KR

& R

© B

rach

man

& L

eves

que

200

51

Kn

ow

led

ge

Rep

rese

nta

tio

nan

dR

easo

nin

g

KR

& R

© B

rach

man

& L

eves

que

200

52

from

the

book

of t

he s

ame

nam

eby

Ron

ald

J. B

rach

man

and

Hec

tor

J. L

eves

que

Mor

gan

Kau

fman

n P

ublis

hers

, San

Fra

ncis

co, C

A, 2

004

KR

& R

© B

rach

man

& L

eves

que

200

53

1.

Intr

oduc

tion

KR

& R

© B

rach

man

& L

eves

que

200

54

Wh

at is

kn

ow

led

ge?

Eas

ier

ques

tion:

how

do

we

talk

abo

ut it

?

We

say

“Jo

hn k

now

s th

at ..

.” a

nd fi

ll th

e bl

ank

with

a p

ropo

sitio

n–

can

be tr

ue /

fals

e,

rig

ht /

wro

ng

Con

tras

t: “

John

fear

s th

at ..

.”–

sam

e co

nten

t, d

iffer

ent a

ttitu

de

Oth

er fo

rms

of k

now

ledg

e:•

know

how

, who

, wha

t, w

hen,

...

•se

nsor

imot

or:

typi

ng, r

idin

g a

bicy

cle

•af

fect

ive:

dee

p un

ders

tand

ing

Bel

ief:

not

nec

essa

rily

true

and

/or

held

for

appr

opria

te r

easo

nsan

d w

eake

r ye

t: “

John

sus

pect

s th

at ..

.”

Her

e: n

o di

stin

ctio

nta

king

the

wor

ld to

be

one

way

and

not

ano

ther

the

mai

n id

ea

KR

& R

© B

rach

man

& L

eves

que

200

55

Wh

at is

rep

rese

nta

tio

n?

Sym

bols

sta

ndin

g fo

r th

ings

in th

e w

orld

"Joh

n"

"Joh

n lo

ves

Mar

y"

firs

t aid

wom

en

John

the

prop

ositi

on th

at

John

love

s M

ary

Kno

wle

dge

repr

esen

tatio

n:sy

mbo

lic e

ncod

ing

of p

ropo

sitio

ns b

elie

ved

(by

som

e ag

ent)

KR

& R

© B

rach

man

& L

eves

que

200

56

Wh

at is

rea

son

ing

?

Man

ipul

atio

n of

sym

bols

enc

odin

g pr

opos

ition

s to

pro

duce

re

pres

enta

tions

of n

ew p

ropo

sitio

ns

Ana

logy

: ar

ithm

etic

“101

1” +

“10”

→→→→

“110

1”⇓

⇓

⇓

elev

en

tw

o

thirt

een

“Joh

n is

Mar

y's

fath

er”

⇓

“Joh

n is

an

adul

t m

ale”

⇓J

M

J

KR

& R

© B

rach

man

& L

eves

que

200

57

Wh

y kn

ow

led

ge?

For

suf

ficie

ntly

com

plex

sys

tem

s, it

is s

omet

imes

use

ful t

o de

scrib

e sy

stem

s in

term

s of

bel

iefs

, goa

ls, f

ears

, int

entio

ns

e.g.

in

a ga

me-

play

ing

prog

ram

“bec

ause

it b

elie

ved

its q

ueen

was

in d

ange

r, b

ut w

ante

d to

stil

l co

ntro

l the

cen

ter

of th

e bo

ard.

”

mor

e us

eful

than

des

crip

tion

abou

t act

ual t

echn

ique

s us

ed fo

rde

cidi

ng h

ow to

mov

e“b

ecau

se e

valu

atio

n pr

oced

ure

P u

sing

min

imax

ret

urne

d a

valu

e of

+7

for

this

pos

ition

= t

akin

g an

inte

ntio

nal s

tanc

e (

Dan

Den

nett)

Is K

R ju

st a

con

veni

ent w

ay o

f tal

king

abo

ut c

ompl

ex s

yste

ms?

•so

met

imes

ant

hrop

omor

phiz

ing

is in

appr

opria

tee.

g. t

herm

osta

ts

•ca

n al

so b

e ve

ry m

isle

adin

g!fo

olin

g us

ers

into

thin

king

a s

yste

m k

now

s m

ore

than

it d

oes

KR

& R

© B

rach

man

& L

eves

que

200

58

Wh

y re

pre

sen

tati

on

?

Not

e: in

tent

iona

l sta

nce

says

not

hing

abo

ut w

hat i

s or

is n

ot

repr

esen

ted

sym

bolic

ally

e.

g. i

n ga

me

play

ing,

per

haps

the

boar

d po

sitio

n is

rep

rese

nted

, but

the

goal

of

getti

ng a

kni

ght o

ut e

arly

is n

ot

KR

Hyp

othe

sis:

(B

rian

Sm

ith)

“Any

mec

hani

cally

em

bodi

ed in

telli

gent

pro

cess

will

be

com

pris

ed o

f str

uctu

ral

ingr

edie

nts

that

a)

we

as e

xter

nal o

bser

vers

nat

ural

ly ta

ke to

rep

rese

nt a

pr

opos

ition

al a

ccou

nt o

f the

kno

wle

dge

that

the

over

all p

roce

ss e

xhib

its, a

nd b

) in

depe

nden

t of s

uch

exte

rnal

sem

antic

attr

ibut

ion,

pla

y a

form

al b

ut c

ausa

l and

es

sent

ial r

ole

in e

ngen

derin

g th

e be

havi

our

that

man

ifest

s th

at k

now

ledg

e.”

Tw

o is

sues

: ex

iste

nce

of s

truc

ture

s th

at•

we

can

inte

rpre

t pro

posi

tiona

lly

•de

term

ine

how

the

syst

em b

ehav

es

Kno

wle

dge-

base

d sy

stem

: on

e de

sign

ed th

is w

ay!

KR

& R

© B

rach

man

& L

eves

que

200

59

Tw

o e

xam

ple

s

Exa

mpl

e 1

printColour(snow) :- !, write("It's white.").

printColour(grass) :- !, write("It's green.").

printColour(sky) :- !, write("It's yellow.").

printColour(X) :- write("Beats me.").

Exa

mpl

e 2

printColour(X) :- colour(X,Y), !,

write("It's "), write(Y), write(".").

printColour(X) :- write("Beats me.").

colour(snow,white).

colour(sky,yellow).

colour(X,Y) :- madeof(X,Z), colour(Z,Y).

madeof(grass,vegetation).

colour(vegetation,green).

Bot

h sy

stem

s ca

n be

des

crib

ed in

tent

iona

lly.

Onl

y th

e 2n

d ha

s a

sepa

rate

col

lect

ion

of s

ymbo

lic

stru

ctur

es à

la K

R H

ypot

hesi

s

its k

now

ledg

e ba

se (

or K

B)

∴

a sm

all k

now

ledg

e-ba

sed

syst

em

KR

& R

© B

rach

man

& L

eves

que

200

510

KR

an

d A

I

Muc

h of

AI i

nvol

ves

build

ing

syst

ems

that

are

kno

wle

dge-

base

dab

ility

der

ives

in p

art f

rom

rea

soni

ng o

ver

expl

icitl

y re

pres

ente

d kn

owle

dge

–la

ngua

ge u

nder

stan

ding

,

–pl

anni

ng,

–di

agno

sis,

–“e

xper

t sys

tem

s”,

etc.

Som

e, to

a c

erta

in e

xten

tga

me-

play

ing,

vis

ion,

e

tc.

Som

e, to

a m

uch

less

er e

xten

tsp

eech

, mot

or c

ontr

ol,

etc.

Cur

rent

res

earc

h qu

estio

n:ho

w m

uch

of in

telli

gent

beh

avio

ur is

kno

wle

dge-

base

d?

Cha

lleng

es: c

onne

ctio

nism

, oth

ers

KR

& R

© B

rach

man

& L

eves

que

200

511

Wh

y b

oth

er?

Why

not

“co

mpi

le o

ut”

know

ledg

e in

to s

peci

aliz

ed p

roce

dure

s?•

dist

ribut

e K

B to

pro

cedu

res

that

nee

d it

(as

in E

xam

ple

1)

•al

mos

t alw

ays

achi

eves

bet

ter

perf

orm

ance

No

need

to th

ink.

