FA15 CS188 Lecture 08 PL Semantics and Inference.pptx-2

7/23/2019 FA15 CS188 Lecture 08 PL Semantics and Inference.pptx-2

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* 188: A"ticial Intelligence

'"opositional ogic

*emantics and In.e"ence

Inst"ucto": *tua"t ;ussell

Uni/e"sit# o. ali.o"nia, <e"kele#



=noledge

=noledge ase > set o. sentences in a .o"m

language ecla"ati/e app"oach to uilding an agent (

s#stem): Tell it hat it needs to kno (o" ha/e it Learn t

knoledge)

Then it can Ask itsel. hat to do?anse"s shou."om the =<

Agents can e /ieed at the knowledge lei@e@, hat the# know, "ega"dless o. ho

implemented

Knowledge base

Inference engine

Domain-specific facts

Generic code



ogic

Syntax : What sentences a"e alloedD

Semantics: What a"e the possible worldsD

Which sentences a"e true in hich o"ldsD (i@e@defnition o. t"uth)

α1

α2 α3

*#ntaEland *emanti



FEamples

'"opositional logic *#ntaE: ' ∨ (¬G ∧ ;)7 X1 ⇔ (;aining ⇒ *unn

'ossile o"ld: '>t"ue,G>t"ue,;>.alse,*>t"ue

*emantics: α ∧ β is t"ue in a o"ld iJ is α t"ue an(etc@)

!i"st+o"de" logic *#ntaE: ∀E ∃# '(E,#) ∧ ¬G(Koe,.(E)) ⇒ .(E)>.(#)

'ossile o"ld: Lects o1, o%, o37 ' holds .o" Mo1

holds .o" Mo3N7 . (o1)>o17 Koe>o37 etc@

*emantics: φ(σ) is t"ue in a o"ld i. σ>o and φ h



In.e"ence: entailment

Entailment : α O> β (Bα entails βC o" Bβ .ollo

αC) iJ in e/e"# o"ld he"e α is t"ue, β is als I@e@, the α+o"lds a"e a suset o. the β+o"lds Pm⊆ models(β)Q

In the eEample, α% O> α1

(*a# α% is ¬G ∧ ; ∧ * ∧ W α1 is ¬G )

α1

α2



In.e"ence: p"oo.s

A p"oo. is a demonstration o. entailment e

and β 6ethod 1: model-checking

!o" e/e"# possile o"ld, i. α is t"ue make su"e thatoo

L= .o" p"opositional logic (nitel# man# o"lds)7 n"st+o"de" logic

6ethod %: theorem-proving *ea"ch .o" a seuence o. p"oo. steps (applications

inerence rules) leading ."om α to β

F@g@, ."om ' ∧ (' ⇒ G), in.e" G # odus !onens



In.e"ence ("eminde")

6ethod 1: model-checking

!o" e/e"# possile o"ld, i. α is t"ue make su"e thatoo

6ethod %: theorem-proving *ea"ch .o" a seuence o. p"oo. steps (applications

inerence rules) leading ."om α to β Sound algo"ithm: e/e"#thing it claims to p"o/

entailed

"omplete algo"ithm: e/e"# that is entailed ca

p"o/ed

' iti l l i t Th



'"opositional logic s#ntaE: The g"udetails

Ri/en: a set o. p"oposition s#mols X1,X%,S

(e o.ten add T"ue and !alse .o" con/enience)

Xi is a sentence

I. α is a sentence then ¬α is a sentence

I. α and β a"e sentences then α ∧ β is a sent

I. α and β a"e sentences then α ∨ β is a sent

I. α and β a"e sentences then α ⇒ β is a sen

I. α and β a"e sentences then α ⇔ β is a sen

And p@s@ the"e a"e no othe" sentences2

' iti l l i ti



'"opositional logic semantics: un/a"nished t"uth

function '+T;UFD(α,model) returns t"ue o" .a

if α is a s#mol then return ookup(α, mode if Lp(α) > ¬ then return not('+T;UFD(A"g1(α),model))

if Lp(α) > ∧ then return and('+T;UFD

(A"g1(α),model),'+T;UFD

(A"g%(α),model))

etc@



FEample

; ⇒ ( ⇔ U) (i. it0s "aining, #ou sta# d"# i. and onl# i. #

um"ella) 6odel >t"ue, ;>.alse, U>t"ue S@

.alse ⇒ (t"ue ⇔ t"ue)

