THE PREDICATE CALCULUS140.126.122.189/upload/1052/B02312A2017131717201.pdf · 2017-01-03 · Predicate calculus symbols may represent either variables, constants functions, or predicates

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⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

THE PREDICATE CALCULUS

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ The truth tables

Def: Two expressions are equivalent if they have the same truth value.

syntactically different

but logically equivalent.

It is important when using inference rules that require

expressions to be in a specific form.

The Predicate Calculus

Propositional calculus Predicate calculus

"it rained on Tuesday" weather(tuesday, rain)

What's the difference?

Def: Predicate calculus symbols

1. A, B, C, ... , Z, a, b, c, ..., z.

2. 0, 1, ..., 9.

3. _.

Symbols: 1. Begin with a letter 2. Followed by any sequence of these legal

characters. Example:

legitimate characters: a R 6 9 p _ z

illegitimate characters: # % @ / & ""

4


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legitimate symbols: George fire3 bill XXXX

friends_of tom_and_jerry

illegitimate symbols: 3jack ab%cd

duck!!! ***71

"no blanks allowed"

Predicate calculus symbols may represent either variables, constants functions, or predicates.

1. Constants:

1) Specific objects or properties. 2) Begin with a lowercase letter.

Ex: george, tree, tall, blue Note: The constants true and false are reserved as

truth symbols. 2. Variables:

1. General classes of objects or properties.

2. Begin with an uppercase letter.

Ex: legal variables: George, BILL, KAte illegal variables: geORGE, bill 3. Functions:

1) Begin with a lowercase letter.

2) Mappings between nonnumeric domain as well as common arithmetic functions.

Def:

arity: The number of elements in the domain mapped onto each element of the range.

Ex: father: arity 1 plus: arity n


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Def: function expression: a function symbol followed by its arguments. Example:

f(X,Y) father(david)

price(bananas) Notation:

mapping domain ⎯⎯⎯⎯⎯⎯⎯→ range arguments evaluation value

Def:

term : a constant, a variable, or function expression.

→ It denotes objects and properties in the problem domain.

Example: cat times(2,3) blue X mother(jane) kate

• symbols:

1. Truth symbols: true, false (reserved symbols)

2. Constant symbols:

3. Variable symbols:

4. Function symbols: • function expression:

Consists of a function of arity n. • term:

Either a constant, a variable, or function expression.


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Atomic sentence (atomic expression, atom, proposition)

Remark:

1. arity: number of argument

2. Both name and arity equals ⇒ the same relation.


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From atomic sentence to sentence

The variable quantifiers

variable

X,Y,Zvariable quantifier

universal ∀∃existential

generalize constrain

atomic sentence

sentence

∃Y Yfriends( ,peter) ( )∃YS

∀X Xlikes( ,ice_ cream) ( )∀XS

atomic sentences

¬ ∧ ∨ ⇒ ∀ ∃= X

predicate calculus sentences

Example of predicate calculus sentences:

plus(two,three) equal(plus(two,three),five)

∃∧

X Xfoo( , two,plus(two, three))equal(plus(two, three), five)

(

(foo(two, two,plus(two, three)))

equal(plus(two, three),five) = true)⇒

equal(plus(2,3),seven)


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A simple applied example

mother(eve,abel) mother(eve,cain) father(adam,abel) father(adam,cain)

∀ ∀ ∨⇒

X Y X Y X YX Y

father( , ) mother( , )parent( , )

∀ ∀ ∀ ∧⇒

X Y Z X Y XY Zparent( , ) parent( ,Z)

sibling( ),

How to apply these sentences to infer new fact such as

sibling(cain,abel)

The semantics of predicate calculus

. A semantics for the predicate calculus

atomic sentences

¬ ∧ ∨ ⇒ ∀ ∃= X

predicate calculus sentences

interpretation

truth value assignment


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Relationships between ¬ ∀ ∃

¬∃ = ∀ ¬Xp X X p X( ) ( ) ¬∀ = ∃ ¬Xp X X p X( ) ( ) ∃ = ∃Xp X Yp Y( ) ( ) ∀ = ∀Xp X Yp Y( ) ( )

∀ ∧ = ∀ ∧∀X p X q X Xp X Yq Y( ( ) ( )) ( ) ( ) ∃ ∨ = ∃ ∨ ∀X p X q X Xp X Yq Y( ( ) ( )) ( ) ( )

Examples: English → Predicate calculus sentences

If it doesn't rain tomorrow, Tom will go to the mountains.

