Metric Temporal Logic for Ontology-Based Data Access over Log Data

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by  Diego Calvanese

Metric Temporal Logic for Ontology-Based Data Access over Log Data
D. Calvanese, E. Gu¨zel Kalaycı,V.Ryzhikov,G. Xiao and M. Zakharyaschev
FacultyofComputerScience DepartmentofComputerScience
FreeUniversityofBozen-Bolzano,Italy Birkbeck,UniversityofLondon,UK
{calvanese,kalayci,ryzhikov,xiao}@inf.unibz.it [email protected]
7
1
Abstract these periods are smaller and have a fixed length, the def-
0
2 initionabovecanstillbe adjustedbyusingtheconjunction
WepresentanewmetrictemporallogicHornMTLoverdense oftheformHFW∧X−HFW∧X−X−HFW∧.... How-
n time and its datalog extension datalogMTL. The use of
ever, first, having the data with fixed-period timestamps is
a datalogMTLisdemonstratedinthecontextofontology-based
not always a realistic assumption, and, second, doing the
J data access over meteorological data. We show decidabil-
4 ityofansweringontology-mediatedqueriesforapractically adjustmentabovecontradictstheOBDAphilosophy,where
relevantnon-recursivefragmentofdatalogMTL. Finally,we the ontologyuser is not requiredto have knowledgeof the
] discussdirectionsofthefuturework,includingthepotential structureofthedatasources. Therefore,thefollowingdefi-
O use-casesinanalyzinglogdataofenginesanddevices. nitionwouldbemorenatural
L Hurricane←⊟61hHurricaneForceWind,
>0
. Introduction
s where ⊟61h is a metric temporal operator during
c The aim of ontology-based data access >0
[ the previous hour. The logic required to express
(OBDA) (Poggietal.2008) is, on one hand, to repre-
1 sent the information from various heterogeneous data such statements is a kind of metric temporal logic
or modal logic of metric spaces; see (Koymans1990;
v sources in a unified and conceptually transparent way
Kurucz,Wolter,andZakharyaschev2005) for surveys and
6 by means of mappings. On the other hand, the ontology
furtherreferences.
7 language allows one to define concepts in terms of other
In this paper, we introduce a metric temporal logic
9 concepts, and thereby represent frequently used query
0 patterns as reusable concepts. The end-user, in that case, HornMTLwith the operator ⊟✁✄de, where ✄ is either > or
0 > (and similarly for ✁) and e,d are time distances, its fu-
can obtain the required information by means of simple
1. conceptual queries and is not required to know neither tureanalogue⊞✁✄de,aswellastheirduals ✁✄de and ✁✄de. We
0 the structure of the source data nor the definitions of the interpretthislogicoveradensetemporaldomain. Therea-
7 conceptsheisusing. son for not considering a discrete domain is that we want
1 Due to up-to-date requirements of industry (see, e.g., toabstractfromthegranularitiesoftime(periodsoftimes-
v: (Kharlamovetal.2014)) the OBDA approach is being ac- tamps)inthedatasources. Inourlogic,weallowthestate-
i tively adopted in the context of the temporal data of mentsoftheformP@ι,whereιisanintervalspecifiedbya
X streams and logs. Initially, onlythe classical non-temporal pairoftimeinstants,torepresenttheconceptualizedtempo-
r ontology languages were considered to mediate the ac- raldata. Themeaningof,say,P@(t1,t2]isthatP holdsat
a cesstotemporaldata(Gutie´rrez-BasultoandKlarman2012; alltimestbetweent1(notincludingit)andt2(includingit).
O¨zcepetal.2013;Baader,Borgwardt,andLippmann2013; We assume that we can convertdata from any source with
KlarmanandMeyer2014). Later, the ontology lan- timestamped tuples to this format by means of mappings.
guages with temporalized concepts were studied in Forexample,ifasourcecontainstheinformationoftemper-
this context (Artaleetal.2015; Kontchakovetal.2016; aturemeasurementstakeneveryhour,suchas13:21: -1◦C,
Basulto,Jung,andKontchakov2016). Such concepts are 14:21: 2◦C,15:21: -1◦C,etc.,wecanconceptualizethem
definedbymeansoflineartemporallogic(LTL);forexam- asthestatementsPositiveTemp@(13:21,14:21],etc. Note
ple,theaxiom thatwhethertoincludetheendsofintervalsornot,aswellas
whethertoconsider2◦Ctobethecaseinthehourpreceding
Hurricane←HurricaneForceWind∧ or following14:21, is the choice of the mappingdesigner.
