On Metric Temporal Description Logics

On Metric Temporal Description Logics

V´ıctor Guti´errez-Basulto

and Jean Christoph Jung

and Ana Ozaki

Abstract. We introduce metric temporal description logics (mTDLs) as combinations of the classical description logic ALC with (a) LTL^bin, an extension of the temporal logic LTL with suc- cinctly represented intervals, and (b) metric temporal logic MTL, ex- tending LTL^bin with capabilities to quantitatively reason about time delays. Our main contributions are algorithms and tight complexity bounds for the satisfiability problem in thesemTDLs: For mTDLs based on (fragments of) LTL^bin, we establish complexity bounds ranging from EXPTIMEto 2EXPSPACE. FormTDLs based on (frag- ments of) MTL interpreted over the naturals, we establish complexity bounds ranging from EXPSPACEto 2EXPSPACE.

1 Introduction

Classical Description logics (DLs) are fragments of first-order logic aiming at the representation of and reasoning about knowledge. The importance of DLs lies in the fact that they are, arguably, the prime formalism to encode ontologies, e.g., they underpin the web ontology language OWL 2, the medical ontology SNOMEDCT and the the- saurus of the US national cancer institute. It has been observed that in many domains where ontologies are used an implicit or explicit notion oftimeplays a central role [25]. As an instance, many terms in the medical domain are described making reference to temporal patterns; for example, the description of the autoimmune disease di- abetes must specify that it might lead to glaucoma in the future. On the other hand, DLs were initially developed with the aim of captur- ingstaticknowledge. To address this shortcoming, over the last 20 years a plethora of temporal DLs (TDLs), extensions of DLs with an explicit temporal component, have been proposed [5, 25].

The most popular approach to constructing TDLs is to combine classical DLs with temporal logics such as LTL and CTL, and to provide a two-dimensional product-like semantics [29, 15, 25]–one dimension for time and the other for DL quantification. Temporal DLs of this kind support the definition of terms using, e.g., the temporal operators ‘at the next/previous point’ or ‘somewhere in the future/past’. As an example, in TDLs based on CTL we can use∃has.DiabetesvE3∃develops.Glaucomato say that ‘a patient with diabetes may develop glaucoma in the future’. The importance of TDLs based on LTL and CTL is witnessed by the vast amount of research conducted on the topic in the last decade; in particu- lar, TDLs using expressive as well as lightweight DLs, with differ- ent levels of interaction between the components, have been investi- gated [31, 7, 25, 13, 17, 8, 18, 19]. Moreover, this sort of TDLs have been already successfully used in applications, e.g., to describe con- ceptual models capturing the evolution of databases over time [8].

However, their temporal constructs does not seem to always ade- quately carter for the needs of ontology designers and users. Indeed,

1University of Bremen, Germany;{victor,jeanjung}@tzi.de

2University of Liverpool, UK; anaozaki@liverpool.ac.uk

temporal primitives such as ‘eventually in the future’ might not be sufficiently precise for temporal conceptual modeling in an ontol- ogy. As an instance, in the medical domain ontological modelling often requires reference to concrete durations. Consider, for exam- ple, lyme disease: Affected patients develop a rash within 3-32 days after the infection. Since the infection can only occur after exposure to ticks, the concrete temporal interval of 3-32 days can be used to rule out lyme disease as a cause of rash.

Observe that, although it is possible to express eventuality within an interval by ‘unfolding’ all the timepoints represented in an in- terval, allowing intervals in the language with end-points in bi- nary would result in an exponentially more succinct statement.

For instance, for expressing that a patient “develops a rash even- tually within 3-32 days”, in TDLs based on plain LTL, the ontol- ogy designer has to write#³(∃has.Rash)t. . .t#³²(∃has.Rash), whereas the same can be expressed elegantly, and more succinctly by3[3,32]∃has.Rashin the logics studied in this paper. Despite the need of this feature, TDLs based on temporal logics succinctly cap- turing time intervals (to the best of our knowledge) have not yet been considered in the literature. It is important to note that TDLs based on Halpern and Shoham’s (HS) interval logic of Allen’s relations have been recently investigated [3, 9]. However, these TDLs are or- thogonal to the ones investigated here because they are interval-based logics, i.e., intervals (instead of time points) are the basic time units.

The purpose of this paper is to initiate the study of metric TDLs (mTDLs) allowing for quantitative temporal reasoning. In particu- lar, we are interested in TDLs merging qualitative temporal asser- tions together with quantitative constraints so as to get the benefits of the qualitative and quantitative abstraction levels. To this end, we consider TDLs based on (a) LTL^bin, the extension of LTL with suc- cinctly represented intervals, and (b) the real-time metric temporal logic MTL, extending LTL^binwith capabilities to quantitatively rea- son about time delays. We look at TDLs that might prove useful for applications: we consider the traditional DLALC, make the (most- general) constant domain assumption, and apply temporal operators to concepts and, in the second part of the paper, to TBox statements.

We do not apply temporal operators to roles since this typically leads to undecidability [25]. Our main contributions are algorithms for the satisfiability problem and complexity bounds.

Our study starts withmTDLs based on LTL^binand temporal op- erators applied only to concepts. For example, in LTL^binALCwe can use

∃exposedTo.Tickv2[3,32](∃has.Rash→♦[0,3]∃gets.LymeDTest) to say that ‘persons exposed to ticks whom develop a rash within 3- 32 days after that must be tested for Lyme disease within three days.’

We have argued above that LTL^bin_ALCis not more expressive than LTLALC; however, the (exponential!) translation does not give tight complexity bounds. More specifically, the translation yields a 2EX-

propositional combined withALC combined withALC

temporal operators on concepts temporal operators on concepts and TBoxes

LTL PSPACE[30] EXPTIME[25] EXPSPACE[15]

LTL^bin EXPSPACE[1] EXPSPACE[Thm. 1] 2EXPSPACE[Thm. 5]

LTL^0,∞ PSPACE[24] EXPTIME[Thm. 2] EXPSPACE[Thm. 7]

MTL EXPSPACE[1] 2EXPSPACE[Thm. 3] 2EXPSPACE[Thm. 6]

MTL^0,∞ PSPACE[24] EXPSPACE[Thm. 4] EXPSPACE[Thm. 8]

Table 1: Overview of previous and new complexity results.

PTIMEupper bound because satisfiability in LTLALC can be done in EXPTIME[25]. In contrast, in our first main result, we develop an algorithm based on quasimodels in order to obtain EXPSPACE- completeness for satisfiability in LTL^binALC. Recall that satisfiability in LTL^binis also EXPSPACE-complete [1], thus, the combination with ALCis “for free.”

As the next step, we consider as temporal component metric tem- poral logic MTL [23], which extends LTL^binby explicitly associating to each state a timestamp, allowing then for quantitative reasoning of time-delays. MTL has been intensively studied in the last 20 years;

in particular, different semantics, syntactic restrictions and underly- ing time domains have been considered, for an overview see [27]. We will consider here MTL over the naturals with so-calledpointwise se- mantics. Under this semantics we can see states asobservations, say of a real-time system, that have an explicit discrete timestamp and consequently think of the time difference between two consecutive observations as a time-delay. For example, in MTLALCwe use

PhDStudentu ¬∃pays.Feesv#[1,3](∃gets.Reminderu (♦[0,7]∃pays.Feest(¬∃access.LabU[7,∞)∃pays.Fees))

to express that, if the system observes that a PhD student has not paid the fees, then it should issue a reminder in the next system cycle (that is, observation) which is necessarily occurring within the next three time units; moreover, the student then should pay the fees within seven time units (for example days) or she does not have access to the lab until she pays.

