3.4 Related Work
3.4.2 Querying for Temporal Data
Query answering with the goal of retrieving data is in focus of recent research in DLs, in general, and this is reflected in the latest explorations on reasoning about temporal knowledge. The different works are primarily discerned regarding the ontology con-sidered, depending on whether it is also temporal or written in a classical DL. The former approaches offer more expressivity but, on the other hand, tend to lead to higher reasoning complexities. For this reason, they are usually studied w.r.t. lightweight DLs.
[Art+15a] investigate different temporal extensions of the DL-Lite logics described in Section 2.1.10 The goal of the work are first-order rewritability results regarding the temporal query answering problem (see Definition 2.22). The queries are arbitrary combinations of temporalized concept and role atoms using the operators of first-order temporal logic and thus very expressive, which is why epistemic semantics are employed (i.e., it is well known that queries with negation are not tractable, even in the atemporal setting [Gut+15]). However, the investigations then mainly consider a very restricted setting where roles are disregarded and the ABoxes contain a single individual name only (the name itself may occur multiple times in an ABox). Though, these results can, amongst others, be applied to show first-order rewritability of instance query answering in TDL-Lite#coreF and TDL-Lite2coreF . In particular, [Art+15a] define specific temporal canonical interpretations for TKBs in these logics, which can be used to construct the rewritings.
Also for TKBs where the ontology is written in T EL#∗ (i.e., the logic allows for both #F and #P), there are canonical models, which can be used to show that the satisfiability problem is tractable w.r.t. data complexity if rigid roles are disallowed [GJK16]. The latter paper also identifies a certain periodicity of the ontology to ensure decidability and proposes several acyclicity notions for ontologies in temporal EL that yield tractable combined and data complexity.
Research in the second direction—temporal query answering regarding classical onto-logies—has been first considered in a general way in [GK12] regarding expressive op-erators on both the temporal and the DL side (i.e., first-order formulas). Temporal conjunctive queries have first been studied in [BBL13], focusing on the complexity of the entailment problem. That paper and follow-up investigations [BBL15b; BBL15a]
focus onALC and extensions such asSHOQ—under open world semantics—, which is reflected in the rather high complexity results. Yet, for many of the considered DLs the data complexity is in co-NP, as in the atemporal case, even in the presence of rigid concept names. Observe that the exact complexity w.r.t. rigid roles is still open, but containment in ExpTime is known. The combined complexity is generally as in the atemporal case if rigid symbols are not considered, but w.r.t. rigid symbols increases up to2-ExpTime.
Together with [Kla13; KM14b] we regard TCQ answering and entailment w.r.t. the most prominent lightweight description logics [BLT15; BT15b; BT15a]. Our complexity results on TCQ entailment [BT15b; BT15a] are detailed in Chapters 5, 6, and 7 regard-ing EL, DL-Litecore to DL-LiteHhorn, and extensions of DL-LiteHhorn, respectively; note that especially w.r.t. combined complexity, the latter logics have however turned out to be not very “lightweight” any more. In Chapter 8, we describe the initial parts of the work in [BLT15], by generalizing TCQs and proposing a practical approach for temporal query answering; for details about algorithms, especially including rigid concept names, we refer to the paper. In a nutshell, the feasibility of rewriting is achieved by dropping the negation operator. [Kla13; KM14b] also describe an approach for implementa-tion w.r.t.DL-Litecore, but achieve the first-order rewritability by considering epistemic semantics. Note that there are also some works on non-standard reasoning problems
re-10Note that [Art+15a] actually study fragments that are slightly more expressive than those described in Section 2.1 (e.g., they consider more expressive RIs that may even model rigid roles).
3.4 Related Work
garding TCQs that are different from query answering and entailment [KM13; KM14a].
There, the focus is on ABox abduction, which roughly is the question of how the given ABox sequence can be extended so that a specific TCQ is entailed.
Observe that, although the above described queries considered in [Art+15a] are very expressive, they do not subsume TCQs. TCQs allow to existentially quantify variables occurring in conjunctions of atoms, whereas the other queries can only express tree-shaped CQs, via concept expressions.
A similar observation can be made regarding the logics where the axioms are tempor-alized. While formulas containing only temporalized assertions can be directly regarded as TCQs, the latter cannot express local CIs. On the other hand, CQs that are not tree-shaped (e.g., see the introductory example CQ (1.1)) cannot be modeled through concepts in logics such asALC-LTL.
There are also some recent proposals of interval temporal description logics that allow for tractable query answering [Art+14a; Art+15b]. Specifically, they combine fragments of the Halpern-Shoham interval logic [HS91a] with DL-Lite. Note that this setting is rather different from the one we consider. Nevertheless, these approaches follow an interesting direction of research, given that, for example, the possibility to associate data with intervals has been recently incorporated into the SQL standard [KM12].
