First Order Logic First Order Logic Course: CS40002 Course: - - PowerPoint PPT Presentation

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First Order Logic First Order Logic Course: CS40002 Course: - - PowerPoint PPT Presentation

Sep 26, 2023 402 likes •580 views

First Order Logic First Order Logic Course: CS40002 Course: CS40002 Instructor: Dr. Pallab Dasgupta Pallab Dasgupta Instructor: Dr. Department of Computer Science & Engineering Department of Computer Science & Engineering Indian

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First Order Logic First Order Logic

Course: CS40002 Course: CS40002 Instructor: Dr. Instructor: Dr. Pallab Dasgupta Pallab Dasgupta

Department of Computer Science & Engineering Department of Computer Science & Engineering Indian Institute of Technology Indian Institute of Technology Kharagpur Kharagpur

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Knowledge and Reasoning Knowledge and Reasoning

  • Representation, Reasoning and Logic

Representation, Reasoning and Logic

  • Propositional

Propositional Logic Logic

  • First

First-

  • Order Logic

Order Logic

  • Inference in first

Inference in first-

  • order logic
  • rder logic
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CSE, IIT CSE, IIT Kharagpur Kharagpur

First First-

  • order Logic
  • rder Logic
  • Constant

Constant → → A | 5 | A | 5 | Kolkata Kolkata | | … …

  • Variable

Variable → → a | x | s | a | x | s | … …

  • Predicate

Predicate → → Before | Before | HasColor HasColor | Raining | | Raining | … …

  • Function

Function → → Mother | Cosine | Mother | Cosine | Headoflist Headoflist | | … …

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CSE, IIT CSE, IIT Kharagpur Kharagpur

First First-

  • order Logic
  • rder Logic
  • Sentence

Sentence → → AtomicSentence AtomicSentence | Sentence Connective Sentence | Sentence Connective Sentence | Quantifier Variable, | Quantifier Variable, … … Sentence Sentence | | ¬ ¬ Sentence | (Sentence) Sentence | (Sentence)

  • AtomicSentence

AtomicSentence → → Predicate(Term, Predicate(Term, … …) | Term = Term ) | Term = Term

  • Term

Term → → Function(Term, Function(Term, … …) | Constant | Variable ) | Constant | Variable

  • Connective

Connective → → ⇒ ⇒ | | ∧ ∧ | | ∨ ∨ | | ⇔ ⇔

  • Quantifier

Quantifier → → ∀ ∀ | | ∃ ∃

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Examples Examples

  • Not all students take both History & Biology

Not all students take both History & Biology

  • Only one student failed History

Only one student failed History

  • Only one student failed both History & Biology

Only one student failed both History & Biology

  • The best score in History is better than the

The best score in History is better than the best score in Biology best score in Biology

  • No person likes a professor unless the

No person likes a professor unless the professor is smart professor is smart

  • Politicians can fool some of the people all the

Politicians can fool some of the people all the time, and they can fool all the people some of time, and they can fool all the people some of the time, but they cant fool all the people all the time, but they cant fool all the people all the time the time

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Examples Examples

  • Russel’s

Russel’s Paradox: Paradox:

  • There is a single barber in town.

There is a single barber in town.

  • Those and only those who do not shave

Those and only those who do not shave themselves are shaved by the barber. themselves are shaved by the barber.

  • Who shaves the barber?

Who shaves the barber?

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Inference rules Inference rules

  • Universal elimination:

Universal elimination:

  • ∀

∀ x Likes( x, x Likes( x, IceCream IceCream ) ) with the with the substitution substitution {x / Einstein} {x / Einstein} gives us gives us Likes( Einstein, Likes( Einstein, IceCream IceCream ) )

  • The substitution has to be done by a

The substitution has to be done by a ground term ground term

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Inference rules Inference rules

  • Existential elimination:

Existential elimination:

  • From

From ∃ ∃ x Likes( x, x Likes( x, IceCream IceCream ) ) we may we may infer infer Likes( Man, Likes( Man, IceCream IceCream ) ) as long as as long as Man does not appear elsewhere in the Man does not appear elsewhere in the Knowledge base Knowledge base

  • Existential introduction:

Existential introduction:

  • From

From Likes( Likes( Monalisa Monalisa, , IceCream IceCream ) ) we can we can infer infer ∃ ∃ x Likes( x, x Likes( x, IceCream IceCream ) )

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Reasoning in first Reasoning in first-

  • order logic
  • rder logic
  • The law says that it is a crime for a Gaul

The law says that it is a crime for a Gaul to sell potion formulas to hostile nations. to sell potion formulas to hostile nations.

