1 FOL CS 331/531 Dr M M Awais Unification “It is an algorithm for determining the substitutions needed to make two predicate calculus expressions match”

CS 331/531 Dr M M Awais 1

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Unification

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

expressions match”

Reference: Luger’s Book chapter 3, 2nd edition


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Implementation of Unification

Using“List format” (PNF have already been discussed)

PC Syntax List Syntaxp(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))


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Unification: Examples

Unify(

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

).

Complete Substitution

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


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Logic – Based Financial Advisor

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


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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


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Defining ConstraintsDefining Constraints

Adequate Savings/Income:Income: Rs 15,000/month

extra Rs 4000/ dependent/month

Saving: Rs 25,000/yearextra Rs

5000/dependent/year


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Calculating AdequacyDefine Functions to calculate the minimum

income and savings

Minimum Savings: min_saving(X)min_saving(X)=25000+5000*X

Minimum Income: min_income(X)

min_income(X)=15000+4000*X

where X are the number of dependents


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Outputs

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

How can we represent outputs?


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Rules saving_acc (inadequate) investment (savings)

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

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


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What next? Define saving adequate Define saving inadequate Define income adequate Define income inadequate


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Savings

Check if saved amount is greater than the minimum saving required

)(_

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

adequateaccsaving

YsavingXgreaterYdependantsYXsavedamtX

)(_

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

inadequateaccsaving

YsavingXgreaterYdependantsYXsavedamtX


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Income

Check if the income is greater than the minimum income

)(

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

adequateincome

YincomemixgreaterYdependantsYsteadyXearningsX

)(

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

inadequateincome

YincomemixgreaterYdependantsYsteadyXearningsX

)(),( inadequateincomeunsteadyXearningsX


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Try firing rules with 9, 10, 11


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Example knowledge baseThe 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

Solution presented using alternate syntaxReference: Norviq and Russell


FOLExample 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) (skolemization)… 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)


FOLExample 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 WestR3R3: 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) ^ Missile(M1)

West, who is American …American(West)

The country Nono, an enemy of America …Enemy(Nono,America)


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•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)


R2: x Owns(Nono,x) Mssile(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


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R4{M1/x}

R3{M1/x}

R5{Nono/x}




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R1{West/x , M1/y , Nono/z}




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2. BACKWARD CHAINING

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




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Backward chaining example

West/x


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West/x


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West/x


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West/x , M1/y

M1/y


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West/x , M1/y , Nono/z

M1/y

Nono/z


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M1/y

Nono/z


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M1/y

Nono/z

Documents

1 FOL CS 331/531 Dr M M Awais Unification “It is an algorithm for determining the substitutions needed to make two predicate calculus expressions match”