Based on the aforementioned result for LTL^binALCand the limited interaction of the dimensions, one could conjecture that the com- plexity of MTLALCis not higher than that of the components, and therefore EXPSPACE-complete. Surprisingly, we show that this is not the case by establishing a 2EXPSPACElower bound, which is later shown to be tight.

We then turn our attention to the case where temporal opera- tors are additionally applied to TBoxes. As for the basic case, we start by looking at LTL^binALC. Most interestingly, it can be shown that the aforementioned 2EXPSPACE-hardness result can be lifted to LTL^binALCtemporal TBox satisfiabilty. Matching upper bounds for LTL^binALCand MTLALCfollow from the translation to the qualitative case LTLALC, where TBox satisifiability is known to be EXPSPACE- complete [25].

Based on similar observations for the propositional case [24], we finally looked at the restrictions LTL^0,∞_ALC and MTL^0,∞_ALC where in- tervals are only of the form[0, c]or[c,∞]. This is still expressive enough to succinctly model, for instance, time limits. We show that, indeed, the quasimodel technique can be leveraged to show that this leads to better complexity in many cases.

An overview of existing and new results is given in Table 1.

Missing proofs are provided in an extended version, available at www.informatik.uni-bremen.de/tdki/research/papers/GJO16.pdf.

2 Preliminaries

Intervals.We use standard notation for (open and closed) intervals, e.g.,[c1, c2)is the set of alln∈Nwithc1≤n < c2. It is thus clear what is meant withn∈IandI⊆I⁰for intervalsI, I⁰.

LTL^binALCsyntax.LTL^binALCis a TDL based on LTL and the classical DLALC. LetNCandNRbe countably infinite sets ofconceptand role names, respectively. LTL^binALC-conceptsC, Dare formed accord- ing to the rule:

C, D::=A| ¬C|CuD| ∃r.C|#C|CUID

whereA ∈NC,r∈NR, andIis anintervalof the form[c1, c2]or [c1,∞)withc1, c2 ∈ Ngiven inbinary. We use standard Boolean and temporal abbreviations:CtD, ∀r.C,>,♦^IC, and2^IC for

¬(¬Cu ¬D),¬∃r.¬C,At ¬A,>UIC, and¬♦I¬C, respectively.

We omit intervals of the form[0,∞) and writeCUD instead of CU[0,∞)D, and use the subscript·cto refer to intervals of the form [c, c].

An LTL^binALCTBoxis a finite set ofconcept inclusions (CIs)C v DwithC, DLTL^bin_ALC-concepts. We useC ≡Dto refer to the two concept inclusionsC v DandD vC. Thesizeof a TBoxT (a conceptC) is the number of symbols required to writeT (C).

LTL^binALCsemantics. The semantics of LTL^binALCis given in terms of interpretations, that is, structuresI = (∆^I,(In)n∈N), where each Inis a classical DL interpretation with domain∆^I: we haveA^In⊆

∆^Iandr^In ⊆∆^I×∆^I. We usually writeA^I,nandr^I,ninstead of A^Inandr^In, respectively. For instance,d∈A^I,nmeans that in the interpretationI, the objectdis an instance of the concept nameA attime pointn. The stipulation that all time points share the same domain∆^Iis called theconstant domain assumption(meaning that objects are not created or destroyed over time), and it is the most general choice in the sense that increasing, decreasing, and varying domains can all be reduced to it [15].

We now define the semantics of LTL^binALC-concepts. To this end, we extend the mapping·^I,nfrom concept names to complex LTL^bin_ALC- concepts as follows:

(¬C)^I,n = ∆^I\C^I,n, (CuD)^I,n = C^I,n∩D^I,n,

(∃r.C)^I,n = {d∈∆^I| ∃e∈C^I,nwith(d, e)∈r^I,n}, (#C)^I,n = {d∈∆^I|d∈C^I,n+1},

(CUID)^I,n = {d∈∆^I| ∃k > n: d∈D^I,k∧k−n∈I

∧ ∀m∈(n, k) : d∈C^I,m}.

An interpretationIis amodelof a conceptC ifC^I,0 6= ∅; it is a model of a CICvD, writtenI|=CvD, ifC^I,n⊆D^I,n, for all n∈ N. We callIamodel of a TBoxT, writtenI |=T, ifI |=α for allα∈ T. Note that TBoxes are interpretedgloballyin the sense that all CIs must be satisfied at every time point.

Reasoning problem.We are interested in the reasoning problem of satisfiability relative to global TBoxes (throughout the paper only called satisfiability), that is, given an LTL^binALC-conceptCand TBox T, decide whetherCandT have a common model.

Sequences.Throughout the paper we use sequences with the follow- ing notation. For a (possibly infinite) sequenceσ = σ(0)σ(1). . ., we write σ^≤n and σ^>n for the head σ(0)σ(1). . . σ(n) and tail σ(n+ 1)σ(n+ 2). . .ofσ, respectively. We also writeσ^>i,≤jfor the subsequenceσ(i+ 1). . . σ(j)ofσ. For a finite sequenceσ1and a sequenceσ2, we denote withσ1∗σ2, or just σ1σ2 if no confu- sion is possible, theconcatenationofσ1andσ2. As usual, we define σ¹=σ,σⁿ⁺¹=σσⁿandσ^ω=σσσ . . ..

3 LTL

We aim at devising algorithms and establishing tight complexity bounds for the satisifiability problem. We first concentrate on de- veloping an algorithm for satisfiability in LTL^binALC, yielding a tight EXPSPACEupper bound. The lower bound is a consequence of the following:(i)allowing the abbreviation#ⁿ(meaningnconsecutive

‘next’ operators) withnencoded in binary in LTL makes satisfiabil- ity checking EXPSPACE-hard [1, 2] and(ii)#ⁿcan be expressed in LTL^bin_ALCwith♦n.

In the second part of this section, we show that satisfiability in the restriction LTL^0,∞_ALCof LTL^binALCto intervals of the form[0, c]or [c,∞)is complexity-wise better-behaved. In particular, it is EXP- TIME-complete and thus not harder than inALC.

The main structure underlying our decision procedure are so- calledquasimodels, which have been used for studying the satisfi- ability in various TDLs [31, 15, 7, 17]. In a nutshell, a quasimodel is an abstraction of an interpretationI = (∆^I,(In)n∈N)in which each (possibly infinite)Inis replaced by aquasistate, that is, afinite set oftypes.

We show that quasimodels exhibit a monotonic behavior and apply regularity arguments to show membership in EXPSPACEand EXP- TIME, respectively. We show that to check satisfiability it suffices to consider quasimodels of the form:

S(0)S(1). . . S(n)^ω, (1) with S(i) ⊇ S(i+ 1), for all 0 ≤ i < n, and n double- and single-exponentiallybounded in the size ofC andT for LTL^binALC

and LTL^0,∞_ALC, respectively.