X−HurricaneForceWind We then extendHornMTLtodatalogMTLthat also allows
forstandardDatalogreasoningaboutobjectsoftheapplica-
definesahurricaneashurricaneforcewindlastingfor1hour tiondomain(weatherstations,cities,sensors,etc.).
(X− istheprevioustimeLTLoperator). Oneeasilynotices We present a few preliminary results on datalogMTL.
that this definition works only if the temporal data arrives First, we describe a use-case of OBDA over meteorologi-
strictly in hourly periods, such as 13:21, 14:21, etc. If cal data with SQL mappings to a large real-world weather
database and datalogMTLas an ontology language. Sec- toλasfollows:
ond, we develop an ontology-mediated query answering M,t|=⊞✁dλ iff M,t′ |=λforallt′suchthat
algorithm for a non-recursive fragment datalogMTL✷ of ✄e
datalogMTL. Finally, we report some preliminary evnarlua- t′−t✄eandt′−t✁d, (2)
tionresultsshowingthefeasibilityofourapproach. M,t|=⊟✁dλ iff M,t′ |=λforallt′suchthat
✄e
t−t′✄eandt−t′✁d, (3)
HornMTLanddatalogMTL
M,t|= ✁dλ iff M,t′ |=λforsomet′suchthat
✄e
Syntax. We consider a propositional temporal logic t′−t✄eandt′−t✁d, (4)
HovoerrntMheTtLemwpitohrathldeosmetaoinfpTroispoomsiotiropnhailcvtoar(iRab,l6es)Pw0i,tPh10,a.n.d. M,t|= ✁✄deλ iff M,t′ |=λforsomet′suchthat
arithmeticoperations+,−. Thatis,weassumedensetime. t−t′✄eandt−t′✁d. (5)
Letint(T)be the setof (non-empty)intervalson T, which
We say thatM satisfies a data assertion P@ι if M,t |= P
are of the form[t1,t2], [t1,t2), (t1,t2], and (t1,t2), where forallt∈ι. WesaythatMsatisfiesanontologyaxiomλ←
tdio∈noTt∪di{s−tin∞gu,i∞sh}b,ehtwiseeeinthtehre(ionrte[r,vaanldshiti1s,e∞ith]earn)dohrt1].,(∞W)e, λfo1r∧a·ll·i·∧=λk1,(.re.s.p,ekc,tiimveplyly,⊥M←,tλ|=1∧λ·(·r·e∧spλ.,kM),i,ftM|=,t⊥|=),fλoir,
considerh∞,∞itobeempty,andanalogouslyfor−∞. We
every t ∈ T. Thus, the ontology axioms are global. We
also assume that ≤ is defined on T∪{−∞,∞} and +,−
say that M satisfies a data instance D (resp., ontology Ø)
are defined on pairsof elementsfromT and{−∞,∞}, in
if it satisfies each statement in it. Finally, we say that M
a standardway.) Define a datainstance D as a non-empty
satisfiesaknowledgebase(Ø,D)andwriteM|=(Ø,D)if
finitesetofdataassertions(orfacts)oftheform:
MsatisfiesbothØandD.
Ourmainreasoningproblemisqueryanswering. Define
Pi@ι,
an atomic query (AQ) as an expression P@δ, where P is
wherePiisapropositionalvariableandι∈int(T). a proposition and δ is an interval variable. An ontology
Ø and an AQ P@δ constitute an ontology-mediatedquery
Weusethetemporaloperatorsoftheform:
(OMQ)Q(δ) = (Ø,P@δ). AcertainanswertoQ(δ)over
– ⊞✁d(alwaysbetweeneanddinthefuture), Disanyintervalι ∈ int(T)suchthatM |= (Ø,D)implies
✄e M,t|=P forallt∈ι.