Just

do

it!–

ridin

g a

bike

–dr

ivin

g a

car

–pl

ayin

g ch

ess?

–do

ing

mat

h?

–st

ayin

g al

ive?

?

Ski

lls (

Hub

ert D

reyf

us)

•no

vice

s th

ink;

exp

erts

rea

ct

•co

mpa

re to

cur

rent

“ex

pert

sys

tem

s”:

know

ledg

e-ba

sed

!

KR

& R

© B

rach

man

& L

eves

que

200

512

Ad

van

tag

e

Kno

wle

dge-

base

d sy

stem

mos

t sui

tabl

e fo

r op

en-e

nded

tas

ksca

n st

ruct

ural

ly is

olat

e re

ason

s fo

r pa

rtic

ular

beh

avio

ur

Goo

d fo

r

•ex

plan

atio

n an

d ju

stifi

catio

n–

“Bec

ause

gra

ss is

a fo

rm o

f veg

etat

ion.

”

•in

form

abili

ty: d

ebug

ging

the

KB

–“N

o th

e sk

y is

not

yel

low

. It's

blu

e.”

•ex

tens

ibili

ty: n

ew r

elat

ions

–“C

anar

ies

are

yello

w.”

•ex

tens

ibili

ty: n

ew a

pplic

atio

ns–

retu

rnin

g a

list o

f all

the

whi

te th

ings

–pa

intin

g pi

ctur

es

KR

& R

© B

rach

man

& L

eves

que

200

513

Co

gn

itiv

e p

enet

rab

ility

Hal

lmar

k of

kno

wle

dge-

base

d sy

stem

:th

e ab

ility

to b

e to

ldfa

cts

abou

t the

wor

ld a

nd a

djus

t our

beh

avio

ur

corr

espo

ndin

gly

for

exam

ple:

rea

d a

book

abo

ut c

anar

ies

or r

are

coin

s

Cog

nitiv

e pe

netr

abili

ty (

Zen

on P

ylys

hyn)

actio

ns th

at a

re c

ondi

tione

d by

wha

t is

curr

ently

bel

ieve

d an

exa

mpl

e:

we

norm

ally

leav

e th

e ro

om if

we

hear

a fi

re a

larm

we

do n

ot le

ave

the

room

on

hear

ing

a fir

e al

arm

if

we

belie

ve th

at th

e al

arm

is b

eing

test

ed /

tam

pere

dca

n co

me

to th

is b

elie

f in

very

man

y w

ays

so th

is a

ctio

n is

cog

nitiv

ely

pene

trab

le

a no

n-ex

ampl

e:

blin

king

ref

lex

KR

& R

© B

rach

man

& L

eves

que

200

514

Wh

y re

aso

nin

g?

Wan

t kno

wle

dge

to a

ffect

act

ion

not

do a

ctio

n A

if se

nten

ce P

is in

KB

but

do a

ctio

n A

if w

orld

bel

ieve

d in

sat

isfie

s P

Diff

eren

ce:

Pm

ay n

ot b

e ex

plic

itly

repr

esen

ted

Nee

d to

app

ly w

hat i

s kn

own

in g

ener

al

to th

e pa

rtic

ular

s of

a g

iven

situ

atio

n

Exa

mpl

e: “P

atie

ntx

is a

llerg

ic to

med

icat

ion

m.”

“Any

body

alle

rgic

to m

edic

atio

n m

is a

lso

alle

rgic

to m

'.”

Is it

OK

to p

resc

ribe

m'

for

x?

Usu

ally

nee

d m

ore

than

just

DB

-sty

le r

etrie

val o

f fac

ts in

the

KB

KR

& R

© B

rach

man

& L

eves

que

200

515

En

tailm

ent

Sen

tenc

esP

1, P

2, ..

., P

nen

tail

sen

tenc

e P

iff t

he tr

uth

of P

isim

plic

it in

the

trut

h of

P1,

P2,

...,

Pn.

If th

e w

orld

is s

uch

that

it s

atis

fies

the

Pi t

hen

it m

ust a

lso

satis

fy P

.

App

lies

to a

var

iety

of l

angu

ages

(la

ngua

ges

with

trut

h th

eorie

s)

Infe

renc

e: th

e pr

oces

s of

cal

cula

ting

enta

ilmen

ts•

soun

d: g

et o

nly

enta

ilmen

ts

•co

mpl

ete:

get

all

enta

ilmen

ts

Som

etim

es w

ant u

nsou

nd /

inco

mpl

ete

reas

onin

gfo

r re

ason

s to

be

disc

usse

d la

ter

Logi

c: s

tudy

of e

ntai

lmen

t rel

atio

ns•

lang

uage

s

•tr

uth

cond

ition

s

•ru

les

of in

fere

nce

KR

& R

© B

rach

man

& L

eves

que

200

516

Usi

ng

log

ic

No

univ

ersa

l lan

guag

e / s

eman

tics

•W

hy n

ot E

nglis

h?

•D

iffer

ent t

asks

/ w

orld

s

•D

iffer

ent w

ays

to c

arve

up

the

wor

ld

No

univ

ersa

l rea

soni

ng s

chem

e•

Gea

red

to la

ngua

ge

•S

omet

imes

wan

t “ex

tral

ogic

al”

reas

onin

g

Sta

rt w

ith fi

rst-

orde

r pr

edic

ate

calc

ulus

(F

OL)

•in

vent

ed b

y ph

iloso

pher

Fre

ge fo

r th

e fo

rmal

izat

ion

of m

athe

mat

ics

•bu

t will

con

side

r su

bset

s / s

uper

sets

and

ver

y di

ffere

nt lo

okin

g re

pres

enta

tion

lang

uage

s

KR

& R

© B

rach

man

& L

eves

que

200

517

Kn

ow

led

ge

leve

l

Alle

n N

ewel

l's a

naly

sis:

•K

now

ledg

e le

vel:

dea

ls w

ith la

ngua

ge, e

ntai

lmen

t

•S

ymbo

l lev

el:

deal

s w

ith r

epre

sent

atio

n, in

fere

nce

Pic

king

a lo

gic

has

issu

es a

t eac

h le

vel

•K

now

ledg

e le

vel:

expr

essi

ve a

dequ

acy,

theo

retic

al c

ompl

exity

, ...

•S

ymbo

l lev

el:

arch

itect

ures

,da

ta s

truc

ture

s,

algo

rithm

ic c

ompl

exity

, ...

Nex

t: w

e be

gin

with

FO

L at

the

know

ledg

e le

vel

KR

& R

© B

rach

man

& L

eves

que

200

518

2.

The

Lan

guag

e of

Firs

t-or

der

Logi

c

KR

& R

© B

rach

man

& L

eves

que

200

519

Dec

lara

tive

lan

gu

age

Bef

ore

build

ing

syst

embe

fore

ther

e ca

n be

lear

ning

, rea

soni

ng, p

lann

ing,

expl

anat

ion

...

need

to b

e ab

le to

exp

ress

kno

wle

dge

Wan

t a p

reci

se d

ecla

rativ

e la

ngua

ge•

decl

arat

ive:

bel

ieve

P=

hol

d P

to b

e tr

ueca

nnot

bel

ieve

P w

ithou

t som

e se

nse

of

wha

t it w

ould

mea

n fo

r th

e w

orld

to s

atis

fy P

•pr

ecis

e: n

eed

to k

now

exa

ctly

w

hat s

trin

gs o

f sym

bols

cou

nt a

s se

nten

ces

wha

t it m

eans

for

a se

nten

ce to

be

true

(b

ut w

ithou

t hav

ing

to s

peci

fy w

hich

one

s ar

e tr

ue)

Her

e: l

angu

age

of fi

rst-

orde

r lo

gic

agai

n: n

ot th

e on

ly c

hoic

e

KR

& R

© B

rach

man

& L

eves

que

200

520

Alp

hab

et

Logi

cal s

ymbo

ls:

•P

unct

uatio

n: (

,),.

•C

onne

ctiv

es:

¬,∧

,∨,∀

, ∃, =

•V

aria

bles

:x,

x1,

x 2, .

.., x

', x"

, ...,

y, .

.., z

, ...

