Description
Log
ics—
Basics,App
lications,andMore
IanHorrocks
Inform
ationManagem
entGroup
Universityof
Manchester,UK
UlrikeSattler
Institut
furTheoretischeInform
atik
TU
Dresden,Germany
TU
Dre
sden
Ger
man
y1
Overview
oftheTutorial
•HistoryandBasics:Syntax,
Sem
antics,ABoxes,Tboxes,InferenceProblem
s
andtheirinterrelationship,
andRelationshipwithother(log
ical)form
alisms
•ApplicationsofDLs:
ER-diagram
swithi.com
demo,
ontologies,etc.
includ
ing
system
demon
stration
•ReasoningProcedures:
simpletableaux
andwhy
they
work
•ReasoningProceduresII:morecomplex
tableaux,no
n-standard
inferenceprob
-
lems
•Complexityissues
•Implementing/OptimisingDLsystems
TU
Dre
sden
Ger
man
y2
Description
Log
ics
•family
oflogic-basedknow
ledg
erepresentation
form
alismswell-suited
forthe
representation
ofandreason
ingab
out
➠terminologicalknowledge
➠configurations
➠ontologies
➠databaseschemata
–schemadesign
,evolution,
andqu
eryop
timisation
–source
integrationin
heterogeneou
sdatabases/data
warehou
ses
–conceptual
modelling
ofmultidimension
alaggregation
➠...
•descendentsof
semantics
networks,fram
e-basedsystem
s,andKL-O
NE
•akaterm
inolog
ical
KRsystem
s,conceptlang
uages,etc.
TU
Dre
sden
Ger
man
y3
Architectureof
aStand
ardDLSystem
. . .
Con
cret
eSitua
tion
Ter
min
olog
y
Father
=Manu∃
haschild.>...
Human
=Mammalu
Biped
. . .
John:Humanu
Father
John
haschild
Bill
Kno
wle
dge
Bas
eI N F E R E N C E S Y S T E M
I N T E R F A C E
Des
crip
tion
Log
ic
TU
Dre
sden
Ger
man
y4
Introdu
ctionto
DLI
ADescription
Log
ic-mainlycharacterisedby
asetofconstructors
that
allow
tobu
ildcomplexconcepts
androlesfrom
atom
icon
es,
concepts
correspon
dto
classes/areinterpretedassets
ofob
jects,
rolescorrespon
dto
relation
s/areinterpretedasbinaryrelationson
objects,
Example:HappyFatherintheDLALC
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�������
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�������
�������
�������
Manu(∃
has-child.Blue)u
(∃has-child.Green)u
(∀has-child.HappytRich)
TU
Dre
sden
Ger
man
y5
Introdu
ctionto
DL:SyntaxandSem
antics
ofALC
Sem
antics
givenby
means
ofaninterpretationI=(∆
I,·I):
Con
structor
Syntax
Example
Sem
antics
atom
icconcept
AHum
anAI⊆∆I
atom
icrole
Rlikes
RI⊆∆I×∆I
ForC,D
concepts
andR
arole
name
conjun
ction
CuD
Hum
anu
Male
CI∩DI
disjun
ction
CtD
Nicet
Rich
CI∪DI
negation
¬C
¬Meat
∆I\CI
exists
restrict.∃R.C
∃has-child.Hum
an{x|∃y.〈x,y〉∈RI∧y∈CI}
valuerestrict.∀R.C
∀has-child.Blond{x|∀y.〈x,y〉∈RI⇒
y∈CI}
TU
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sden
Ger
man
y6
Introdu
ctionto
DL:Other
DLCon
structors
Con
structor
Syntax
Example
Sem
antics
number
restriction
(≥nR)
(≥7has-child){x||{y.〈x,y〉∈RI}|≥n}
(≤nR)
(≤1has-mother){x||{y.〈x,y〉∈RI}|≤n}
inverserole
R−
has-child−
{〈x,y〉|〈y,x〉∈RI}
trans.
role
R∗
has-child∗
(RI)∗
concrete
domain
u1,...,u
n.P
h-father·age,age.>
{x|〈uI 1,...,uI n〉∈P}
etc.
ManydifferentDLs/DLconstructors
have
beeninvestigated
TU
Dre
sden
Ger
man
y7
Introdu
ctionto
DL:Knowledg
eBases:TBoxes
For
term
inolog
ical
know
ledg
e:TBoxcontains
Conceptdefinitions
A=
C(A
aconceptname,C
acomplex
concept)
Father=
Manu∃has-child.Hum
an
Hum
an=
Mam
malu∀has-child−.H
uman
;introdu
cemacros/names
forconcepts,canbe(a)cyclic
Axioms
C1v
C2
(Cicomplex
concepts)
∃favourite.Breweryv∃drinks.B
eer
;restrict
your
models
AninterpretationIsatisfies
aconceptdefinition
A. =C
iffAI=CI
anaxiom
C1vC
2iff
CI 1⊆CI 2
aTBox
TiffI
satisfies
alldefinition
sandaxiomsinT
;I
isamodelofT
TU
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sden
Ger
man
y8
Introdu
ctionto
DL:Knowledg
eBases:ABoxes
For
assertionalknow
ledg
e:ABoxcontains
Conceptassertions
a:C
(aan
individu
alname,C
acomplex
concept)
John:Manu∀has-child.(Maleu
Happy)
Roleassertions
〈a1,a
2〉:R
(aiindividu
alnames,R
arole)
〈Joh
n,Bill〉:has-child
AninterpretationIsatisfies
aconceptassertion
a:C
iffaI∈CI
aroleassertion
〈a1,a
2〉:R
iff〈aI 1,aI 2〉∈RI
anABox
AiffI
satisfies
allassertions
inA
;I
isamodelofA
TU
Dre
sden
Ger
man
y9
Introdu
ctionto
DL:Basic
InferenceProblem
s
Subsumption:CvD
IsCI⊆DIin
allinterpretation
sI?
w.r.t.TBoxT:CvTD
IsCI⊆DIin
allmodelsI
ofT?
;structureyour
know
ledg
e,compu
tetaxono
my
Consistency:IsC
consistent
w.r.t.T?
IsthereamodelI
ofTwithCI6=∅?
ofABoxA:IsA
consistent?
IsthereamodelofA?
ofKB
( T,A
):Is
(T,A
)consistent?
Isthereamodelof
bothTandA?
InferenceProblem
sarecloselyrelated:
CvTD
iffCu¬D
isinconsistent
w.r.t.T,
(nomodelofI
hasan
instance
ofCu¬D
)
Cisconsistent
w.r.t.T
iffnotCvTAu¬A
;DecisionProcduresforconsistency(w.r.t.TBoxes)suffice
TU
Dre
sden
Ger
man
y10
Introdu
ctionto
DL:Basic
InferenceProblem
sII
For
mostDLs,thebasicinferenceprob
lemsaredecidable,
withcomplexitiesbetween
Pand
Exp
Tim
e.
Whyisdecidabilityimportant?
Why
does
semi-decidabilityno
tsuffice?