– ⊟✁d(alwaysbetweeneanddinthepast),
✄e
HornMTL✷ fragment. We consider one importantfrag-
✁d
– ✄e (sometimebetweeneanddinthefuture), ment HornMTL✷ of HornMTL, where the operators ✁d
✄e
– ✁✄de (sometimebetweeneanddinthepast), and ✁✄de aredisallowedintheheadsoftherules. Notethat
eachHornMTL✷ KBcanbeconvertedtoKBthathas⊞✁d
✄e
where✁iseither<or6,e,daredistances,thatis,positive and⊟✁d operatorsonly,andtheoriginalKBisaconserva-
elements of T, and ✄ is either > or >. Thus, e.g., ⊞<>ed tiveex✄teensionofit. Forexample,anaxiomR←P ∧ ✁dQ
expresses‘alwaysbetweeneanddinthefutureincludinge can be replaced by the pair of axioms R ← P ∧ Q′✄aend
and excludingd’ andsimilarly forthe otheroperators. We
⊞✁dQ′ ← Q. Finally, we consider a non-recursive frag-
alsoimposethefollowingconsistencyrequirementonevery ✄e
mentHornMTL✷ ofHornMTL✷ by adoptingthe simplest
operator O✁d (henceforth we assume O ∈ {⊞,⊟, , }, nr
✄e definition of non-recursivivity: consider the relation ≺ on
✷∈{⊞,⊟},and✸∈{ , }):
the symbols of Ø defined as P ≺ Q iff there is an axiom
– thereexistst∈Tsuchthatt✄eandt✁d. inØ,whereP occursintheheadandQinthebody(P de-
pendsonQ). We requirethatP ≺∗ P fornosymbolP in
Propositionalliteralsaredefinedbythefollowinggrammar: Ø,where≺∗ isatransitiveclosureof≺.
λ ::= Pi | O✁✄deλ. datalogMTL. ConsiderthepredicatesymbolsP0,P1,...,
each of some arity m ≥ 0, and a set of object vari-
Anontology,Ø,isafinitesetofaxiomsoftheform:
ablesx0,x1,.... Data instances D here contain assertions
P(c)@ι, where P is an m-ary predicate symbol, c an m-
λ←λ1∧···∧λk, ⊥←λ1∧···∧λk. (1) tupleofindividualconstants,andι∈int(T). Thisassertion
saysthatP(c) istrueatι. We denotebyind(D) theset of
Aknowledgebase(KB)isapair(Ø,D).
allindividualconstantsin D. AnontologyØ is a finite set
ofaxiomsoftheform(1)withthe literalsλ definedbythe
grammar:
Semantics. ConsideraninterpretationM = (T,·M)such
that PiM ⊆ T for each propositionalvariable Pi and write λ ::=(τ 6=τ′) | (τ =τ′) | P(x) | O✁✄deλ,
M,t|=Pi whent∈PiM fort∈T. Asusual,itisassumed whereP is a predicatesymbolof arity m, x is a vector of
thatM,t 6|= ⊥forallt ∈ T. Weextendthedefinitionof|= mvariables,andτ,τ′ areindividualterms,i.e.,variablesor
constants. We also impose other standard datalog restric- SELECT ID AS x,
tions on our programs, and forbid (in)equality predicates lag(TIME) over (partition
in the heads. We call the predicates occurring in D exten- by ID order by TIME) AS t1,
sional and those occurring in the head of the axioms of Ø TIME AS t2, "(" AS h, "]" AS i
intentional. An interpretation, M, is based on the domain FROM Weather
∆=ind(D)(fortheindividualvariablesandconstants)and WHERE P01I > lag(P01l)
T.Foranym-arypredicateP,m-tuplecfrom∆andt∈T, over(partition by ID order by TIME)
M specifies whether P is true on c at t, in which case we
writeM,t |= P(c). Letν beanassignmentofelementsof Thatis, weextracttheintervalsoftheshape(t1,t2], where
t1andt2arethetwonexttimestampsforagivenstation.The
∆toindividualterms(weadoptthestandardnameassump-
ends of the interval are chosen to reflect the fact that, e.g.,
tion:ν(c)=c,foreveryindividualconstantc). Weset:
theprecipitationismeasuredaccumulativelyandthedevice
M,t|=ν τ 6=τ′ iff ν(x0)6=ν(x1), producestheoutputintheendofthemeasurementinterval.