Fix

ed m

eani

ng a

nd u

se

like

keyw

ords

in a

pro

gram

min

g la

ngua

ge

Non

-logi

cal s

ymbo

ls•

Pre

dica

te s

ymbo

ls (

like

Dog

)N

ote

: not

trea

ting

= as

a p

redi

cate

•F

unct

ion

sym

bols

(li

ke b

estF

rien

dOf)

Dom

ain-

depe

nden

t mea

ning

and

use

like

iden

tifie

rs in

a p

rogr

amm

ing

lang

uage

Hav

e ar

ity:

num

ber

of a

rgum

ents

arity

0 p

redi

cate

s: p

ropo

sitio

nal s

ymbo

ls

arity

0 fu

nctio

ns: c

onst

ant s

ymbo

ls

Ass

ume

infin

ite s

uppl

y of

eve

ry a

rity

KR

& R

© B

rach

man

& L

eves

que

200

521

Gra

mm

ar

Ter

ms

1.E

very

var

iabl

e is

a te

rm.

2.If

t 1, t

2, ..

., t n

are

term

s an

d fi

s a

func

tion

of a

rity

n,th

enf(

t 1, t

2, ..

., t n

)is

a te

rm.

Ato

mic

wffs

(w

ell-f

orm

ed fo

rmul

a)

1.If

t 1, t

2, ..

., t n

are

term

s an

d P

is a

pre

dica

te o

f arit

y n,

then

P(t

1, t 2

, ...,

t n)

is a

n at

omic

wff.

2.If

t 1 a

nd t 2

are

term

s, th

en (

t 1=t

2) is

an

atom

ic w

ff.

Wffs 1

.Eve

ry a

tom

ic w

ff is

a w

ff.

2.If

α an

d β

are

wffs

, and

v is

a v

aria

ble,

then

¬α,

(α∧β

),(α

∨β),

∃v.α

, ∀v.

α ar

e w

ffs.

The

pro

posi

tiona

l sub

set:

no

term

s, n

o qu

antif

iers

Ato

mic

wffs

: on

ly p

redi

cate

s of

0-a

rity:

(p∧

¬(q

∨r))

KR

& R

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522

No

tati

on

Occ

asio

nally

add

or

omit

(,),

.

Use

[,] a

nd {

,} a

lso.

Abb

revi

atio

ns:

(α ⊃

β)

for

(¬α

∨ β)

safe

r to

rea

d as

dis

junc

tion

than

as

“if

... th

en ..

.”

(α ≡

β)

for

((α⊃

β) ∧

(β⊃

α))

Non

-logi

cal s

ymbo

ls:

•P

redi

cate

s:

mix

ed c

ase

capi

taliz

ed

Pers

on, H

appy

, Old

erT

han

•F

unct

ions

(an

d co

nsta

nts)

: mix

ed c

ase

unca

pita

lized

fath

erO

f, s

ucce

ssor

, jo

hnSm

ith

KR

& R

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523

Var

iab

le s

cop

e

Like

var

iabl

es in

pro

gram

min

g la

ngua

ges,

the

varia

bles

in F

OL

have

a s

cope

det

erm

ined

by

the

quan

tifie

rs

Lexi

cal s

cope

for

varia

bles

P(x

)∧

∃x[P

(x)

∨ Q

(x)]

free

boun

d

occu

rren

ces

of v

aria

bles

A s

ente

nce:

wff

with

no

free

var

iabl

es (

clos

ed)

Sub

stitu

tion:

α[v/

t] m

eans

α w

ith a

ll fr

ee o

ccur

renc

es o

f the

vre

plac

ed b

y te

rm t

Not

e: w

ritte

n α

els

ewhe

re (

and

in b

ook)

Als

o:α[

t 1,..

.,tn]

mea

nsα[

v 1/t

1,...

,vn/

t n]

v t

KR

& R

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Sem

anti

cs

How

to in

terp

ret s

ente

nces

?•

wha

t do

sent

ence

s cl

aim

abo

ut th

e w

orld

?

•w

hat d

oes

belie

ving

one

am

ount

to?

With

out a

nsw

ers,

can

not u

se s

ente

nces

to r

epre

sent

kno

wle

dge

Pro

blem

:ca

nnot

fully

spe

cify

inte

rpre

tatio

n of

sen

tenc

es b

ecau

se n

on-lo

gica

l sy

mbo

ls r

each

out

side

the

lang

uage

So:

mak

e cl

ear

depe

nden

ce o

f int

erpr

etat

ion

on n

on-lo

gica

l sym

bols

Logi

cal i

nter

pret

atio

n:sp

ecifi

catio

n of

how

to u

nder

stan

d pr

edic

ate

and

func

tion

sym

bols

Can

be

com

plex

!

Dem

ocra

ticC

ount

ry, I

sAB

ette

rJud

geO

fCha

ract

erT

han,

favo

urite

IceC

ream

Flav

ourO

f, p

uddl

eOfW

ater

27

KR

& R

© B

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525

Th

e si

mp

le c

ase

The

re a

re o

bjec

ts.

som

e sa

tisfy

pre

dica

te P

; so

me

do n

ot

Eac

h in

terp

reta

tion

settl

es e

xten

sion

of P

.bo

rder

line

case

s ru

led

in s

epar

ate

inte

rpre

tatio

ns

Eac

h in

terp

reta

tion

assi

gns

to fu

nctio

n f

a m

appi

ng fr

om o

bjec

ts

to o

bjec

ts.

func

tions

alw

ays

wel

l-def

ined

and

sin

gle-

valu

ed

The

FO

L as

sum

ptio

n:th

is is

all

you

need

to k

now

abo

ut th

e no

n-lo

gica

l sym

bols

to

und

erst

and

whi

ch s

ente

nces

of F

OL

are

true

or

fals

e

In o

ther

wor

ds, g

iven

a s

peci

ficat

ion

of»

wha

t obj

ects

ther

e ar

e

»w

hich

of t

hem

sat

isfy

P

»w

hat m

appi

ng is

den

oted

by

f

it w

ill b

e po

ssib

le to

say

whi

ch s

ente

nces

of F

OL

are

true

KR

& R

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Inte

rpre

tati

on

s

Tw

o pa

rts:

ℑ=

〈 D,I

〉

Dis

the

dom

ain

of d

isco

urse

can

be a

ny n

on-e

mpt

y se

t

not j

ust f

orm

al /

mat

hem

atic

al o

bjec

ts

e.g.

peop

le, t

able

s, n

umbe

rs, s

ente

nces

, uni

corn

s, c

hunk

s of

pea

nut b

utte

r,

situ

atio

ns, t

he u

nive

rse

I is

an

inte

rpre

tatio

n m

appi

ng

IfP

is a

pre

dica

te s

ymbo

l of a

rity

n,

I[P

] ⊆

D×D

× ...×

D

an n

-ary

rel

atio

n ov

er D

for

prop

ositi

onal

sym

bols

,

I[p]

= {

} o

rI[

p] =

{〈〉

}

In p

ropo

sitio

nal c

ase,

con

veni

ent t

o as

sum

e

ℑ

= I

∈ [

prop

. sym

bols

→ {

true

, fal

se}]

Iff

is a

func

tion

sym

bol o

f arit

y n,

I[f]

∈ [

D×D

× ...×

D→

D]

an n

-ary

func

tion

over

D

for

cons

tant

s,I[

c] ∈

D

KR

& R

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Den

ota

tio

n

In te

rms

of in

terp

reta

tion

ℑ, t

erm

s w

ill d

enot

e el

emen

ts o

f the

do

mai

nD

.

will

writ

e el

emen

t as

||t|| ℑ

For

term

s w

ith v

aria

bles

, the

den

otat

ion

depe

nds

on th

e va

lues

of

varia

bles

will

writ

e as

||t|| ℑ

,µ

whe

reµ

∈ [

Var

iabl

es→

D],

calle

d a

varia

ble

assi

gnm

ent

Rul

es o

f int

erpr

etat

ion:

1.||v

|| ℑ,µ

= µ

(v).

2.||

f(t 1

, t2,

...,

t n) |

| ℑ,µ

= H

(d1,

d2,

...,

d n)

whe

reH

= I[

f]

and

d i=

||t i

|| ℑ,µ,

recu

rsiv

ely

KR

& R

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528

Sat

isfa

ctio

n

In te

rms

of a

n in

terp

reta

tion

ℑ, s

ente

nces

of F

OL

will

be

eith

er

true

or

fals

e.

For

mul

as w

ith fr

ee v

aria

bles

will

be

true

for

som

e va

lues

of t

he

free

var

iabl

es a

nd fa

lse

for

othe

rs.