↓ ¬weather(rain,tomorrow)⇒go(tom,mountains).

All basketball players are tall.

↓ ∀X(basketball-player(X)⇒tall(X)).

Some people like anchovies.

↓ ∃X(person(X)∧likes(X,anchovies)).

If wishes were horses, beggars would ride.

↓ equal(wishes,horses)⇒ride(beggars).

Nobody likes taxes.

↓ ¬∃Xlikes(X,taxes)


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A real-world example: ◎ To stack a block: 1. Both blocks must be clear. 2. Hand is empty. 3. To perform pick-up and put-down. 1. Both blocks must be clear. → ∀ ¬∃ ⇒X Y Y X X( ( , ) ( ))on clear The action:

∀ ∀ − ∧ ∧∧ − ∧ −⇒

X Y X YX X Y

X Y

(( ( ( )( ) ( , ))

( , ))

hand empty clear ) clearpick up put down

stack

Note:

1. The actions must be performed in the order in which they appear in a rule premise.

2. The block-world figure is an interpretation.

3. A different set of blocks in another location → Another interpretation (I).


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AI in Practice:

Using Inference Rules to Product Predicate Calculus Expressions

Satisfy: I satisfies

sentences ⎯⎯⎯⎯⎯→ true

logically following: SFX (X logically follows from S)

I I In1 2, ,..., satisfy Set of expressions (S) ⎯⎯⎯⎯⎯⎯⎯→ true

⇓ I I In1 2, ,..., satisfy

X ⎯⎯⎯⎯⎯⎯⎯→ true


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logic inference:

To product new sentences that logically follow a given set of predicate calculus sentences.

The inference rules:

modus ponens: P and P Q⇒ true → Q modus tolens: P Q⇒ true, Q false → ¬P elimination: QP∧ true → P, Q introduction: P and Q true → QP∧ universal instantiation: ∀XP X( ) true → XaaP ∈ )(

Example: modus ponens

All men are mortal Socrates is a man ))()(( xmortalxmanx ⇒∀ man(Socrates)

(unify)

man(Socrates) mortal(Socrates)

Unification

What: To determine the substitutions needed to make two expressions match.

Why: To apply inference rule. How: To replace the quantified variable by terms

from the domain. Example:

1. ∀ ⇒X X X( ( ) ))man mortal( man(socrates) mortal(socrates) 2. P(X) P(Y) equivalent


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3. ∃X Xparent( , tom) ⇒ parent(bob,tom) or parent(mary,tom)

Assume bob and mary are tom‘s parents 4. ∀ ∃X Ymother(X,Y) ⇒ mother(X,f(X)) Y depends on X

The substitution issues 1. Any constant is not replaceable. 2.

constant 1constant 2

variable X

3. occurs check:

X P(X)

otherwise P(P(P(...(X)...)

4. consistency:

( ( ) ))man mortal(X X⇒ all occurrence of the variable 5. Time-invariant

IF X t X t1 2( ) ( ) THEN constant constant1X t X t( ) ( )+ → +α α2 Ex: Let p(a,X) unifies the primise of p(Y,Z) ⇒ q(Y,Z) i.e. (a/Y,X/Z) THEN q(a,X) Now, let q(W,b) ⇒ r(W,b) It is claimed that (a/W,b/X) So we may infer r(a,b)


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6. Composition:

Ex:

Let {X/Y,W/Z}, {V/X}, {a/V,f(b)/W} ⇓ {V/Y,W/Z}, {a/V,f(b)/W} ⇓ {a/Y,f(b)/Z} 7. Generality

Ex:

Unify P(X) and P(Y):

What's the difference between

{fred/X,fred/Y} and {Z/X,Z/Y} a constant a variable

. An unification example LIST syntax unify ((parents X (father X) (mother bill)), (parents bill (father bill) y)) PC syntax unify parent(X,father(X),mother(bill)) parent(bill,father(bill),Y)


