'&

$%

The

logic

ofSoft

Com

putin

gIV

Ostrava

,C

zechR

epublic,

Octo

ber

5-7

,2005

Fuzzy

Descrip

tion

Logics

and

the

Sem

antic

Web

Um

berto

Stra

ccia

I.S.T

.I.-C

.N.R

.P

isa,Ita

ly

straccia@isti.cnr.it

“Calla

isa

verylarge,

long

white

flow

eron

thick

stalks”

1

'&

$%

Outlin

e

•In

troductio

nto

Descrip

tion

Logics

(DLs)

•Sem

antic

Web

,O

nto

logies

and

DLs

•Fuzzy

DLs

•C

onclu

sions

&Futu

reW

ork

2

'&

$%

Intro

ductio

nto

Descrip

tion

Logics

3

'&

$%

What

Are

Descrip

tion

Logics?

(http://dl.kr.org/)

•A

family

oflogic-b

asedknow

ledge

represen

tationform

alisms

–D

escendan

tsof

seman

ticnetw

orks

and

KL-O

NE

–D

escribe

dom

ainin

terms

ofcon

cepts

(classes),roles

(prop

erties,

relationsh

ips)

and

indiv

iduals

•D

istingu

ished

by:

–Form

alsem

antics

(typically

model

theoretic)

∗D

ecidab

lefragm

ents

ofFO

L

∗C

loselyrelated

toP

roposition

alM

odal

&D

ynam

icLogics

∗C

loselyrelated

toG

uard

edFragm

ent

ofFO

L,L2

and

C2

–P

rovision

ofin

ference

services

∗D

ecisionpro

cedures

forkey

prob

lems

(e.g.,satisfi

ability,

subsu

mption

)

∗Im

plem

ented

system

s(h

ighly

optim

ised)

4

'&

$%

DLs

Basics

•C

oncep

tnam

esare

equivalen

tto

unary

pred

icates

–In

general,

concep

tseq

uiv

toform

ulae

with

one

freevariab

le

•R

olenam

esare

equivalen

tto

bin

arypred

icates

–In

general,

roleseq

uiv

toform

ulae

with

two

freevariab

les

•In

div

idual

nam

esare

equivalen

tto

constan

ts

•O

perators

restrictedso

that:

–Lan

guage

isdecid

able

and,if

possib

le,of

lowcom

plex

ity

–N

oneed

forex

plicit

use

ofvariab

les

∗R

estrictedform

of∃an

d∀

–Featu

ressu

chas

countin

gcan

be

succin

ctlyex

pressed

5

'&

$%

Descrip

tion

Logic

Syste

m

Inferen

ceSystem

⇔U

serIn

terface

m

Know

ledge

Base

TB

ox

“D

efines

termin

olo

gy

ofth

eapplica

tion

dom

ain

”

Man

=HumanuMale

HappyFather

=Manu∃haschild.Female

AB

ox

“Sta

tesfa

ctsabout

asp

ecific

world

”

John:HappyFather

(John,Mary):h

aschild

6

'&

$%

The

DL

Fam

ily

•A

givenD

Lis

defi

ned

by

setof

concep

tan

drole

formin

gop

erators

•Sm

allestprop

ositionally

closedD

LisA

LC( A

ttributive

Lan

guage

with

Com

plem

ent)

–C

oncep

tscon

structed

usin

gu

,t,¬

,∃an

d∀

Synta

xE

xam

ple

C,D

−→

>|

(top

concep

t)

⊥|

(botto

mco

ncep

t)

A|

(ato

mic

concep

t)Human

Cu

D|

(concep

tco

nju

nctio

n)

HumanuMale

Ct

D|

(concep

tdisju

nctio

n)

NiceuRich

¬C

|(co

ncep

tneg

atio

n)

¬Meat

∃R

.C|

(existen

tialquantifi

catio

n)

∃haschild.Blond

∀R

.C(u

niv

ersalquantifi

catio

n)

∀haschild.Human

7

'&

$%

DL

Sem

antic

s

•Sem

antics

isgiv

enin

terms

ofan

interp

retatio

nI

=(∆

I,·I),

where

–∆I

isth

edom

ain

(anon-em

pty

set)

–·I

isan

interp

retatio

nfu

nctio

nth

at

maps:

∗C

oncep

t(cla

ss)nam

eA

into

asu

bset

AI

of∆I

∗R

ole

(pro

perty

)nam

eR

into

asu

bset

RI

of∆I×

∆I

∗In

div

idualnam

ea

into

an

elemen

tof∆I×

∆I

–In

terpreta

tion

functio

n·I

isex

tended

toco

ncep

tex

pressio

ns:

>I

=∆I

⊥I

=∅

(C1u

C2)I

=C

1I∩

C2I

(C1t

C2)I

=C

1I∪

C2I

(¬C

)I

=∆I\

CI

(∃R

.C)I

={x∈

∆I|∃y.〈x

,y〉∈

RI∧

y∈

CI}

(∀R

.C)I

={x∈

∆I|∀y.〈x

,y〉∈

RI⇒

y∈

CI}

8

'&

$%

DLs

and

FO

L

•ALC

map

pin

gto

FO

L:in

troduce

–a

unary

pred

icateA

foran

atomic

concep

tA

–a

bin

arypred

icateR

fora

roleR

•Tran

slatecon

cepts

Can

dD

asfollow

s

t(>,x)

=true

t(⊥,x)

=false

t(A,x)

7→A

(x)

t(C1u

C2,x)

=t(C

1,x)∧

t(C2,x)

t(C1t

C2,x)

7→t(C

1,x)∨

t(C2,x)

t(¬C

,x)