Not

atio

n:

will

writ

e as

ℑ,µ

= α

“α is

sat

isfie

d by

ℑ a

nd µ

”

whe

reµ

∈ [

Var

iabl

es→

D],

as b

efor

e

orℑ

= α

, w

hen

α is

a s

ente

nce

“

α is

true

und

er in

terp

reta

tion

ℑ”

orℑ

= S

, w

hen

S i

s a

set o

f sen

tenc

es

“the

ele

men

ts o

f S a

re tr

ue u

nder

inte

rpre

tatio

n ℑ

”

And

now

the

defin

ition

...

KR

& R

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529

Ru

les

of

inte

rpre

tati

on

1.ℑ

,µ=

P(t

1, t 2

, ...,

t n)

iff

〈d1,

d2,

...,

d n〉

∈ R

whe

reR

= I

[P]

and

d i=

||t

i|| ℑ

,µ,

as o

n de

nota

tion

slid

e

2.ℑ

,µ=

(t 1

= t 2

) i

ff|| t

1|| ℑ

,µ i

s th

e sa

me

as ||

t 2|| ℑ

,µ

3.ℑ

,µ=

¬α

iff

ℑ,µ

≠α

4.ℑ

,µ=

(α∧

β)

iffℑ

,µ=

α a

ndℑ

,µ=

β

5.ℑ

,µ=

(α∨

β)

iffℑ

,µ=

α o

rℑ

,µ=

β

6.ℑ

,µ=

∃vα

iff

for

som

e d

∈ D

,ℑ

,µ{d

;v}

=α

7.ℑ

,µ=

∀vα

iff

fo

r al

l d∈

D,

ℑ,µ

{d;v

}=

αw

here

µ{d;

v} is

just

like

µ, e

xcep

t tha

t µ(v

)=d.

For

pro

posi

tiona

l sub

set:

ℑ=

piff

I[p]

≠ {

}an

d th

e re

st a

s ab

ove

KR

& R

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530

En

tailm

ent

def

ined

Sem

antic

rul

es o

f int

erpr

etat

ion

tell

us h

ow to

und

erst

and

all w

ffs

in te

rms

of s

peci

ficat

ion

for

non-

logi

cal s

ymbo

ls.

But

som

e co

nnec

tions

am

ong

sent

ence

s ar

e in

depe

nden

t of t

he

non-

logi

cal s

ymbo

ls in

volv

ed.

e.g.

If α

is tr

ue u

nder

ℑ ,

then

so

is ¬

(β∧¬

α),

no m

atte

r w

hat ℑ

is,

why

α i

s tr

ue,

wha

t β is

, ...

S|=

α if

f fo

r ev

ery

ℑ ,

if ℑ

|=S

then

ℑ|=

α.S

ay th

at S

enta

ilsα

or α

is a

logi

cal c

onse

quen

ce o

f S:

In o

ther

wor

ds:

for

noℑ

,ℑ

|=S

∪ {

¬α}

.S

∪ {

¬α}

is u

nsat

isfia

ble

Spe

cial

cas

e w

hen

Sis

em

pty:

|=α

iff

for

ever

yℑ

,ℑ

|=α.

Say

that

α is

val

id.

Not

e:{α

1, α 2

, ...,

αn}

|=α

iff

|=(α

1 ∧ α

2∧

... ∧

αn)

⊃ α

finite

ent

ailm

ent r

educ

es to

val

idity

KR

& R

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531

Wh

y d

o w

e ca

re?

We

do n

ot h

ave

acce

ss to

use

r-in

tend

ed in

terp

reta

tion

of n

on-

logi

cal s

ymbo

ls

But

, with

ent

ailm

ent ,

we

know

that

if S

is tr

ue in

the

inte

nded

in

terp

reta

tion,

then

so

is α

.If

the

user

's v

iew

has

the

wor

ld s

atis

fyin

g S

,the

n it

mus

t als

o sa

tisfy

α.

The

re m

ay b

e ot

her

sent

ence

s tr

ue a

lso;

but

α is

logi

cally

gua

rant

eed.

So

wha

t abo

ut o

rdin

ary

reas

onin

g?D

og(f

ido)

�

Mam

mal

(fid

o) ?

?

Not

ent

ailm

ent!

The

re a

re lo

gica

l int

erpr

etat

ions

whe

reI[

Dog

]⊄

I[M

amm

al]

incl

ude

such

con

nect

ions

exp

licitl

y in

S

∀x[

Dog

(x)

⊃ M

amm

al(x

)]

Get

:S

∪ {

Dog

(fid

o)}

|=M

amm

al(f

ido)

Key

idea

of K

R:

the

rest

is ju

stde

tails

...

KR

& R

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que

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532

Kn

ow

led

ge

bas

es

KB

is s

et o

f sen

tenc

esex

plic

it st

atem

ent o

f sen

tenc

es b

elie

ved

(incl

udin

g an

y as

sum

ed

conn

ectio

ns a

mon

g no

n-lo

gica

l sym

bols

)

KB

|=α

α is

a fu

rthe

r co

nseq

uenc

e of

wha

t is

belie

ved

•ex

plic

it kn

owle

dge:

K

B

•im

plic

it kn

owle

dge:

{ α

| K

B|=

α }

Ofte

n no

n tr

ivia

l: e

xplic

it�

impl

icit

Exa

mpl

e: Thr

ee b

lock

s st

acke

d.

Top

one

is g

reen

.

Bot

tom

one

is n

ot g

reen

.

Is th

ere

a gr

een

bloc

k di

rect

ly o

n to

p of

a n

on-g

reen

blo

ck?

A B C

gree

n

non-

gree

n

KR

& R

© B

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man

& L

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que

200

533

A f

orm

aliz

atio

n

S=

{On(

a,b)

,O

n(b,

c),

Gre

en(a

),¬

Gre

en(c

)}al

l tha

t is

requ

ired

α =

∃x∃y

[Gre

en(x

)∧

¬G

reen

(y)

∧ O

n(x,

y)]

Cla

im:

S|=

α

Pro

of:

Letℑ

be

any

inte

rpre

tatio

n su

ch th

at ℑ

|=S

.

Cas

e 1:

ℑ|=

Gre

en(b

).C

ase

2:ℑ

|≠ G

reen

(b).

∴ ℑ

|= G

reen

(b) ∧

¬G

reen

(c)

∧ O

n(b,

c).

∴ ℑ

|=¬

Gre

en(b

)

∴ ℑ

|=α

∴ ℑ

|=G

reen

(a) ∧

¬G

reen

(b)

∧ O

n(a,

b).

∴ ℑ

|=α

Eith

er w

ay,

for

any

ℑ,

ifℑ

|=S

then

ℑ|=

α.

So

S|=

α.

QE

D

KR

& R

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que

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534

Kn

ow

led

ge-

bas

ed s

yste

m

Sta

rt w

ith (

larg

e) K

B r

epre

sent

ing

wha

t is

expl

icitl

y kn

own

e.g.

wha

t the

sys

tem

has

bee

n to

ld o

r ha

s le

arne

d

Wan

t to

influ

ence

beh

avio

ur b

ased

on

wha

t is

impl

icit

in th

e K

B(o

r as

clo

se a

s po

ssib

le)

Req

uire

s re

ason

ing

dedu

ctiv

e in

fere

nce:

proc

ess

of c

alcu

latin

g en

tailm

ents

of K

B

i.e g

iven

KB

and

any

α, d

eter

min

e if

KB

|=α

Pro

cess

is s

ound

if w

hene

ver

it pr

oduc

es α

, the

n K

B |=

αdo

es n

ot a

llow

for

plau

sibl

e as

sum

ptio

ns th

at m

ay b

e tr

uein

the

inte

nded

inte

rpre

tatio

n

Pro

cess

is c

ompl

ete

if w

hene

ver

KB

|=α,

it p

rodu

ces

αdo

es n

ot a

llow

for

proc

ess

to m

iss

som

e α

or b

e un

able

to

dete

rmin

e th

e st

atus

of α

KR

& R

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que

200

535

3.

Exp

ress

ing

Kno

wle

dge

KR

& R

© B

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man

& L

eves

que

200

536

Kn

ow

led

ge

eng

inee

rin

g

KR

is fi

rst a

nd fo

rem

ost a

bout

kno

wle

dge

mea

ning

and

ent

ailm

ent

find

indi

vidu

als

and

prop

ertie

s, t

hen

enco

de fa

cts

suffi

cien

t for

ent

ailm

ents

Bef

ore

impl

emen

ting,

nee

d to

und

erst

and

clea

rly•

wha

t is

to b

e co

mpu

ted?

