[PPT] - AIMA: Chapter 8 (Setions 8.1 and 8.2) Choueiry Setion PowerPoint Presentation

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Title: First-Order Logi AIMA: Chapter 8 (Se tions 8.1 and 8.2) Se tion 8.3, dis ussed briey , is also required reading In tro du tion to Arti ial In telligen e CSCE 476-876, Spring 2016 URL:

www. se.unl.edu/

ho uei ry/ S1 6-4 76- 87 6 Berthe Y. Choueiry (Sh u-w e-ri) (402)472-5444 B.Y. Choueiry 1 Instru tor's notes #13 April 8, 2016

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Outline

First-order

logi :

basi

elemen ts

syn

tax

seman

ti s

Examples

B.Y. Choueiry 2 Instru tor's notes #13 April 8, 2016

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Pros and

ns
f

prop

sitional

logi

Prop
sitional

logi is de larativ e : pie es

f

syn tax

rresp
nd

to fa ts

Prop
sitional

logi allo ws partial/disjun tiv e/negated information (unlik e most data stru tures and databases)

Prop
sitional

logi is

mp
sitional

: meaning

f B1,1 ∧ P1,2

is deriv ed from meaning

f B1,1

and

f

P1,2

Meaning

in prop

sitional

logi is

n

text-indep enden t (unlik e natural language, where meaning dep ends

n
n

text)

but...

Prop

sitional

logi has v ery limited expressiv e p

w

er E.g., annot sa y pits ause breezes in adja en t squares ex ept b y writing

ne

sen ten e for ea h square B.Y. Choueiry 3 Instru tor's notes #13 April 8, 2016

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Prop

sitional

Logi

is

simple

illustrates

imp

rtan

t p

in

ts: mo del, inferen e, v alidit y , satisabilit y , ..

is

restri tiv e: w

rld

is a set

f

fa ts

la

ks expressiv eness:

→

In PL, w

rld
n

tains fa ts First-Order Logi

more

sym b

ls

(ob je ts, prop erties, relations)

more
nne tiv

es (quan tier) B.Y. Choueiry 4 Instru tor's notes #13 April 8, 2016

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First Order Logi

→

F OL pro vides more "primitiv es" to express kno wledge:

b

je ts (iden tit y & prop erties)

relations

among

b

je ts (in luding fun tions) Ob je ts: p eople, houses, n um b ers, Einstein, Husk ers, ev en t, .. Prop erties: smart, ni e, large, in telligen t, lo v ed,

urred,

.. Relations: brother-of, bigger-than, part-of,

urred-after,

.. F un tions: father-of, b est-friend, double-of, .. Examples: (ob je ts? fun tion? relation? prop ert y?)

ne

plus t w

equals

four [si ℄

squares

neigh b

ring

the wumpus are smelly B.Y. Choueiry 5 Instru tor's notes #13 April 8, 2016

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Logi A ttra ts: mathemati ians, philosophers and AI p eople A dv an tages:

allo

ws to represen t the w

rld

and reason ab

ut

it

expresses

an ything that an b e programmed Non- ommittal to:

sym

b

ls
uld

b e

b

je ts

r

relations (e.g., King(Gusta v e), King(Sw eden, Gusta v e), Mer iless(King))

lasses,

ategories, time, ev en ts, un ertain t y .. but amenable to extensions: OO F OL, temp

ral

logi s, situation/ev en t al ulus, mo dal logi , et .

− →

Some p eople think F OL *is* the language

f

AI true/false? donno :( but it will remain around for some time.. B.Y. Choueiry 6 Instru tor's notes #13 April 8, 2016

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T yp es

f

logi Logi s are hara terized b y what they

mmit

to as primitiv es On tologi al

mmitmen

t : what existsfa ts?

b

je ts? time? b eliefs? Epistemologi al

mmitmen

t : what states

f

kno wledge?

Language Ontological Commitment Epistemological Commitment (What exists in the world) (What an agent believes about facts) Propositional logic facts true/false/unknown First-order logic facts, objects, relations true/false/unknown Temporal logic facts, objects, relations, times true/false/unknown Probability theory facts degree of belief 0…1 Fuzzy logic degree of truth degree of belief 0…1

Higher-Order Logi : views relations and fun tions

f

F OL as

b

je ts B.Y. Choueiry 7 Instru tor's notes #13 April 8, 2016

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Syn tax

f

F OL: w

rds

and grammar The w

rds:

sym b

ls
Constan

t sym b

ls

stand for

b

je ts: QueenMary , 2, UNL, et .

