1 fol cs 331 dr m m awais inferencing finding solution to queries propositionalization –(for...

CS 331 Dr M M Awais 1

FOL

Inferencing

Finding solution to queries

• Propositionalization – (For reference only, may leave slides 2 to 7)

• Unification– Forward Chaining– Backward Chaining

• Resolution

Luger’s Book:

Chapter 2and

Chapter 12(upto 12.2)


FOL

Propositionalization

1. Brute Force Approach

2. Apply all possible bindings to the variables.

3. Add every clause in the sentence as a proposition to the knowledge base.

4. Search for the required proposition (Goal proposition)


FOL Reduction to propositional inferenceSuppose the KB contains just the following:

X wealthy(X) greedy(X) bad(X)wealthy(arif)greedy(arif)cousins(shahid,arif)

• Instantiating the universal sentence in all possible ways, we have:wealthy(arif) greedy(arif) bad(arif)wealthy(shahid) greedy(shahid) bad(shahid)wealthy(arif)greedy(arif)cousins(shahid,arif)

Universal Instantiation

•The new KB is propositionalized: proposition symbols are

wealthy(arif), greedy(arif), bad(arif), wealthy(shahid), greedy(shahid), bad(shahid), cousins(shahid,arif).


FOL

Reduction contd.

• Every FOL KB can be propositionalized so as to preserve entailment (entailments are dependencies)

• (A ground sentence is entailed by new KB iff entailed by original KB)

• Idea: propositionalize KB and query, apply resolution, return result

• Problem: with function symbols, there are infinitely many ground terms,– e.g., Father(Father(Father(John)))

–•

•

•


FOL

Reduction contd.

Theorem: Herbrand (1930). If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the propositionalized KB

Idea: For n = 0 to ∞ do create a propositional KB by instantiating with depth-$n$ terms see if α is entailed by this KB

Problem: works if α is entailed, loops if α is not entailed

Theorem: Turing (1936), Church (1936) Entailment for FOL is

semidecidable (algorithms exist that say yes to every entailed sentence, but no algorithm exists that also says no to every nonentailed sentence.)

•


FOL

Problems with propositionalization

• Propositionalization seems to generate lots of irrelevant sentences.

• E.g., from:x King(x) Greedy(x) Evil(x)King(John)y Greedy(y)Brother(Richard,John)

• it seems obvious that Evil(John), but propositionalization produces lots of facts such as Greedy(Richard) that are irrelevant

• With p k-ary predicates and n constants, there are p·nk instantiations.

••

••


FOL

Unification

“It is an algorithm for determining the substitutions needed to make two predicate

calculus expressions match”

The process of universal/existential instantiation require substitutions


FOL

Unification

• Unify(α,β) = θ if αθ = βθ p q θ knows(ali,X) knows(ali,sana) knows(ali,X) knows(Y,hira) knows(ali,X) knows(Y,mother(Y))knows(ali,X) knows(X,hira)

• Standardizing apart eliminates overlap of variables, e.g., the fourth sentence can have knows(Z,hira)

•

•


FOL

Unification

• Unify(α,β) = θ if αθ = βθ

p q θ

knows(ali,X) knows(ali,sana) {sana/X}

knows(ali,X) knows(Y,hira)

knows(ali,X) knows(Y,mother(Y))

knows(ali,X) knows(X,hira)


•

•


FOL

Unification


p q θ


knows(ali,X) knows(Y,hira) {hira/X,ali/Y}

knows(ali,X) knows(Y,mother(Y))



•


FOL

Unification


p q θ



knows(ali,X) knows(Y,mother(Y)) {ali/Y,mother(ali)}



•


FOL

Unification


p q θ



knows(ali,X) knows(Y,mother(Y)) {ali/Y,mother(ali)}

knows(ali,X) knows(X,hira) {fail}


•


FOL

Unification - MGU

• To unify knows(ali,X) and knows(Y,Z)θ = {ali/Y, Z/X } or θ = {ali/Y, ali/X, ali/Z}

(if only ali is present in the data base of facts)

• The first unifier is more general than the second.

• There is a single Most General Unifier (MGU) that is unique up to renaming of variables.

MGU = { ali/Y, Z/X}••


FOL

Implementation of Unification

Using“List format”

PC Syntax List Syntax

p(a,b) (p a b)

p(f(a), g(X,Y)) (p(f a)(g X Y))

p(x) ^ q(y) ((p x) ^ (q y))


FOL

Unification Examples

Unify((parents X(father X) (mother bill)), (parents bill (father bill)Y)).