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AI in Practice:


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Application: A Logic-Based Financial Advisor

The advices:

Saving account income saving stocks ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ inadequate investment adequate adequate investment adequate inadequate combination ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

In terms of PC: savings_account(inadequate) ⇒ investment(savings) savings_account(adequate)∧income(adequate) ⇒ investment(stocks) savings_account(adequate)∧income(inadequate) ⇒ investment(combination)


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To determine whether savings and income are adequate or inadequate ⇓

The minimum adequate savings and the minimum income must be defined

The advices: 1. To have at least $5000 in the bank for each dependent. 2. An adequate income must be a steady income and supply at least $15,000 per

year plus an additional $4000 for each dependent.

In terms of PC: minsavings(X) = 5000*X minincome(X) = 15000+(4000*X)


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The adequacy of savings is determined by the rule:

∀Xamount_saved(X)∧∃Y(dependents(Y)∧greater(X,minsavings(Y))) ⇒ savings_account(adequate)

∀Xamount_saved(X)∧∃Y(dependents(Y)∧¬greater(X,minsavings(Y))) ⇒ savings_account(inadequate)

The adequacy of income is determined by the rule: ∀Xearnings(X,steady)∧∃Y(dependents(Y)∧greater(X,minincome(Y))) ⇒ income(adequate)

∀Xearnings(X,steady)∧∃Y(dependents(Y)∧¬greater(X,minincome(Y))) ⇒ income(inadequate) ∀Xearnings(X,unsteady) ⇒ income(inadequate)


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To perform a consultation for an individual with three independents, $22,000 in savings, and a steady income of $25,000. Three PC sentences are stated:

amount_saved(22000)

earnings(25000,steady)

dependents(3)


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The final logic system: 1 savings_account(inadequate) ⇒ investment(savings) 2 savings_account(adequate)∧income(adequate) ⇒ investment(stocks) 3 savings_account(adequate)∧income(inadequate) ⇒ investment(combination) 4 ∀Xamount_saved(X)∧∃Y(dependents(Y)∧greater(X,minsavings(Y))) ⇒ savings_account(adequate) 5 ∀Xamount_saved(X)∧∃Y(dependents(Y)∧¬greater(X,minsavings(Y))) ⇒ savings_account(inadequate) 6 ∀Xearnings(X,steady)∧∃Y(dependents(Y)∧greater(X,minincome(Y))) ⇒ income(adequate) 7 ∀Xearnings(X,steady)∧∃Y(dependents(Y)∧¬greater(X,minincome(Y))) ⇒ income(inadequate) 8 ∀Xearnings(X,unsteady) ⇒ income(inadequate) 9 amount_saved(22000) 10 earnings(25000,steady) 11 dependents(3) where minsavings(X) = 5000*X; minincome(X) = 15000+(4000*X)


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Now using unification and modus ponens, (10 & 11 → 7) Application

earnings(25000,steady)∧(dependents(3) unifies with

earnings(X,steady)∧(dependents(Y) under the substitution

{25000/X, 3/Y} It yields earnings(25000,steady)∧dependents(3)∧¬greater(25000,minincome(3)) ⇒ income(inadequate)

Evaluating the function minincome yield: earnings(25000,steady)∧dependents(3)∧¬greater(25000,27000)) ⇒ income(inadequate) It results a true conclusion:

12 income(inadequate)


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Similarly, amount_saved(22000)∧dependents(3) unifies with 4 under the substitution

{22000/X, 3/Y}

It yields amount_saved(22000)∧dependents(3)∧greater(22000,minsavings(3)) ⇒ savings_account(adequate)

Evaluating the function minincome yield: amount_saved(22000)∧dependents(3)∧greater(22000,15000) ⇒ savings_account(adequate) It also results a true conclusion:

13 savings_account(adequate)


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Examine the three expressions 3, 12, and 13, and apply modus ponens again we get the final conclusion

investment(combination)

How to arrange the combination?

Fuzzy Expert Systems


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AI in Practice:

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THE PREDICATE CALCULUS140.126.122.189/upload/1052/B02312A2017131717201.pdf · 2017-01-03 · Predicate calculus symbols may represent either variables, constants functions, or predicates