Note that a similar regularity condition holds for LTL in the sense that every satisfiable LTL formula has a regular model like (1) (with S(i) propositional valuations) [26]. The main difference is thatnis exponentially-bounded (satisfiability is thus PSPACE) and that a larger suffix could be the regular part repeating infinitely; in LTL^binALC, due to monotonicity,S(n)is the only periodic set.

Throughout the section, we assume without loss of generality that the TBoxT is of the form{> vCT}and denote withsub(C,T) the set of all subconcepts ofCandCT.

3.1 Full LTL

We start with introducing some required notation. Denote with cl(C,T)the closure under single negations of the set:

sub(C,T)∪ {DUE|DU[c,∞)E∈sub(C,T)}. (‡) As usual, atype forCandT is a subsett⊆cl(C,T)such that:

• D∈tiff¬D6∈t, for all¬D∈cl(C,T);

• DuE∈tiff{D, E} ⊆t, for allDuE∈cl(C,T);

• CT ∈t.

We will usetp(C,T)to denote the set of all types forCandT and ]C,T to denote the number of types,|tp(C,T)|.

We now describe a set of types that appropriately abstracts a clas- sical description logic interpretationIn. Aquasistate forCandT is a setQ⊆tp(C,T)of types such that:

• ift∈Qand∃r.D∈t, then there ist⁰∈Qsuch that{D}∪{¬E|

¬∃r.E∈t} ⊆t⁰.

We next show how to temporally relate types in different quasistates;

most importantly, regarding how temporal formulas of the form#D andDUIEare captured. Lett=t(0)t(1). . .∈tp(C,T)be a (pos- sibly infinite) sequence of types. We say thattrealizes DUIE if there ism∈Isuch thatE∈t(m)and, for all0< l < m, we have D ∈ t(l). From here on, we useS = S(0)S(1). . .to denote an infinite sequence of quasistates forCandT. Arunr=r(0)r(1). . . throughSis a sequence of types forCandT such that for alln≥0:

(R1) r(n)∈S(n);

(R2) #D∈r(n)iffD∈r(n+ 1), for all#D∈cl(C,T);

(R3) DUIE ∈ r(n) iff r^≥n realizes DUIE, for all DUIE ∈ cl(C,T).

Intuitively, a run is a sequence of types which characterizes the tem- poral evolution of a domain element.

We now have the ingredients to formally define a quasimodel. A quasimodel forCandT is a pair(S,R)withRa set of runs through Ssuch that:

(Q1) C∈tfor somet∈S(0); and

(Q2) for allt∈S(n),n≥0there is a runr∈Rsuch thatr(n) =t.

Intuitively,(Q1)ensures thatCis witnessed at time point 0, and(Q2) ensures that each type has an appropriate temporal evolution through the quasimodel. We show in the appendix that concept satisfiability is characterized by the existence of a quasimodel forCandT: Lemma 1. There is a model ofCandT iff there is a quasimodel for CandT.

This characterization, however, does not serve yet as the basis of an algorithm as bothS andRare infinite. In the next step, we show that quasimodels can be assumed to have a certain regular shape.

Henceforth, letKdenote the largest constant occurring inCandT (or1if none exist), and let`1 = (]C,T)^K+K. We then have the following normal form of quasimodels.

Lemma 2. There is a quasimodel forCandT iff there is a quasi- model(S,R)forCandT whereSis of the form

S =Sⁿ₀⁰. . . S_m−1^nm−1S_m^ω

for quasistatesS0, . . . , SmwithSi )Si+1,0≤i < m, and num- bersn0, . . . , nm−1< `1.

The proof of Lemma 2 (cf. appendix) proceeds in two steps.(i)We first show that we can extend any quasimodel such that a quasistate at timei+ 1is contained in the quasistate at timei, that is,Si+1⊆Si, i≥0.(ii)We then show that if`1consecutive quasistates coincide, that is,S(i) =S(i+ 1) =. . . =S(i+`1)for somei≥0, then

we can assume that all subsequent quasistates coincide as well, that is,S(j) =S(i)for allj≥i.

Obviously, the (strict!) containment condition on the Si in Lemma 2 implies thatm is at most]C,T since the S(i) are non- empty sets of types. Moreover, note that, due to`1, the length of the initial irregular part ofSis double-exponentially bounded in the size ofCandT. Lemmas 1 and 2 give thus rise to the following non- deterministic procedure for checking concept satisfiability.

1. Non-deterministically choosem < ]C,T non-empty sets of types S0). . .)Smand a sequencen0, . . . , nm−1of binary numbers such thatni< `1for all0≤i < m.

2. Verify that the sequenceSdefined as S=Sⁿ₀⁰. . . S_m−1^nm−1S_m^ω

can be extended to a quasimodel forCandT, that is, check that:

(a) eachSi,0≤i≤m, is a quasistate;

(b) there is at∈S0withC∈t;

(c) for eachi≥0andt∈S(i), there is a runrthroughSsuch thatr(i) =t.

The procedure is obviously correct (given Lemmas 1 and 2), but in- volves a non-effective step: in 2(c), infinitely many tests have to be performed. It thus remains to show how to effectively execute 2(c).

To this end, we show, in Lemma 3, that it suffices to check 2(c) for alli≤]C,T·`1, a double exponential number; then, in Lemma 4, we identify a certain regular form of runs, lending itself to implementa- tion. For both Lemmas, letSbe as in Lemma 2.

Lemma 3. If the condition in 2(c) is satisfied for alli≤]C,T ·`1, then it is satisfied for alli≥0.

Proof. Similar to the proof of (ii) in Lemma 2. o Lemma 4. If there is arrun throughS withr(i) = t, for some i≥0, then there is a runr⁰throughSwhich satisfiesr⁰(i) =tand is of the shape

r⁰=s(0)· · ·s(k1)∗(s⁰(0)· · ·s⁰(k2))^ω

for typess(0), . . . , s(k1), s⁰(0), . . . , s⁰(k2)andk1≤i+ (]C,T)^K, andk2≤ |cl(C,T)| ·(]C,T)^K.

Proof. We are going to use the following Claim.

Claim.Ifris a run throughSandr^{≥p−K,≤p}=r^{≥q−K,≤q}for some p < q, thenr⁰=r^≤p∗r^>qis a run throughS.

Proof of the Claim.We need to show that Conditions(R1)to(R3) hold for r⁰. Condition (R1) is an immediate consequence of the construction of r⁰ and S. For (R2), we only need to check that for every#D ∈ cl(C,T),#D ∈r⁰(n)iffD ∈ r⁰(n+ 1), for all n≥0. This follows from the fact thatr(p) =r(q)andr∈R.

For(R3), we check concepts of the formDUIE∈cl(C,T). Note that sincer^{≥p−K,≤p} = r^{≥q−K,≤q}, we haver^0≥p−K = r^≥q−K. Then, forn≥p−K,

DUIE∈r⁰(n)iffr^0≥nrealizesDUIE.

From now on assumen < p−K. IfI= [c1, c2]then, sincec1, c2∈ [0, K], we cannot exceedp. Then,(R3)holds.