M,t|=ν τ =τ′ iff ν(x0)=ν(x1), Analogously to Precipitation, we populate by the data the
otherextensionalpredicates,suchasPositiveTemp(temper-
M,t|=ν P(x)iff M,t|=ν P(ν(x)), aturewellabove0◦C),HurricaneForceWind(windwiththe
and use inductively the formulas (2)–(5) with |=ν instead speedabove118km/h),TempAbove24andTempAbove41
of |= for the cases O✁dλ. We say M satisfies an ontology (temperatureabove24and41◦C).
✄e
axiomλ←λ1∧···∧λk (respectively,⊥←λ1∧···∧λk), Considertheontologycontainingtheaxioms:
ifM,t|=ν λiforeachiimpliesM,t|=ν λ(resp.,M,t|=ν Rain(x)←PositiveTemp(x)∧Precipitation(x),
⊥), foreveryt ∈ Tandassignmentν. Finally,Msatisfies
adataassertionP(c)@ιifM,t |= P(c)foreacht ∈ι,and ⊟6>10hHurricane(x)←⊟6>10hHurricaneForceWind(x),
M|=(Ø,D)isdefinedstraightforwardly. ⊟624hExcessiveHeat(x)←⊟624hTempAbove24(x)∧
AQsaredefinedasP(x)@δ,whereP isapredicatesym- >0 >0
bol of arity m, and δ is an intervalvariable. An ontology- 624hTempAbove41(x),
>0
mediated query is defined Q(x,δ) = (Ø,P(x)@δ). A
The second axiom is already discussed in the introduction
certain answer to Q(x,δ) over D is any pair (c,ι), such
(here we use a slightly modified version to say that hurri-
that c = ν(x) for some ν, and M |= (Ø,D) implies
caneholdsalso atthe time point, when the hurricaneforce
M,t|=P(c)forallt∈ι.
windbegins),whereasthelastaxiomformalizesthedefini-
NotethatHornMTLisafragmentofdatalogMTL(where
tionofthesituationwhenanexcessiveheatwarningshould
all predicates have arity 0). We also consider the frag-
msaemnetssydnattaaclotigcMreTsLtr✷ictaionndsdaastHaloorgnMMTTLL✷n✷radnedfiHneodrnwMiTthLt✷nhre. bhmeoauixsrissmuweadlitahabcotchvoeerd4mi1ni◦ngiCtmo)a.tlheteUmSpeWraetautrheerabFoovreec2a4st◦OCffiacneds(t2h4e
We can also populate the binary predicate
WeatherUseCase
LocationOf(x,y)@ht1,t2ibyusing:
OurOBDAapproachcanbeusedtoanalyzemeteorological
data through ontology-mediated queries. The MesoWest1 SELECT COUNTY AS x, ID AS y,
project makes publicly available historical records of the −∞ AS t1, ∞ AS t2, "(" AS h, ")" AS i
FROM Metadata
weatherstationsacrosstheUS.Thisdataisavailableinthe
relationaltablesWeathercontainingthefollowingfields: Note that we assume that LocationOf holds between a
ID. StationID.Example:KHYS. countyandastationglobally.Itisnowpossibletodefine:
TIME. Timestamp.Example:11-11-20158:55CST. HurricaneAffectedCounty(x)←
TMP. Temperature.Example:15.6◦C. LocationOf(x,y)∧Hurricane(y),
SKNT. WindSpeed.Example:9.2km/h. SpreadRainCounty(x)←LocationOf(x,y)∧
P01I. Precipitationin1hour.Example:0.09cm. LocationOf(x,z)∧(y 6=z)∧Rain(y)∧Rain(z).
Moreover, there are metadata tables Metadata containing, ✷
Query Answering indatalogMTL
inparticular,locationinformationofstationsinthefields: nr
In this section we first present an algorithm for computing
ID. StationID.Example:KHYS.
certainanswerstoanHornMTL✷ OMQQ(δ)=(Ø,P@δ)
nr
COUNTY. Example:Ellis.
overD.