Ifsubsum
ption(and
henceconsistency)
isun
decidable,
and
➠subsum
ptionissemi-decidable,
then
consistencyisnotsemi-decidable
➠consistencyissemi-decidable,
then
subsum
ptionisnotsemi-decidable
➠Quest
fora“h
ighlyexpressive”DLwith“practicable”inferenceprob
lems
whereexpressiveness
depends
ontheapplication
practicabilitychangedover
thetime
TU
Dre
sden
Ger
man
y11
Introdu
ctionto
DL:History
Com
plexityof
Inferences
provided
byDLsystem
sover
thetime
late
’80s
earl
y’9
0s’9
0sm
id’9
0sla
te
Un
dec
idab
le
Exp
Tim
e
PS
pac
e
NP
PT
ime
Inve
stig
atio
n of
Com
plex
ity o
f Inf
eren
ce P
robl
ems/
Alg
orith
ms
star
ts
Cra
ck, K
ris
Cla
ssic
(A
T&
T)
Lo
om
KL
-ON
EN
IKL
Fac
t,
D
LP
, Rac
e
TU
Dre
sden
Ger
man
y12
Introdu
ctionto
DL:State-of-the-im
plem
entation
-art
Inthelast
5years,DL-based
system
swerebu
iltthat
✔canhand
leDLsfarmoreexpressive
thanALC
(close
relativesof
converse-D
PDL)
•Num
ber
restrictions:“p
eoplehaving
atmost2cats
andexactly1do
g”
•Com
plex
roles:
inverse(“has-child”—
“child-of”),
transitive
closure(“off
spring
”—
“has-child”),
role
inclusion(“has-daug
hter”—
“has-child”),etc.
✔im
plem
entprovably
soun
dandcompleteinferencealgorithms
(for
ExpTim
e-completeprob
lems)
✔canhand
lelargeknow
ledg
ebases
(e.g.,Galen
medical
term
inolog
yon
tology:2,74
0concepts,41
3roles,1,21
4axioms)
✔arehigh
lyop
timised
versions
oftableau-basedalgorithms
✔perform
(surprisinglywell)on
benchmarks
formodallogicreason
ers
(Tableaux’98
,Tableaux’99
)
TU
Dre
sden
Ger
man
y13
RelationshipwithOther
Log
ical
Formalisms:
First
Order
Predicate
Log
ic
MostDLsaredecidablefragmentsofFOL:Introdu
ce
aun
arypredicateA
foraconceptnameA
abinary
relationR
forarole
nameR
Translate
complex
conceptsC,D
asfollows:
t x(A)=A(x),
t y(A)=A(y),
t x(CuD)=t x(C)∧t x(D),
t y(CuD)=t y(C)∧t y(D),
t x(CtD)=t x(C)∨t x(D),
t y(CtD)=t y(C)∨t y(D),
t x(∃R.C)=∃y.R(x,y)∧t y(C),
t y(∃R.C)=∃x.R(y,x)∧t x(C),
t x(∀R.C)=∀y.R(x,y)⇒
t y(C),
t y(∀R.C)=∀x.R(y,x)⇒
t x(C).
ATBoxT={C
i. =D
i}istranslated
as
ΦT=∀x.
∧
1≤
i≤n
t x(C
i)⇔
t x(D
i)
TU
Dre
sden
Ger
man
y14
RelationshipwithOther
Log
ical
Formalisms:
First
Order
Predicate
Log
icII
Cisconsistent
iffitstranslationt x(C)issatisfiable,
Cisconsistent
w.r.t.T
iffitstranslationt x(C)∧ΦT
issatisfiable,
CvD
ifft x(C)⇒
t x(D)isvalid
CvTD
iffΦ
t⇒∀x.(t x(C)⇒
t x(D))
isvalid.
;ALC
isafragmentof
FOLwith2variables(L2),know
nto
bedecidable
;ALC
withinverserolesandBooleanop
eratorson
rolesisafragmentof
L2
;furtheradding
number
restrictions
yields
afragmentof
C2
(L2with“cou
ntingqu
antifiers”),know
nto
bedecidable
✦in
contrast
tomostDLs,adding
transitive
roles(binaryrelation
s/
transitive
closureop
erator)to
L2leadstoundecidability
✦manyDLs(likemanymodallogics)arefragments
oftheGuardedFragment
✦mostDLsareless
complex
than
L2:
L2isNExpTim
e-complete,
mostDLsarein
ExpTim
e
TU
Dre
sden
Ger
man
y15
RelationshipwithOther
Log
ical
Formalisms:
ModalLog
ics
DLsandModalLog
icsarecloselyrelated:
ALCÀ
multi-m
odalK:
CuDÀ
C∧D,
CtDÀ
C∨D
¬CÀ¬C
,
∃R.CÀ〈R〉C
,∀R.CÀ[R]C
transitive
rolesÀ
transitive
fram
es(e.g.,inK4)
regu
larexpression
son
rolesÀ
regu
larexpression
son
prog
rams(e.g.,in
PDL)
inverserolesÀ
converse
prog
rams(e.g.,in
C-PDL)
number
restrictionsÀ
determ
inisticprog
rams(e.g.,in
D-PDL)
➫no
TBoxes
availablein
modallogics
;“internalise”
axiomsusingaun
iversalroleu:C
. =DÀ[u](C⇔
D)
➫no
ABox
availablein
modallogics
;useno
minals
TU
Dre
sden
Ger
man
y16
Ap
plic
atio
ns
of
Des
crip
tio
nL
og
ics
App
licat
ions
–p.
1/9
Ap
plic
atio
nA
reas
I
☞Te
rmin
olog
ical
KR
and
Ont
olog
ies
•D
Lsin
itial
lyde
sign
edfo
rte
rmin
olog
ical
KR
(and
reas
onin
g)•
Nat
ural
tous
eD
Lsto
build
and
mai
ntai
non
tolo
gies
☞S
eman
ticW
eb•
Sem
anti
cm
arku
pw
illbe
adde
dto
web
reso
urce
s➙
Aim
is“m
achi
neun
ders
tand
abili
ty”
•M
arku
pw
illus
eO
nto
log
ies
topr
ovid
eco
mm
onte
rms
ofre
fere
nce
with
clea
rse
man
tics
•R
equi
rem
entf
orw
ebba
sed
onto
logy
lang
uage
➙W
elld
efine
dse
man
tics
➙B
uild
son
exis
ting
Web
stan
dard
s(X
ML,
RD
F,R
DF
S)
•R
esul
ting
lang
uage
(DA
ML+
OIL
)is
bas
edo
na
DL
(SHIQ
)•
DL
reas
on
ing
can
beus
edto
,e.g
.,➙
Sup
port
onto
logy
desi
gnan
dm
aint
enan
ce➙
Cla
ssify
reso
urce
sw
.r.t.
onto
logi
es
App
licat
ions
–p.
2/9
Ap
plic
atio
nA
reas
II
☞C
onfig
urat
ion
•C
lass
icsy
stem
used
toco
nfigu
rete
leco
ms
equi
pmen
t•
Cha
ract
eris
tics
ofco
mpo
nent
sde
scrib
edin
DL
KB
•R
easo
ner
chec
ksva
lidity
(and
pric
e)of
confi
gura
tions
☞S
oftw
are
info
rmat
ion
syst
ems
•La
SS
IEsy
stem
used
DL
KB
for
flexi
ble
softw
are
docu
men
tatio
nan
dqu
ery
answ
erin
g
☞D
atab
ase
appl
icat
ions
☞..