V

ariable sym b

ls

stand for

b

je ts: x , y , et .

Predi ate

sym b

ls

stand for relations: Odd, Ev en, Brother, Sibling, et .

F

un tion sym b

ls

stand for fun tions (viz. relation) F ather-of, Square-ro

t,

LeftLeg, et .

Quan

tiy ers ∀ , ∃

Conne tiv

es: ∧ , ∨ , ¬ , ⇒ , ⇔ ,

(Sometimes)

equalit y = Predi ates and fun tions an ha v e an y arit y (n um b er

f

argumen ts) B.Y. Choueiry 8 Instru tor's notes #13 April 8, 2016

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Basi elemen ts in F OL (i.e., the grammar) In prop

sitional

logi , ev ery expression is a sen ten e In F OL,

T

erms

Sen

ten es:

atomi

sen ten es

mplex

sen ten es

Quan

tiers:

Univ

ersal quan tier

Existen

tial quan tier B.Y. Choueiry 9 Instru tor's notes #13 April 8, 2016

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T erm logi al expression that refers to an

b

je t

built

with:

nstan

t sym b

ls,

v ariables, fun tion sym b

ls

T erm =

function(term1, . . . , termn)

r
nstan

t

r

v ariable

ground

term: term with no v ariable B.Y. Choueiry 10 Instru tor's notes #13 April 8, 2016

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A tomi sen ten es state fa ts built with terms and predi ate sym b

ls

A tomi sen ten e =

predicate(term1, . . . , termn)

r term1 = term2

Examples: Brother (Ri hard, John) Married (F atherOf(Ri hard), MotherOf(John)) B.Y. Choueiry 11 Instru tor's notes #13 April 8, 2016

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Complex Sen ten es built with atomi sen ten es and logi al

nne tiv

es

¬S S1 ∧ S2 S1 ∨ S2 S1 ⇒ S2 S1 ⇔ S2

Examples: Sibling(KingJohn,Ri hard) ⇒ Sibling(Ri hard,KingJohn)

>(1, 2) ∨ ≤(1, 2) >(1, 2) ∧ ¬>(1, 2)

B.Y. Choueiry 12 Instru tor's notes #13 April 8, 2016

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T ruth in rst-order logi : Seman ti Sen ten es are true with resp e t to a mo del and an in terpretation Mo del

n

tains

b

je ts and relations among them In terpretation sp e ies referen ts for

nstant

symb

ls →
b

je ts pr e di ate symb

ls →

relations fun tion symb

ls →

fun tional relations An atomi sen ten e predicate(term1, . . . , termn) is true i the

b

je ts referred to b y term1, . . . , termn are in the relation referred to b y predicate B.Y. Choueiry 13 Instru tor's notes #13 April 8, 2016

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Mo del in F OL: example

R J

$ left leg

n head

brother brother person person king crown left leg

The domain

f

a mo del is the set

f
b

je ts it

n

tains: v e

b

je ts In tended in terpretation: Ri hard refers Ri hard the Lion Heart, John refers to Evil King John, Brother refers to brotherho

d

relation, et . B.Y. Choueiry 14 Instru tor's notes #13 April 8, 2016

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Mo dels for F OL: Lots! W e an en umerate the mo dels for a giv en KB v

abulary:

F

r

ea h n um b er

f

domain elemen ts n from 1 to ∞ F

r

ea h k

ary

predi ate Pk in the v

abulary

F

r

ea h p

ssible k
ary

relation

n n
b

je ts F

r

ea h

nstan

t sym b

l C

in the v

abulary

F

r

ea h hoi e

f

referen t for C from n

b

je ts . . . Computing en tailmen t b y en umerating mo dels is not going to b e easy! There are man y p

ssible

in terpretations, also some mo del domain are not b

unded

− →

Che king en tailmen t b y en umerating is not an

ption

B.Y. Choueiry 15 Instru tor's notes #13 April 8, 2016

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Quan tiers allo w to mak e statemen ts ab

ut

en tire

lle tions
f
b

je ts

univ

ersal quan tier: mak e statemen ts ab

ut

ev erything

existen

tial quan tier: mak e statemen ts ab

ut

some things B.Y. Choueiry 16 Instru tor's notes #13 April 8, 2016

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Univ ersal quan ti ation

∀ variables sentence

Example: all dogs lik e b

nes ∀ xDog(x) ⇒ LikeBones(x)

x = Indy is a dog x = Indiana Jones is a p erson

∀ x P

is equiv alen t to the

njun tion
f

instan tiations

f P

Dog(Indy) ⇒ LikeBones(Indy) ∧ Dog(Rebel) ⇒ LikeBones(Rebel) ∧ Dog(KingJohn) ⇒ LikeBones(KingJohn) ∧ . . .