Now, considerI = [c1,∞), wherec1 ∈ [0, K]. Asr^0≥p−K = r^≥q−K, if alreadyr^0≥n,≤prealizesDUIE, thenDUIE ∈ r⁰(n).

Otherwise, assume thatr^0≥n,≤pdoes not realizeDUIE. Then, for n < p−K, we have thatDUIE∈r(n)iffD∈r(n), . . . , r(p)and DUE∈r(p) =r⁰(p).

Asr^0≤p =r^≤pandr⁰(p) = r(q), we haveDUIE ∈ r⁰(n)iff D∈r⁰(n), . . . , r⁰(p)andr^0≥prealizesDUE. Then, forn < p−K, DUIE∈r⁰(n)iffr^0≥nrealizesDUIE.That is,(R3)holds.

This finishes the proof of the Claim.

An n-sequence of types is just a finite sequence of types s(0)· · ·s(n−1). Fork ∈ N∪ {∞}, we say that ann-sequence s(0)· · ·s(n−1)appearsktimes inrif there arekdistinctj ≥0 such thatr(j+l) =s(l), for all0≤l < n.

Letrbe a run throughSwithr(i) =tand choosek1≥iminimal such that everyK-sequence appearing inr^≥k1appears infinitely of- ten. We argue that it is without loss of generality that, betweeniand k1, everyK-sequence appears at most once: by the Claim, we can cut sequences that appear more than once. As there are at most(]C,T)^K suchK-sequences, we can assume thatk1≤i+ (]C,T)^K.

Now, letU be the set ofCUID ∈ r(k1)that are not realized in r^{≥k1,≤k1+K}, and choosen≥k1minimal such thatr^{≥k1,≤k1+K} = r^≥n,≤n+Kand eachCUID ∈U is realized inr^≥k1,<n. Letm1be minimal such thatr^≥k1,≤m1 realizes someDUIE ∈U. Reasoning as above using the Claim yields that we can assume without loss of generality thatm1 ≤k1+ (]C,T)^K. Let nowm2 > m1be mini- mal such thatr^≥k1,≤m2realizestwoCUID∈U. As before, we can show that without loss of generality m2≤k1+ 2(]C,T)^K. Contin- uing this reasoning, we can conclude that, without loss of generality, n≤k1+|cl(C,T)| ·(]C,T)^K.

It is now routine to verify thatr⁰ = r^<k1 ∗(r^≥k1,<k2)^ω, with

k2=n−k1, is a run throughS. o

We are now in a position to show that the above procedure can be implemented using only (non-deterministic) exponential space. Ob- viously, the setsS0, . . . , Sm and the numbers n0, . . . , nm−1 can be stored in exponential space. Moreover, steps 2(a) and 2(b) can clearly be checked in exponential space. For 2(c), Lemma 3 implies that at most]C,T ·]C,T ·`1, that is, double-exponentially many, pairs(t, i) have to be considered, but only one at a time. Finally, (the proof of) Lemma 4 enables the following algorithm for check- ing the existence of a run. First, guess binary numbersk1, k2as in Lemma 4; then, guess a run in the form of the lemma. For the latter, proceed in a “sliding window” fashion: keepK consecutive types and verify(R1)-(R3) for the first type in the sequence, then drop that type, guess the next type, and continue. For detecting the loop, store the sequencer(k1), . . . , r(k1+K)and verify that it appears again atr(k1+k2), . . . , r(k1+k2+K)and, moreover, that each DUIE∈r(k1)is realized beforek2. We conclude the desired result.

Theorem 1. Satisfiability inLTL^binALCisEXPSPACE-complete.

3.2 LTL

In this section, we consider LTL^0,∞_ALC, a well-behaved, yet expressive, fragment of LTL^bin_ALCin which intervals can only be of the form[0, c]

or[c,∞). This sort of intervals is useful to set maximum (deadline) points and minimum (initial) ones. For example, the CI

PhDStudentu ∃defends.Thesisv♦[0,4]∃submits.RevisedThesis says that ‘PhD students who defend their thesis must submit a revised version within 4 weeks ’.

We show that allowing intervals of this restricted form does not in- crease the complexity of satisfiability compared toALCor LTLALC, for both of which concept satisfiability is EXPTIME-complete [25].

We concentrate again on the upper bound since EXPTIME-hardness follows from satisfiability inALC[12].

Our algorithm relies again on quasimodels; however, we will slightly adapt the definition of types to address the restricted inter- vals. As a consequence, it will suffice to consider quasimodels of the form (1) wherenis onlysingle-exponentiallybounded, finally yielding an EXPTIMEdecision procedure.

We first adapt the notion of a type. Instead of (‡), we define cl(C,T) as the closure under single negations ofsub(C,T) ex- tended with

{DU[0,c]E, DU[c,∞)E|c∈[0, K], DUIE∈sub(C,T)}.

Based on this, it is straightforward to show that Lemma 2 remains true for LTL^0,∞_ALC.

Note that there are now double-exponentially many types which typically prohibits an EXPTIMEdecision procedure based on type elimination [28]. However, it is easy to see that a typetappearing in some quasimodel satisfies the following property.

(P) For everyDUI1E, DUI2E∈cl(C,T)withI1 ⊆I2, we have DUI₁E∈timpliesDUI₂E∈t.

To see(P), fix some quasimodel(S,R)forCandT and assume that DUI₁E∈tfor somet∈S(n),n≥0. By Condition(Q2), there is a runr∈Rsuch thatr(n) =t. AsDUI1E∈t, by Condition(R3), r^≥nrealizesDUI₁E. SinceI1⊆I2,r^≥nalso realizesDUI₂E. By Condition(R3)again,DUI2E∈r(n).

Thus, it suffices to consider only types that satisfy(P), whose num- ber]C,T is bounded by(2·K)^2|sub(C,T)|, that is, exponential. From now on assume w.l.o.g. thattp(C,T)is the set of types in which(P) holds. The next lemma shows that we can assume that our quasi- models reach a periodic quasistate after at most exponentially many quasistates.

Lemma 5. There is a quasimodel forCandT iff there is a quasi- model(S,R)forCandT of the form

S=S0. . . Sn−1(Sn)^ω,

for quasistatesS0, . . . , SnwithSi)Si+1,0≤i < n≤]C,T. The proof proceeds in two steps, as in Lemma 2. In the first step we modify our quasimodel so that each quasistate at timei+ 1is contained in the quasistate at timei. But now, in the second step, we show that iftwoconsecutive quasistates coincide, that is,S(i) = S(i+ 1)for somei ≥0, then we can assume that all subsequent quasistates coincide as well, that is,S(j) =S(i)for allj≥i.

Based on Lemma 5, we now present an algorithm that performs type elimination, similar to what has been done for LTLALC[25].

Defineρ(n) =min{]C,T, n}, for alln≥0, and, moreover, define an operation “−1” on intervals as follows:[0, c]−1 = [0, c−1]and [c+ 1,∞)−1 = [c,∞), for allc≥0, and[0,∞)−1 = [0,∞).