.alse ⇒ t"ue

t"ue



*imple theo"em p"o/ing: !o"a"d c

!o"a"d chaining applies 6odus 'onens to g

ne .acts: Ri/en X1 ∧ X% ∧ S Xn ⇒ and X1, X%, S, Xn

In.e"

!o"a"d chaining keeps appl#ing this "ule, a

ne .acts, until nothing mo"e can e added ;eui"es =< to contain onl# defnite clause

(onunction o. s#mols) ⇒ s#mol7 o"

A single s#mol (note that X is eui/alent to T"u



!o"a"d chaining algo"ithm

function '+!+FTAI*D(=<, ) returns t"ue o" .alse count V a tale, he"e countPcQ is the nume" o. s#mols in in.e""ed V a tale, he"e in.e""edPsQ is initiall# .alse .o" all s

agenda V a ueue o. s#mols, initiall# s#mols knon to e

while agenda is not empt# do p V 'op(agenda)

if p > then return t"ue

if in.e""edPpQ > .alse then in.e""edPpQVt"ue

for each clause c in =< he"e p is in c@p"emise dodec"ement countPcQ

if countPcQ > & then add c@conclusion to agreturn .alse



Q

P

M

L

A

!o"a"d chaining eEample: '"o/

' ⇒ G

∧ 6 ⇒ '

< ∧ ⇒ 6

A ∧ ' ⇒

A ∧ < ⇒

A

1

%%

%

%

&

&

A

.alse< .alse

.alse6 .alse

' .alse

G

.alse

CLAUSES

AGENDA

A B

INFERREDCOUNT

Lx

xxxx true

1

1

x

xxxx true

1

!

x

xxxx true

1

!

"x

xxxx true

!

#x

xxxx true

!

!

L $x x

xxxx true



'"ope"ties o. .o"a"d chainin

Theo"em: ! is sound and complete .o" deni

=<s *oundness: .ollos ."om soundness o. 6odus

(eas# to check)

ompleteness p"oo.:

1@ ! "eaches a Eed point he"e no ne atomic sentences %@ onside" the nal inferred tale as a model m, assigning

s#mols

3@ F/e"# clause in the o"iginal =< is t"ue in m

'"oo.: *uppose a clause a1∧@@@ ∧ak ⇒ is .alse in m

Then a1∧@@@ ∧ak is t"ue in m and is .alse in m



*imple model checking

function TT+FTAI*D(=<, ) returns t"ue o" .alse return TT+HF=+A(=<,,s#mols(=<) U s#mols(),)function TT+HF=+A(=<,,s#mols,model) returns t"ue o

if empt#D(s#mols) then if '+T;UFD(=<,model) then return '+T;UFD(,moelse return t"ue

else

' V "st(s#mols) "est V "est(s#mols) return and ( TT+HF=+A(=<,,"est,model Y ' >

TT+HF=+A(=<,,"est,model Y '



*imple model checking, cont

*ame "ecu"sion as

ackt"acking L(%n) time, linea" space

We can do much ette"2

#1%true #1%false

#2%true #2%

#n%true

1 1 1

1 1 &

1

KB'

α'