•w

hy a

nd w

here

infe

renc

e is

nec

essa

ry?

Exa

mpl

e do

mai

n: s

oap-

oper

a w

orld

peop

le, p

lace

s, c

ompa

nies

, mar

riage

s, d

ivor

ces,

han

ky-p

anky

, dea

ths,

ki

dnap

ping

s, c

rimes

, ...

Tas

k: K

B w

ith a

ppro

pria

te e

ntai

lmen

ts•

wha

t voc

abul

ary?

•w

hat f

acts

to r

epre

sent

?

KR

& R

© B

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man

& L

eves

que

200

537

Vo

cab

ula

ry

Dom

ain-

depe

nden

t pre

dica

tes

and

func

tions

mai

n qu

estio

n: w

hat a

re th

e in

divi

dual

s?

here

: pe

ople

, pla

ces,

com

pani

es, .

..

nam

ed in

divi

dual

s jo

hn, s

leez

yTow

n, f

aulty

Insu

ranc

eCor

p, f

ic, j

ohnQ

smith

, ...

basi

c ty

pes

Pers

on, P

lace

, Man

, Wom

an, .

..

attr

ibut

esR

ich,

Bea

utif

ul, U

nscr

upul

ous,

...

rela

tions

hips

Liv

esA

t, M

arri

edT

o, D

augh

terO

f, H

adA

nAff

airW

ith,

Bla

ckm

ails

, ...

func

tions

fath

erO

f, c

eoO

f, b

estF

rien

dOf,

...

KR

& R

© B

rach

man

& L

eves

que

200

538

Bas

ic f

acts

Usu

ally

ato

mic

sen

tenc

es a

nd n

egat

ions

type

fact

sM

an(j

ohn)

,

Wom

an(j

ane)

,

Com

pany

(fau

ltyIn

sura

nceC

orp)

prop

erty

fact

sR

ich(

john

),

¬H

appi

lyM

arri

ed(j

im),

Wor

ksFo

r(jim

,fic

)

equa

lity

fact

sjo

hn =

ceo

Of(

fic)

,

fic

= f

aulty

Insu

ranc

eCor

p,

best

Frie

ndO

f(jim

) =

john

Like

a s

impl

e da

taba

se (

can

stor

e in

a ta

ble)

KR

& R

© B

rach

man

& L

eves

que

200

539

Co

mp

lex

fact

s

Uni

vers

al a

bbre

viat

ions

∀y[

Wom

an(y

)∧

y≠

jane

⊃ L

oves

(y,jo

hn)]

∀y[

Ric

h(y)

∧ M

an(y

)⊃

Lov

es(y

,jane

)]

∀x∀

y[L

oves

(x,y

)⊃

¬B

lack

mai

ls(x

,y)]

Inco

mpl

ete

know

ledg

eL

oves

(jan

e,jo

hn)

∨ L

oves

(jan

e,jim

)w

hich

?

∃x[A

dult(

x)∧

Bla

ckm

ails

(x,jo

hn)]

who

?

Clo

sure

axi

oms

∀x[

Pers

on(x

)⊃

x=

jane

∨ x=

john

∨ x=

jim ..

.]

∀x∀

y[M

arri

edT

o(x,

y)⊃

...

]

∀x[

x=fi

c ∨

x=ja

ne∨

x=jo

hn∨

x=jim

...]

also

use

ful t

o ha

ve j

ane

≠ jo

hn

...

poss

ible

to e

xpre

ss

with

out q

uant

ifier

s

cann

ot w

rite

dow

na

mor

e co

mpl

ete

vers

ion

limit

the

dom

ain

of d

isco

urse

KR

& R

© B

rach

man

& L

eves

que

200

540

Ter

min

olo

gic

al f

acts

Gen

eral

rel

atio

nshi

ps a

mon

g pr

edic

ates

. F

or e

xam

ple:

disj

oint

∀x[

Man

(x)

⊃ ¬

Wom

an(x

)]

subt

ype

∀x[

Sena

tor(

x)⊃

Leg

isla

tor(

x)]

exha

ustiv

e∀

x[A

dult(

x)⊃

Man

(x) ∨

Wom

an(x

)]

sym

met

ry∀

x∀y

[Mar

ried

To(

x,y)

⊃ M

arri

edT

o(y,

x)]

inve

rse

∀x∀

y[C

hild

Of(

x,y)

⊃ P

aren

tOf(

y,x)

]

type

res

tric

tion

∀x∀

y[M

arri

edT

o(x,

y)⊃

Pe

rson

(x)

∧ P

erso

n(y)

∧ O

ppSe

x(x,

y)]

Usu

ally

uni

vers

ally

qua

ntifi

ed c

ondi

tiona

ls o

r bi

cond

ition

als

som

etim

es

KR

& R

© B

rach

man

& L

eves

que

200

541

En

tailm

ents

: 1

Is th

ere

a co

mpa

ny w

hose

CE

O lo

ves

Jane

?

∃x[C

ompa

ny(x

)∧

Lov

es(c

eoO

f(x)

,jane

)] ?

?

Sup

pose

ℑ|=

KB

.T

hen

ℑ|=

Ric

h(jo

hn),

Man

(joh

n),

and

ℑ|=

∀y[

Ric

h(y)

∧ M

an(y

)⊃

Lov

es(y

,jane

)]

soℑ

|= L

oves

(joh

n,ja

ne).

Als

oℑ

|= j

ohn

= c

eoO

f(fi

c),

soℑ

|= L

oves

( ce

oOf(

fic)

,jane

).F

inal

lyℑ

|= C

ompa

ny(f

aulty

Insu

ranc

eCor

p),

and

ℑ|=

fic

= f

aulty

Insu

ranc

eCor

p,

soℑ

|= C

ompa

ny(f

ic).

Thu

s,ℑ

|= C

ompa

ny(f

ic)

∧ L

oves

( ce

oOf(

fic)

,jane

),

and

so

ℑ|=

∃x[C

ompa

ny(x

)∧

Lov

es(c

eoO

f(x)

,jane

)].

Can

ext

ract

iden

tity

of c

ompa

ny fr

om th

is p

roof

KR

& R

© B

rach

man

& L

eves

que

200

542

En

tailm

ents

: 2

If no

man

is b

lack

mai

ling

John

, the

n is

he

bein

g bl

ackm

aile

d by

so

meb

ody

he lo

ves?

∀x[

Man

(x)

⊃ ¬

Bla

ckm

ails

(x,jo

hn)]

⊃∃y

[Lov

es(j

ohn,

y)∧

Bla

ckm

ails

(y,jo

hn)]

??

Not

e:

KB

|=(α

⊃ β

) i

ff K

B ∪

{α}

|=β

Let:

ℑ|=

KB

∪ {

∀x[

Man

(x)

⊃¬

Bla

ckm

ails

(x,jo

hn)]

}S

how

:ℑ

|=∃y

[Lov

es(j

ohn,

y)∧

Bla

ckm

ails

(y,jo

hn)

Hav

e:∃x

[Adu

lt(x)

∧ B

lack

mai

ls(x

,john

)]an

d∀

x[A

dult(

x)⊃

Man

(x)

∨ W

oman

(x)]

so∃x

[Wom

an(x

)∧

Bla

ckm

ails

(x,jo

hn)]

.

The

n:∀

y[R

ich(

y)∧

Man

(y)

⊃ L

oves

(y,ja

ne)]

and

Ric

h(jo

hn)

∧ M

an(j

ohn)

soL

oves

(joh

n,ja

ne)!

But

:∀

y[W

oman

(y)

∧y

≠ ja

ne⊃

Lov

es(y

,john

)]an

d∀

x∀y[

Lov

es(x

,y)

⊃¬

Bla

ckm

ails

(x,y

)]so

∀y[

Wom

an(y

)∧

y≠

jane

⊃¬

Bla

ckm

ails

(y,jo

hn)]

and

Bla

ckm

ails

(jan

e,jo

hn)!

!

Fin

ally

:L

oves

(joh

n,ja

ne)

∧ B

lack

mai

ls(j

ane,

john

)so

:∃y

[Lov

es(j

ohn,

y)∧

Bla

ckm

ails

(y,jo

hn)]

KR

& R

© B

rach

man

& L

eves

que

200

543

Wh

at in

div

idu

als?