=¬

t(C,x)

t(∃R

.C,x)

=∃y.R

(x,y)∧

t(C,y)

t(∀R

.C,x)

=∀y.R

(x,y)⇒

t(C,y)

9

'&

$%

DL

Know

ledge

Base

•A

DL

Know

ledge

Base

isa

pairK

=〈T

,A〉,

where

–T

isa

TB

oxco

nta

inin

ggen

eralin

clusio

naxio

ms

ofth

efo

rmCv

D(“

concep

tD

subsu

mes

concep

tC

”),

I|=

Cv

Diff

CI⊆

DI

∗co

ncep

tdefi

nitio

ns

are

ofth

efo

rmA

=C

(equiv

toAv

C,Cv

A)

∗prim

itive

concep

tdefi

nitio

ns

are

ofth

efo

rmAv

C

∗Som

etimes,

aT

Box

can

conta

inprim

itive

and

concep

tdefi

nitio

ns

only,

where

no

ato

mca

nbe

defi

ned

more

than

once

and

no

recursio

nis

allow

ed7→

Com

puta

tionalco

mplex

itych

anges

dra

matica

lly

–A

isa

AB

oxco

nta

inin

gassertio

ns

ofth

efo

rma:C

(“in

div

iduala

isan

insta

nce

of

concep

tC

)and

(a,b):R

(indiv

idualb

isan

R-fi

llerofin

div

iduala”)

I|=

a:C

iffaI∈

CI

I|=

(a,b):R

iff〈aI,bI〉∈

RI

10

'&

$%

Note

on

DL

nam

ing

AL:

C,D

−→

>|⊥|A|Cu

D|¬

A|∃R

.>|∀

R.C

C:

Concept

negatio

n,¬

C.

Thus,ALC

=AL

+C

S:

Use

dfo

rALC

with

transitiv

ero

lesR

+

U:

Concept

disju

nctio

n,

C1t

C2

E:

Existe

ntia

lquantifi

catio

n,∃R

.C

H:

Role

inclu

sion

axio

ms,

R1v

R2

N:

Num

ber

restric

tions,

(≥n

R)

and

(≤n

R),

e.g

.(≥

3hasChild)

(has

at

least

3ch

ildre

n)

Q:

Qualifi

ed

num

ber

restric

tions,

(≥n

R.C

)and

(≤n

R.C

),e.g

.(≤

2hasChild.Adult)

(has

at

most

2adult

child

ren)

O:

Nom

inals

(single

ton

cla

ss),{a},e.g

.∃haschild.{mary}.

Note:

a:C

equiv

to{a}v

Cand

(a,b):R

equiv

to{a}v∃R

.{b}

I:

Inverse

role

,R−

,e.g

.

F:

Functio

nalro

le,

f

For

insta

nce,

SHIF

=S

+H

+I

+F

=ALCR

+HIF

SHOIN

=S

+H

+O

+I

+N

=ALCR

+HOIN

11

'&

$%

Short

Histo

ryofD

escrip

tion

Logics

Phase

1:

Mostly

system

develop

men

t(early

eighties)

•In

complete

system

s(B

ack,C

lassic,Loom

,K

andor,

...)

•B

asedon

structu

ralalgorith

ms

Phase

2:

first

tableau

xalgorith

ms

and

complex

ityresu

lts(m

id-eigh

ties-

mid

-nin

eties)

•D

evelopm

ent

oftab

leaux

algorithm

san

dcom

plex

ityresu

lts

•Tab

leaux-b

asedsy

stems

(Kris,

Crack

)

•In

vestigationof

optim

izationtech

niq

ues

Phase

3:

Optim

izedsy

stems

forvery

expressive

DLs

(mid

-nin

eties-

...)

•Tab

leaux

algorithm

sfor

veryex

pressive

DLs

•H

ighly

optim

isedtab

leausy

stems

(FaC

T,D

LP,R

acer)

•R

elationsh

ipto

modal

logican

ddecid

able

fragmen

tsof

FO

L

12

'&

$%

Late

stdevelo

pm

ents

Phase

4:

•M

ature

implem

entation

s

•M

ainstream

application

san

dTools

–D

atabases

∗con

sistency

ofcon

ceptu

alsch

ema

(EE

R,U

ML)

usin

ge.g.D

LR(n

-aryD

L)

∗sch

ema

integration

∗Q

uery

subsu

mption

(w.r.t.

acon

ceptu

alsch

ema)

–O

ntologies

and

the

Sem

antic

Web

(and

Grid

)

∗O

ntology

engin

eering

(design

,m

ainten

ance,

integration

)

∗R

easoing

with

ontology

-based

mark

up

(metad

ata)

∗Serv

icedescrip

tionan

ddiscovery

–C

omm

ercialim

plem

enation

s

∗C

erebra,

Racer

13

'&

$%

The

Sem

antic

Web

14

'&

$%

The

Sem

antic

Web

Visio

nand

DLs

•T

he

WW

Was

we

know

itnow

–1st

generation

web

mostly

han

dw

rittenH

TM

Lpages

–2n

dgen

eration(cu

rrent)

web

oftenm

achin

egen

erated/active

–B

othin

tended

fordirect

hum

anpro

cessing/in

teraction

•In

nex

tgen

erationw

eb,resou

rcessh

ould

be

more

accessible

toau

tomated

pro

cesses

–To

be

achieved

via

seman

ticm

arkup

–M

etadata

annotation

sth

atdescrib

econ

tent/fu

nction

15

'&

$%

Onto

logie

s

•Sem

antic

mark

up

must

be

mean

ingfu

lto

autom

atedpro

cesses

•O

ntologies

will

play

akey

role

–Sou

rceof

precisely

defi

ned

terms

(vocab

ulary

)

–C

anbe

shared

acrossap

plication

s(an

dhum

ans)

•O

ntology

typically

consists

of:

–H

ierarchical

descrip

tionof

importan

tcon

cepts

indom

ain

–D

escription

sof

prop

ertiesof

instan

cesof

eachcon

cept

•O

ntologies

canbe

used

,e.g.