.
App
licat
ions
–p.
3/9
Dat
abas
eS
chem
aan
dQ
uer
yR
easo
nin
g
☞DLR
(n-a
ryD
L)ca
nca
ptur
ese
man
tics
ofm
any
conc
eptu
alm
odel
ling
met
hodo
logi
es(e
.g.,
EE
R)
☞S
atis
fiabi
lity
pres
ervi
ngm
appi
ngto
SHIQ
allo
ws
use
ofD
Lre
ason
ers
(e.g
.,Fa
CT,
RA
CE
R)
☞D
LA
box
can
also
capt
ure
sem
antic
sof
conj
unct
ive
quer
ies
•C
anre
ason
abou
tque
ryco
ntai
nmen
tw.r.
t.sc
hem
a
☞D
Lre
ason
ing
can
beus
edto
supp
ort
•S
chem
ade
sign
,evo
lutio
nan
dqu
ery
optim
isat
ion
•S
ourc
ein
tegr
atio
nin
hete
roge
neou
sda
taba
ses/
data
war
ehou
ses
•C
once
ptua
lmod
ellin
gof
mul
tidim
ensi
onal
aggr
egat
ion
☞E
.g.,
I.CO
MIn
telli
gent
Con
cept
ualM
odel
ling
tool
(Enr
ico
Fran
coni
)•
Use
sFa
CT
syst
emto
prov
ide
reas
onin
gsu
ppor
tfor
EE
R
App
licat
ions
–p.
4/9
I.CO
MD
emo
App
licat
ions
–p.
5/9
Term
ino
log
ical
KR
and
On
tolo
gie
s
☞G
ener
alre
quire
men
tfor
med
ical
term
inol
ogie
s
☞S
tatic
lists
/taxo
nom
ies
diffi
cult
tobu
ildan
dm
aint
ain
•N
eed
tobe
very
larg
ean
dhi
ghly
inte
rcon
nect
ed•
Inev
itabl
yco
ntai
nm
any
erro
rsan
do
mis
sio
ns
☞G
alen
proj
ecta
ims
tore
plac
est
atic
hier
arch
yw
ithD
L•
Des
crib
eco
ncep
ts(e
.g.,
spira
lfra
ctur
eof
left
fem
ur)
•U
seD
Lcl
assi
fier
tobu
ildta
xon
om
y
☞N
eede
dex
pres
sive
DL
and
effic
ient
reas
onin
g•
Des
crip
tions
use
tran
sitiv
e/in
vers
ero
les,
GC
Iset
c.•
Ver
yla
rge
KB
s(t
ens
ofth
ousa
nds
ofco
ncep
ts)
➙E
ven
prot
otyp
eK
Bis
very
larg
e(≈
3,00
0co
ncep
ts)
➙E
xist
ing
(inco
mpl
ete)
clas
sifie
rto
ok≈
24h
ou
rsto
clas
sify
KB
➙Fa
CT
syst
em(s
ound
and
com
plet
e)ta
kes≈
60se
con
ds A
pplic
atio
ns–
p.6/
9
Rea
son
ing
Su
pp
ort
for
On
tolo
gy
Des
ign
☞D
Lre
ason
erca
nbe
used
tosu
ppor
tdes
ign
and
mai
nten
ance
☞E
xam
ple
isO
ilEd
onto
logy
edito
r(f
orD
AM
L+O
IL)
•Fr
ame
base
din
terf
ace
(like
Pro
tegé
,Ont
oEdi
t,et
c.)
•E
xten
ded
tocl
arify
sem
antic
san
dca
ptur
ew
hole
DA
ML+
OIL
lang
uage
➙S
lots
expl
icitl
yex
iste
ntia
lor
valu
ere
stric
tions
➙B
oole
anco
nnec
tives
and
nest
ing
➙P
rope
rtie
sfo
rsl
otre
latio
ns(t
rans
itive
,fun
ctio
nale
tc.)
➙G
ener
alax
iom
s
☞R
easo
ning
supp
ortf
orO
ilEd
prov
ided
byFa
CT
syst
em•
Fram
ere
pres
enta
tion
tran
slat
edin
toSHIQ
•C
omm
unic
ates
with
FaC
Tvi
aC
OR
BA
inte
rfac
e•
Indi
cate
sin
cons
iste
ncie
san
dim
plic
itsu
bsum
ptio
ns•
Can
mak
eim
plic
itsu
bsum
ptio
nsex
plic
itin
KB
App
licat
ions
–p.
7/9
DA
ML
+OIL
Med
ical
Term
ino
log
yE
xam
ple
s
E.g
.,D
AM
L+O
ILm
edic
alte
rmin
olog
yon
tolo
gy
☞Tr
ansi
tive
role
sca
ptur
etr
ansi
tive
part
onom
y,ca
usal
ity,e
tc.
Sm
okin
gv
∃ca
uses
.C
ance
rpl
usC
ance
rv
∃ca
uses
.D
eath
⇒C
ance
rv
Fata
lThi
ng
☞G
CIs
repr
esen
tadd
ition
alno
n-de
finiti
onal
know
ledg
e
Sto
mac
h-U
lcer
. =U
lcer
u∃ha
sLoc
atio
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tom
ach
plus
Sto
mac
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v∃ha
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-Of-
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ero
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ure
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es/c
ause
dBy
rela
tions
hip
Dea
thu∃ca
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By.
Sm
okin
gv
Pre
mat
ureD
eath
⇒S
mok
ingv
Cau
seO
fPre
mat
ureD
eath
☞C
ardi
nalit
yre
stric
tions
add
cons
iste
ncy
cons
trai
nts
Blo
odP
ress
urev
∃ha
sVal
ue.(H
ight
Low
)u
61h
asV
alue
plus
Hig
hv
¬Lo
w⇒
Hig
hLow
Blo
odP
ress
urev
⊥
App
licat
ions
–p.
8/9
OilE
dD
emo
App
licat
ions
–p.