T ypi ally: ⇒ is the main

nne tiv

e with ∀ Common mistak e: using ∧ as the main

nne tiv

e with ∀ Example: ∀ x Dog(x) ∧ LikeBones(x) all

b

je ts in the w

rld

are dogs, and all lik e b

nes

B.Y. Choueiry 17 Instru tor's notes #13 April 8, 2016

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Existen tial quan ti ation

∃ variables sentence

Example: some studen t will talk at the T e hF air

∃ xStudent(x) ∧ TalksAtTechFair(x)

P at, Leslie, Chris are studen ts

∃ x P

is equiv alen t to the disjun tion

f

instan tiations

f P

Student(Pat) ∧ TalksAtTechFair(Pat) ∨ Student(Leslie) ∧ TalksAtTechFair(Leslie) ∨ Student(Chris) ∧ TalksAtTechFair(Chris) ∨ . . .

T ypi ally: ∧ is the main

nne tiv

e with ∃ Common mistak e: using ⇒ as the main

nne tiv

e with ∃

∃ x Student(x) ⇒ TalksAtTechFair(x)

is true if there is an y

ne

who is not Studen t B.Y. Choueiry 18 Instru tor's notes #13 April 8, 2016

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Prop erties

f

quan tiers (I)

∀x ∀y

is the same as ∀y ∀x

∃x ∃y

is the same as ∃y ∃x

∃x ∀y

is not the same as ∀y ∃x

∃x ∀y Loves(x, y)

There is a p erson who lo v es ev ery

ne

in the w

rld

∀y ∃xLoves(x, y)

Ev ery

ne

in the w

rld

is lo v ed b y at least

ne

p erson Quan tier dualit y: ea h an b e expressed using the

ther

∀x Likes(x, IceCream) ¬ ∃x ¬Likes(x, IceCream) ∃x Likes(x, Broccoli) ¬ ∀x ¬Likes(x, Broccoli)

P arsimon y prin ipal: ∀ , ¬ , and ⇒ are su ien t B.Y. Choueiry 19 Instru tor's notes #13 April 8, 2016

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Prop erties

f

quan tiers (I I) Nested quan tier:

∀ x(∃ y(P(x, y)):

ev ery

b

je t in the w

rld

has a parti ular prop ert y , whi h is the prop ert y to b e related to some

b

je t b y the relation P

∃ x (∀ y(P(x, y)):

there is some

b

je t in the w

rld

that has a parti ular prop ert y , whi h is the prop ert y to b e related to ev ery

b

je t b y the relation P Lexi al s oping: ∀ x[Cat(x) ∨ ∃ xBrother(Richard, x)] W ell-formed form ulas (WFF): (kind

f
rre t

sp elling) ev ery v ariable m ust b e in tro du ed b y a quan tier

∀ xP(y)

is not a WFF B.Y. Choueiry 20 Instru tor's notes #13 April 8, 2016

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Examples Brothers are siblings . Sibling is symmetri . One's mother is

ne's

female paren t . A rst

usin

is a hild

f

a paren t's sibling B.Y. Choueiry 21 Instru tor's notes #13 April 8, 2016

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Examples .

∀x, y Brother(x, y) ⇒ Sibling(x, y)

.

∀x, y Sibling(x, y) ⇒ Sibling(y, x)

.

∀x, y Mother(x, y) ⇒ (Female(x) ∧ Parent(x, y))

.

∀x, y FirstCousin(x, y) ⇔ ∃a, b Parent(a, x) ∧ Sibling(a, b) ∧ Parent(b, y)

B.Y. Choueiry 22 Instru tor's notes #13 April 8, 2016

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T ri ky example Someone is lo v ed b y ev ery

ne

∃ x ∀ y Loves(y, x)

Someone with red-hair is lo v ed b y ev ery

ne

∃ x ∀ y Redhair(x) ∧ Loves(y, x)

Alternativ ely:

∃ x Person(x) ∧ Redhair(x) ∧ (∀ y Person(y) ⇒ Loves(y, x))

