The original proof again targets TCQs but similarly applies in our more general setting.
To sum up, the TQ satisfiability problem (w.r.t. a TKB) is reduced to two separate ones in [BGL12; BBL15b]: one in LTL and one or several “atemporal” ones in DL—
assuming that the set W and mapping ι, which link the two parts, are given. The intuition is thatW contains exactly the worlds occurring in the LTL model Wand that ι designates the worlds of the first n+ 1 time points. Each world induces a set of QL query literals. The DL part then checks whether these sets are consistent (w.r.t. certain classical KBs), to ensure that a DL-LTL structure satisfying the QL queries (and the TKB) according to Wcan indeed exist.
3.3 Problem Analysis and Technical Contributions
In this work, we focus on the temporal query languages and reasoning problems intro-duced in this chapter. We consider various different settings:
• We investigate the combined complexity of satisfiability in DL-LTL, both com-bined and data complexity of TCQ entailment, and rewritability of temporal query answering for various query languagesQL.
• In all three parts, we considerELand differentDL-Litefragments as instantiations of DL. In the last chapter, we also regard various other lightweight logics.
• We consider different settings w.r.t. the rigid symbols and distinguish if no rigid names, only rigid concept names, or both rigid concept and role names are allowed.
For solving satisfiability and entailment, we generally apply the approach presented in the previous section (see Lemma 3.13). That is, for deciding the DL-LTL and the TCQ satisfiability problem, we focus on the following three problems of
(i) obtainingW and ι,
(ii) solving the LTL satisfiability test (t-satisfiability), and (iii) solving the DL satisfiability test(s) (r-satisfiability).
Regarding the choice of methods to obtain W and ι for (i), we have that each can be obtained by enumeration, guessing, or direct construction, depending on the complexity class we target. The size ofW is exponential in that of the considered query Φ, and the size of the mappingιis linear inn, but there are exponentially many possible such map-pings. Note that, in the special case where we have the trivial ABox sequence (∅)0≤i≤n, a mapping ιcan always be obtained easily if there is a set W that satisfies all but the last conditions of t-satisfiability and r-satisfiability, respectively. To see this, observe that W must not be empty, and that the last condition in Definition 3.12 requires the worldWι(0), an element ofW, to be consistent w.r.t. the knowledge basehO,∅i, which is true for every world in W ifW satisfies the other conditions in that definition. Given that the other conditions are met, the last condition in Definition 3.11 can thus easily be satisfied by defining ι: [0,0]→ [1, k] such that it maps 0 to the index of the world from W that exists because of the first condition. We state that as a fact for future reference.
Fact 3.15 For a TQ Φ, TKB hO,(∅)0≤i≤ni, and set W that satisfies all but the last conditions in Definitions 3.12 and 3.11, there exists a mapping ι: [0, n] → [1, k] such that the remaining conditions are also satisfied.
Problem (ii) is generally independent of the query languageQL and DLDL. More-over, it can be solved in exponential time w.r.t. combined complexity and in polynomial time w.r.t. data complexity, by using a reduction to the emptiness of a B¨uchi automa-ton [BBL15b, Lem. 4.12].
Regarding Problem (iii), we have two characterizations of r-satisfiability, Defini-tion 3.12 and Lemma 3.14. The critical point with the former is the requirement that the interpretations J1, . . . ,Jk,I0, . . . ,Inshare a common domain, because the satisfia-bility tests for different interpretations cannot be done independently of each other, if rigid symbols are considered. Observe that J1, . . . ,Jk characterize the satisfiability of the conjunctions χi w.r.t.hO,∅i,i∈[1, k], respectively. Moreover, the functions of the common domain are mainly two, which gets more evident if the additionally required respect of rigid symbols is considered:
(F1) Synchronize the interpretation of rigid symbols regarding the named individuals.
(F2) Guarantee that the satisfiability of the conjunctions χi, i∈[1, k], which is repre-sented by the respective interpretations Ji, is not contradicted by the interpreta-tion of the rigid names in the other interpretainterpreta-tions.
Problem (iii) thus depends onQLandDL, and especially on which symbols are allowed to be rigid. This can be similarly seen by considering Lemma 3.14. If rigid names are not considered, then the k conjunctionsχ(i) with i∈[1, k] in the conjunction χW,ι
do not share any symbols. That is, their satisfiability can be tested independently of each other. This is important considering the fact that the size of χW,ι depends on the exponential number k.
Lemma 3.16 ([BGL12, Lem. 5.1])Let χ1, . . . , χ` be conjunctions ofQLquery liter-als,O1, . . . ,O`be ontologies, andA1, . . . ,A` be ABoxes such that elements with different index i∈[1, `]do not share concept and role names. Then the conjunction χ1∧ · · · ∧χ`
is satisfiable w.r.t. the KB hS1≤i≤`Oi,S1≤i≤`Aii iff, for each i∈[1, `],χi is satisfiable w.r.t. hOi,Aii.