–To

facilitateagen

t-agent

comm

unication

ine-com

merce

–In

seman

ticbased

search

–To

prov

ide

richer

service

descrip

tions

that

canbe

more

flex

ibly

interp

retedby

intelligen

tagen

ts16

'&

$%

Exam

ple

Onto

logy

•V

ocab

ulary

and

mean

ing

(defi

nition

s)

–E

lephan

tis

acon

cept

whose

mem

bers

area

kin

dof

anim

al

–H

erbivore

isa

concep

tw

hose

mem

bers

areex

actlyth

osean

imals

who

eaton

lyplan

tsor

parts

ofplan

ts

–A

dult

Elep

han

tis

acon

cept

whose

mem

bers

areex

actlyth

ose

elephan

tsw

hose

ageis

greaterth

an20

years

•B

ackgrou

nd

know

ledge/con

straints

onth

edom

ain(gen

eralax

ioms)

–A

dult

Elep

han

tsw

eighat

least2,000

kg

–A

llE

lephan

tsare

either

African

Elep

han

tsor

Indian

Elep

han

ts

–N

oin

div

idual

canbe

both

aH

erbivore

and

aC

arnivore

17

'&

$%

Exam

ple

Onto

logy

(Pro

tege)1

8

'&

$%

Onto

logy

Descrip

tion

Languages

•Shou

ldbe

suffi

ciently

expressive

tocap

ture

most

usefu

lasp

ectsof

dom

ain

know

ledge

represen

tation

•R

easonin

gin

itsh

ould

be

decid

able

and

efficien

t

•M

any

diff

erent

langu

ageshas

been

prop

osed:

RD

F,R

DFS,O

IL,

DA

ML+

OIL

•O

WL

( Ontology

Web

Lan

guage)

isth

ecu

rrent

emergin

glan

guage:

it’sa

standard

19

'&

$%

OW

LLanguage

•T

hree

species

ofO

WL

–O

WL

full

isunio

nofO

WL

synta

xand

RD

F(b

ut,

undecid

able)

–O

WL

DL

restrictedto

FO

Lfra

gm

ent

(reaso

nin

gpro

blem

inN

EX

PT

IME

)

–O

WL

Lite

isea

sierto

implem

ent

subset

ofO

WL

DL

(reaso

nin

gpro

blem

in

EX

PT

IME

)

•Sem

antic

layerin

g

–O

WL

DL≡

OW

Lfu

llw

ithin

Descrip

tion

Logic

fragm

ent

–D

Lsem

antics

offi

cially

defi

nitiv

e

•O

WL

DL

based

onSHIQ

Descrip

tion

Logic

(ALCHIQR

+)

–In

fact

itis

equiva

lent

toSHOIN

(D)

•O

WL

Lite

based

onSHIF

Descrip

tion

Logic

(ALCHIFR

+)

•B

enefi

tsfro

mm

any

yea

rsofD

Lresea

rch

–Form

alpro

perties

well

understo

od

(com

plex

ity,decid

ability

)

–K

now

nrea

sonin

galg

orith

ms

–Im

plem

ented

system

s(h

ighly

optim

ised)

20

'&

$%

Abstra

ct

Synta

xD

LSynta

xE

xam

ple

Desc

riptio

ns

(C)

A(U

RI

refe

rence)

AConference

owl:Thing

>

owl:Nothing

⊥

intersectionOf(C

1C

2...)

C1u

C2

Referenceu

Journal

unionOf(C

1C

2...)

C1t

C2

Organizationt

Institution

complementOf(C

)¬

C¬

MasterThesis

oneOf(o

1...)

{o1,...}

{"WISE","ISWC",...}

restriction(R

someValuesFrom(C

))∃R

.C∃parts.InCollection

restriction(R

allValuesFrom(C

))∀R

.C∀date.Date

restriction(R

hasValue(o

))R

:o

date

:2005

restriction(R

minCardinality(n

))(≥

nR

)>

1location

restriction(R

maxCardinality(n

))(≤

nR

)6

1publisher

restriction(U

someValuesFrom(D

))∃U

.D∃issue.integer

restriction(U

allValuesFrom(D

))∀U

.D∀name.string

restriction(U

hasValue(v

))U

:v

series

:"LNCS"

restriction(U

minCardinality(n

))(≥

nU

)>

1title

restriction(U

maxCardinality(n

))(≤

nU

)6

1author

21

'&

$%

Abstra

ct

Synta

xD

LSynta

xE

xam

ple

Axio

ms

Class(A

partial

C1

...

