09 Inference

7/25/2019 09 Inference

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Artificial Intelligence

Inference in First-Order Logic

Readings: Chapter 9 of Russell &

Norvig.


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Universal instantiation (UI)

Every instantiation of a universally quantified sentence isentailed by it:

v

SUBST({v/g}, )

for any variablev and ground termg

E.g., x King(x) Greedy(x) = Evil(x)yields

King(John)Greedy(John) = Evil(John)

King(Richard)Greedy(Richard) = Evil(Richard)

King(Father(John)) Greedy(Father(John)) = Evil(Fathe...


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Existential instantiation (EI)

For any sentence, variablev, and constant symbolkthat does not appear elsewhere in the knowledge base:

v SUBST({v/k}, )

E.g., x Crown(x) OnHead(x,John)yields

Crown(C1) OnHead(C1,John)

providedC1 is a new constant symbol, called a Skolemconstant.


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UI versus EI

Fromxy(y+x=y)we obtain

y+a=y

From x y(y+x=y)we obtain

y+e=y

providedeis a new constant symbol.

UI can be applied several times to addnew sentences;

the new KB is logically equivalent to the old.EI can be applied once toreplacethe existentialsentence; the new KB isnotequivalent to the old, but is

satisfiable iff the old KB was satisfiable.


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Reduction to Propositional Inference

Suppose the KB contains just the following:

x King(x) Greedy(x) = Evil(x)

King(John)

Greedy(John)

Brother(Richard, John)

Instantiating the universal sentence inall possibleways, we have

King(John) Greedy(John) = Evil(John)

King(Richard) Greedy(Richard) = Evil(Richard)

King(John)

Greedy(John)

Brother(Richard, J ohn)

The new KB is propositionalized: proposition symbols are


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Reduction to Propositional Inference

Claim: a ground sentence is entailed by new KB iffentailed by original KB.

Claim: every FOL KB can be propositionalized so as to

preserve entailment.Idea: propositionalize KB and query, apply resolution,return result.

Problem: with function symbols, ground terms areinfinitely many, e.g.,Father(Father(Father(John))).

Theorem: Herbrand (1930). If a sentenceis entailed

by an FOL KB, it is entailed by afinitesubset of thepropositional KB.

Idea: Forn= 0 to, create a propositional KB by

instantiating with depth-nterms see ifis entailed bythis KB.

Problem: works if is entailed, loo s ifis not entailed


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Unification

We can get the inference immediately if we can find asubstitution such thatKing(x)andGreedy(x)matchKing(John)andGreedy(y). That is,={x/John, y/John}works

UNIFY(, ) =if =

p q

Knows(John,x) Knows(John,Jane)

Knows(John,x) Knows(y,OJ)Knows(John,x) Knows(y,Mother(y))

Knows(John,x) Knows(x,OJ)


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Unification


UNIFY(, ) =if =

p q

Knows(John,x) Knows(John,Jane) {x/Jane}

Knows(John,x) Knows(y,OJ)Knows(John,x) Knows(y,Mother(y))



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Unification


UNIFY(, ) =if =

p q


Knows(John,x) Knows(y,OJ) {x/OJ, y/John}Knows(John,x) Knows(y,Mother(y))



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Unification


UNIFY(, ) =if =

p q


Knows(John,x) Knows(y,OJ) {x/OJ, y/John}Knows(John,x) Knows(y,Mother(y)) {y/John, x/M other(J



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Unification

We can get the inference immediately if we can find asubstitution such thatKing(x)andGreedy(x)matchKing(John)andGreedy(y). That is,={x/John, y/John}works.

UNIFY(, ) =if =

p q


Knows(John,x) Knows(y,OJ) {x/OJ, y/John}Knows(John,x) Knows(y,Mother(y)) {y/John, x/M other(J

Knows(John,x) Knows(x,OJ) fail

Standardizing apart eliminates overlap of variables, e.g.,Knows(z17, OJ)


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Conversion to CNF

Everyone who loves all animals is loved by someone:

x [ y Animal(y) = Loves(x, y)] = [ y Loves(y, x)]

1. Eliminate biconditionals and implications

x [ y Animal(y) Loves(x, y)] [ y Loves(y, x)]

2. Moveinwards: x, p x p, x, p x p:

x [ y (Animal(y) Loves(x, y))] [ y Loves(y, x)]




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Conversion to CNF contd.

3. Standardize variables: each quantifier should use adifferent one

x [ y Animal(y) Loves(x, y)] [ z Loves(z, x)]

4. Skolemize: a more general form of existentialinstantiation. Each existential variable is replaced by aSkolem function of the enclosing universally quantified

variables:

x [Animal(F(x)) Loves(x, F(x))] Loves(G(x), x)


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Conversion to CNF contd.

5. Drop universal quantifiers:[Animal(F(x)) Loves(x, F(x))] Loves(G(x), x)

6. Distributeover:[Animal(F(x))Loves(G(x), x)][Loves(x, F(x))Loves(G(x), x

G li d M d P (GMP)


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Generalized Modus Ponens (GMP)

p1, p2

, . . . , pn, (p1p2 . . .pnq)

q wherepi

=pi

p1 isKing(John) p1 isKing(x)

p2 isGreedy(y) p2 isGreedy(x)

is{x/John, y/John} qisEvil(x)

qisEvil(John)

GMP used with KB of definite clauses (those clauseshavingexactlyone positive literal).