B.Y. Choueiry 23 Instru tor's notes #13 April 8, 2016

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Equalit y

term1 = term2

is true under a giv en in terpretation if and

nly

if term1 and term2 refer to the same

b

je t Examples

F

ather(John)=Henry

1 = 2

is satisable

2 = 2

is v alid

Useful

to distinguish t w

b

je ts:

Denition
f

(full) Sibling in terms

f Parent :

∀x, y Sibling(x, y) ⇔ [¬(x = y) ∧ ∃m, f¬(m = f) ∧ Parent(m, x) ∧ Parent(f, x) ∧ Parent(m, y) ∧ Parent(f, y)]

Sp
t

has at least t w

sisters:

... AIMA, Exer ise 8.4. W rite: All Germans sp eak the same languages, where Speaks(x, l) means that p erson x sp eaks language l . B.Y. Choueiry 24 Instru tor's notes #13 April 8, 2016

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Kno wledge represen tation (KR) Domain: a se tion

f

the w

rld

ab

ut

whi h w e wish to express some kno wledge Example: F amily relations (kinship):

Ob

je ts: p eople

Prop

erties: gender, married, div

r ed,

single, wido w ed

Relations:

paren tho

d,

brotherho

d,

marriage.. Unary predi ates: Male, F emale Binary relations: P aren t, Sibling, Brother, Child, et . F un tions: Mother, F ather

∀ m, c, Mother(c) = m ⇔ Female(m) ∧ Parent(m, c)

B.Y. Choueiry 25 Instru tor's notes #13 April 8, 2016

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In Logi (informally)

Basi

fa ts: axioms (denitions)

Deriv

ed fa ts: theorems Indep enden t axiom an axiom that annot b e deriv ed from the rest

− →

Goal

f

mathemati ians: nd the minimal set

f

indep enden t axioms In AI

Assertions:

sen ten es added to a KB using TELL

Queries
r

goals: sen ten es ask ed to KB using ASK B.Y. Choueiry 26 Instru tor's notes #13 April 8, 2016

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In tera ting with F OL KBs Supp

se

a wumpus-w

rld

agen t is using an F OL KB and p er eiv es a smell and a breeze (but no glitter) at t = 5 :

Tell(KB, Percept([Smell, Breeze, None], 5)) Ask(KB, ∃aAction(a, 5))

I.e., do es the KB en tail an y parti ular a tions at t = 5 ? Answ er: Y es, {a/Shoot}

←

substitution (binding list) Giv en a sen ten e S and a substitution σ ,

Sσ

denotes the result

f

plugging σ in to S ; e.g.,

S = Smarter(x, y) σ = {x/Hillary, y/Bill} Sσ = Smarter(Hillary, Bill) Ask(KB, S)

returns some/all σ su h that KB |

= Sσ

B.Y. Choueiry 27 Instru tor's notes #13 April 8, 2016

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Prepare for next le ture: AIMA, Exer ise 8.24, page 319 T ak es(x, c, s): studen t x tak es

urse c

in semester s P asses(x, c, s): studen t x passes

urse c

in semester s S ore(x, c, s): the s ore

btained

b y studen t x in

urse c

in semester s

x > y

: x is greater that y

F

and G : sp e i F ren h and Greek

urses

Buys(x, y, z ): x buys y from z Sells(x, y, z ): x sells y from z Sha v es(x, y ): p erson x sha v es p erson y Born(x, c): p erson x is b

rn

in

un

try c P aren t(x, y ): p erson x is paren t

f

p erson y Citizen(x, c, r ): p erson x is itizen

f
un

try c for reason r Residen t(x, c): p erson x is residen t

f
un

try c

f

p erson y F

ls(x, y, t ):

p erson x fo

ls

p erson y at time t Studen t (x ), P erson(x ), Man(x ), Barb er(x ), Exp ensiv e(x ), Agen t(x ), Insured(x ), Smart(x ), P

liti ian(x

), B.Y. Choueiry 28 Instru tor's notes #13 April 8, 2016

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AI Limeri k If y

ur

thesis is utterly v a uous Use rst-order predi ate al ulus With su ien t formalit y The sheerest banalit y Will b e hailed b y the riti s: "Mira ulous!" Henry Kautz In Canadian A rti ial Intel ligen e, Septemb er 1986 he ad

f

AI at A T&T L abs-R ese ar h Pr

gr

am

- hair
f

AAAI-2000 Pr

fessor

at University

f

W ashington, Se attle F

unding

Dir e tor

f

Institute for Data S ien e and Pr

fessor

at University

f

R

hester

B.Y. Choueiry 29 Instru tor's notes #13 April 8, 2016