4 LTL over Lightweight Description Logic Axioms
In this chapter, we focus on an abstract lightweight DL DL, regard a formula Φ in DL-LTL, and investigate the combined complexity of the satisfiability problem in various settings. DL-LTL formulas allow to temporalize ontological axioms and hence, for example, to express rules about temporal knowledge. The following EL-LTL formula describes that it always must hold that, if all occupants in a room are sleeping at two consecutive observation moments, then all lights and screens are switched off at the second time of observation:
2F(RoomOccupantvSleeping)∧#F(RoomOccupantvSleeping)→
#F (Lightv ∃HasState.SwitchedOff)∧ #F (Screenv ∃HasState.SwitchedOff). Recall that EL-LTL formulas can also contain assertions, next to CIs; for instance, the fact that Ann is sleeping at two consecutive observation moments should always imply that her devices are not fully powered at the second time of observation:
2F
Sleeping(ann)∧#FSleeping(ann)→
#F ¬∃HasDevice.∃HasState.FullPowerMode(ann).
In line with the examples where DL=EL we specifyDL by requiring it to satisfy the following properties:
(P1) It allows for conjunction, >, and either⊥or qualified existential restriction.
(P2) Satisfiability of conjunctions ofDL literals can be decided inNP.
Note that DL-Litehorn is expressive enough to satisfy the first property. In Section 4.1, we show thatDL-LiteHbool is “lightweight” enough to satisfy the second one and thatEL also represents an instance of DL. We then prove in Section 4.2 that satisfiability in DL-LTL is inPSpaceand hence not harder than satisfiability in LTL, if rigid symbols are disregarded. The rather negative result that rigid symbols lead to NExpTime -completeness is shown in Section 4.3 based on the NExpTime-hardness proof of ALC-LTL [BGL12, Lem. 6.2]; note that satisfiability inALC-LTL is however2-ExpTime-hard if rigid roles are considered. Since complexities of this magnitude nevertheless seem to be out of the scope of usual applications of the lightweight DLs, we in the remainder of the chapter consider restricted formulas, with global CIs. Indeed, for the concrete temporal DLs EL-LTL and DL-LiteHhorn-LTL, the satisfiability problem then also is in PSpacein the presence of rigid symbols.
Our containment proofs basically follow Lemma 3.13. The goal is thus to find a corresponding setW of LTL worlds within specific time and space constraints, although the size of this set is exponential in that of Φ. Note that ι is irrelevant by Fact 3.15.
Throughout the chapter, we use the notation of Section 3.2.
4.1 Satisfiability of Conjunctions of DL Literals
Since conjunctions of DL literals do not contain temporal operators, we do neither have to focus on a specific time point nor to consider an entire DL-LTL structure.
Satisfiability can be decided by regarding a single interpretation (i.e., the remaining parts of the structure are irrelevant and can be chosen arbitrarily, which also means that rigid names have no effects).
ForEL, we can therefore use a reduction to the knowledge base consistency problem in the DL ELO⊥, by creating EL axioms capturing the semantics of the respective literals.
Lemma 4.1 Satisfiability of conjunctions of EL literals can be decided inP.
Proof. The proof is by reduction to the consistency problem in the DL ELO⊥. That is, for a given conjunction Ψ of EL literals, we construct an ELO⊥ knowledge base K =hO,Ai that is consistent iff Ψ is satisfiable.
The initial sets O and A contain all GCIs and, respectively, assertions that occur non-negated in Ψ. For each negative literal ¬α occurring in Φ, K is then extended according to the kind of ¬αas follows:
• Negated concept assertion¬C(a): the axiomsC(a) andCuCv ⊥, whereC is a fresh concept name, are added to K.
• Negated role assertion ¬R(a, b): the axiom {a} u ∃R.{b} v ⊥is added to O.
• Negated GCI ¬(C v D): the axioms C(a), D(a), and DuD v ⊥, where ais a fresh individual name and D is a fresh concept name, are added toK.
It is easy to see that K is consistent iff Ψ is satisfiable. Since consistency ofELO⊥ KBs is decidable in polynomial time [BBL05, Thm. 4], the claim thus holds.
RegardingDL-Lite, the proof is even more easy. This is because the complexity of KB consistency is given in [Art+09] for KBs where the ABoxes may contain negated asser-tions, as in our setting (see Section 2.1.1). A KB is thus obtained from a conjunction of DL-Liteliterals by replacing each negated CI¬(B1u. . .uBm vBm+1t. . .tBm+n) simi-lar as above by CIsBm+1uB v ⊥, . . . , Bm+nuB v ⊥,and assertionsB1(a), . . . , Bm(a), B(a), whereBandaare fresh symbols. From the results in [Art+09, Thm. 8.2] we then get the following.
Lemma 4.2 Satisfiability of conjunctions of DL-Lite literals can be decided
• in P in DL-LiteHhorn and
• in NP in DL-LiteHbool.