Cn)

Av

C1u

...u

Cn

Humanv

AnimaluBiped

Class(A

complete

C1

...C

n)

A=

C1u

...u

Cn

Man

=HumanuMale

EnumeratedClass(A

o1

...o

n)

A={o1}t

...t{o

n}

RGB

={r}t{g}t{b}

SubClassOf(C

1C

2)

C1v

C2

EquivalentClasses(C

1...C

n)

C1

=...=

Cn

DisjointClasses(C

1...C

n)

Ciu

Cj

=⊥

,i6=

jMalev¬Female

ObjectProperty(R

super

(R1)...

super

(Rn))

Rv

Ri

HasDaughterv

hasChild

domain(C

1)

...domain(C

n)

(≥1

R)v

Ci

(≥1hasChild)v

Human

range(C

1)

...range(C

n)

>v∀R

.Di

>v∀hasChild.Human

[inverseof(R

0)]

R=

R−0

hasChild

=hasParent−

[symmetric]

R=

R−

similar

=similar−

[functional]

>v

(≤1

R)

>v

(≤1hasMother)

[Inversefunctional]

>v

(≤1

R−

)

[Transitive]

Tr(R

)T

r(ancestor)

SubPropertyOf(R

1R

2)

R1v

R2

EquivalentProperties(R

1...R

n)

R1

=...=

Rn

cost

=price

AnnotationProperty(S

)

22

'&

$%

Abstra

ct

Synta

xD

LSynta

xE

xam

ple

DatatypeProperty(U

super

(U1)...super

(Un))

Uv

Ui

domain(C

1)

...domain(C

n)

(≥1

U)v

Ci

(≥1hasAge)v

Human

range(D

1)

...range(D

n)

>v∀U

.Di

>v∀hasAge.posInteger

[functional]

>v

(≤1

U)

>v

(≤1hasAge)

SubPropertyOf(U

1U

2)

U1v

U2

hasNamev

hasFirstName

EquivalentProperties(U

1...U

n)

U1

=...=

Un

Indiv

iduals

Individual(o

type

(C1)...type

(Cn))

o:C

itim:Human

value(R

1o1)

...value(R

no

n)

(o,o

i ):Ri

(tim,mary):h

asChild

value(U

1v1)

...value(U

nv

n)

(o,v1):U

i(tim,14):h

asAge

SameIndividual(o

1...o

n)

o1

=...=

on

presidentBush

=G.W

.Bush

DifferentIndividuals(o

1...o

n)

oi6=

oj,i6=

jjohn6=

peter

23

'&

$%

XM

Lre

pre

senta

tion

ofO

WL

state

ments

E.g.,

Personu

∀hasChild.(D

octort

∃hasChild.Doctor):

24

'&

$%

Concre

tedom

ain

sin

OW

L

•O

WL

supports

concrete

dom

ain

s:in

tegers,

strings,

...

•C

lean

separa

tion

betw

eenobject

classes

and

concrete

dom

ain

s

–D

isjoin

tin

terpreta

tion

dom

ain

:dI⊆

∆D

,and

∆D∩

∆I

=∅

–D

isjoin

tco

ncrete

pro

perties:

UI⊆

∆I×

∆D

,e.g

.,(tim,14):h

asAge,

(sf,“SoftComputing”):h

asAcronym

•P

hilo

sophica

lrea

sons:

–C

oncrete

dom

ain

sstru

ctured

by

built-in

pred

icates

•P

ractica

lrea

sons:

–O

nto

logy

language

remain

ssim

ple

and

com

pact

–Sem

antic

integ

rityofonto

logy

language

not

com

pro

mised

–Im

plem

enta

bility

not

com

pro

mised

can

use

hybrid

reaso

ner

∗O

nly

need

sound

and

com

plete

decisio

npro

cedure

for

d1I∩

...∩

dnI,w

here

di

isa

(posssib

lyneg

ated

)data

type

•In

the

DL

literatu

re,th

eseare

called

Concrete

Dom

ain

s

•N

ota

tion:

(D).

E.g

.,ALC(D

)isALC

+co

ncrete

dom

ain

s,O

WL

DL

=SHOIN

(D)

25

'&

$%

Reaso

nin

gw

ithO

WL

DL

26

'&

$%

Reaso

nin

g

•W

hat

can

we

do

with

it?

–D

esign

and

main

tenance

ofonto

logies

∗C

heck

class

consisten

cyand

com

pute

class

hiera

rchy

∗Particu

larly

importa

nt

with

larg

eonto

logies/

multip

leauth

ors

(≥210

defi

ned

concep

ts)

–In

tegra

tion

ofonto

logies

∗A

ssertin

ter-onto

logy

relatio

nsh

ips

∗R

easo

ner

com

putes

integ

rated

class

hiera

rchy/co

nsisten

cy

–Q

uery

ing

class

and

insta

nce

data

w.r.t.

onto

logies

∗D

etermin

eif

setoffa

ctsare

consisten

tw

.r.t.onto

logies

∗D

etermin

eif

indiv

iduals

are

insta

nces

ofonto

logy

classes

∗R

etrieve

indiv

iduals/

tuples

satisfy

ing

aquery

expressio

n

∗C

heck

ifone

class

subsu

mes

(ism

ore

gen

eralth

an)

anoth

erw

.r.t.onto

logy

•H

owdo

we

do

it?

–U

seD

Ls

reaso

ner

(OW

LD

L=SHOIN

(D))

27

'&

$%

Basic

Infe

rence

Pro

ble

ms

Consis

tency:

Check

ifknow

ledge

ism

eanin

gfu

l

•IsK

consisten

t?7→

There

exists

som

em

odelI

ofK

•Is

Cco

nsisten

t?7→

CI6=∅

for

som

eso

me

modelI

ofK

Subsum

ptio

n:

structu

reknow

ledge,

com

pute

taxonom

y

•K|=

Cv

D?7→

CI⊆

DI

for

all

models

IofK

Equiv

ale

nce:

check

iftw

ocla

ssesden

ote

sam

eset

ofin

stances

•K|=

C=

D?7→

CI

=DI

for

all

models

IofK

Instantia

tio

n:

check

ifin

div

iduala

insta

nce

ofcla

ssC

•K|=

a:C

?7→

aI∈

CI

for

all

models

IofK

Retrie

val:

retrieve

setofin

div

iduals

that

insta

ntia

teC

•C

om

pute

the

set{a

:K|=

a:C}

Pro

blem

sare

all

reducib

leto

consisten

cy,e.g

.