All variables assumed universally quantified

S d f GMP


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Soundness of GMP

Need to show that

p1, . . . , pn

, (p1 . . .pnq)|=q

provided thatpi =pifor alli

Lemma: For any definite clausep, we havep|=pby UI1. (p1. . .pnq)|= (p1. . .pnq))and

(p1. . .pnq) = (p1. . .pnq)

2. p1

, . . . , pn

|=p1

. . .pn

|=p1

. . .pn

3. From 1 and 2,qfollows by ordinary Modus Ponens

R l ti


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Resolution

Full first-order version:1 k, m1 mn

(1 i1i+1 k m1 mj1 mj+1

where UNIFY(i, mj) = .

For example,

Rich(x) Unhappy(x)

Rich(Ken)

Unhappy(Ken)

with={x/Ken}.

S d f R l ti


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Soundness of Resolution

Need to show1 k, m1 mn|= c

where c

is(1 i1i+1 k m1 mj1mj+1 mn)and UNIFY(i, mj) = .

This is true because

1. 1 k |= (1 k)by UI,

2. m1 mn|= (m1 mn)by UI, and

3. (1 k)and(m1 mn)imply cbypropositional resolution.

Using Resol tion


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Using Resolution

1. ObtainCN F(KB )

2. Apply resolution steps to the CNF

3. If the empty clause is generated, thenKB |=.

Theorem: Resolution is a refutationally complete inferencesystem forKB|=.

Example Knowledge Base


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Example Knowledge Base

The law says that it is a crime for an American to sellweapons to hostile nations. The country Nono, an enemy ofAmerica, has some missiles, and all of its missiles were

sold to it by Colonel West, who is American.

Prove that Col. West is a criminal

Example Knowledge Base contd


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Example Knowledge Base contd.

. . .it is a crime for an American to sell weapons to hostilenations:



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American(x) Weapon(y) Sells(x,y,z)Hostile(z) =

Criminal(x)Nono. . .has some missiles



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Criminal(x)Nono. . .has some missiles, i.e., x Owns(Nono,x) M issile(x):

Owns(Nono,M1)andMissile(M1). . .all of its missiles were sold to it by Colonel West



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x M issile(x) Owns(Nono,x) = Sells(West,x,Nono)Missiles are weapons:



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. . .it is a crime for an American to sell weapons to hostile

nations:





Missile(x)Weapon(x)An enemy of America counts as hostile:



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. . .it is a crime for an American to sell weapons to hostile

nations:





Missile(x)Weapon(x)An enemy of America counts as hostile:

Resolution Proof: Definite Clauses


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Resolution Proof: Definite Clauses

American(West)

Missile(M1)

Missile(M1)

Owns(Nono,M1)

Enemy(Nono,America) Enemy(Nono,America)

Criminal(x)Hostile(z)L

Sells(x,y,z)L

Weapon(y)L

American(x)L > > > >

Weapon(x)Missile(x)L >

Sells(West,x,Nono)Missile(x)L

Owns(Nono,x)L> >

Hostile(x)Enemy(x,America)L >

Sells(West,y,z)L

Weapon(y)L

American(West)L > >

Hostile(z)L>

Sells(West,y,z)L

Weapon(y)L >

Hostile(z)L>

Sells(West,y,z)L>

Hostile(z)L>L

Missile(y)

Hostile(z)L>L

Sells(West,M1,z)

> > LHostile(Nono)

LOwns(Nono,M1)

LMissile(M1)

> LHostile(Nono)

LOwns(Nono,M1)

LHostile(Nono)

Criminal(West)L

Logic programming


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Logic programming

Sound bite: computation as inference on logical KBs

Logic programming Ordinary programming

1. Identify problem Identify problem

2. Assemble information Assemble information

3. Tea break Figure out solution

4. Encode information in KB Program solution

5. Encode problem instance as facts Encode problem instance as data

6. Ask queries Apply program to data

7. Find false facts Debug procedural errors

Should be easier to debugCapital(N ewY ork, U S)thanx:=x+ 2!

Prolog Systems


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Prolog Systems

A resolution inference system on Horn clauses + bells & whistles

Widely used in Europe, Japan (basis of 5th Generation project)

Compilation techniques 60 million LIPS

Program = set of defi nite clauses, which are of form:

head :- literal1, . . . literaln.

weapon(X) :- missile(X).

Effi cient unifi cation by open coding

Effi cient retrieval of matching clauses by direct linking

Depth-fi rst, left-to-right backward chaining

Built-in predicates for arithmetic etc., e.g.,X is Y*Z+3

Closed-world assumption (negation as failure) e.g., given

alive(X) :- not dead(X). alive(joe)succeeds if

dead(joe) fails.

Resolution Proof in Prolog


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Resolution Proof in Prolog

% it is a crime to sell weapons to hostile nations:

criminal(X):-american(X),weapon(Y),sells(X,Y,Z),hostile(Z)

% Nono ... has some missiles,

owns(nono,m1).

missile(m1).% all of its missiles were sold to it by Colonel West

sells(west,X,nono) :- missile(X), owns(nono,X).

% Missiles are weapons

weapon(X) :- missile(X).

% An enemy of America counts as hostile:

hostile(X) :- enemy(X,america).

% The country Nono, an enemy of America ...\alenemy(nono,america).

% West, who is American ...\al

american(west).

?- criminal(Who).

Who = west.

Prolog Examples


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Prolog Examples

Depth-first search from a start state X:

dfs(X) :- goal(X).

dfs(X) :- successor(X,S),dfs(S).

No need to loop over S: successorsucceeds for eachAppending two lists to produce a third:

append([],Y,Y).

append([X|L],Y,[X|Z]) :- append(L,Y,Z).

query: append(A,B,[1,2]) ?

answers: A=[] B=[1,2]

A=[1] B=[2]

A=[1,2] B=[]

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09 Inference

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