STATE. Example:Kansas.
We can conceptualize this raw data by means of the SQL NormalformforHornMTL✷ . Ourprocedureworkson
nr
mappings. For example, to extract the data for the exten- theontologyØcontainingonlytheclausesoftheshape:
sional predicate Precipitation(x)@ht1,t2i (with the mean- P ←Q∧R, ⊥←Q∧R,
ingprecipitationoccursatxduringht1,t2i),wecanusethe
followingSQLquery: ⊞✁✄deP ←Q, ⊟✁✄de P ←Q,
1http://mesowest.utah.edu/ P ←⊞✁✄deQ, P ←⊟✁✄deQ
It is an easy exercise to verify that everyHornMTL✷nr can and (t2,t3). To compute P∗ in general we first produce a
bebroughttothenormalformbyperformingthefollowing closureQ′ofQ∗ astheminimaltablesatisfying:
operations:
∗ ′
Q (cid:0)t1,t2,h,i(cid:1)→Q(cid:0)t1,t2,h,i(cid:1),
– Substitutetheaxiomsoftheshapeλ←λ1∧···∧λk for ∗ ′ ′ ′ ′ ′ ′
k ≥ 3 by k −1 axioms with binary conjunctions using Q (cid:0)t1,t2,h,i(cid:1)∧Q(cid:0)t1,t2,h,i(cid:1)∧(t2 ≤t2)∧
freshsymbols. Analogouslyfortheaxiomswith⊥inthe ints(cid:0)t1,t2,h,i,t′1,t′2,h′,i′(cid:1)→Q′(cid:0)t′1,t2,h′,i(cid:1).
head.
AfterthatP∗canbeobtainedby:
– Remove ✸✁dλ literals in the body of the axioms as
sketchedin✄Pereliminaries. Q′(cid:0)t1,t2,h,i(cid:1)∧fit(cid:0)t1,t2,h,i,e,d,✄,✁(cid:1)→
– Removethenestedmodalities✷✁✄deλbysubstitutingthem P∗(cid:0)t1−e,t2−d,de(cid:0)h,✄(cid:1),de(cid:0)i,✁(cid:1)(cid:1),
for✷✁✄dePλ,forafreshsymbolsPλ,andadding: where fit(cid:0)t1,t2,h,i,e,d,✄,✁(cid:1) is ⊤, if there exists t ∈ T
– Pλ ←λ,if✷✄✁deλoccurredinthebodyoftheaxiom, such that {t + t′ | t′ ✄ eandt′ ✁ d} ⊆ ht1,t2i, and
– λ←Pλ,if✷✄✁deλoccurredintheheadoftheaxiom. t⊥′✄otehearnwdits′e✁. dE}ssceanntibaellsyh,iffittedhsooldtshaitfitthfietssiengsimdeenhtt1{,tt′2i|.
– Removetheaxiomsoftheshapeλ0 ← λ1 ∧λ2, ifλi = Finally, anotheredgefunctionde is neededto computethe
✷✁✄deP forsome0 ≤ i ≤ 2,asdescribedintheprevious ends of the resulting interval. Here de(cid:0)h,✄(cid:1) is [, if either
step. Analogouslyfortheaxiomswith⊥inthehead. h is ( and ✄ is >, or h is [ and ✄ is >; otherwise de(cid:0)h,✄(cid:1)
It can be readily verified that the resulting ontology in the is (. The definition of de(cid:0)i,✁(cid:1) is symmetric. The axiom
normalformisinHornMTL✷nr. ⊟✁✄deP ←Qishandledanalogously. Observethatthecom-
putationofQ′requiresrecursion.