•K|=

Cv

Diff〈T

,A∪{a:Cu¬

D}〉

not

consisten

t

•K|=

a:C

iff〈T

,A∪{a:¬

C}〉

not

consisten

t

28

'&

$%

Reaso

nin

gin

DLs:

Basic

s

•Tablea

ux

alg

orith

mdecid

ing

concep

tco

nsisten

cy

•Try

tobuild

atree-lik

em

odelI

ofth

ein

put

concep

tC

•D

ecom

pose

Csy

nta

ctically

–A

pply

tablea

uex

pansio

nru

les

–In

ferco

nstra

ints

on

elemen

tsofm

odel

•Tablea

uru

lesco

rrespond

toco

nstru

ctors

inlo

gic

(u,t

,...)

–Som

eru

lesare

nondeterm

inistic

(e.g.,t

,≤)

–In

pra

ctice,th

ism

eans

search

•Sto

pw

hen

no

more

rules

applica

ble

or

clash

occu

rs

–C

lash

isan

obvio

us

contra

dictio

n,e.g

.,A

(x),¬

A(x

)

•C

ycle

check

( blo

ckin

g)

may

be

need

edfo

rterm

inatio

n

•C

satisfi

able

iffru

lesca

nbe

applied

such

that

afu

llyex

panded

clash

freetree

is

constru

cted

29

'&

$%

Table

aux

check

ing

consiste

ncy

ofanALC

concept

•W

ork

son

atree

(semantics

thro

ugh

view

ing

treeas

an

AB

ox)

–N

odes

represen

telem

ents

of∆I,la

belled

with

sub-co

ncep

tsof

C

–E

dges

represen

tro

le-successo

rship

sbetw

eenelem

ents

of∆I

•W

ork

son

concep

tsin

neg

atio

nnorm

alfo

rm:

push

neg

atio

nin

side

usin

gde

Morg

an’

laws

and

¬(∃

R.C

)7→

∀R

.¬C

¬(∀

R.C

)7→

∃R

.¬C

•It

isin

itialised

with

atree

consistin

gofa

single

(root)

node

x0

with

L(x

0)={C}

•A

treeT

conta

ins

acla

shif,

for

anode

xin

T,{A

,¬A}⊆L

(x)

•R

eturn

s“C

isco

nsisten

t”if

rules

can

be

applied

s.t.th

eyyield

acla

sh-free,

com

plete

(no

more

rules

apply

)tree

30

'&

$%

ALC

Table

au

rule

s

x•{

C1 u

C2 ,...}

−→u

x•{

C1 u

C2 ,C

1 ,C2 ,...}

x•{

C1 t

C2 ,...}

−→t

x•{

C1 t

C2 ,C

,...}for

C∈{C

1 ,C2 }

x•{∃

R.C

,...}−→

∃x•{∃

R.C

,...}R↓y•{

C}

x•{∀

R.C

,...}R↓y•{

...}

−→∀

x•{∃

R.C

,...}R↓y•{

..., C}

31

'&

$%

Soundness

and

Com

ple

teness

Theore

m1

1.

The

tablea

ualgo

rithm

isa

PSPA

CE

(usin

gdep

th-fi

rstsea

rch)

decisio

n

proced

ure

for

consisten

cy(a

nd

subsu

mptio

n)

ofA

LCco

ncep

ts.

2.ALC

has

the

tree-mod

elpro

perty

The

tableau

canbe

modifi

edto

aP

SPA

CE

decision

pro

cedure

for

•e.g.A

LCNan

dALCI

•T

Box

with

acyclic

concep

tdefi

nition

susin

glazy

unfold

ing

(unfold

ing

on

dem

and)

•N

ote:ALC

with

general

inclu

sionax

ioms,

Cv

D,ju

mps

toE

XP

TIM

E

32

'&

$%

Exte

nsio

ns:

genera

lin

clu

sion

axio

ms

•T

Box

esw

ithgen

eralin

clusion

axiom

sCv

D

–E

achnode

must

be

labeled

with

¬Ct

D(recall,

Cv

D≡

FO

L∀x.¬

t(C,x

)∨t(D

,x)

–H

owever,

termin

ationnot

guaran

teed

Given

,a:A

and

Av∃R

.Ath

en

a•{A

,¬At∃R

.A}

a•{A

,¬At∃R

.A,∃

R.A}

R↓

y1•{A

,¬At∃R

.A,∃

R.A}

R↓

y2•{A

,¬At∃R

.A,∃

R.A}

...

Note:

y1,y

2sh

are

sam

epro

perties

33

'&

$%

Blo

ckin

g

•W

hen

creating

new

node,

check

ancestors

foreq

ual

(superset)

label

•If

such

anode

isfou

nd,new

node

isblo

cked

Given

,a:A

and

Av∃R

.Ath

en

a•{A

,¬At∃R

.A,∃

R.A}

R↓y

1 •{A

,¬At∃R

.A,∃

R.A}

Node

isblo

cked

Note

:

•B

lock

ing

may

not

work

with

more

com

plex

DLs

(e.g.,

usin

gR−

)