Complete Substitution

{bill/X, mother (bill)/Y}


FOL


FOL

Logic – Based Financial Advisor

To help a user decide whether to invest in a saving A/C or the stock market or both


FOL

Policy:

Saving inadequate - should invest in saving A/c regardless of income

Saving adequateand adequate income - More profitable option of stock investment

Low Income& adequate savings - Split surplus between saving and stock investment


FOL

Defining ConstraintsDefining Constraints

Adequate Savings/Income:

Income: Rs 15000/month

extra Rs 4000/ dependent/month

Saving: Rs 25,000/year

extra Rs 5000/dependent/year


FOL

Calculating AdequacyDefine Functions to calculate the minimum income and

savings

Minimum Savings: min_saving(X)

min_saving(X)=25000+5000*X

where X are the number of dependents

Minimum Income: min_income(X)

min_income(X)=15000+4000*X


FOL

Outputs

Investment (savings)Investment (stocks)Investment (combination)

How can we represent outputs?


FOL

Rules saving_acc (inadequate) investment (savings)

saving_acc(adequate) ^ income(adequate) investment(stocks)

saving_acc (adequate) ^ income (inadequate) investment (combination)


FOL

What next?

• Define Saving adequate

• Define saving inadequate

• Define income adequate

• Define income inadequate


FOL

Savings

Check if saved amount is greater than the minimum saving required

)(_

))(min_,(^)((^)(_

adequateaccsaving

YsavingXgreaterYdependantsYXsavedamtX

)(_

))(min_,(^)((^)(_

inadequateaccsaving

YsavingXgreaterYdependantsYXsavedamtX


FOL

Income

Check if the income is greater than the minimum income

)(

))(_,(^)((^),(

adequateincome

YincomemixgreaterYdependantsYsteadyXearningsX

)(

))(_,(^)((^),(

inadequateincome

YincomemixgreaterYdependantsYsteadyXearningsX

)(),( inadequateincomeunsteadyXearningsX


FOL

Try firing rules with 9, 10, 11


FOL

Example knowledge base

The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American.

Prove that Col. West is a criminalProve that Col. West is a criminal•

•


FOL

Example knowledge base contd.... it is a crime for an American to sell weapons to hostile nations:

american(X) weapon(Y) sells(X,Y,z) hostile(z) criminal(X)Nono … has some missiles, i.e., X Owns(Nono,X) Missile(X):

owns(nono,m1) and missile(m1)… all of its missiles were sold to it by Colonel West

missile(X) owns(nono,X) sells(west,X,nono)Missiles are weapons:

missile(X) weapon(X)An enemy of America counts as "hostile“:

enemy(X,america) hostile(X)West, who is American …

american(west)The country Nono, an enemy of America …

enemy(nono,america)

–••••


FOL

Example knowledge base contd.Rule Base:Rule Base:... it is a crime for an American to sell weapons to hostile nations:

R1: american(X) weapon(Y) sells(X,Y,z) hostile(z) criminal(X)Nono … has some missiles, i.e.,

R2R2: X owns(nono,X) missile(X):… all of its missiles were sold to it by Colonel West

R3R3: missile(X) owns(nono,X) sells(west,X,nono)Missiles are weapons:

R4:R4: missile(X) weapon(X)An enemy of America counts as "hostile“:

R5R5: enemy(X,america) hostile(X)

•Database of Facts:Database of Facts:owns(nono,m1) and missile(m1)

West, who is American …american(west)

The country Nono, an enemy of America …enemy(nono,america)

•••


FOL

•START FROM THE FACTS•THINK WHICH RULES APPLY

Be careful about the convention constants start with capital letters and variables with small

(applicable to next few slides)

R1: american(X) weapon(Y) sells(X,Y,Z) hostile(Z) criminal(X) R2: X owns(nono,X) missile(X):R3: missile(X) owns(nono,X) sells(west,X,nono)R4: missile(X) weapon(X)R5: enemy(X,america) hostile(X)

•

1. FORWARD CHAINING


FOL R1: american(X) weapon(Y) sells(X,Y,Z) hostile(Z) criminal(X) R2: X owns(nono,X) missile(X):R3: missile(X) owns(nono,X) sells(west,X,nono)R4: missile(X) weapon(X)R5: enemy(X,america) hostile(X)

•

R4{m1/X}

R3{m1/X}

R5{nono/X}


FOL

R1{west/X , m1/Y , nono/Z}

R1: american(X) weapon(Y) sells(X,Y,Z) hostile(Z) criminal(X) R2: X owns(nono,X) missile(X):R3: missile(X) owns(nono,X) sells(west,X,nono)R4:: missile(X) weapon(X)R5: enemy(X,america) hostile(X)

•


FOL

2. BACKWARD CHAINING

•START FROM THE CONCLUSION•THINK IS THE RULE SUPPORT AVAILABLE

R1: american(X) weapon(Y) sells(X,Y,Z) hostile(Z) criminal(X) R2: X owns(nono,X) missile(X):R3: missile(X) owns(nono,X) sells(west,X,nono)R4: missile(X) weapon(X)R5: enemy(X,america) hostile(X)