Algorithm. WefirstassumethatthefactsofD arestored Clearly,whenP occursintheheadofseveralaxioms,the
in thetablesofthe shapePi∗(t1,t2,h,i), wheret1,t2 ∈ T, tableP∗ istakenequaltotheunionofthetablescomputed
h is either ( or [, and i is either ) or ]. E.g., for D = above. Infact,foreverysymbolP inØthealgorithmcom-
{Pi@(t1,t2],Pi@[t′1,t′2]}weproducethetablePi∗withtwo putesP∗that,foraconsistentKB(Ø,D),satisfies:
tuples {(cid:0)t1,t2,(,](cid:1),(cid:0)t′1,t′2,[,](cid:1)}. Consider an intentional • foreveryt ∈ T, there exists a certainanswer ι to OMQ
symbol P and assume that for all Q such that P ≺ Q the Q(δ) = (Ø,P@δ)overD suchthatt ∈ ιiffthereexists
tablesQ∗ arecomputed.Considernowthecases: atuple(cid:0)t1,t2,h,i(cid:1)inP∗ suchthatt∈ht1,t2i.
P ← Q∧R. Then P∗ is computedas the minimaltable This correctness follows directly from the semantics of
satisfyingthecondition: HornMTL✷ . Then, if the table ⊥∗ is empty, as an output
nr
∗ ∗ ′ ′ ′ ′ of the OMQ Q(δ) = (Ø,G@δ) over D we produce the
Q (cid:0)t1,t2,h,i(cid:1)∧R (cid:0)t1,t2,h,i(cid:1)∧ table G∗ (otherwise, we return G∗ with one special tuple
′ ′ ′ ′ ∗ ′′ ′′ ′′ ′′
ints(cid:0)t1,t2,h,i,t1,t2,h,i(cid:1)→P (cid:0)t1,t2,h ,i (cid:1), (cid:0)−∞,∞,(,)(cid:1) as (Ø,D) is inconsistent). Clearly, the cor-
rectness above guarantees that G∗ represents the set of all
where ints(t1,t2,h,i,t′1,t′2,h′,i′) is ⊤ if ht1,t2i ∩ certainanswers.
h′t′1,t′2i′ 6= ∅ (the intervals intersect), otherwise it is ⊥, OnecanextendtheapproachpresentedabovetoOMQan-
and h′′t′1′,t′2′i′′ = ht1,t2i ∩ h′t′1,t′2i′ (the result of the sweringindatalogMTL✷ . Indeed,itispossibleto convert
intersection). Note that P∗ is computed as a temporal anarbitrarydatalogMTLn✷r ontologytotheoneinthenormal
join (Gaoetal.2005) of Q∗ and R∗. We also create a ta- formsimilartothatusednarbove.ThetablesP∗ needtocon-
ble⊥∗ fortheaxioms⊥←Q∧R. tain the tuples of the shape (cid:0)c1,...,cm,t1,t2,h,i(cid:1), where
⊞⊞⊞✁dP ← Q. Then P∗ is computed as a minimal table m is the arity of P. The rules for processing the temporal
✄e
satisfying: axiomsessentiallyremainthesame. Therulesforcomput-
ingtheconjunctions(joins)needtobeadjustedtocorrectly
Q∗(cid:0)t1,t2,h,i(cid:1)→P∗(cid:0)t1+e,t2+d,ed(cid:0)h,✄(cid:1),ed(cid:0)i,✁(cid:1)(cid:1) handletheindividualargumentsofthepredicates.
where the edge function ed(h,✄) returns [, if h is [ and ✄ Discussionand Future Work
is>, and(, otherwise. Thened(i,✁)is definedsymmetri-
Initial Experiments. We made experiments to evalu-
cally. Forexample,ifQ∗ = {(cid:0)t1,t2,(,](cid:1)}andtheaxiomis ate the performance of the proposed algorithm on the
⊞<>edP ←Q,thenP∗ ={(cid:0)t1+e,t2+d,(,)(cid:1)}. Theaxiom Hurricane(x)@δ and ExcessiveHeat(x)@δ OMQs with the
⊟✁dP ←Qishandledanalogously. ontologyfromtheweatherusecase. Weimplementedtheal-
✄e
gorithmoftheprevioussection,foragivenOMQ,asanSQL
P ← ⊞⊞⊞✁✄deQ. Consider the following example: let Q∗ = query using WITH clause and the RECURSIVE operator.