•W

ithnum

ber

restrictions

(N),

som

esa

tisfiable

concep

tshav

eonly

non-fi

nite

models:

e.g.,

testing¬

Cw

ithT

={>v∃R

.C,>

v(≤

1R−

)}

•Sophistica

tedm

ethods

has

been

dev

eloped

todetect

∞rep

etition

of

substru

ctures

34

'&

$%

Focus

ofD

LR

ese

arch

•decid

ability

/co

mplex

ityof

rea-

sonin

g

•req

uires

restricteddescrip

tion

language

•sy

stemand

com

plex

ityresu

lts

availa

ble

forva

rious

com

bin

atio

ns

ofco

nstru

ctors

•applica

tion

relevant

concep

ts

must

be

defi

nable

•so

me

applica

tions

dom

ain

sre-

quire

very

expressiv

eD

Ls

•effi

cient

alg

orith

ms

inpra

cticefo

r

very

expressiv

eD

Ls

Rea

sonin

gfea

sible

versu

s⇐⇒

Expressiv

itysu

fficien

t

35

'&

$%

Som

eC

om

ple

xity

Resu

lts

36

'&

$%

Fuzzy

Descrip

tion

Logics

“Calla

isa

verylarge,

long

white

flow

eron

thick

stalks”

37

'&

$%

Obje

ctive

•To

exten

dclassical

DLs

toward

sth

erep

resentation

ofan

dreason

ing

with

vague

concep

ts

•A

pplication

:Sem

antic

Web

•D

evelopm

ent

ofpractical

reasonin

galgorith

ms

•System

implem

entation

s

38

'&

$%

Exam

ple

(fuzzy

DL-L

ite,

Curre

nt

work)

Hotelv

∃hasLocation

Conferencev

∃hasLocation

Hotelv

¬Conference

LocationI⊆

GISCoordinates

distanceI

:GISCoord×

GISCoord→

N

dis

tan

ce(x

,y)

=...

closeI

:N→

[0,1]

clo

se(x

)=

max(0

,1−

x1000)

hasLocation

hasLocation

distance

hl1

cl1

300

hl1

cl2

500

hl2

cl1

750

hl2

cl2

750

......

HotelID

hasLocation

h1

hl1

h2

hl2

......

ConferenceID

hasLocation

c1

cl1

c2

cl2

......

HotelID

closenessdegree

h1

0.7

h2

0.25

......

“Fin

dhote

lclo

seto

confe

rencec1”:Query(c1,h)←

Hotel(h

),hasLocation(h

,h

l),Conference(c1),hasLocation(c1,cl),

distance(h

l,cl,

d),close(d

)

39

'&

$%

Exam

ple

(Logic-b

ase

din

form

atio

nre

trievalm

odel)

Birdv

Animal

Dogv

Animal

snoopy

:Dog

woodstock

:Bird

ImageRegion

ObjectID

isAboutdegree

o1

snoopy

0.8

o2

woodstock

0.7

......

Qu

ery

=ImageRegionu∃isAbout.Animal

Query(ir

)←

ImageRegion(ir

),isAbout(ir

,x),Animal(x

)

40

'&

$%

Exam

ple

(Gra

ded

Enta

ilment)

auditt

mg

ferrarienzo

Car

speed

auditt

243

mg

≤170

ferrarienzo≥

350

SportsCar

=Caru∃hasSpeed.very(High)

K|=

〈ferrarienzo:SportsCar,1〉

K|=

〈auditt:SportsCar,0.9

2〉

K|=

〈auditt:¬

SportsCar,0.7

2〉

41

'&

$%

Exam

ple

(Gra

ded

Subsu

mptio

n)

Minor

=Personu

∃hasAge.≤

18

YoungPerson

=Personu

∃hasAge.Young

K|=〈Minorv

YoungPerson,0

.2〉

Note:

with

out

explicit

mem

bersh

ipfu

nction

ofYoung,in

ference

cannot

be

draw

n

42

'&

$%

Basic

prin

ciple

sofFuzzy

DLs

•In

classicalD

Ls,

acon

cept

Cis

interp

retedby

anin

terpretation

Ias

aset

ofin

div

iduals

•In

fuzzy

DLs,

acon

cept

Cis

interp

retedbyI

asa

fuzzy

setof

indiv

iduals

•E

achin

div

idual

isin

stance

ofa

concep

tto

adegree

in[0

,1]

•E

achpair

ofin

div

iduals

isin

stance

ofa

roleto

adegree

in[0

,1]

43

'&

$%

Fuzzy

ALC

conce

pts

Interpretatio

n:

I=

∆I

CI

:∆I→

[0,1]

RI

:∆I×

∆I→

[0,1]

t=

t-norm

s=

s-norm

n=

negatio

n

i=

implic

atio

n

Concepts:

Syn

tax

Sem

an

tics

C,D

−→

>|>I(x

)=

1

⊥|⊥I(x

)=

0

A|

AI(x

)∈

[0,1]

Cu

D|

(C1u

C2)I(x

)=

t(C1I(x

),C

2I(x

))

Ct

D|

(C1t

C2)I(x

)=

s(C

1I(x

),C

2I(x

))

¬C|

(¬C

)I(x

)=

n(CI(x

))

∃R

.C|

(∃R

.C)I(x

)=

sup

y∈∆I

t(RI(x

,y),

CI(y

))

∀R

.C(∀

R.C

)I(u

)=

infy∈∆I

i(RI(x

,y),

CI(y

)}

Assertio

ns:〈a:C

,n〉,I|=〈a:C

,n〉

iffCI(aI)≥

n(sim

ilarly

for

role

s)

•in

div

idual

ais

insta

nce

ofconcept

Cat

least

todegre

en,

n∈

[0,1]∩

Q

Inclu

sio

naxio

ms:

Cv

D,

•I|=

Cv

Diff∀x∈

∆I

.CI(x

)≤

DI(x

),(a

ltern

ativ

e,∀x∈

∆I

.i(CI(x

),DI(x

))=

1)

44

'&

$%

Basic

Infe

rence

Pro

ble

ms

Consis

tency:

Check

ifknow

ledge

ism

eanin

gfu

l

•IsK

consisten

t?