9/9
Reasoning
Procedu
res:
DecidingCon
sistency
ofALCN
Con
cepts
Asawarm-up,
wedescribeatableau-basedalgorithm
that
•decidesconsistencyofALCN
concepts,
•triesto
build
a(tree)
modelI
forinpu
tconceptC
0,
•breaks
downC
0syntactically,inferringconstraintson
elem
ents
inI,
•usestableaurulescorrespon
ding
toop
eratorsinALCN
(e.g.,→u,→∃)
•works
non-determ
inistically,in
PSpace
•stop
swhenclash
occurs
•term
inates
•returns“C
0isconsistent”iffC
0isconsistent
TU
Dre
sden
Ger
man
y17
Reasoning
Procedu
res:
Tableau
Algorithm
•works
onatree
(sem
antics
throug
hview
ingtree
asan
ABox):
nodes
representelem
ents
of∆I,labelledwithsub-concepts
ofC
0
edges
representrole-successorshipsbetweenelem
ents
of∆I
•works
onconcepts
innegationnormalform
:pu
shnegation
inside
usingde
Morgan’
lawsand
¬(∃R.C)
;∀R.¬C
¬(∀R.C)
;∃R.¬C
¬(≤
nR)
;(≥
(n+1)R)
¬(≥
nR)
;(≤
(n−1)R)(n≥1)
¬(≥
0R)
;Au¬A
•isinitialised
withatree
consisting
ofasing
le(root)no
dex
0with
L(x
0)={C
0}:
•atreeT
contains
aclash
if,forano
dex
inT,
{A,¬A}⊆
L(x)or
{(≥
mR),(≤
nR)}⊆
L(x)forn<m
•returns“C
0isconsistent”if
rulescanbeapplieds.t.
they
yield
clah-free,
complete(nomorerulesapply)
tree
TU
Dre
sden
Ger
man
y18
Reasoning
Procedu
res:ALC
Tableau
Rules
����
����
����
����
��
��
��
����
����
����
�����������������������������������������������������������������
�����������������������������������������������������������������
����
{C
1t
C2,C
,...}
x{C
1u
C2,...}
x{C
1u
C2,C
1,C
2,...}
x{C
1t
C2,...}
→u
x{∃R
.C,...}
x
{C}
{∃R
.C,...}
R y x R y{...,C}
yRx{∀R
.C,...}
{...}
{∀R
.C,...}
→∃
→∀
→t
forC∈{C
1,C
2}
x
TU
Dre
sden
Ger
man
y19
Reasoning
Procedu
res:N
Tableau
Rules
����
����
����
����
��
��
���������
���������
���������
���������
���������
����
����
����
����
����
����
����
����
����
���� �
��������
���������
���������
���������
���������
���������
���������
���������
���������
���������
����
����
����
����
����
����
���������
���������
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���������
���������
���������
���������
���������
���������
!!!!
!!!!
!!!!
!!!!
!!!! "
""""""""
"""""""""
"""""""""
"""""""""
"""""""""
#########
#########
#########
#########
#########
$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$%$
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((
)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*%*
x{(≥
nR
),...}
xha
sno
R-s
ucc.
x
R
mer
getw
oR
-suc
cs.
{(≤
nR
),...}
x{(≤
nR
),...}
...
R
>n
→≤
x R y
→≥
{}
{(≥
nR
),...}
TU
Dre
sden
Ger
man
y20
Reasoning
Procedu
res:
Sou
ndness
andCom
pleteness
LemmaLetC
0beanALCN
conceptandT
obtained
byapplying
the
tableaurulestoC
0.Then
1.therule
applicationterminates,
2.ifT
isclash-free
andcomplete,
thenT
defines(canon
ical)(tree)
modelforC
0,and
3.ifC
0hasamodelI,then
therulescanbeappliedsuch
that
they
yield
aclash-free
andcompleteT.
Corollary
(1)The
tableaualgorithm
isa(P
Space)decision
procedu
refor
consistency(and
subsum
ption)
ofALCN
concepts
(2)ALCN
hasthetree
modelprop
erty
TU
Dre
sden
Ger
man
y21
Reasoning
Procedu
res:
Sou
ndness
andCom
pletenessII
ProofoftheLemma
1.(Termination)
The
algorithm
“mon
oton
ically”constructs
atree
who
se
depth
islinearin|C
0|:
quantifier
depthdecreasesfrom
node
tosuccs.
breadth
islinearin|C
0|(evenifnu
mber
inNRsarecodedbinarily)
2.(C
anon
ical
model)Com
plete,
clash-free
treeT
definesa(tree)
pre-modelI:
nodesx
correspon
dto
elem
entsx∈∆I
edgesx
R →y
define
role-relationship
x∈AI
iffA∈
L(x)forconceptnamesA
;Easyto
thatC∈
L(x)⇒
x∈CI—
ifC6=(≥
nR)
If(≥
nR)∈
L(x),
thenx
might
havelessthannR-successors,bu
t
the→≥-ruleensuresthat
thereis≥1R-successor...
TU
Dre
sden
Ger
man
y22
Reasoning
Procedu
res:
Sou
ndness
andCom
pletenessIII
copysomeR-successors(including
sub-trees)
toob
tainnR-successors:
����
����
����
����
��
��
��
����
����
���������
���������
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���������
���������
���������
���������
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���������
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����
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����
����
����
����
����
����
���� �
���������
����������
����������
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����������
���������
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���������
���������
������������������������
������������������������
������������������������
������������������������
������������������������
�����������������������
�����������������������
�����������������������
�����������������������
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����
����
����
����
����
����
����
����
����
���� �
��������
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���������
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!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!
""##
=n
x
...
R{(≥
nR
),...}
x
...
R{(≥
nR
),...}
...
<n
;canonicaltreemodelforinputconcept
3.(C
ompleteness)
Use
modelI
ofC
0tosteerapplicationof
non-determ
isticrules
(→t,→≤)viamapping
π:Nodesof
Tree−→∆I
with
C∈
L(x)⇒
π(x)∈CI.
Thiseasily
impliesclash-freenesof
thetree
generated.
TU
Dre
sden
Ger
man
y23
MaketheTableau
Algorithm
runin
PSpace:
TomakethetableaualgorithmruninPSpace:
①ob
servethat
branches
areindependent
from
each
other
②ob
servethat
each
node
(lab
el)requ
ires
linearspaceon
ly
③recallthat
pathsareof
leng
th≤|C
0|
④construct/search
thetreedepthfirst
⑤re-use
spacefrom
alreadyconstructedbranches
;spacepolynom
ialin|C
0|sufficesforeach
branch/for
thealgorithm
;tableaualgorithm
runs
inNPspace(Savitch:NPspace=
PSpace)
TU
Dre
sden
Ger
man
y24
Reasoning
Procedu
res:
Extensibility
Thistableaualgorithm
canbemodified
toaPSpace
decision
procedu
refor
✔ALC
withqualifyingnumberrestrictions
(≥nRC)and(≤
nRC)
✔ALC
withinverseroleshas-child−
✔ALC
withroleconjunction
∃(RuS).C
and∀(RuS).C
✔TBoxeswithacyclicconceptdefinitions:
unfolding
(macro
expansion)
iseasy,bu
tsubop
timal:
may
yieldexpon
ential
blow
-up
lazyunfolding
(unfolding
ondemand)
isop
timal,consistencyin
PSpace
decidable
TU
Dre
sden
Ger
man
y25
Reasoning
Procedu
res:
ExtensibilityII
Langu
ageextensions
that
requ
iremoreelab
oratetechniqu
esinclud
e
➠TBoxeswithgeneralaxiomsC
ivD
i:
each
node
mustbelabelledwith¬C
itD
i
quantifier
depthno
long
erdecreases
;term
inationno
tgu
aranteed
➠Transitiveclosureofroles:
node
labels(∀R∗.C)yieldsC
inallR
n-successor
labels
quantifier
depthno
long
erdecreases
;term
inationno
tgu
aranteed
Useblocking(cycle
detection)
toensure
term
ination
(but
therigh
tblockingto
retain
soun
dnessandcompleteness)
TU
Dre
sden
Ger
man
y26
Rea
son
ing
Pro
ced
ure
sII
Rea
soni
ngPr
oced
ures
II–
p.1/
9
No
n-T
erm
inat
ion
☞A
sal
read
ym
entio
ned,
forALC
with
gen
eral
axio
ms
basi
cal
gorit
hmis
no
n-t
erm
inat
ing
☞E
.g.