*atisailit# and entailmen

A sentence is satisfable i. it is t"ue in at le

o"ld (c. *'s2) *uppose e ha/e a h#pe"+e-cient *AT sol/

can e use it to test entailmentD *uppose α O> β

Then α ⇒ β is t"ue in all o"lds Hence ¬(α ⇒ β) is .alse in all o"lds

Hence α ∧ ¬β is .alse in all o"lds, i@e@, unsatisa

*o, add the negated conclusion to hat #ou

test .o" (un)satisailit#7 also knon as "edu



onuncti/e no"mal .o"m (

F/e"# sentence can e eEp"essed as a con#

clauses Fach clause is a dis#unction o. literals

Fach lite"al is a s#mol o" a negated s#mo

on/e"sion to ! # a seuence o. standa

t"ans.o"mations: ; ⇒ ( ⇔ U) (i. it0s "aining, #ou sta# d"# i. and onl#

um"ella)

; ⇒ (( ⇒ U) ∧ (U ⇒))

¬; / ((¬ / U) ∧ (¬U / ))

(eplace biconditional b) two implications

(eplace α ⇒ β b) ¬α * β

Distribute * o*er ∧



F-cient *AT sol/e"s

' (a/is+'utnam+ogemann+o/eland) is the co

mode"n sol/e"s Fssentiall# a ackt"acking sea"ch o/e" models

some eEt"as: Early termination: stop i.

all clauses a"e satised7 e@g@, (A ∨ <) ∧ (A ∨ ¬) is satisA>t"ue

an# clause is .alsied7 e@g@, (A ∨ <) ∧ (A ∨ ¬) is satisedA>.alse,<>.alse

!ure literals: i. all occu""ences o. a s#mol in as+unsatised clauses ha/e the same sign, then gi/e that /alue



' algo"ithm

function '(clauses,s#mols,model) returns t"ue o" .alsif e/e"# clause in clauses is t"ue in model then return t"

if some clause in clauses is .alse in model then return .a ',/alue V!I+'U;F+*6<L(s#mols,clauses,model) if ' is non+null then return '(clauses, s#molsZ',modelY'>/alue)

',/alue V!I+UIT+AU*F(clauses,model)

if ' is non+null then return '(clauses, s#molsZ',modelY'>/alue)

' V !i"st(s#mols)7 "est V ;est(s#mols) return or('(clauses,"est,modelY'>t"ue), '(clauses,"est,modelY'>.alse))

- i



F-cienc#

a[/e implementation o. ': sol/e \1&& /a"iale

FEt"as: ]a"iale and /alue o"de"ing (."om *'s)

i/ide and conue"

aching unsol/ale sucases as eEt"a clauses to a/oid "e

ool indeEing and inc"emental "ecomputation t"icks so th

step o. the ' algo"ithm is e-cient (t#picall# L(1)) IndeE o. clauses in hich each /a"iale appea"s ^/e$+/e

=eep t"ack nume" o. satised clauses, update hen /a"iales

=eep t"ack o. nume" o. "emaining lite"als in each clause

;eal implementation o. ': sol/e \1&&&&&&& /a"

*AT l i i



*AT sol/e"s in p"actice

i"cuit /e"ication: does this ]*I ci"cuit com

"ight anse"D *o.ta"e /e"ication: does this p"og"am com

"ight anse"D

*o.ta"e s#nthesis: hat p"og"am compute

anse"D '"otocol /e"ication: can this secu"it# p"oto

"okenD

'"otocol s#nthesis: hat p"otocol is secu"e .

taskD

*



*umma"#

Lne possile agent a"chitectu"e: knoledge

in.e"ence ogics p"o/ide a .o"mal a# to encode kno

A logic is dened #: s#ntaE, set o. possile o"condition

ogical in.e"ence computes entailment "elatamong sentences

eEt: a logical 'acman agent

< k i



<"eak ui_

enite clauses a"e a

special kind o. <ooleanconst"aint

In.e"ence ith deniteclauses # ! (o" ') islinea"+time

In.e"ence in t"ee+st"uctu"ed *'s is linea"+time

Is the"e a connectionD

ot uite2

A complete sea"c

ala#s nds a soeEists

A complete in.e"ealgo"ithm ala#sp"oo. i. the ue"#

an e use compalgo"ithms to uiin.e"ence algo"ith

es2 *&>=<, R > ue"#

added

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FA15 CS188 Lecture 08 PL Semantics and Inference.pptx-2