We say that typestandt⁰arecompatibleif the following holds:

• #D∈tiffD∈t⁰, for all#D∈cl(C,T); and

• DUIE ∈ t iff either (the sequence) t t⁰ realizes DUIE, or {D, DUI−1E} ⊆t⁰, for allDUIE∈cl(C,T).

The algorithm starts with sets

S0, . . . , Sn−1, Sn

wheren = ]C,T and eachSiis initially set totp(C,T). We then exhaustively eliminate typestfrom someSi,0≤i≤niftviolates one of the following conditions:

(T1) for all∃r.D ∈ t, there is t⁰ ∈ Si such that{D} ∪ {¬E |

¬∃r.E∈t} ⊆t⁰;

(T2) there ist⁰∈Sρ(i+1)such thattandt⁰are compatible;

(T3) ifi >0, there ist⁰∈Si−1such thatt⁰andtare compatible;

(T4) for allDUIE∈t, there isk≥0and a sequence t1 ∈Sρ(i+1), . . . , tk∈Sρ(i+k)

such thatt0· · ·tk(witht0 =t) realizesDUIE, andtlandtl+1

are compatible, for all0≤l < k.

Before giving details on how to implement the conditions, espe- cially(T4), we finish the description of the algorithm and show cor- rectness. The algorithm stops when no further types can be elimi- nated. It returns ‘satisfiable’ if there is a survivingt∈S0withC∈t, and ‘unsatisfiable’, otherwise.

Lemma 6. The algorithm returns ‘satisfiable’ iff there is a quasi- model forCandT.

Proof. For(⇒), letS₀^∗, . . . , S_n^∗be the result of the type elimination procedure. Define(S^∗,R)withS^∗ =S^∗0. . . Sn−1^∗ (S^∗n)^ωandRas the set of all sequencesrof types such that, for alli≥0:

1. r(i)∈S^∗(i);

2. r(i)andr(i+ 1)are compatible; and

3. DUIE∈r(i)iffr^≥irealizesDUIE, for allDUIE∈cl(C,T).

We now argue that (S^∗,R) is a quasimodel. By (T1), the sets S0^∗, . . . , S^∗ngenerated by the algorithm are quasistates, so S^∗is a sequence of quasistates. By assumption, there ist∈S₀^∗withC∈t, which gives us(Q1). By definition ofR, we have that everyr∈R is a run throughS^∗(see(R1)-(R3)). Then, for(Q2), we only need to see that for everyt∈ S^∗(j)there isr ∈Rsuch thatr(j) = t, j∈N. Letr⁰=r^≤jbe a sequence of types such that all consecutive types inr⁰are compatible andr⁰(j) =t. By(T3)such sequence ex- ists. We now extend this run using(T2)and(T4). Assume that there is noDUIE∈r⁰(j). Then by(T2)there ist⁰∈S^∗(j+ 1)such that tandt⁰are compatible. So we extendr⁰witht⁰. Now assume there isDUIE ∈ r⁰(j). By(T4)there is a minimal sequencet0t1· · ·tk

of types that realizes allDUIE∈r⁰(j). We extendr⁰witht1. Con- tinuing with this process one can ensure the existence of an infinite sequence satisfying all conditions of the sequences inR.

For the other direction(⇐), assume there is a quasimodel, which is without loss of generality of the formS⁰ = S₀⁰. . . Sn−1⁰ (S_n⁰)^ω, by Lemma 5. LetS0^∗, . . . , S^∗nbe the result of the type elimination.

It is routine to verify thatS_i⁰ ⊆ S_i^∗,0 ≤ i ≤ nby showing that no type inS⁰iviolates(T1)-(T4). Clearly, each type satisfies(T1), since(S⁰,R)is a quasimodel. Moreover, conditions(T2)-(T4)are consequences of the existence of runs through each type.

Observe finally that, by(Q1), there is somet∈S₀⁰, thust∈S^∗₀, withC∈t, that is, the algorithm returns ‘satisfiable’. o It is not hard to see that the algorithm runs in exponential time.

The maintained sets of types have initally exponential size and in ev- ery step some type is eliminated. Conditions(T1)-(T3)can clearly be checked in exponential time. Finally,(T4)can be cast as a reach- ability problem, which can be solved in polynomial time, in the fol- lowing (exponentially sized) graph: vertices(t, i)for allt∈Siand edges between(t, i)and(t⁰, ρ(i+ 1))ifftandt⁰ compatible. We thus conclude:

Theorem 2. Satisfiability inLTL^0,∞_ALCisEXPTIME-complete.

4 MTL

In this section, we investigate a TDL that emerges from combining the real-time logic MTL (over the naturals) and ALC. MTLALC- conceptsare formed according to the following rule

C, D::=A| ¬C|CuD| ∃r.C|#^IC|CUID, whereA∈NC,r∈NR, andIis an interval.

Note that MTLALC-concepts are formed just like LTL^binALC- concepts except for the constructor#I. The main difference between LTL^binALCand MTLALClies in their semantics: MTLALCis a timed extension of LTL^binALC, in other words, each interpretation in(In)n∈N

explicitly refers to its time (think of it as the reading of a‘fictitious discrete clock’) allowing to quantitatively reason about time delays.

Formally, a timed interpretation I is a tuple (∆^I,(In)n∈N, τ) withτ :N→Na mapping withτ(n)< τ(n+ 1), for alln ∈N, which specifies that the n-th interpretation happens to be at time pointτ(n). Note that there might begapsbetween two interpreta- tions, e.g., whenτ(3) = 8andτ(4) = 10, then there is no inter- pretation at time point9. Intuively, we view(In, τ(n))n≥0as ase- quence of observations, for instance, in a real-time system, and then understand the differenceτ(n+ 1)−τ(n)as the time delay between observationsnandn+ 1.

The interpretation function·^I,nis lifted to complex concepts as in Section 2 for the constructors¬,u, and∃r.C. For#IandUI, it is defined as follows:

(#IC)^I,n = {d|d∈C^I,n+1∧τ(n+ 1)−τ(n)∈I}, (CUID)^I,n = {d| ∃k > n:d∈D^I,k∧τ(k)−τ(n)∈I

∧ ∀m∈(n, k) :d∈C^I,m}.

One could expect that, just like for LTL^bin_ALC, the complexity of sat- isfiability in MTLALCis not higher than in the components; in par- ticular, EXPSPACE-complete as in MTL [1]. Surprisingly, we prove that there is an exponential jump in the complexity; the main reason for such an increase is that, due to slightly different semantics, the independence of elements in eachInis lost.

Theorem 3. Satisfiability inMTLALCis2EXPSPACE-complete.

We prove here only the lower bound. The upper bound will follow from a more general result, see Theorem 6 in Section 5.

Proof. We reduce the word problem of a double-exponentially space-bounded deterministic Turing machine. Fix that TM A = (Q,Σ,Γ, δ, q0, F)withδ:Q×Γ→Q×Γ× {l, r}and assume that Ais2²ⁿ-space bounded on inputs of lengthn. LetQ⁰ =Q∪ {q}, k=|Γ×Q⁰|+ 1and fix some bijectionπ: [1, k−1]→Γ×Q⁰. We are going to use the following symbols:

• Tape, to mark the tape cells;

• Aa,q,a∈Γ, q∈Q⁰, to label cells with a symbolaand a stateq;

qexpresses that the head is somewhere else.