ifhu
man
v∃ha
s-m
othe
r.hu
man
∈T
,the
n¬
hum
ant∃ha
s-m
othe
r.hu
man
adde
dto
ever
yno
de
L(w
)=
{hu
man
,(¬
hum
ant∃
has-
mot
her.
hum
an),∃
has-
mot
her.
hum
an}
w y
has-
mot
her
xL
(x)
={hu
man
,(¬
hum
ant∃
has-
mot
her.
hum
an),∃
has-
mot
her.
hum
an}
has-
mot
her
L(y
)=
{hu
man
,(¬
hum
ant∃
has-
mot
her.
hum
an),∃
has-
mot
her.
hum
an}
Rea
soni
ngPr
oced
ures
II–
p.2/
9
Blo
ckin
g
☞W
hen
crea
ting
new
node
,che
ckan
cest
ors
for
equa
l(su
pers
et)
labe
l
☞If
such
ano
deis
foun
d,ne
wno
deis
blo
cked
L(w
)=
{hu
man
,(¬
hum
ant∃
has-
mot
her.
hum
an),∃
has-
mot
her.
hum
an}
w x
has-
mot
her
L(x
)=
{hu
man
,(¬
hum
ant∃
has-
mot
her.
hum
an)}
Blo
cked
Rea
soni
ngPr
oced
ures
II–
p.3/
9
Blo
ckin
gw
ith
Mo
reE
xpre
ssiv
eD
Ls
☞S
impl
esu
bset
bloc
king
may
notw
ork
with
mor
eco
mpl
exlo
gics
☞E
.g.,
reas
onin
gw
ithin
vers
ero
les
•E
xpan
ding
node
labe
lcan
affe
ctpr
edec
esso
r•
Labe
lofb
lock
ing
node
can
affe
ctpr
edec
esso
r•
E.g
.,te
stin
gC
u∃S.C
w.r.
t.T
box
T=
{>v
∀R−
.(∀S−
.¬C
),>
v∃R
.C}
w
x
S
y
R
L(x
)=
{C
,∀R−
.(∀S−
.¬C
),L
(y)
={C
,∀R−
.(∀S−
.¬C
),
L(w
)=
{C
,∃S.C
,∀R−
.(∀S−
.¬C
),
∃R
.C}
∃R
.C}
∃R
.C}
Blo
cked
Blo
cked
Rea
soni
ngPr
oced
ures
II–
p.4/
9
Dyn
amic
Blo
ckin
g
☞S
olut
ion
(for
inve
rse
role
s)is
dyn
amic
blo
ckin
g•
Blo
cks
can
bees
tabl
ishe
dbr
oken
and
re-e
stab
lishe
d•
Con
tinue
toex
pand
∀R
.Cte
rms
inbl
ocke
dno
des
•C
heck
that
cycl
essa
tisfy
∀R
.Cco
ncep
ts
z
w
x
S
R
y
R
∃R
.C,∀S−
.¬C
,¬
C}
L(x
)=
{C
,∀R−
.(∀S−
.¬C
),
∃R
.C,∀S−
.¬C}
L(z
)=
{C
,∀R−
.(∀S−
.¬C
),
∃R
.C}
L(y
)=
{C
,∀R−
.(∀S−
.¬C
),
L(w
)=
{C
,∃S.C
,∀R−
.(∀S−
.¬C
),
∃R
.C}
Cla
sh
Rea
soni
ngPr
oced
ures
II–
p.5/
9
No
n-fi
nit
eM
od
els
☞W
ithnu
mbe
rre
stric
tions
som
esa
tisfia
ble
conc
epts
have
only
non-
finite
mod
els
☞E
.g.,
test
ing¬
Cw
.r.t.T
={>
v∃R
.C,>
v6
1R−
}
w yx
R R
L(w
)=
{¬
C,∃R
.C,6
1R−
}
L(x
)=
{C
,∃R
.C,6
1R−
}
L(y
)=
{C
,∃R
.C,6
1R−
}
R
mod
elm
ust
beno
n-fin
ite R
easo
ning
Proc
edur
esII
–p.
6/9
Inad
equ
acy
of
Dyn
amic
Blo
ckin
g
☞W
ithno
n-fin
item
odel
s,ev
endy
nam
icbl
ocki
ngno
teno
ugh
☞E
.g.,
test
ing¬
Cw
.r.t.T
={>
v∃R
.(C
u∃R−
.¬C
),>
v6
1R−
}
w yx
R R−L(w
)=
{¬
C,∃R
.(C
u∃R−
.¬C
),6
1R−
}
L(x
)=
{(C
u∃R−
.¬C
),∃R
.(C
u∃R−
.¬C
),6
1R−
,C
,∃R−
.¬C}
Blo
cked
L(y
)=
{(C
u∃R−
.¬C
),∃R
.(C
u∃R−
.¬C
),6
1R−
,C
,∃R−
.¬C}
But
∃R−
.¬C
∈L
(y)
not
sati
sfied
Inco
nsis
tenc
ydu
eto
61R
−
∈L
(y)
and
C∈
L(x
)
Rea
soni
ngPr
oced
ures
II–
p.7/
9
Do
ub
leB
lock
ing
I
☞P
robl
emdu
eto
∃R−
.¬C
term
on
lysa
tisfie
din
pre
dec
esso
rof
bloc
king
node
w x
R
L(w
)=
{¬
C,∃R
.(C
u∃R−
.¬C
),6
1R−
}
L(x
)=
{(C
u∃R−
.¬C
),∃R
.(C
u∃R−
.¬C
),6
1R−
,C
,∃R−
.¬C}
☞S
olut
ion
isD
ou
ble
Blo
ckin
g(p
airw
ise
bloc
king
)•
Pre
dece
ssor
sof
bloc
ked
and
bloc
king
node
sal
soco
nsid
ered
•In
part
icul
ar,∃
R.C
term
ssa
tisfie
din
pred
eces
sor
ofbl
ocki
ngno
dem
usta
lso
besa
tisfie
din
pred
eces
sor
ofbl
ocke
dno
de¬
C∈
L(w
)
Rea
soni
ngPr
oced
ures
II–
p.8/
9
Do
ub
leB
lock
ing
II
☞D
ueto
pairw
ise
cond
ition
,blo
ckno
long
erho
lds
☞E
xpan
sion
cont
inue
san
dco
ntra
dict
ion
disc
over
ed
w yx
R R
L(w
)=
{¬
C,∃R
.(C
u∃R−
.¬C
),6
1R−
}
L(x
)=
{(C
u∃R−
.¬C
),∃R
.(C
u∃R−
.¬C
),6
1R−
,C
,∃R−
.¬C
,¬
C}
L(y
)=
{(C
u∃R−
.¬C
),∃R
.(C
u∃R−
.¬C
),6
1R−
,C
,∃R−
.¬C}
Cla
sh Rea
soni
ngPr
oced
ures
II–
p.9/
9
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
ALC+
GC
IsALCIQ
R+
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
· R+
dec
lare
role
sas
tran
sitive
Ffe
ature
chai
n(d
is)a
gree
men
t
uan
d∀
only
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
Iin
vers
ero
les:
h-c
hild−
NN
Rs:
(≥n
h-c
hild
)
TU
Dre
sden
Ger
man
y27
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
ALC+
GC
IsALCIQ
R+
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
NN
Rs:
(≥n
h-c
hild
)
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
· R+
dec
lare
role
sas
tran
sitive
Ffe
ature
chai
n(d
is)a
gree
men
t
uan
d∀
only
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
TU
Dre
sden
Ger
man
y28
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
ALC+
GC
IsALCIQ
R+
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
· R+
dec
lare
role
sas
tran
sitive
Ffe
ature
chai
n(d
is)a
gree
men
t
uan
d∀
only
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
TU
Dre
sden
Ger
man
y29
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
ALC+
GC
IsALCIQ
R+
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
· R+
dec
lare
role
sas
tran
sitive
Ffe
ature
chai
n(d
is)a
gree
men
t
uan
d∀
only
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