{(cid:0)t1,t2,(,](cid:1),(cid:0)t2,t3,(,)(cid:1)}andtheaxiomP ←⊟<>edQsuch Thatis,theintermediatetablesofthealgorithmaredefined
thatd−e<t3−t1. Then,accordingtothesemantics,P∗ = as a sequence of virtual SQL tables. The configuration of
{(cid:0)t1−e,t3−d,(,](cid:1)}.InordertocomputeP∗correctlywe thecomputerthatwasusedfortheexperimentsisIntelCore
need to consider the concatenation of the intervals (t1,t2] [email protected],8GBRAMwith1867MHzDDR3andOSX
ElCapitanoperatingsysteminversion10.11.4.Theweather 2015. First-order rewritability of temporal ontology-
dataisstoredin64bitPostgreSQLversion9.4.5.Weranthe mediatedqueries. InProceedingsofthe Twenty-FourthIn-
queries over a table including 140881 rows. It took 3199 ternational Joint Conference on Artificial Intelligence, IJ-
ms for Hurricane and 481876 ms for ExcessiveHeat to re- CAI2015,BuenosAires,Argentina,July25-31,2015,2706–
trieve the results. We interpret this outcome as a positive 2712.
indicationofthefeasibilityofourapproach:evenastraight- [Baader,Borgwardt,andLippmann2013] Baader, F.; Borg-
forward implementation appears to work. We foresee the wardt, S.; and Lippmann, M. 2013. Temporalizing
followingthreedirectionsofthefuturework:
ontology-baseddataaccess.InProc.ofthe24thInt.Conf.on
Automated Deduction (CADE-24), volume 7898 of LNCS,
New Use Cases. Our language is capable of expressing 330–344. Springer.
complexpatternsofeventsthatareofinterestforsuchpur- [Basulto,Jung,andKontchakov2016] Basulto, V. G.; Jung,
posesasdiagnosticsofenginesordevices.Theaxiom J.; andKontchakov,R. 2016. TemporalizedELontologies
foraccessingtemporaldata: Complexityofatomicqueries.
SmoothShutDown←IdleRPM∧⊟><015minIntermRPM∧ In Proceedings of the 25th International Joint Conference
625minRunningRPM, onArtificialIntelligence(IJCAI-16). AAAIPress.
>15min
[Gaoetal.2005] Gao, D.; Jensen, C. S.; Snodgrass, R. T.;
forinstance, describestheeventof smoothshutdownofan
andSoo,M.D.2005.Joinoperationsintemporaldatabases.
engine as being in an idle state after having intermediate
TheVLDBJournal14(1):2–29.
speed(RpM)for15minutesandhavingarunningspeedbe-
[Gutie´rrez-BasultoandKlarman2012] Gutie´rrez-Basulto,
forethat(notfurtherthan25minutes).Theaxiom:
V., and Klarman, S. 2012. Towards a unifying approach
ConsHighVibration←⊟650sec 610secHighVibration to representing and querying temporal data in description
>0 >0
logics. InProc.ofthe6thInt.Conf.onWebReasoningand
describesconsistenthighvibrationashighvibrationoccur- Rule Systems (RR 2012), volume 7497 of LNCS, 90–105.
ringevery10secondsduringaminute.OurOBDAapproach Springer.
seemstobeabletocapturemanyindustrialuse-cases.Inthe
[HirshfeldandRabinovich2005] Hirshfeld, Y., and Rabi-
future,weplantoinvestigatesuchpotentialapplications.
novich, A. 2005. Timer formulas and decidable metric
temporal logic. Information and Computation 198(2):148
Open Theoretical Problems. At the moment, we do –178.
not know whether OMQ answering in HornMTLis decid- [Kharlamovetal.2014] Kharlamov, E.; Solomakhina, N.;
able. In fact, this question is open even for the fragment O¨zc¸ep, O¨.; Zheleznyakov, D.; Hubauer, T.; Lamparter, S.;
HornMTL✷. We plan to obtain complexity results for Roshchin,M.;Soylu,A.;andWatson,S.2014.Howseman-
those languages, and we are particularly interested in tictechnologiescanenhancedataaccessatSiemensEnergy.
data complexity (that is, the complexity in the size of D InProc. ofthe 13thInt.SemanticWeb Conf.(ISWC2014),
when Q(δ) is assumed to be fixed). It is also important PartI,volume8796ofLNCS,601–619. Springer.
to understand how the complexity results for HornMTL
[KlarmanandMeyer2014] Klarman, S., and Meyer, T.
carry over to datalogMTL. To achieve our goal, we
2014.QueryingtemporaldatabasesviaOWL2QL.InProc.
plan to study various techniques developed in the area
ofthe8thInt.Conf.onWebReasoningandRuleSystems(RR
of metric temporal logics (OuaknineandWorrell2005;
2014),volume8741ofLNCS,92–107. Springer.