Recall that we use the abbreviations3ⁱand#ⁱinstead of3[i,i]and

#[i,i], and just3instead of3[0,∞). For inputs of lengthn, we will construct a TBoxTn, whose basic ingredients are the following con- cept inclusions:

Tapev3[0,k]Tapeu2^[0,k−1]¬Tape (2)

Tapev#^[0,k−1]> (3)

#ⁱ> ≡Aπ(i), for alli∈[1, k−1] (4)

Intuitively, using CI (2), we enforce that everyk-th time point is la- beled withTape. By CI (3), we express that, ifTapeis observed, the next observation is due within1tok−1time points, but there is a choice. Finally, using CI (4), wegloballymark all domain elements in a world, depending on the delay of the next observation, with some Aa,q, that is, information about state and tape symbol.

It remains to show how to synchronize consecutive configura- tions. Basically, the technique goes back to the following well-known lemma [21, Lemma 3.3], which is based on [20, Lemma 4.1] itself.

Lemma 7. For eachn ≥ 1, there is a satisfiable formulaϕn in propositional temporal logic extended with#ⁿ,nin binary, of size O(n), and someM ≥0such thatϕn|=#^mp2iffm=M+j·2ⁿ· 2²ⁿ, for somej≥0.

Using (the proof of) this well-known result, one can define a concept Cnthat satisfies an analogous property, namely

Tn|=Cnv3mP2 iff m=j·k·2ⁿ·2²ⁿ,for somej.

We use this conceptCn(without giving details on the shape ofCn) to describe the remaining relevant parts inTn. We include the following concept inclusions:

• Fora∈Γandq:

TapeuAa,qv ∃r.(Cnu ¬P2U(P2u_q

t

• Fora∈Γ,q∈Q, andδ(q, a) = (, b, ):

TapeuAa,qv ∃r.(Cnu ¬P2U(P2uAb,q)) (6)

• Fora∈Γ,q∈Q, andδ(q, a) = (q⁰, , r):

TapeuAa,qv3k∃r.(Cnu ¬P2U(P2u

t

• Fora∈Γ,q∈Q, andδ(q, a) = (q⁰, , l):

Tapeu3^kAa,qv ∃r.(Cnu ¬P2U(P2u

t

LetN =k·2ⁿ·2²ⁿ. Intuitively, CI (5) states that a world labeled withqis labeled with the same symbol in the next configuration, that is,N tape cells later. CI (6) ensures that, if a world is labeled with (a, q), then the corresponding worldNtape cells later is labeled with bwhenδ(q, a) = (, b, ); the corresponding state isqas the head moves left or right. Finally, CIs (7) and (8) make sure that the head is moved according to the transition. It remains to ensure that the non- head worlds are labeled withq. For this, one has to take into account the environment of a cell, as illustrated by the following CI:

TapeuAa₁,q₁u3^k(¬P2uAa₂,q₂u3^k(¬P2uAa₃,q₃)))v 3^k∃r.(Cnu ¬P2U(P2u

t

ifq1=q2=q3=qorδ(q1, a1) = (, , l)orδ(q3, a3) = (, , r).

The remaining cases are similar. In particular, at cells close to the left or right border of a configuration, it suffices to take a smaller environment into account.

Now, letw=a1· · ·anbe some input word forA. Define a con- ceptCwby taking:

Cw=TapeuCnuAa₁,q₀u

n−1

l

i=1

3^ikAa_i+1,qu 3^(n−1)k(A6b,qUP2)u3_a∈Γ,q∈F

t

Intuitively,Tapeensures a computation is initiated,Cnensures that the tape is separated into configurations,Aa1,q0and the big conjunc- tion enforces that the input word is written on the tape andA6b,qUP2

ensures that the remaining cells are labeled with blank 6b and are non-head states. Finally, the last conjunct expresses that a final state is reachable. Based on the construction, it is not hard to verify the following claim, which finishes the reduction.

Claim.Aacceptswof lengthnif there is a model ofCwandTn. o

4.1 MTL

Restricting the intervals to the form[0, c]and[c,∞)leads to bet- ter complexity also for MTLALC; however, not to EXPTIMEas for LTL^0,∞_ALC. To see this, we sketch here how to adapt the reduction used in the previous theorem to get an EXPSPACE-lower bound. A match- ing upper bound follows from Theorem 8 below.

Recall that CIs (2)-(4) provide the central idea of the reduction.

While (2) and (3) are already in MTL^0,∞_ALC, we replace (4) with CIs

¬3[0,i]> u

k−1

l

l=i+2

¬Aπ(l)vA_π(i+1), and (9) A_π(i)v ¬3^[0,i−1]>, (10) for all0≤i < k−1. Intuitively, (9) expresses that if there is a gap of at leasti(realized by¬3[0,i]>) and allA_π(l)forl > i+ 1are not satisfied, that is, there is no larger gap, concludeAπ(i+1). Together with (10), this implies that, again, a uniqueAa,qis satisfied forall domain elements in a world. Note that, as there is a fixed Turing machine with an EXPSPACE-hard word problem,kis fixed and we do not require succinct encoding here.

The remainder of the above proof deals with synchronizing infor- mation between consecutive configurations. While the conceptCn

can certainly not be defined in MTL^0,∞_ALC, we can use the succinct in- tervals to communicate between tape cells that areexponentiallyfar away. For instance, we can mark everyN=k·2ⁿ-th time point with a concept nameXbyXv♦[0,N]Xu2^[0,N−1]¬X. We thus get:

Theorem 4. Satisfiability inMTL^0,∞_ALCisEXPSPACE-complete.

5 Temporal TBoxes

We now take a look at the case where temporal operators can also be applied to concept inclusions in the TBox, which adds means for expressing dynamics of global information, e.g., in norms.

5.1 Temporal TBoxes in LTL

TemporalLTL^bin_ALC-TBoxesare defined by the following grammar:

ϕ, ψ::=CvD| ¬ϕ|ϕ∧ψ|#ϕ|ϕUIψ,

whereC, Dare LTL^bin_ALC-concepts,Ian interval. We define the truth relationI, n|=ϕ(withIan interpretation andn∈Na time point) by starting withI, n|=C vDiffC^I,n ⊆D^I,n, and extending it to the complex TBox formulas analogously to Section 2; e.g.,I, n|=

#ϕiffI, n+ 1|=ϕ.Iis amodelof a temporal LTL^binALC-TBoxϕif I,0|=ϕ.

We are concerned with the problem oftemporal TBox satisfiability, that is, the problem of deciding whether a given temporal TBoxϕhas a model. Note that, in contrast to Section 3, a concept is not part of the input because there is a model of a conceptCand a temporal TBox ϕif and only if the temporal TBox¬(> v ¬C)∧ϕis satisfiable.

Temporal TBoxes are useful to set the dynamics of protocols or norms. For example, the temporal LTL^binALC-TBox

♦[4,4](PhDStudu ∃defends.Thesisu ¬∃has.ConfPubv

#∃takes.ValidationExam) says that after 4 years there will be a norm stating that all PhD stu- dents who defend their thesis and do not have a conference publica- tion will need to take a validation exam the year after that.