TU
Dre
sden
Ger
man
y30
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
ALC+
GC
IsALCIQ
R+
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
· R+
dec
lare
role
sas
tran
sitive
Ffe
ature
chai
n(d
is)a
gree
men
t
uan
d∀
only
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
TU
Dre
sden
Ger
man
y31
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
ALC+
GC
IsALCIQ
R+
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
· R+
dec
lare
role
sas
tran
sitive
Ffe
ature
chai
n(d
is)a
gree
men
t
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
uan
d∀
only
TU
Dre
sden
Ger
man
y32
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
f1...f
n↓
g1...g
man
df
1...f
n↑
g1...g
m
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
ALC+
GC
IsALCIQ
R+
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
· R+
dec
lare
role
sas
tran
sitive
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
uan
d∀
only
Ffe
ature
chai
n(d
is)a
gree
men
t
fi,
gi
funct
ional
role
s
TU
Dre
sden
Ger
man
y33
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
ALC+
GC
Is
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
uan
d∀
only
Ffe
ature
chai
n(d
is)a
gree
men
t
ALCIQ
R+
· R+
dec
lare
role
sas
tran
sitive
TU
Dre
sden
Ger
man
y34
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
· R+
dec
lare
role
sas
tran
sitive
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
uan
d∀
only
Ffe
ature
chai
n(d
is)a
gree
men
t
ALCIQ
R+
ALC
TU
Dre
sden
Ger
man
y35
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
· R+
dec
lare
role
sas
tran
sitive
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
uan
d∀
only
Ffe
ature
chai
n(d
is)a
gree
men
t
ALCIQ
R+
ALC
TU
Dre
sden
Ger
man
y36
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
· R+
dec
lare
role
sas
tran
sitive
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
uan
d∀
only
Ffe
ature
chai
n(d
is)a
gree
men
t
ALCIQ
R+
ALC
TU
Dre
sden
Ger
man
y37
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
·¬,∩
,∪B
oole
anop
son
role
s
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
uan
d∀
only
Ffe
ature
chai
n(d
is)a
gree
men
t
ALCIQ
R+
ALC
· R+
dec
lare
role
sas
tran
sitive
TU
Dre
sden
Ger
man
y38
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
uan
d∀
only
Ffe
ature
chai
n(d
is)a
gree
men
t
ALCIQ
R+
ALC
· R+
dec
lare
role
sas
tran
sitive
·¬,∩
,∪B
oole
anop
son
role
s
TU
Dre
sden
Ger
man
y39
Com
plexityof
DLs:
Overview
oftheCom
plexityof
Con
cept
Con
sistency
(co-
)NP
PPSpa
ceExp
Tim
eN
Exp
Tim
e
ALC
reg
ALCNO
add
regu
lar
role
s
add
univ
ersa
lro
leALC
u
ALCO
+QI
still
inE
xpT
ime
wrt
gener
alT
Box
es
ual
.N
Rs:
(≥n
h-c
hild
Blo
nd)
Iin
vers
ero
les:
h-c
hild−
Onom
inal
s:”J
ohn”
isa
conce
pt
withou
tt
ALN
subsu
mption
of
wrt
acyc
.T
Box
es
subsu
mption
ofFL
0
wrt
acyc
.T
Box
esALCF
ALCF
ALCIO
ALC¬
ALCIQO
ALC¬
,∩,∪
ALCN
(wrt
acyc.
TB
oxes
)
FL
0(c
o-N
P)
ALUN
(NP
)w
ithou
t∃
,on
ly¬
A
ALE
(co-
NP
)
only¬
Aw
ithou
tt
and
NR
s,
add
role
hie
rarc
hie
sALCHIQ
R+
NN
Rs:
(≥n
h-c
hild
)
uan
d∀
only
Ffe
ature
chai
n(d
is)a
gree
men
t
ALCIQ
R+
ALC
· R+
dec
lare
role
sas
tran
sitive
·¬,∩
,∪B
oole
anop
son
role
s
TU
Dre
sden
Ger
man
y40
Com
plexityof
DLs:
Whatwas
left
out
Weleft
outavarietyof
complexityresultsfor
➫conceptconsistency
ofotherDLs
(e.g.,thosewith“con
cretedo
mains”)
➫otherstandardinferences
(e.g.,deciding
consistencyof
ABoxes
w.r.t.TBoxes)
➫“non-standard”inferencessuch
as
–matchingandun
ification
ofconcepts
–rewriting
concepts
–leastcommon
subsum
er(ofasetof
concepts)
–mostspecificconcept(ofan
ABox
individu
al)
TU
Dre
sden
Ger
man
y41
Imp
lem
enti
ng
DL
Sys
tem
s
Impl
emen
tatio
n–
p.1/
14
Nai
veIm
ple
men
tati
on
s
Pro
blem
sin
clud
e:
☞S
pace
usag
e●
Sto
rage
requ
ired
for
tabl
eaux
data
stru
ctur
es●
Rar
ely
ase
rious
prob
lem
inpr
actic
e
☞T
ime
usag
e●
Sea
rch
requ
ired
due
tono
n-de
term
inis
ticex
pans
ion
●S
erio
us
prob
lem
inpr
actic
e●
Miti
gate
dby
:➙
Car
eful
cho
ice
of
alg
ori
thm
➙H
ighl
yo
pti
mis
edim
ple
men
tati
on
Impl
emen
tatio
n–
p.2/
14
Car
efu
lCh
oic
eo
fA
lgo
rith
m
☞Tr
ansi
tive
role
sin
stea
dof
tran
sitiv
ecl
osur
e●
Det
erm
inis
ticex
pans
ion
of∃R
.C,e
ven
whe
nR
∈R
+
●(R
elat
ivel
y)si
mpl
ebl
ocki
ngco
nditi
ons
●C
ycle
sal
way
sre
pres
ent(
part
of)
cycl
ical
mod
els
☞D
irect
algo
rithm
/impl
emen
tatio
nin
stea
dof
enco
ding
s●
GC
Iaxi
oms
can
beus
edto
“enc
ode”
addi
tiona
lop
erat
ors/
axio
ms
●P
ower
fult
echn
ique
,par
ticul
arly
whe
nus
edw
ithF
Lcl
osur
e●
Can
enco
deca
rdin
ality
cons
trai
nts,
inve
rse
role
s,ra
nge/
dom
ain,
...