Note that [BGL12, Lem. 5.1] refers to the language QLof ALC axioms, but the result obviously also holds in our more abstract setting.
Rigid names, on the other hand, generally require the consideration of the whole conjunction. We thus get only rather high upper bounds from Lemma 3.14.
Lemma 3.17 If the satisfiability problem for conjunctions of QLquery literals w.r.t. a DLKB is contained inNP, then the satisfiability problem for temporalQLqueries w.r.t.
a DL TKB is contained in NExpTime(NP), and the entailment problem is contained in co-NExpTime (co-NP), w.r.t. combined (data) complexity.
Proof. We focus on the satisfiability problem and regard the conditions in Lemma 3.13.
Given the above observations, (i) both the set W ⊆ 2{p1,...,pm} of worlds and the map-ping ι: [0, n]→ [1, k] can be nondeterministically guessed in exponential (polynomial)
3.3 Problem Analysis and Technical Contributions
time, and (ii) t-satisfiability can be decided in exponential (polynomial) time in the input (data). Regarding (iii) r-satisfiability, we have that χW,ι is of size exponential in
|Φ|because of the firstk conjuncts. The size of each of the other nconjuncts is log(n) times (i.e., for the representation of the index) the sum of a number linear in |Φ|, for the conjuncts χ(k+i+1), i∈[0, n]. The size of OW,ι is exponential in |Φ|, and linear in
|O|and n. The size of Ais log(n) times the sum of all |Ai|,i∈[1, n], and thus polyno-mial in the data. Given the assumption (and the fact that |Φ|and |O|are constants), the satisfiability of the conjunction in Lemma 3.14 can be tested in nondeterministic exponential (polynomial) time. TQ satisfiability is thus contained inNExpTime(NP), and entailment in co-NExpTime(co-NP).
The above observations show that, in many cases, we cannot directly follow the ap-proach of [BGL12; BBL15b] because we target considerably lower complexity results.
While the size ofι is only an obstacle for designing algorithms of sublinear complexity, the exponential size ofW makes it impossible to guess (and store) this set by using only a polynomial amount of space. And known results only allow to solve Problems (ii) and (iii) in exponential time. The trivial approach of Lemma 3.13 does hence neither provide an upper bound of PSpacew.r.t. combined complexity, the lower bound we have from LTL satisfiability, nor tractable data complexity, if rigid symbols are considered. Yet, forELand the Horn fragments ofDL-Lite, we usually have such reasoning complexities.
Indeed, we obtain suchPSpaceand tractability results, even in some settings with rigid symbols. More specifically, our contributions are as follows:
• We propose a new, general procedure for solving the TQ satisfiability problem in polynomial space w.r.t. combined complexity based on adapting the algorithm originally proposed for solving the LTL satisfiability problem, which is specified in Algorithm 2.1. Recall that the latter algorithm iteratively constructs a model for a given LTL formulaϕ: it considers a sequence of exponentially many time pointsi, iteratively regards each of them, guesses a world wi, and describes polynomial-space tests ensuring the adequacy of wi (i.e., regarding the worlds guessed in previous steps) and relying on a polynomial amount of information that is kept and updated during the iteration. We extend that algorithm to iteratively con-struct a model for a given TQ w.r.t. a TKB. That is, Problems (i) and (ii) are solved in general for Φpa, independently of QL and DL; specifically, we use the worldwiguessed in the original algorithm7 to determine the elementW ofW that represents theQLqueries satisfied at i, and ι.8 But our procedure still has to be tailored to the specific problem under consideration (e.g., TCQ entailment in EL) regarding Problem (iii). Our approach is sketched in Algorithm 3.1. The high-lighted parts represent the critical extensions of Algorithm 2.1 (i.e., extensions that may influence correctness or complexity); the functions which target the r-satisfiability testing are specified later in this work for each considered problem individually.
7Recall thatwiis determined byFpres.
8To determineιin this way, we have to ensure that the firstn time points are considered. This can be done by guessing the starts of the period such that it is large enough. We also have to require that Φpa∈ Fpreswheni=ninstead of wheni= 0.