OuaknineandWorrell2008;
HirshfeldandRabinovich2005) and modal [Kontchakovetal.2016] Kontchakov, R.; Pandolfo, L.;
logics over metric spaces (Kutzetal.2003; Pulina, L.; Ryzhikov, V.; and Zakharyaschev, M. 2016.
Sheremet,Wolter,andZakharyaschev2010; Temporalandspatialobdawithmany-dimensionalhalpern-
WolterandZakharyaschev2005). shohamlogic.InProceedingsofthe25thInternationalJoint
Conference on Artificial Intelligence (IJCAI-16). AAAI
Press.
Implementation and Optimizations. The proposed
query answering algorithm for datalogMTL✷ clearly [Koymans1990] Koymans, R. 1990. Specifying real-time
nr
properties with metric temporal logic. Real-Time Systems
allows for optimizations. For example, computing the
transitive closure of the table Q∗ when processing the 2(4):255–299.
axiom P ← ⊞✁dQ seems to be avoidable. Moreover, our [Kurucz,Wolter,andZakharyaschev2005] Kurucz, A.;
✄e
algorithmdoesnotmakeanyassumptionregardingthetem- Wolter,F.; andZakharyaschev,M. 2005. Modallogicsfor
poralorderingofthetuples. Ifsucharealisticassumptionis metric spaces: Open problems. In We Will Show Them!
made,wemaybeabletodevelopmoreefficientalgorithms, EssaysinHonourofDovGabbay,VolumeTwo,193–108.
inparticular,byusingindexesontimestamps. [Kutzetal.2003] Kutz, O.; Wolter, F.; Sturm, H.; Suzuki,
N.;andZakharyaschev,M. 2003. Logicsofmetricspaces.
References ACMTrans.Comput.Log.4(2):260–294.
[Artaleetal.2015] Artale, A.; Kontchakov, R.; Kovtunova, [OuaknineandWorrell2005] Ouaknine, J., and Worrell, J.
A.; Ryzhikov, V.; Wolter, F.; and Zakharyaschev, M. 2005. Onthedecidabilityofmetrictemporallogic. InPro-
ceedings of the 20th Annual IEEE Symposium on Logic in
Computer Science, LICS ’05, 188–197. Washington, DC,
USA:IEEEComputerSociety.
[OuaknineandWorrell2008] Ouaknine, J., and Worrell, J.
2008. Some recent results in metric temporal logic. In
Proceedingsofthe6thInternationalConferenceonFormal
Modeling and Analysis of Timed Systems, FORMATS ’08,
1–13. Berlin,Heidelberg:Springer-Verlag.
[O¨zcepetal.2013] O¨zcep, O.; Mo¨ller, R.; Neuenstadt, C.;
Zheleznyakov, D.; and Kharlamov, E. 2013. A semantics
fortemporalandstream-basedqueryansweringinanOBDA
context. Technical report, Deliverable D5.1, FP7-318338,
EU.
[Poggietal.2008] Poggi, A.; Lembo, D.; Calvanese, D.;
DeGiacomo,G.;Lenzerini,M.;andRosati,R. 2008. Link-
ingdatatoontologies. J.onDataSemanticsX:133–173.
[Sheremet,Wolter,andZakharyaschev2010] Sheremet, M.;
Wolter, F.; and Zakharyaschev, M. 2010. A modal logic
framework for reasoning about comparative distances and
topology. Ann.PureAppl.Logic161(4):534–559.
[WolterandZakharyaschev2005] Wolter, F., and Za-
kharyaschev,M. 2005. Alogicformetricandtopology. J.
Symb.Log.70(3):795–828.