Som

etim

es u

sefu

l to

redu

ce n

-ary

pre

dica

tes

to 1

-pla

ce

pred

icat

es a

nd 1

-pla

ce fu

nctio

ns•

invo

lves

rei

fyin

g pr

oper

ties:

new

indi

vidu

als

•ty

pica

l of d

escr

iptio

n lo

gics

/ fr

ame

lang

uage

s

(la

ter)

Fle

xibi

lity

in te

rms

of a

rity:

Purc

hase

s(jo

hn,s

ears

,bik

e)or

Purc

hase

s(jo

hn,s

ears

,bik

e,fe

b14)

orPu

rcha

ses(

john

,sea

rs,b

ike,

feb1

4,$1

00)

Inst

ead:

intr

oduc

e pu

rcha

se o

bjec

ts

Purc

hase

(p)

∧ a

gent

(p)=

john

∧ o

bj(p

)=bi

ke∧

sour

ce(p

)=se

ars

∧ ..

.al

low

s pu

rcha

se to

be

desc

ribed

at v

ario

us le

vels

of d

etai

l

Com

plex

rel

atio

nshi

ps:

Mar

ried

To(

x,y)

vs.

ReM

arri

edT

o(x,

y)vs

. ...

Inst

ead

def

ine

mar

ital s

tatu

s in

term

s of

exi

sten

ce o

fm

arria

ge a

nd d

ivor

ce e

vent

s.

Mar

riag

e(m

)∧

hus

band

(m)=

x∧

wif

e(m

)=y

∧ d

ate(

m)=

...∧.

..

KR

& R

© B

rach

man

& L

eves

que

200

544

Ab

stra

ct in

div

idu

als

Als

o ne

ed in

divi

dual

s fo

r nu

mbe

rs, d

ates

, tim

es, a

ddre

sses

, etc

.ob

ject

s ab

out w

hich

we

ask

wh-

ques

tions

Qua

ntiti

es a

s in

divi

dual

sag

e(su

zy)

= 1

4

age-

in-y

ears

(suz

y) =

14

age-

in-m

onth

s(su

zy)

= 1

68

perh

aps

bette

r to

hav

e an

obj

ect f

or “

the

age

of S

uzy”

, who

se v

alue

in y

ears

is 1

4

year

s(ag

e(su

zy))

= 1

4

mon

ths(

x)=

12*

year

s(x)

cent

imet

ers(

x)=

100

*met

ers(

x)

Sim

ilarly

with

loca

tions

and

tim

esin

stea

d of

time(

m)=

"Jan

5 2

006

4:47

:03E

ST"

can

use

time(

m)=

t∧

yea

r(t)

=20

06∧

...

KR

& R

© B

rach

man

& L

eves

que

200

545

Oth

er s

ort

s o

f fa

cts

Sta

tistic

al /

prob

abili

stic

fact

s•

Hal

f of t

he c

ompa

nies

are

loca

ted

on th

e E

ast S

ide.

•M

ost o

f the

em

ploy

ees

are

rest

less

.

•A

lmos

t non

e of

the

empl

oyee

s ar

e co

mpl

etel

y tr

ustw

orth

y,

Def

ault

/ pro

toty

pica

l fac

ts•

Com

pany

pre

side

nts

typi

cally

hav

e se

cret

arie

s in

terc

eptin

g th

eir

phon

e ca

lls.

•C

ars

have

four

whe

els.

•C

ompa

nies

gen

eral

ly d

o no

t allo

w e

mpl

oyee

s th

at w

ork

toge

ther

to b

e m

arrie

d.

Inte

ntio

nal f

acts

•Jo

hn b

elie

ves

that

Hen

ry is

tryi

ng to

bla

ckm

ail h

im.

•Ja

ne d

oes

not w

ant J

im to

thin

k th

at s

he lo

ves

John

.

Oth

ers

...

KR

& R

© B

rach

man

& L

eves

que

200

546

4.

Res

olut

ion

KR

& R

© B

rach

man

& L

eves

que

200

547

Go

al

Ded

uctiv

e re

ason

ing

in la

ngua

ge a

s cl

ose

as p

ossi

ble

to fu

ll F

OL

¬,

∧, ∨

, ∃,

∀

Kno

wle

dge

Leve

l:gi

ven

KB

, α,

det

erm

ine

if K

B |=

α.

orgi

ven

an o

pen

α[x 1

,x2,

...x n

], fi

nd t 1

,t 2,..

.t n s

uch

that

KB

|=α[

t 1,t 2

,...t n

]

Whe

n K

B is

fini

te {

α 1, α

2, ...

, αk}

KB

|=α

iff|=

[(α 1

∧ α

2 ∧ ..

. ∧ α

k) ⊃

α]

iffK

B∪

{¬

α} i

s un

satis

fiabl

e

iff

KB

∪ {

¬α}

|=

FA

LSE

whe

re F

ALS

E is

som

ethi

ng li

ke ∃

x.(x

≠x)

So

wan

t a p

roce

dure

to te

st fo

r va

lidity

, or

satis

fiabi

lity,

or

for

enta

iling

FA

LSE

.

Will

now

con

side

r su

ch a

pro

cedu

re (

first

with

out q

uant

ifier

s)

KR

& R

© B

rach

man

& L

eves

que

200

548

Cla

usa

l rep

rese

nta

tio

n

For

mul

a =

set

of c

laus

es

Cla

use

= s

et o

f lite

rals

Lite

ral

= a

tom

ic s

ente

nce

or it

s ne

gatio

npo

sitiv

e lit

eral

and

neg

ativ

e lit

eral

Not

atio

n:If

ρis

a li

tera

l, th

en ρ

is it

s co

mpl

emen

t

p⇒

¬p

¬p

⇒ p

To

dist

ingu

ish

clau

ses

from

form

ulas

:

[ an

d ]

for

clau

ses:

[p, r

, s]

{ an

d }

for

form

ulas

:{

[p, r

, s],

[p,

r, s

], [

p]

}

[] i

s th

e em

pty

clau

se{}

is th

e em

pty

form

ula

So

{} is

diff

eren

t fro

m{[

]}!

Inte

rpre

tatio

n:F

orm

ula

und

erst

ood

as c

onju

nctio

n of

cla

uses

Cla

use

unde

rsto

od a

s di

sjun

ctio

n of

lite

rals

Lite

rals

und

erst

ood

norm

ally

{[p,

¬q]

, [r]

, [s]

}

repr

esen

ts

((p

∨ ¬

q)∧

r∧

s)

[ ]

repr

esen

ts

FAL

SE

KR

& R

© B

rach

man

& L

eves

que

200

549

CN

F a

nd

DN

F

Eve

ry p

ropo

sitio

nal w

ff α

can

be c

onve

rted

into

a fo

rmul

a α′

inC

onju

nctiv

e N

orm

al F

orm

(C

NF

) in

suc

h a

way

that

|=

α ≡

α′.

1.el

imin

ate

⊃ a

nd≡

usi

ng (

α ⊃

β)

� (

¬α

∨ β)

etc

.

2.pu

sh ¬

inw

ard

usi

ng¬

(α ∧

β)

� (

¬α

∨ ¬

β) e

tc.

3.di

strib

ute

∨ ov

er∧

usi

ng (

(α ∧

β)

∨ γ)

� (

(α ∨

γ)

∧ (β

∨ γ

))

4.co

llect

term

s u

sing

(α

∨ α)

� α

etc

.

Res

ult i

s a

conj

unct

ion

of d

isju

nctio

n of

lite

rals

.an

ana

logo

us p

roce

dure

pro

duce

s D

NF

, a

disj

unct

ion

of c

onju

nctio

n of

lite

rals

We

can

iden

tify

CN

F w

ffs w

ith c

laus

al fo

rmul

as(p

∨ ¬

q∨

r)∧

(s ∨

¬r)

�

{ [p

,¬q,

r],

[s,

¬r]

}

So:

giv

en a

fini

te K

B, t

o fin

d ou

t if K

B |=

α, i

t will

be

suffi

cien

t to

1.pu

t (K

B ∧

¬α)

into

CN

F, a

s ab

ove

2.de

term

ine

the

satis

fiabi

lity

of th

e cl

ause

s

KR

& R

© B

rach

man

& L

eves

que

200

550

Res

olu

tio

n r

ule

of

infe

ren

ce

Giv

en tw

o cl

ause

s, in

fer

a ne

w c

laus

e:F

rom

cla

use

{p

}∪

C1,

and

{¬

p}

∪ C

2,

infe

r cl

ause

C1

∪ C

2.