The first result here is that the complexity of temporal TBox satis- fiability is exponentially higher than for concept satisfiability relative to global TBoxes; notably, the lower bound is a consequence of The- orem 3 above.

Theorem 5. Satisfiability of temporal LTL^binALC-TBoxes is 2EXPSPACE-complete.

Membership in 2EXPSPACE is a consequence of the follow- ing:(i) satisfiability of temporal LTLALC-TBoxes is EXPSPACE- complete [25], and(ii)any LTL^binALCtemporal TBox can be trans- lated into an equivalent though exponentially larger LTLALC-TBox, by expanding the succinctly encoded intervals.

For the lower bound, we reduce the satisfiability problem for MTLALC, which is 2EXPSPACE-hard cf. Theorem 3. Introduce a fresh concept nameGap, which intuitively models the “gaps” be- tween consecutive observations in MTLALC, and define the map·^† inductively by taking:

A^†=A (¬C)^†=¬(C^†) (CuD)^†=C^†uD^†

(∃r.C)^†=∃r.C^†

(#IC)^†=GapUI(¬GapuC^†) (CUID)^†= (GaptC^†)UI(¬GapuD^†) (CvD)^†= (¬GapuC^†vD^†)

It is routine to verify that:

Lemma 8. AnMTLALC-conceptCand TBoxT are satisfiable iff the following temporalLTL^binALC-TBox is satisfiable:

¬(> vGapt ¬C^†)∧ 2 ^

α∈T

α^†∧2(> vGap∨ > v ¬Gap)∧23(> v ¬Gap).

5.2 Temporal TBoxes in MTL

The syntax oftemporalMTLALC-TBoxesis obtained from the syn- tax of LTL^binALC-TBoxes by just replacing#ϕwith#Iϕ. The seman- tics is adapted accordingly as discussed in Section 4; for instance

I, n|=#Iϕ iff I, n+ 1|=ϕandτ(n+ 1)−τ(n)∈I.

Theorem 6. Satisfiability of temporal MTLALC-TBoxes is 2EXPSPACE-complete.

The lower bound is inherited from Theorem 3. For the upper bound, we lift the mapping·^† given in the proof of Theorem 5 to tempo- ral MTLALC-TBoxes. For a temporal MTLALC-TBoxϕ, defineϕ^† inductively as follows:

(ϕ∧ψ)^†=ϕ^†∧ψ^† (¬ϕ)^†=¬(ϕ^†)

(#Iϕ)^†= (> vGap)UI(> v ¬Gap∧ϕ^†) (ϕUIψ)^†= (> vGap∨ϕ^†)UI(> v ¬Gap∧ψ^†) The following Lemma, which is proved similar to Lemma 8, together with the fact that satisfiability of temporal LTL^binALC-TBoxes can be checked in 2EXPSPACEconcludes the upper bound.

Lemma 9. A temporal MTLALC-TBox ϕ is satisfiable iff ϕ^† ∧ 2(> vGap∨ > v ¬Gap)∧23(> v ¬Gap)is satisfiable.

5.3 Restriction to intervals [0, c], [c, ∞)

We have seen in Theorem 2 that the restriction to intervals of the form[0, c],[c,∞)leads to better complexity in the case of (classical) satisfiability. We show here that this in fact also applies to temporal TBoxes. In fact, the observations made in Section 3.2 apply here as well and it is fairly straightforward to extend it to this more general setting. The upper bound is then obtained by adapting a strategy that has been used for monodic first-order temporal logic,QT LU21in [15, Theorem 11.30].

Due to this proximity, we sketch only the necessary changes. We need to extend the definition of a type to reflect the information about the TBox formulas as follows. For a TBox formulaϕ, denote with sub(ϕ)the set of all subformulas ofϕtogether with all subconcepts appearing in some of these subformulas; in particular,sub(ϕ)can contain both a concept inclusionC vDand a conceptC. Similar to Section 3.2,cl(ϕ)is the closure under single negation ofsub(ϕ) extended with the set

{αU[0,c]β, αU[c,∞)β|c∈[0, K], αUIβ∈sub(ϕ)},

whereKis the largest constant inϕandα, βcould be concepts or TBox formulae. Now, a type is a subsett⊆cl(ϕ)such that:

• α∈tiff¬α6∈t, for all¬α∈cl(ϕ);

• ψ∧χ∈tiff{ψ, χ} ⊆t, for allψ∧χ∈cl(ϕ);

• DuE∈tiff{D, E} ⊆t, for allDuE∈cl(ϕ);

• CvD∈tandC∈timpliesD∈t.

As argued in Section 3.2, we only need to consider those (exponen- tially many) types, which satisfy property(P), appropriately lifted to include TBox formulas. Aquasistate forϕis a set of types with the additional requirement that the typesagree on the TBox formulas, that is,ψ ∈ tiffψ ∈ t⁰for typest, t⁰ in the same quasistate, and all TBox subformulasψ. After lifting also the run condition(R3)to apply to TBox formulas, the notion of a quasimodel remains (almost) identical: aquasimodel forϕis a pair(S,R)such that:

(Q1) ϕ∈tfor somet∈S(0); and

(Q2) for allt∈S(n),n≥0there is a runr∈Rsuch thatr(n) =t.

As before, the existence of a quasimodel forϕcharacterizes satisfia- bility ofϕ. Moreover, if there is a quasimodel, then there is a quasi- model(S,R)of the regular form

S=S(0). . . S(n−1)(S(n). . . S(n+m−1))^ω

withn≤]qs_ϕandm≤ |sub(ϕ)|·]qs_ϕ·(]ϕ)²+]qs_ϕ, where]ϕand ]qs_ϕdenote the number of types and quasistates forϕ, respectively.

Thus, both the lengthnof the initial part and the lengthmof the cycle are double exponentially bounded.

Based on this, one can devise the following algorithm, similar to [15, Lemma 11.30] and the algorithm for the proof of Theo- rem 1: guess numbersn,mwithin the mentioned bounds, and step by step the sequenceS keeping always only two consecutive qua- sistates. While guessing the sequence, verify on the fly that each type inS(i)has a compatible type inS(i+ 1), and vice versa. At time pointn, storeS(n)and continuemmore steps until reaching S(m+n) = S(n). Moreover, verify that all αUIβappearing in some type inS(n)are realized on the way toS(n+m). It should be clear that this can be done in (non-deterministic) exponential-space, yielding:

Theorem 7. Satisfiability of temporal LTL^0,∞_ALC-TBoxes is EXPSPACE-complete.

As a consequence of Lemma 9, we additionally obtain:

Theorem 8. Satisfiability of temporal MTL^0,∞_ALC-TBoxes is EXPSPACE-complete.

6 Conclusions and Future Work

In this paper, we have launched the study ofmetricTDLs allowing for quantitative temporal reasoning, and established a fairly com- plete landscape of the complexity of satisfiability for LTL^binALCand MTLALC (over the naturals). Most interestingly, we have shown that the ability to reason explicitly about timestamps of observations brings additional computational complexity. In particular, the com- plexity of concept satisfiability is then the same as that of temporal TBox satisfiability, c.f. Table 1.