Subsum

ptio

n:

structu

reknow

ledge,

com

pute

taxonom

y

•K|=

Cv

D?

Equiv

ale

nce:

check

iftw

ofu

zzyco

ncep

tsare

the

sam

e

•K|=

C=

D?

Graded

instantia

tio

n:

Check

ifin

div

iduala

insta

nce

ofcla

ssC

todeg

reeat

least

n

•K|=〈a

:C,n〉

?

BT

VB

:B

estTru

thV

alu

eB

ound

pro

blem

•glb(K

,a:C

)=

sup{n|K|=〈a

:C,n〉}

?

Retrie

val:

Rank

setofin

div

iduals

that

insta

ntia

teC

w.r.t.

best

truth

valu

ebound

•R

ank

the

setR

(K,C

)={〈a

,glb(K

,a:C

)〉}

45

'&

$%

Som

eN

ote

son

...

•V

alue

restrictions:

–In

classicalD

Ls,∀

R.C

≡¬∃

R.¬

C

–T

he

same

isnot

true,

ingen

eral,in

fuzzy

DLs

(dep

ends

onth

e

operators’

seman

tics,not

true

inG

odel

logic).

∀hasParent.Human6≡¬∃

hasParent.¬Human

??

•M

odels:

–In

classicalD

Ls>

v¬

(∀R

.A)u

(¬∃R

.¬A

)has

no

classicalm

odel

–In

God

ellogic

ithas

no

finite

model,

but

has

anin

finite

model

•T

he

choice

ofth

eap

prop

riatesem

antics

ofth

elogical

connectives

is

importan

t.

–Shou

ldhave

reasonab

lelogical

prop

erties

–C

ertainly

itm

ust

have

efficien

talgorith

ms

solvin

gbasic

inferen

ce

prob

lems

46

'&

$%

Tow

ard

sfu

zzyO

WL

Lite

and

OW

LD

L

•R

ecallth

atO

WL

Lite

and

OW

LD

Lrelate

toSH

IF(D

)an

dSH

OIN

(D),

respectively

•W

eneed

toex

tend

the

seman

ticsof

fuzzy

ALC

tofu

zzy

SHOIN

(D)

=ALCR

+ HOIN

R(D

)

•A

ddition

ally,w

ead

dm

odifi

ers(e.g.,

very)

•A

ddition

ally,w

ead

dcon

cretefu

zzycon

cepts

(e.g.,Young)

47

'&

$%

Concre

tefu

zzy

concepts

•E

.g.,Small,Young,High,e

tc.w

ithex

plicit

mem

bersh

ipfu

nction

•U

seth

eid

eaof

concrete

dom

ains:

–D

=〈∆

D ,ΦD 〉

–∆

Dis

anin

terpretation

dom

ain

–ΦD

isth

eset

ofcon

cretefu

zzydom

ainpred

icatesd

with

apred

efined

arityn

and

fixed

interp

retationdD:∆

nD→

[0,1]

–For

instan

ce,

Minor

=Personu

∃hasAge.≤

18

YoungPerson

=Personu

∃hasAge.Young

48

'&

$%

Modifi

ers

•V

ery,m

oreOrL

ess,sligh

tly,etc.

•A

pply

tofu

zzysets

toch

ange

their

mem

bersh

ipfu

nction

–very(x

)=

x2

–slightly(x

)=√

x

•For

instan

ce,

SportsCar

=Caru

∃speed.very(High)

49

'&

$%

Num

ber

Restric

tions

•M

aybe

aprob

lemcom

putation

ally

•T

he

seman

ticsof

the

concep

t(≥

nS

)

(≥n

R)I(x

)=

sup{y1,...,y

n}⊆

∆I

∧

ni=1RI(x

,yi )

•Is

the

result

ofview

ing

(≥n

R)

asth

eop

enfirst

order

formula

∃y1 ,...,y

n.

n∧

i=1

R(x

,yi )∧

∧

1≤

i<j≤

n

yi 6=

yj

.

•T

he

seman

ticsof

the

concep

t(≤

nR

)

(≤n

R)I(x

)=¬

(≥n

+1

R)I(x

)

•N

ote:(≥

1R

)≡∃R

.>

50

'&

$%

Reaso

nin

g

•For

full

fuzzy

SHOIN

(D)

orSHIF

(D):

does

not

exists

yet!!

•E

xists

forfu

zzyALC

(D)

+m

odifi

ers+

fuzzy

concrete

concep

ts(u

nder

Lukasiew

iczsem

antics,

alsounder

“Zad

ehsem

antics”)

•U

sual

fuzzy

tableau

xcalcu

lus

does

not

work

anym

ore(p

roblem

sw

ith

modifi

ersan

dcon

cretefu

zzycon

cepts)

•U

sual

fuzzy

tableau

xcalcu

lus

does

not

solveth

eB

TV

Bprob

lem

•N

ewalgorith

muses

bou

nded

Mix

edIn

tegerP

rogramm

ing

oracle,as

for

Man

yV

alued

Logics

–R

ecall:th

egen

eralM

ILP

pro

blemis

tofind

x∈

Qk,y

∈Z

m

f(x

,y)

=m

in{f(x

,y):A

x+

By≥

h}A

,Bin

tegerm

atrixes

51

'&

$%

Require

ments

•W

ork

sfo

rusu

alfu

zzyD

Lsem

antics

(Zadeh

semantics)

and

Luka

siewicz

logic

•M

odifi

ersare

defi

nable

as

linea

rin

-equatio

ns

over

Q,Z

(e.g.,

linea

rhed

ges)