•



•

Backward chaining example



•



FOL

3. Modus Ponen Based Solution

1. 6 with 3 suggests, through Modus Ponens:

weapons(m1) :{m1/X}

2. From 4 and 9

hostile(nono) : {nono/X}

3. From 2, 5, 6, Modus Ponens

sells(west,nono,m1):m1/X

4. From 2,3,4,5,6, and 9

criminal(west):criminal(west):

{west/X, m1/Y, nono/Z}{west/X, m1/Y, nono/Z}

1. X,Y,Z[american(X)^weapon(Y)^nation(Z)^hostile(Z)^sells(X,Z,Y) criminal(X)]

2. X owns(nono,X)^missiles(X) sells(west,nono,X)

3. X missiles(X) weapons (X)4. X enemy(X,america) hostile(X)5. owns(nono,m1)6. missiles(m1)7. american(west)8. nation(nono)9. enemy(nono,america)10. nation(america)


FOL

Composition

• Find Overall Substitutions

• Given two or more sets of substitutions


FOL

Example• S1={A/K}

• S2={a/A, H/J}

What set would represent overall substitutions

• S1.S2={a/K, H/J}

• Algorithm?


FOL

Example• S1={A/K} S2={a/A, H/J}

• S1.S2={a/K, H/J}

• Algorithm:

1. Apply S2 to S1 (find overall substitution)

2. Add S2 to S1

3. Drop all substitutions in S2 which have common denominators with S1


FOL

Example• S1={A/K} S2={a/A, H/J}• Algorithm:Apply S2 to S1 (find overall substitution)

Apply a/A to A/K would lead to a/KApply H/J to A/K cannot be appliedS1.S2={a/K}

Add S2 to S1S1.S2={a/K a/A,H/J}

Drop all substitutions in S2 which have common denominators with S1nothing gets dropped, hence the answer is {a/K,a/A,H/J}


FOL

Example

• S1={D/A}• S2={a/D}• S12={a/A, a/D}

• S1={D/A,S/G}• S2={x/X, f/D, q/S,w/A}• S1.S2={f/A,q/G, x/X, f/D, q/S,w/A}• S1.S2={f/A,q/G,x/X,q/S} (w/A gets dropped)S1.S2={f/A,q/G, x/X,q/S}


FOL

Is S1.S2 same as S2.S1

• S1={D/A}• S2={a/D}• S21={a/A, D/A}{a/A} where as S12={a/A, a/D}

• S1={D/A,S/G}• S2={x/X, f/D, q/S,w/A}• S2.S1={x/X, f/D, q/S, w/A D/A,S/G}• S2.S1={x/X, f/D, q/S, w/A, S/G} (D/A gets

dropped)

Hence generally speaking S1.S2 not equal to S2.S1


FOL

Is S12 same as S21• S1={D/A}

• S2={X/A}

• S12={D/A, X/A}

• S12={D/A}

• S21={X/A,D/A}

• S21={X/A}


FOL

Example• S1={D/A, d/F,S/E}

• S2={g/D}

• S12={g/A, d/F,S/E, g/D}

• S12={g/A, d/F,S/E, g/D}

• S21={g/D, D/A, d/F,S/E}


FOLSubstitution: Discussion

• Domain: Variables in the substitution together form the domain (denominators)

• Replacements: Terms are the replacements (numerators)

• Pure Vs Impure Substitutions:

– A pure substitution is if all replacements are free of the variables in the domain of the substitution. Otherwise impure

– {s/X, t/I, j/G} (pure)

– The composition of pure substitutions may be impure and vice versa

• Composability:

– A substitution S1 and S2 are composable if and only if the variables in the domain of S1 do not appear among the replacements of S2, otherwise they are noncomposable.

– S1={a/X,b/Y,z/V} S2={U/X,b/V} composable

– S1={a/X,b/Y,z/V} S2={X/U, b/V} noncomposable


FOL Application ExampleKnowledge base:

knows(ali,sana) knows(X,shahid) knows(X,Y)

We also know

knows(ali,X) hates(ali,X)

Unify the conditions to find ali hates whom

unify(knows(ali,X) , knows(ali,sana)) = {sana/X}Using the binding and the rule: ali hates sana

unify(knows(ali,X) , knows(X,shahid)) = {ali/X,shahid/X} This would fail as X cannot have two values at the same time

As the composition of {ali/X} with {shahid/X} or vice versa will result in only one valid substitution

unify(knows(ali,X) , knows(X,Y)) = {ali/X,X/Y}This would mean that ali knows everyone including himself, therefore hates everyone.

1 fol cs 331 dr m m awais inferencing finding solution to queries propositionalization –(for...

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