As immediate future work, we will investigate TDLs based on MTL withcontinuous-semantics(over the reals). For some appli- cations, the continuous-semantics seems to be more appropriate in the sense that a real-time system is continuously observed instead of only when an event or action happens. The change from pointwise to continuous semantics is not for free since full MTL becomes un- decidable; however, several decidable fragments have been already identified [27]. We plan to build on these results and study TDLs based on decidable fragments of MTL with continuous-semantics.

We will also look at quantitative TDLs in the context ofontology- based data access (OBDA)[14] over temporal databases. We believe that the present paper lays important foundations for understand- ing thecombined complexityof thequery answering problemwith mTDLs. However, fordata complexity, i.e., when only the data is considered as part of the input, TBoxes with succinctly represented intervals can be used for free. In this case, an interesting problem is to consider data timestamped with intervals, succinctly representing its validity time. In this scenario, it would be fruitful to study restric- tions ofmTDLs based on ‘data-tractable’ DLs such as those inDL- Lite[4] orEL[11], whose temporal extensions to access temporal (timestamped) data have been recently investigated [10, 6, 16, 22].

However, none of these works studies interval encoding of times- tamps.

REFERENCES

[1] R. Alur and T. A. Henzinger, ‘Real-time logics: Complexity and ex- pressiveness’,Inf. Comput.,104(1), 35–77, (1993).

[2] R. Alur and T. A. Henzinger, ‘A really temporal logic’,J. ACM,41(1), 181–204, (1994).

[3] A. Artale, D. Bresolin, A. Montanari, G. Sciavicco, and V. Ryzhikov,

‘DL-Lite and interval temporal logics: a marriage proposal’, inProc. of ECAI-14, pp. 957–958, (2014).

[4] A. Artale, D. Calvanese, R. Kontchakov, and M. Zakharyaschev, ‘The DL-Lite family and relations’,J. Artif. Intell. Res. (JAIR),36, 1–69, (2009).

[5] A. Artale and E. Franconi, ‘A survey of temporal extensions of descrip- tion logics’,Ann. Math. Artif. Intell.,30(1-4), 171–210, (2000).

[6] A. Artale, R. Kontchakov, A. Kovtunova, V. Ryzhikov, F. Wolter, and M. Zakharyaschev, ‘First-order rewritability of temporal ontology- mediated queries’, inProc. of IJCAI-15, pp. 2706–2712, (2015).

[7] A. Artale, R. Kontchakov, C. Lutz, F. Wolter, and M. Zakharyaschev,

‘Temporalising tractable description logics’, in Proc. of TIME-07, (2007).

[8] A. Artale, R. Kontchakov, V. Ryzhikov, and M. Zakharyaschev, ‘A cookbook for temporal conceptual data modelling with description log- ics’,ACM Trans. Comput. Log.,15(3), 25, (2014).

[9] A. Artale, R. Kontchakov, V. Ryzhikov, and M. Zakharyaschev,

‘Tractable interval temporal propositional and description logics’, in Proc. of AAAI-15, pp. 1417–1423, (2015).

[10] A. Artale, R. Kontchakov, F. Wolter, and M. Zakharyaschev, ‘Temporal description logic for ontology-based data access’, inProc. of IJCAI-13, (2013).

[11] F. Baader, S. Brandt, and C. Lutz, ‘Pushing the EL envelope’, inProc.

of IJCAI-05, pp. 364–369, (2005).

[12] F. Baader, D. Calvanese, D. L. McGuinness, D. Nardi, and P. F. Patel- Schneider, eds. The Description Logic Handbook: Theory, Implemen- tation, and Applications. Cambridge University Press, 2003.

[13] F. Baader, S. Ghilardi, and C. Lutz, ‘LTL over description logic ax- ioms’,ACM Trans. Comput. Log.,13(3), 21, (2012).

[14] M. Bienvenu and M. Ortiz, ‘Ontology-mediated query answering with data-tractable description logics’, inProc. of Reasoning Web, pp. 218–

307, (2015).

[15] D. M. Gabbay, A. Kurucz, F. Wolter, and M. Zakharyaschev,Many- dimensional modal logics: theory and applications, volume 148, North Holland, 2003.

[16] V. Guti´errez-Basulto, J. C. Jung, and R. Kontchakov, ‘TemporalizedEL ontologies for accessing temporal data: Complexity of atomic queries’, inProc. of IJCAI-16, (2016).

[17] V. Guti´errez-Basulto, J. C. Jung, and C. Lutz, ‘Complexity of branching temporal description logics’, inProc. of ECAI-12, (2012).

[18] V. Guti´errez-Basulto, J. C. Jung, and T. Schneider, ‘Lightweight de- scription logics and branching time: a troublesome marriage’, inProc.

of KR-14, (2014).

[19] V. Guti´errez-Basulto, J.C. Jung, and T. Schneider, ‘Lightweight tempo- ral description logics with rigid roles and restricted TBoxes’, inProc.

of IJCAI-15, pp. 3015–3021, (2015).

[20] J. Y. Halpern and M. Y. Vardi, ‘The complexity of reasoning about knowledge and time. i. lower bounds’,J. Comput. Syst. Sci.,38(1), 195–

237, (1989).

[21] I. M. Hodkinson, R. Kontchakov, A. Kurucz, F. Wolter, and M. Za- kharyaschev, ‘On the computational complexity of decidable fragments of first-order linear temporal logics’, inProc. of TIME-ICTL 2003, pp.

91–98, (2003).

[22] R. Kontchakov, L. Pandolfo, L. Pulina, V. Ryzhikov, and M. Za- kharyaschev, ‘Temporal and spatial OBDA with many-dimensional Halpern-Shoham logic’, inProc. of IJCAI-16, (2016).

[23] R. Koymans, ‘Specifying real-time properties with metric temporal logic’,Real-Time Systems,2(4), 255–299, (1990).

[24] C. Lutz, D. Walther, and F. Wolter, ‘Quantitative temporal logics over the reals: PSpace and below’,Inf. Comput.,205(1), 99–123, (2007).

[25] C. Lutz, F. Wolter, and M. Zakharyaschev, ‘Temporal description log- ics: A survey.’, inProc. of TIME-07, pp. 3–14, (2008).

[26] Z. Manna and P. Wolper, ‘Synthesis of communicating processes from temporal logic specifications’,ACM Trans. Program. Lang. Syst.,6(1), 68–93, (1984).

[27] J. Ouaknine and J. Worrell, ‘Some recent results in metric temporal logic’, inProc. of FORMATS-08, pp. 1–13, (2008).

[28] V. Pratt, ‘Models of program logics’, inProc. of 20th Annual Sympo- sium on Foundations of Computer Science, pp. 115–122, (1979).

[29] K. Schild, ‘Combining terminological logics with tense logic’, inProc.

of EPIA, pp. 105–120. Springer, (1993).

[30] A. P. Sistla and E. M. Clarke, ‘The complexity of propositional linear temporal logics’,J. ACM,32(3), 733–749, (1985).

[31] F. Wolter and M. Zakharyaschev, ‘Temporalizing description logics’, in Proc. of FroCoS-98, pp. 379–402. Studies Press/Wiley, (1999).