Algorithm 3.1: Procedure for Deciding TQ Satisfiability Input: TQ Φ, TKBK hO,(Ai)0≤i≤ni
Output: true if Φ is satisfiable w.r.t. K, otherwisefalse
1 d:= GUESSDATA(Φ,K)
2 i:= 0
3 s:= Guess a number ≤2|Φpa|,>0, and such that s≥n
4 p:= Guess a number≤4|Φpa|
5 Fnext :=∅,Fs:=∅,FU :=∅
6 Fpres:= Guess a subset of Clo({Φpa})
7 if not CONSISTENT(Fpres) ornot INITIAL(Fpres) then
8 returnfalse
9 while i≤s+p do
10 Update Fpres and Fnext, and proceed as in Algorithm 2.1 for input Φpa, Lines 9 to 17
11 if i=n and Φpa 6∈ Fpres thenreturnfalse
12 W :=Fpres∩ {p1, . . . , pm}
13 if i > nthen Ai:=∅
14 if not TESTRSAT(Φ,O,Ai, d, i, s, p, W) then
15 return false
16 i:=i+ 1
17 Continue as in Algorithm 2.1, Line 19
In particular, this r-satisfiability testing has to be done as follows for obtaining a concrete, polynomial-space algorithm:
– It can rely on a polynomial amount of additional data dwhich is guessed in the beginning and kept during the iteration.
– It must be done in tests that require only polynomial space and, during the iteration, may consider only one world W—different from the test described in Lemma 3.14, which considers the exponential set W as a whole.
Corollaries 4.5, 4.22, 5.16, and 6.19 are obtained based on this new approach.
• The characterizations of r-satisfiability by such a polynomial amount of informa-tion and condiinforma-tions that can be tested by using only polynomial space presents another main contribution of our work. This is because we also target low com-plexities w.r.t. rigid symbols, regarding satisfiability in EL-LTL with global GCIs (Section 4.4.2), and TCQ entailment in EL (Section 5.1) and Horn fragments of DL-Lite(Section 6.1). The common idea of our characterizations is to adapt Def-inition 3.12: we look for similar interpretations but consider their domains to be
3.3 Problem Analysis and Technical Contributions
disjoint w.r.t. the unnamed individuals; and we use the additional data and test conditions to achieve the effects of the shared domain (Functions (F1) and (F2)).
• For obtaining tractable data complexity regarding TCQ entailment in EL and Horn fragments of DL-Lite, we also integrate the tests described in Lemma 3.13 into the construction ofW andι. The main achievement represents the algorithm we propose for DL-Lite; there, we include rigid symbols and target the sublinear ALogTime complexity. These results are stated in Theorems 5.19 and 6.37.
• The remaining containment results are rather straightforward consequences of Lemma 3.13 and the above observation that, if rigid symbols are disregarded, the r-satisfiability test can be split into exponentially many satisfiability tests, each regarding only polynomially large conjunction ofQLquery literals: Corollaries 4.6, 5.17, and 5.20, and Theorems 7.6 and 7.8.
• Another major contribution are the various hardness results, where we apply the few means these lightweight DLs offer for showing hardness forco-NP,NExpTime, co-NExpTime, and 2-ExpTimein the presence of rigid symbols: Theorems 4.8, 5.18, 5.21, 7.7, and 7.9.
• For TCQ entailment inDL-Litelogics betweenDL-LitekromandDL-LiteHbool, we es-tablish close relationships to TCQ entailment in more expressive DLs: Lemmas 7.2 and 7.11.
• In Chapter 8, we lastly obtain a generic rewritability result for answering TQs without negation, which applies to various concrete query languages QL and lightweight logics—as ontology languages—that have been proposed in the lit-erature.
Note that the following chapters refer to the preliminaries but are self-contained otherwise. This is why explanations in different chapters may overlap due to the above described similarity of the approaches.
We below provide an overview of the assumptions we make throughout Chapters 4 to 7—recall that all of them are without loss of generality:
• Individual names are always rigid.
• Concept and role names occurring in an ABox of a (T)KB also occur in its ontology.
• Individual names occurring in a TCQ also occur in the ABoxes of the (T)KB regarded.
• CQs are connected.
• CQs contained in a TCQ are Boolean and do not share variables; this also applies to multiple occurrences of one CQ.
Especially note that, if we refer to settings without rigid symbols or with only rigid concept names, this does not include the individual names.
Lastly, observe that a rigid concept nameAcan easily be represented using a flexible concept name A0 and a rigid role name RA: by replacing every occurrence of A by A0 and considering the global CIs A0 v ∃RA.> and ∃RA.> v A0. That is, in DL-LTL, the latter are prefixed by 2F and added as conjuncts to the formula and, regarding TCQs, they are added to the ontology. In the context of Chapters 4 to 7, where all the considered DLs allow to express such CIs and global CIs can be expressed in both settings we investigate, DL-LTL satisfiability and TCQ entailment w.r.t. a TKB (with non-empty ontology), the following fact thus holds.
Fact 3.18 TQ satisfiability in the presence of rigid concept names can be linearly re-duced to TQ satisfiability in the presence of rigid role names.
That is, hardness results for the caseNRC 6=∅also apply to the case whereNRR 6=∅, and we can disregard the case where NRC =∅ butNRR 6=∅. Note that we in the following proceed in this way, without explicitly referring to Fact 3.18.