➙E
.g.,
(dom
ain
R.C
)≡
∃R
.>v
C
●(F
L)en
codi
ngs
intr
oduc
e(la
rge
num
bers
of)
axio
ms
●B
UT
even
sim
ple
dom
ain
enco
ding
isd
isas
tro
us
with
larg
enu
mbe
rsof
role
s
Impl
emen
tatio
n–
p.3/
14
Hig
hly
Op
tim
ised
Imp
lem
enta
tio
n
Opt
imis
atio
npe
rfor
med
at2
leve
ls
☞C
ompu
ting
clas
sifi
cati
on
(par
tialo
rder
ing)
ofco
ncep
ts●
Obj
ectiv
eis
tom
inim
ise
num
ber
ofsu
bsum
ptio
nte
sts
●C
anus
est
anda
rdor
der-
theo
retic
tech
niqu
es➙
E.g
.,us
een
han
ced
trav
ersa
ltha
texp
loits
info
rmat
ion
from
prev
ious
test
s●
Als
ous
est
ruct
ural
info
rmat
ion
from
KB
➙E
.g.,
tose
lect
orde
rin
whi
chto
clas
sify
conc
epts
☞C
ompu
ting
sub
sum
pti
on
betw
een
conc
epts
●O
bjec
tive
isto
min
imis
eco
stof
sing
lesu
bsum
ptio
nte
sts
●S
mal
lnum
ber
ofha
rdte
sts
can
dom
inat
ecl
assi
ficat
ion
time
●R
ecen
tDL
rese
arch
has
addr
esse
dth
ispr
oble
m(w
ithco
nsid
erab
lesu
cces
s)
Impl
emen
tatio
n–
p.4/
14
Op
tim
isin
gS
ub
sum
pti
on
Test
ing
Op
tim
isat
ion
tech
niq
ues
broa
dly
fall
into
2ca
tego
ries
☞P
re-p
roce
ssin
gop
timis
atio
ns●
Aim
isto
sim
plif
yK
Ban
dfa
cilit
ate
subs
umpt
ion
test
ing
●La
rgel
yal
gorit
hmin
depe
nden
t●
Par
ticul
arly
impo
rtan
twhe
nK
Bco
ntai
nsG
CIa
xiom
s
☞A
lgor
ithm
icop
timis
atio
ns●
Mai
nai
mis
tore
du
cese
arch
spac
edu
eto
non-
dete
rmin
ism
●In
tegr
alpa
rtof
impl
emen
tatio
n●
But
ofte
nge
nera
llyap
plic
able
tose
arch
base
dal
gorit
hms Im
plem
enta
tion
–p.
5/14
Pre
-pro
cess
ing
Op
tim
isat
ion
s
Use
fult
echn
ique
sin
clud
e
☞N
orm
alis
atio
nan
dsi
mpl
ifica
tion
ofco
ncep
ts●
Refi
nem
ento
ftec
hniq
uefir
stus
edin
KRIS
syst
em●
Lexi
cally
norm
alis
ean
dsi
mpl
ifyal
lcon
cept
sin
KB
●C
ombi
new
ithla
zyun
fold
ing
inta
blea
uxal
gorit
hm●
Faci
litat
esea
rlyde
tect
ion
ofin
cons
iste
ncie
s(c
lash
es)
☞A
bsor
ptio
n(s
impl
ifica
tion)
ofge
nera
laxi
oms
●E
limin
ate
GC
Isby
abso
rbin
gin
to“d
efini
tion”
axio
ms
●D
efini
tion
axio
ms
effic
ient
lyde
altw
ithby
lazy
expa
nsio
n
☞A
void
ance
ofpo
tent
ially
cost
lyre
ason
ing
whe
neve
rpo
ssib
le●
Nor
mal
isat
ion
can
disc
over
“obv
ious
”(u
n)sa
tisfia
bilit
y●
Str
uctu
rala
naly
sis
can
disc
over
“obv
ious
”su
bsum
ptio
n
Impl
emen
tatio
n–
p.6/
14
No
rmal
isat
ion
and
Sim
plifi
cati
on
☞N
orm
alis
eco
ncep
tsto
stan
dard
form
,e.g
.:●
∃R
.C−→
¬∀R
.¬C
●C
tD
−→
¬(¬
Cu¬
D)
☞S
impl
ifyco
ncep
ts,e
.g.:
●(D
uC
)u
(Au
D)−→
Au
Cu
D
●∀R
.>−→
>
●..
.u
Cu
...u¬
Cu
...−→
⊥
☞La
zily
unfo
ldco
ncep
tsin
tabl
eaux
algo
rithm
●U
sena
mes
/poi
nter
sto
refe
rto
com
plex
conc
epts
●O
nly
add
stru
ctur
eas
requ
ired
bypr
ogre
ssof
algo
rithm
●D
etec
tcla
shes
betw
een
lexi
cally
equi
vale
ntco
ncep
ts
{Hap
pyFa
ther
,¬H
appy
Fath
er}−→
clas
h{∀
has-
child
.(D
octo
rtLa
wye
r),∃
has-
child
.(¬
Doc
toru
¬La
wye
r)}−→
sear
ch
Impl
emen
tatio
n–
p.7/
14
Ab
sorp
tio
nI
☞R
easo
ning
w.r.
t.se
tofG
CIa
xiom
sca
nbe
very
cost
ly●
GC
ICv
Dad
dsD
t¬
Cto
ever
yno
dela
bel
●E
xpan
sion
ofdi
sjun
ctio
nsle
ads
tose
arch
●W
ith10
axio
ms
and
10no
des
sear
chsp
ace
alre
ady
2100
●G
AL
EN
(med
ical
term
inol
ogy)
KB
cont
ains
hu
nd
red
sof
axio
ms
☞R
easo
ning
w.r.
t.“p
rimiti
vede
finiti
on”
axio
ms
isre
lativ
ely
effic
ient
●F
orC
Nv
D,a
ddD
on
lyto
node
labe
lsco
ntai
ning
CN
●F
orC
Nw
D,a
dd¬
Do
nly
tono
dela
bels
cont
aini
ng¬
CN
●C
anex
pand
defin
ition
sla
zily
➙O
nly
add
defin
ition
saf
ter
othe
rlo
cal(
prop
ositi
onal
)ex
pans
ion
➙O
nly
add
defin
ition
son
est
epat
atim
e
Impl
emen
tatio
n–
p.8/
14
Ab
sorp
tio
nII
☞Tr
ansf
orm
GC
Isin
topr
imiti
vede
finiti
ons,
e.g.