C1

∪ C

2 is

cal

led

a re

solv

ent o

f inp

ut c

laus

es w

ith r

espe

ct to

p.

Exa

mpl

e:cl

ause

s[w

,r, q

] a

nd[w

, s, ¬

r] h

ave

[w, q

, s]

as

reso

lven

t wrt

r.

Spe

cial

Cas

e:

[p]

and

[¬p]

res

olve

to [

] (

the

C1

and

C2

are

em

pty)

A d

eriv

atio

n of

a c

laus

e c

from

a s

et S

of c

laus

es is

a s

eque

nce

c 1, c

2, ..

., c n

of c

laus

es, w

here

cn

=c,

and

for

each

ci,

eith

er

1. c

i∈

S,

or

2. c

i is

a r

esol

vent

of t

wo

earli

er c

laus

es i

n th

e de

rivat

ion

Writ

e:S

→ c

if th

ere

is a

der

ivat

ion

KR

& R

© B

rach

man

& L

eves

que

200

551

Rat

ion

ale

Res

olut

ion

is a

sym

bol-l

evel

rul

e of

infe

renc

e, b

ut h

as a

co

nnec

tion

to k

now

ledg

e-le

vel l

ogic

al in

terp

reta

tions

Cla

im: R

esol

vent

is e

ntai

led

by in

put c

laus

es.

Sup

pose

ℑ|=

(p

∨ α)

and

ℑ|=

(¬

p∨

β)C

ase

1:ℑ

|=p

then

ℑ|=

β,

soℑ

|= (

α ∨

β).

Cas

e 2:

ℑ|≠

p

then

ℑ|=

α,

soℑ

|= (

α ∨

β).

Eith

er w

ay,

ℑ|=

(α

∨ β)

.

So:

{(p

∨ α)

, (¬

p∨

β)}

|= (

α ∨

β).

Spe

cial

cas

e:[p

] a

nd[¬

p] r

esol

ve to

[ ]

,

so{[

p],[

¬p]

}|=

FA

LSE

that

is:

{[p]

,[¬

p]}

is u

nsat

isfia

ble

KR

& R

© B

rach

man

& L

eves

que

200

552

Der

ivat

ion

s an

d e

nta

ilmen

t

Can

ext

end

the

prev

ious

arg

umen

t to

deriv

atio

ns:

IfS

→ c

the

n S

|= c

Pro

of:

by

indu

ctio

n on

the

leng

th o

f the

der

ivat

ion.

Sho

w (

by lo

okin

g at

the

two

case

s) th

at S

|= c

i.

But

the

conv

erse

doe

s no

t hol

d in

gen

eral

Can

hav

e S

|= c

with

out h

avin

gS

→ c

.

Exa

mpl

e:{[

¬p]

}|=

[¬

p,¬

q]i.e

.¬

p|=

(¬

p∨

¬q)

but n

o de

rivat

ion

How

ever

....

Res

olut

ion

isre

futa

tion

com

plet

e!

Th

eore

m:

S→

[]

iff

S |=

[]

Res

ult w

ill c

arry

ove

r to

qua

ntifi

ed c

laus

es (

late

r)

So

for

any

set S

of c

laus

es: S

is u

nsat

isfia

ble

iff

S→

[].

Pro

vide

s m

etho

d fo

r de

term

inin

g sa

tisfia

bilit

y: s

earc

h al

l der

ivat

ions

for

[].

So

prov

ides

a m

etho

d fo

r de

term

inin

g al

l ent

ailm

ents

soun

d an

d co

mpl

ete

whe

n re

stric

ted

to [

]

KR

& R

© B

rach

man

& L

eves

que

200

553

A p

roce

du

re f

or

enta

ilmen

t

To

dete

rmin

e if

KB

|=α,

•pu

t KB

, ¬α

into

CN

F to

get

S,

as b

efor

e

•ch

eck

if S

→ []

.

Non

-det

erm

inis

tic p

roce

dure

1.C

heck

if []

is in

S.

If ye

s, th

en r

etur

n U

NS

AT

ISF

IAB

LE

2.C

heck

if th

ere

are

two

clau

ses

in S

such

that

they

re

solv

e to

pro

duce

a c

laus

e th

at is

not

alre

ady

in S

.If

no, t

hen

retu

rn S

AT

ISF

IAB

LE

3.A

dd th

e ne

w c

laus

e to

S a

nd g

o to

1.

Not

e: n

eed

only

con

vert

KB

to C

NF

onc

e•

can

hand

le m

ultip

le q

uerie

s w

ith s

ame

KB

•af

ter

addi

tion

of n

ew fa

ct α

, can

sim

ply

add

new

cla

uses

α′ t

o K

B

So:

goo

d id

ea to

kee

p K

B in

CN

F

If K

B =

{},

then

we

are

test

ing

the

valid

ity o

f α

KR

& R

© B

rach

man

& L

eves

que

200

554

Exa

mp

le 1

Sho

w t

hat

KB

|= G

irl

[Fir

stG

rade

] [¬Fi

rstG

rade

, Chi

ld]

[¬C

hild

,¬Fe

mal

e, G

irl]

[¬C

hild

,¬M

ale,

Boy

]

[¬K

inde

rgar

ten,

Chi

ld]

[Fem

ale]

[¬G

irl]

[Chi

ld] [G

irl,

¬Fe

mal

e]

[Gir

l]

[]

nega

tion

of

quer

y

Der

ivat

ion

has

9 cl

ause

s, 4

new

Firs

tGra

de

Firs

tGra

de⊃

Chi

ld

Chi

ld∧

Mal

e ⊃

Boy

Kin

derg

arte

n⊃

Chi

ld

Chi

ld∧

Fem

ale

⊃ G

irl

Fem

ale

KB

KR

& R

© B

rach

man

& L

eves

que

200

555

Exa

mp

le 2

[Rai

n, S

un]

[¬

Sun,

Mai

l] [

¬R

ain,

Mai

l] [

¬M

ail]

[¬Sl

eet,

Mai

l]

[¬R

ain]

[¬Su

n]

[Rai

n]

[]N

ote:

eve

ry c

laus

e no

t in

Sha

s 2

pare

nts

Sho

w K

B |=

Mai

l(R

ain

∨ Su

n)

(Sun

⊃ M

ail)

((R

ain

∨ Sl

eet)

⊃ M

ail)

KB

Sim

ilarly

K

B |≠

Rai

nC

an e

num

erat

e al

l res

olve

nts

give

n ¬

Rai

n,an

d[]

will

not

be

gene

rate

d

KR

& R

© B

rach

man

& L

eves

que

200

556

Qu

anti

fier

s

Cla

usal

form

as

befo

re, b

ut a

tom

isP

(t1,

t 2, .

.., t n

), w

here

t i m

ay

cont

ain

varia

bles

Inte

rpre

tatio

n as

bef

ore,

but

var

iabl

es a

re u

nder

stoo

d un

iver

sally

Exa

mpl

e:{

[P(x

),¬

R(a

,f(b,

x))]

, [Q

(x,y

)] }

inte

rpre

ted

as

∀x∀

y{[R

(a,f(

b,x)

)⊃

P(x

)]∧

Q(x

,y)}

Sub

stitu

tions

:θ

={v

1/t 1

, v 2

/t2,

...,

v n/t n

}

Not

atio

n: I

f ρis

a li

tera

l and

θ is

a s

ubst

itutio

n, th

enρθ

is

the

resu

lt of

the

subs

titut

ion

(and

sim

ilarly

, cθ

whe

rec

is a

cla

use)

Exa

mpl

e:θ

={x

/a,y

/g(x

,b,z

)}

P(x

,z,f(

x,y)

)θ

=P

(a,z

,f(a,

g(x,

b,z)

))

A li

tera

l is

grou

nd if

it c

onta

ins

no v

aria

bles

.

A li

tera

l ρ is

an

inst

ance

of ρ

′, if

for

som

e θ,

ρ =

ρ′θ

.