•Fuzzy

concrete

concep

tsare

defi

nable

as

linea

rin

-equatio

ns

over

Q,Z

(e.g.,

crisp,

triangula

r,tra

pezo

idal,

leftsh

ould

erand

right

should

erm

embersh

ipfu

nctio

ns)

Minor

=Personu∃hasAge.≤

18

YoungPerson

=Personu∃hasAge.Young

Young

=ls(10,30)

•T

hen

glb(K

,a:C

)=

min{x|K∪{〈a

:C≤

x〉

satisfi

able}

glb(K

,Cv

D)

=m

in{x|K∪{〈a

:Cu¬

D≥

1−

x〉

satisfi

able}

–A

pply

tablea

ux

calcu

lus,

then

use

bounded

Mix

edIn

teger

Pro

gra

mm

ing

ora

cle

52

'&

$%

ALC

Table

au

rule

s(e

xcerp

t)x•{〈C

1u

C2,≥

,l〉

,...}

−→u

x•{〈C

1u

C2,≥

,l〉

,〈C

1,≥

,l〉,〈C

2,≥

,l〉,

...}

x•{〈C

1t

C2,≥

,l〉

,...}

−→t

x•{〈C

1t

C2,≥

,l〉

,〈C

1,≥

,x1〉,〈C

2,≥

,x2〉,

x1

+x2

=l,

x1≤

y,x2≤

1−

y,

xi∈

[0,1],

y∈{0,1},...}

x•{〈∃

R.C

,≥,l〉

,...}

−→∃

x•{〈∃

R.C

,≥

,l〉

,...}

〈R

,≥

,l〉↓y•{〈C

,≥

,l〉}

x•{〈∀

R.C

,≥

,l1〉,...}

〈R

,≥

,l2〉↓

y•{...}

−→∀

x•{〈∀

R.C

,≥

,l1〉,...}

〈R

,≥

,l2〉↓

y•{...,〈C

,≥

,x〉

x+

y≥

l1,x≤

y,l1

+l2≤

2−

y,

x∈

[0,1],

y∈{0,1}}

......

...

x•{Av

C,〈A

,≥

,l〉

,...}

−→v

1x•{Av

C,〈C

,≥

,l〉,

...}

x•{Cv

A,〈A

,≤

,l〉

,...}

−→v

2x•{Cv

A,〈C

,≤

,l〉,

...}

......

...53

'&

$%

Exam

ple

•Suppose

K=

Au

Bv

C

〈a:A≥

0.3〉

〈a:B≥

0.4〉

Qu

ery

:=

glb

(K,a:C

)=

min{x|K∪{〈a:C≤

x〉

satisfi

able}

Ste

pTre

e

1.

a•{〈A

,≥

,0.3〉,〈B

,≥

,0.4〉,〈C

,≤

,x〉}

(Hypoth

esis)

2.

∪{〈Au

B,≤

,x〉}

(→v

2)

3.

∪{〈A

,≤

,x1〉,〈B

,≤

,x2〉}

(→u≤

)

∪{x

=x1

+x2−

1,1−

y≤

x1,y≤

x2}

∪{x

i∈

[0,1],

y∈{0,1}}

4.

find

min{x|〈a:A≥

0.3〉,〈a:B≥

0.4〉,

(MIL

PO

racle

)

〈a:C≤

x〉,〈a:A≤

x1〉,〈a:B≤

x2〉,

x=

x1

+x2−

1,1−

y≤

x1,y≤

x2,

xi∈

[0,1],

y∈{0,1}}

5.

MIL

Pora

cle

:x

=0.3

54

'&

$%

Imple

menta

tion

issues

•Several

option

sex

ists:

–Try

tom

apfu

zzyD

Ls

toclassical

DLs

–Try

tom

apfu

zzyD

Ls

tosom

efu

zzy/an

notated

logicprogram

min

g

framew

ork

–B

uild

anad

-hoc

theorem

prover

forfu

zzyD

Ls,

usin

ge.g.,

MIL

P

•A

theorem

prover

forfu

zzyALC

+lin

earhed

ges+

concrete

fuzzy

concep

ts,usin

gM

ILP,has

been

implem

ented

–Look

ing

forvolu

nteers

tocatch

-up

toth

eex

pressive

pow

erof

fuzzy

OW

LLite

orfu

zzyO

WL

DL

(EX

PT

IME

,N

EX

PT

IME

class)

55

'&

$%

Conclu

sions

•C

lassicalD

Ls

areth

ecore

ofstate

ofth

eart

ontology

descrip

tion

langu

ages,as

OW

LD

Lan

dO

WL

Lite

•E

fficien

tim

plem

entation

sex

ists,w

hich

takead

vantage

ofresearch

on

DLs

decision

algorithm

san

dcom

putation

alcom

plex

ityan

alysis

•Fuzzy

DLs

aimat

enhan

cing

the

expressive

pow

erof

DLs

toward

sth

e

represen

tationof

vague

concep

ts

•R

esearchis

stillin

it’sin

fancy

56

'&

$%

Futu

reW

ork

•To

getrid

with

the

subtleties

ofboth

Descrip

tionLogics

and

Fuzzy

Logics,

experts

fromboth

areasare

need

ed

•R

esearchdirection

s:

–C

omputation

alcom

plex

ityof

the

fuzzy

DLs

family

–D

esignof

efficien

treason

ing

algorithm

s

–C

ombin

ing

fuzzy

DLs

with

Logic

Program

min

g

–Lan

guage

exten

sions:

e.g.fu

zzyquan

tifiers

TopCustomer

=Customeru

(Usually)buys.ExpensiveItem

ExpensiveItem

=Itemu

∃price.High

–D

evelopin

ga

system

–...

57