●C
Nu
Cv
D−→
CNv
Dt¬
C
●C
Nt
Cw
D−→
CNw
Du¬
C
☞A
bsor
bin
toex
istin
gpr
imiti
vede
finiti
ons,
e.g.
●C
Nv
A,C
Nv
Dt¬
C−→
CNv
Au
(Dt¬
C)
●C
Nw
A,C
Nw
Du¬
C−→
CNw
At
(Du¬
C)
☞U
sela
zyex
pans
ion
tech
niqu
ew
ithpr
imiti
vede
finiti
ons
●D
isju
nctio
nson
lyad
ded
to“r
elev
ant”
node
labe
ls
☞P
erfo
rman
ceim
prov
emen
tsof
ten
too
larg
eto
mea
sure
●A
tlea
stfo
ur
ord
ers
of
mag
nit
ud
ew
ithG
AL
EN
KB
Impl
emen
tatio
n–
p.9/
14
Alg
ori
thm
icO
pti
mis
atio
ns
Use
fult
echn
ique
sin
clud
e
☞A
void
ing
redu
ndan
cyin
sear
chbr
anch
es●
Dav
is-P
utna
mst
yle
sem
antic
bran
chin
gse
arch
●S
ynta
ctic
bran
chin
gw
ithno
-goo
dlis
t
☞D
epen
denc
ydi
rect
edba
cktr
acki
ng●
Bac
kjum
ping
●D
ynam
icba
cktr
acki
ng
☞C
achi
ng●
Cac
hepa
rtia
lmod
els
●C
ache
satis
fiabi
lity
stat
us(o
flab
els)
☞H
euris
ticor
derin
gof
prop
ositi
onal
and
mod
alex
pans
ion
●M
in/m
axim
ise
cons
trai
nedn
ess
(e.g
.,M
OM
S)
●M
axim
ise
back
trac
king
(e.g
.,ol
dest
first
)
Impl
emen
tatio
n–
p.10
/14
Dep
end
ency
Dir
ecte
dB
ackt
rack
ing
☞A
llow
sra
pid
reco
very
from
bad
bran
chin
gch
oice
s
☞M
ostc
omm
only
used
tech
niqu
eis
bac
kju
mp
ing
●Ta
gco
ncep
tsin
trod
uced
atbr
anch
poin
ts(e
.g.,
whe
nex
pand
ing
disj
unct
ions
)●
Exp
ansi
onru
les
com
bine
and
prop
agat
eta
gs●
On
disc
over
ing
acl
ash,
iden
tify
mos
trec
ently
intr
oduc
edco
ncep
tsin
volv
ed●
Jum
pba
ckto
rele
vant
bran
chpo
ints
wit
ho
ut
exp
lori
ng
alte
rnat
ive
bran
ches
●E
ffect
isto
prun
eaw
aypa
rtof
the
sear
chsp
ace
●P
erfo
rman
ceim
prov
emen
tsw
ithG
AL
EN
KB
agai
nto
ola
rge
tom
easu
re
Impl
emen
tatio
n–
p.11
/14
Bac
kju
mp
ing
E.g
.,if∃R
.¬Au∀R
.(Au
B)u
(C1t
D1)u
...u
(Cnt
Dn)⊆
L(x
)
Bac
kjum
pP
runi
ngt
t
t
R
L(x
)∪{C
1}
L(x
)∪{¬
C1,D
1}
L(x
)∪{¬
C2,D
2}
L(x
)∪{C
n}
L(y
)=
{(A
uB
),¬
A,A
,B}
x
x
x y
x
xL
(x)∪{¬
Cn,D
n}
yL
(y)=
{(A
uB
),¬
A,A
,B}
R
t
t t
L(x
)∪{C
n-1}
Cla
shC
lash
Cla
sh..
.C
lash
Impl
emen
tatio
n–
p.12
/14
Cac
hin
g
☞C
ache
the
satis
fiabi
lity
stat
usof
ano
dela
bel
●Id
entic
alno
dela
bels
ofte
nre
cur
durin
gex
pans
ion
●A
void
re-s
olvi
ngpr
oble
ms
byca
chin
gsa
tisfia
bilit
yst
atus
➙W
hen
L(x
)in
itial
ised
,loo
kin
cach
e➙
Use
resu
lt,or
add
stat
uson
ceit
has
been
com
pute
d●
Can
use
sub/
supe
rse
tcac
hing
tode
alw
ithsi
mila
rla
bels
●C
are
requ
ired
whe
nus
edw
ithbl
ocki
ngor
inve
rse
role
s●
Sig
nific
antp
erfo
rman
cega
ins
with
som
eki
nds
ofpr
oble
m
☞C
ache
(par
tial)
mod
els
ofco
ncep
ts●
Use
tode
tect
“obv
ious
”no
n-su
bsum
ptio
n●
C6v
Dif
Cu¬
Dis
satis
fiabl
e●
Cu¬
Dsa
tisfia
ble
ifm
odel
sof
Can
d¬
Dca
nbe
mer
ged
●If
not,
cont
inue
with
stan
dard
subs
umpt
ion
test
●C
anus
esa
me
tech
niqu
ein
sub-
prob
lem
s
Impl
emen
tatio
n–
p.13
/14
Su
mm
ary
☞N
aive
impl
emen
tatio
nre
sults
inef
fect
ive
non-
term
inat
ion
☞P
robl
emis
caus
edby
non-
dete
rmin
istic
expa
nsio
n(s
earc
h)
●G
CIs
lead
tohu
gese
arch
spac
e
☞S
olut
ion
(par
tial)
is●
Car
eful
choi
ceof
logi
c/al
gorit
hm●
Avo
iden
codi
ngs
●H
ighl
yop
timis
edim
plem
enta
tion
☞M
osti
mpo
rtan
topt
imis
atio
nsar
e●
Abs
orpt
ion
●D
epen
denc
ydi
rect
edba
cktr
acki
ng(b
ackj
umpi
ng)
●C
achi
ng
☞P
erfo
rman
ceim
prov
emen
tsca
nbe
very
larg
e●
E.g
.,m
ore
than
fou
ro
rder
so
fm
agn
itu
de
Impl
emen
tatio
n–
p.14
/14
DLResou
rces
•The
official
DLho
mepage:
http://dl.kr.org/
•The
DLmailinglist:
•Patrick
Lam
brix’svery
useful
DLsite
(including
lots
ofinterestinglinks):
http://www.ida.liu.se/labs/iislab/people/patla/DL/index.html
•The
annu
alDLworksho
p:
DL2002(co-locatedKR20
02):
http://www.cs.man.ac.uk/dl2002
Proceedingson
-lineavailableat:
http://sunsite.informatik.rwth-aachen.de/Publications/CEUR-WS/
•The
OIL
homepage:
http://www.ontoknowledge.org/oil/
•Moreab
outi·com:http://www.cs.man.ac.uk/~franconi/
•Moreab
outFaC
T:http://www.cs.man.ac.uk/~horrocks/
TU
Dre
sden
Ger
man
y42