KR

& R

© B

rach

man

& L

eves

que

200

557

Gen

eral

izin

g C

NF

Res

olut

ion

will

gen

eral

ize

to h

andl

ing

varia

bles

But

to c

onve

rt w

ffs to

CN

F, w

e ne

ed th

ree

addi

tiona

l ste

ps:

1.el

imin

ate

⊃ a

nd≡

2.pu

sh¬

inw

ard

usi

ng a

lso

¬∀

x.α

�

∃x.¬

α e

tc.

3.st

anda

rdiz

e va

riabl

es: e

ach

quan

tifie

r ge

ts it

s ow

n va

riabl

e

e.g.

∃x[P

(x)]

∧ Q

(x)

�

∃z[P

(z)]

∧ Q

(x)

whe

rez

is a

new

var

iabl

e

4. e

limin

ate

all e

xist

entia

ls(d

iscu

ssed

late

r)

5.m

ove

univ

ersa

ls to

the

fron

t us

ing

(∀

xα) ∧

β�

∀

x(α∧

β)

whe

reβ

does

not

use

x

6.di

strib

ute

∨ ov

er∧

7.co

llect

term

s

Get

uni

vers

ally

qua

ntifi

ed c

onju

nctio

n of

dis

junc

tion

of li

tera

lsth

en d

rop

all t

he q

uant

ifier

s...

Igno

re =

for

now

KR

& R

© B

rach

man

& L

eves

que

200

558

Fir

st-o

rder

res

olu

tio

n

Mai

n id

ea: a

lite

ral (

with

var

iabl

es)

stan

ds fo

r al

l its

inst

ance

s; s

o al

low

all

such

infe

renc

es

So

give

n [P

(x,a

),¬

Q(x

)]an

d[¬

P(b

,y),

¬R

(b,f(

y))]

,w

ant t

o in

fer

[¬Q

(b),

¬R

(b,f(

a))]

am

ong

othe

rssi

nce

[P(x

,a),

¬Q

(x)]

has

[P(b

,a),

¬Q

(b)]

and

[¬P

(b,y

),¬

R(b

,f(y

))]

has

[¬P

(b,a

),¬

R(b

,f(a)

)]

Res

olut

ion:

Giv

en c

laus

es:

{ρ1}

∪C

1 a

nd {

ρ 2}

∪C

2.

Ren

ame

varia

bles

, so

that

dis

tinct

in tw

o cl

ause

s.

For

any

θ s

uch

that

ρ1θ

=ρ 2

θ, c

an in

fer

(C1

∪C

2)θ.

We

say

that

ρ1

unifi

es w

ith ρ

2 an

d th

at θ

is a

uni

fier

of t

he tw

o lit

eral

s

Res

olut

ion

deriv

atio

n: a

s be

fore

Th

eore

m:

S→

[] i

ff S

|= [

]iff

S is

uns

atis

fiabl

eN

ote:

The

re a

re p

atho

logi

cal e

xam

ples

whe

re a

slig

htly

mor

e ge

nera

l de

finiti

on o

f Res

olut

ion

is r

equi

red.

We

igno

re th

em fo

r no

w...

KR

& R

© B

rach

man

& L

eves

que

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559

Exa

mp

le 3

[¬H

ardW

orke

r(su

e)]

[¬St

uden

t(su

e)]

[¬G

radS

tude

nt(s

ue)]

[]

x/su

e

x/su

e

[¬St

uden

t(x)

, Har

dWor

ker(

x)]

[¬G

radS

tude

nt(x

), S

tude

nt(x

)]

[Gra

dStu

dent

(sue

)]

Labe

l eac

h st

epw

ith th

e un

ifier

Poi

nt to

rel

evan

tlit

eral

s in

cla

uses

∀x

Gra

dStu

dent

(x)

⊃ S

tude

nt(x

)

∀x

Stud

ent(

x)⊃

Har

dWor

ker(

x)

Gra

dStu

dent

(sue

)

KB

KB

|= H

ardW

orke

r(su

e)?

KR

& R

© B

rach

man

& L

eves

que

200

560

Th

e 3

blo

ck e

xam

ple

[On(

b,c)

]

[On(

a,b)

]

[¬O

n(x,

y),¬

Gre

en(x

), G

reen

(y)] [

Gre

en(a

)]

[¬G

reen

(c)]

[¬G

reen

(a),

Gre

en(b

)]

[¬G

reen

(b),

Gre

en(c

)]

[¬G

reen

(b)]

[Gre

en(b

)]

[]N

ote:

Nee

d to

use

O

n(x,

y)tw

ice,

for

2 ca

ses

{x/b

, y/c

}

{x/a

, y/b

}

KB

= {

On(

a,b)

, O

n(b,

c),

Gre

en(a

),¬

Gre

en(c

)}

Que

ry =

∃x∃y

[On(

x,y)

∧ G

reen

(x)

∧ ¬

Gre

en(y

)]N

ote:

¬Q

has

no

exis

tent

ials

, so

yiel

ds

alre

ady

in C

NF

KR

& R

© B

rach

man

& L

eves

que

200

561

Ari

thm

etic

[¬Pl

us(2

,3,u

)]

[¬Pl

us(1

,3,v

)]

[¬Pl

us(0

,3,w

)]

[]

x/3,

w/3

x/0,

y/3,

v/su

cc(w

),z/

w

x/1,

y/3,

u/su

cc(v

),z/

v

Can

find

the

answ

er in

the

deriv

atio

nu/

succ

(suc

c(3)

)

that

is:

u/5

Can

als

o de

rive

Plus

(2,3

,5)

Ren

ame

varia

bles

to

kee

p th

em d

istin

ct

[¬Pl

us(x

,y,z

), P

lus(

succ

(x),

y,su

cc(z

))]

[Plu

s(0,

x,x)

]

KB

:Pl

us(z

ero,

x,x)

Plus

(x,y

,z)

⊃ P

lus(

succ

(x),

y,su

cc(z

))

Q:

∃uPl

us(2

,3,u

)

For

rea

dabi

lity,

w

e us

e 0 fo

rze

ro,

1 fo

rsu

cc(z

ero)

,2

for

succ

(suc

c(ze

ro))

etc.

KR

& R

© B

rach

man

& L

eves

que

200

562

An

swer

pre

dic

ates

In fu

ll F

OL,

we

have

the

poss

ibili

ty o

f der

ivin

g∃x

P(x

)w

ithou

tbe

ing

able

to d

eriv

e P

(t)

for

any

t.e.

g. th

e th

ree-

bloc

ks p

robl

em

∃x∃y

[On(

x,y)

∧ G

reen

(x)

∧ ¬

Gre

en(y

)]

but c

anno

t der

ive

whi

ch b

lock

is w

hich

Sol

utio

n: a

nsw

er-e

xtra

ctio

n pr

oces

s•

repl

ace

quer

y ∃

xP(x

) by

∃x[

P(x

)∧

¬A

(x)]

whe

reA

is a

new

pre

dica

te s

ymbo

l cal

led

the

answ

er p

redi

cate

•in

stea

d of

der

ivin

g [

], d

eriv

e an

y cl

ause

con

tain

ing

just

the

answ

er p

redi

cate

•ca

n al

way

s co

nver

t to

and

from

a d

eriv

atio

n of

[]

Stud

ent(

john

)

[¬St

uden

t(x)

,¬H

appy

(x),

A(x

)]H

appy

(joh

n)

[¬St

uden

t(jo

hn),

A(j

ohn)

]

[A(j

ohn)

]{x/

john

}

⇓

An

answ

er is

: Joh

n

KB

:St

uden

t(jo

hn)

Stud

ent(

jane

)H

appy

(joh

n)

Q:

∃x[S

tude

nt(x

)∧

Hap

py(x

)]

KR

& R

© B

rach

man

& L

eves

que

200

563

Dis

jun

ctiv

e an

swer

s

[¬H

appy

(joh

n),A

(joh

n)]

[A(j

ane)

, A(j

ohn)

]

{x/jo

hn}

⇓[¬

Stud

ent(

x),¬

Hap

py(x

),A

(x)]

Stud

ent(

jane

)

[¬H

appy

(jan

e),A

(jan

e)]

{x/ja

ne}

[Hap

py(j

ohn)

, Hap

py(j

ane)

]

[Hap

py(j

ohn)

,A(j

ane)

]

Stud

ent(

john

)

An

answ

er is

: ei

ther

Jan

e or

Joh

n

KB

: Stud

ent(

john

)St

uden

t(ja

ne)

Hap

py(j

ohn)

∨ H

appy

(jan

e)

Que

ry:

∃x[S

tude

nt(