and O W,ι contains all copies (i.e., also the j-th) of all axioms in O. So J i satisfies all axioms in O, which yields J = O W,ι. J also satisfies all CQ literals ϕ (i) satisfied in. This is clear if only named individuals are considered ecause and J agree on the interpretation of concept and role names w.r.t. only named individuals and the asic types are maintained. For the other cases, oserve that we can argument ased on single interpretations i /J i, again, since the copies of the CQs only contain one kind of copies of names, and all these interpretations for /J agree on the rigid symols. We consider the case where

15 there is a homomorphism that maps to maximally one named individual. ll the unnamed elements in the range must form a tree structure with a named individual a N (K) N aux as root. ssuming T(a, i ) = (B, M, Q), we must have ϕ Q y Definition 3. From T(a, i ) = T(a, J i ), we get that ϕ (i) also in J is satisfied ased on a and its unnamed successors. f there is a homomorphism π mapping to oth several named and unnamed elements, then a corresponding homomorphism π into J can e otained ased on the type. By Definition 3, for all a (N (K) N aux ) range(π), there is a set of terms V a in T(a, i ) (2) containing exactly the terms t N T (ϕ) with π(t) = a or π(t) eing an unnamed successor of a. From T(a, J i ) = T(a, i ), we get V a T(a, J i ) (2), which means that there must e a corresponding partial homomorphism π a of ϕ into J i, again y Definition 3. Now, we define π : (i) π (t) := π(t) if π(t) N (K N aux ). (ii) π (t) := π a (t) for all a (N (K) N aux ) range(π) and t V a. egarding the negative CQ literals, we proceed y contradiction and assume that J = ϕ (i) j for some p j W ι(i). gain, we can consider single interpretations i /J i. Since we do not adapt the interpretation of the names on only named elements the corresponding homomorphism π must map to unnamed elements. t can neither map to only unnamed elements and maximally to one named element: the arguments correspond to those given aove for the corresponding case for the positive CQ literals if i and J i and and J are switched, respectively. The same holds for the case where π maps to several named and to unnamed elements. ( ) We regard a tuple t that satisfies the conditions from Lemma 1 and construct interpretations 0,..., n, J 1,..., J k as required ased on the KBs in Condition (C1), which contain all the necessary information (i.e., K i yields i for i [0, n] and J i for i [n + 1, n + k]): we have the name-boxes for the named individuals in the TKB, and t tree,i for all i [0, n+k] for those successors (up to some depth d d) of the latter individuals that must exist given O. The shared domain of our interpretations contains all of the names occurring in these KBs and additional unnamed elements, to include the required successors of depths greater than d. The idea is to inductively introduce elements y continuing the repetition we have given the construction of the tree-boxes. The ontology and the Boxes i for i [0, n + k] are clearly satisfied y the respective interpretations, since the KBs K i are consistent and our interpretations are completely defined ased on canonical ones (i.e., the ones used to construct the Boxes t tree,i ; these Boxes descrie the interpretations completely in the sense that they capture all the necessary rigid knowledge and nondeterministic decisions). The additional Boxes Qι(i) ensure that the positive CQ literals are satisfied as required. By contradiction, it can easily e shown that no negative CQ literal is satisfied. By (C2), a corresponding homomorphism π must map to unnamed elements and, since we untangle the interpretations during the construction of t tree,i to depth 2m, it cannot map to elements in N (K) N aux. This yields a contradiction to (C2) ecause there are isomorphic named elements for all those π maps to, since the repeating trees are of depth m. egarding complexity: W and ι can e guessed in polynomial time, and the LTL satisfiaility testing w.r.t. a given W and ι can e done in polynomial time

16 [4, Lem. 4.12]. Similarly, the tuple from Lemma 1 can e guessed in polynomial time; note that the tests if the Boxes for the temporal types are tree-boxes are data independent. Since oth KB consistency and CQ non-entailment are in NP in DL-Lite H krom [1, Thm. 8.2], (C1) and (C2) can e decided nondeterministically in polynomial time. TCQ satisfiaility is thus in NP [4, Lem. 4.7]. Theorem 1. TCQ entailment regarding a TKB in DL-Lite H krom is in co-np. The expressivity of TCQs allows to reduce TCQ entailment in much more expressive DLs to TCQ entailment in DL-Lite H krom [6], without an impact on the data. For example, Cs 7 as. 1 2 and can e encoded using TCQs xy.(x, y) 1 (y) 2 (x) and x. 1 (x) 2 (x) 3 (x) if the ontology is extended y the Cs 2 2 and 3 3. This yields: Corollary 1. TCQ entailment regarding a TKB in LCH is in co-np. 4 Conclusions We have shown that the data complexity of TCQ entailment w.r.t. temporal knowledge ases in expressive DLs is in co-np, even if rigid symols are considered. This result is interesting since already standard conjunctive query entailment is co-np-hard, which means we get the temporal features for free. Yet, it remains to design deterministic algorithms to translate the result into practice. eferences 1. rtale,., Calvanese, D., Kontchakov,., Zakharyaschev, M.: The DL-Lite family and relations. Journal of rtificial ntelligence esearch 36, 1 69 (2009) 2. Baader, F., Borgwardt, S., Lippmann, M.: Temporalizing ontology-ased data access. n: Proc. of CDE. pp (2013) 3. Baader, F., Borgwardt, S., Lippmann, M.: Temporal conjunctive queries in expressive description logics with transitive roles. n: Proc. of. pp (2015) 4. Baader, F., Borgwardt, S., Lippmann, M.: Temporal query entailment in the description logic SHQ. Journal of We Semantics 33, (2015) 5. Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handook: Theory, mplementation, and pplications. Camridge University Press, 2 edn. (2007) 6. Borgwardt, S., Thost, V.: Temporal query answering in DL-Lite with negation. n: Proc. of GC. pp (2015) 7. Borgwardt, S., Thost, V.: Temporal query answering in the description logic EL. n: Proc. of JC. pp (2015) 8. Deutsch,., Nash,., emmel, J.B.: The chase revisited. n: Proc. of PODS. pp (2008) 9. Pnueli,.: The temporal logic of programs. n: Proc. of FOCS. pp (1977) 10. Poggi,., Calvanese, D., De Giacomo, G., Lemo, D., Lenzerini, M., osati,.: Linking data to ontologies. Journal on Data Semantics 10, (2008) 11. Thost, V.: Using Ontology-Based Data ccess to Enale Context ecognition in the Presence of ncomplete nformation. Ph.D. thesis, TU Dresden (2017) 7 The syntax and semantics of many DLs extending DL-Lite H krom are descried in [5].