  • The country Rome, an enemy of Gaul,

The country Rome, an enemy of Gaul, has acquired some potion formulas, and has acquired some potion formulas, and all of its formulas were sold to it by Druid all of its formulas were sold to it by Druid Traitorix Traitorix. .

  • Traitorix

Traitorix is a Gaul. is a Gaul.

  • Is

Is Traitorix Traitorix a criminal? a criminal?

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Generalized Modus Generalized Modus Ponens Ponens

  • For atomic sentences p

For atomic sentences pi

i, p

, pi

i’, and q, where

’, and q, where there is a substitution there is a substitution θ θ such that such that SUBST( SUBST(θ θ, p , pi

i’

’) = SUBST( ) = SUBST(θ θ, p , pi

i),

), for all i:

for all i:

) , ( ) ... ( , ,..., ,

2 1 2 1

q SUBST q p p p p p p

n n

θ ⇒ ∧ ∧ ∧ ′ ′ ′

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Unification Unification

UNIFY(p,q) = UNIFY(p,q) = θ θ where SUBST(

where SUBST(θ θ,p) = SUBST( ,p) = SUBST(θ θ,q) ,q)

Examples: Examples:

UNIFY( Knows( UNIFY( Knows(Erdos Erdos, x),Knows( , x),Knows(Erdos Erdos, , Godel Godel)) )) = {x / = {x / Godel Godel} } UNIFY( Knows( UNIFY( Knows(Erdos Erdos, x), Knows(y, , x), Knows(y,Godel Godel)) )) = {x/ = {x/Godel Godel, y/ , y/Erdos Erdos} }

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Unification Unification

UNIFY(p,q) = UNIFY(p,q) = θ θ where SUBST(

where SUBST(θ θ,p) = SUBST( ,p) = SUBST(θ θ,q) ,q)

Examples: Examples:

UNIFY( Knows( UNIFY( Knows(Erdos Erdos, x), Knows(y, Father(y))) , x), Knows(y, Father(y))) = { y/ = { y/Erdos Erdos, x/Father( , x/Father(Erdos Erdos) } ) } UNIFY( Knows( UNIFY( Knows(Erdos Erdos, x), Knows(x, , x), Knows(x, Godel Godel)) = F )) = F We require the most general unifier We require the most general unifier

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Reasoning with Horn Logic Reasoning with Horn Logic

  • We can convert Horn sentences to a

We can convert Horn sentences to a canonical form and then use generalized canonical form and then use generalized Modus Modus Ponens Ponens with unification. with unification.

  • We

We skolemize skolemize existential formulas and existential formulas and remove the universal ones remove the universal ones

  • This gives us a conjunction of clauses, that

This gives us a conjunction of clauses, that are inserted in the KB are inserted in the KB

  • Modus

Modus Ponens Ponens help us in inferring new help us in inferring new clauses clauses

  • Forward and backward chaining

Forward and backward chaining

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Completeness issues Completeness issues

  • Reasoning with Modus

Reasoning with Modus Ponens Ponens is incomplete is incomplete

  • Consider the example

Consider the example – – ∀ ∀x P(x) x P(x) ⇒ ⇒ Q(x) Q(x) ∀ ∀x x ¬ ¬P(x) P(x) ⇒ ⇒ R(x) R(x) ∀ ∀x Q(x) x Q(x) ⇒ ⇒ S(x) S(x) ∀ ∀x R(x) x R(x) ⇒ ⇒ S(x) S(x)

  • We should be able to conclude S(A)

We should be able to conclude S(A)

  • The problem is that

The problem is that ∀ ∀x x ¬ ¬P(x) P(x) ⇒ ⇒ R(x) cannot R(x) cannot be converted to Horn form, and thus cannot be converted to Horn form, and thus cannot be used by Modus be used by Modus Ponens Ponens

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CSE, IIT CSE, IIT Kharagpur Kharagpur

Godel’s Godel’s Completeness Theorem Completeness Theorem

  • For first

For first-

  • order logic, any sentence that is
  • rder logic, any sentence that is

entailed by another set of sentences can be entailed by another set of sentences can be proved from that set proved from that set

  • Godel

Godel did not suggest a proof procedure did not suggest a proof procedure

  • In 1965 Robinson published his resolution

In 1965 Robinson published his resolution algorithm algorithm

  • Entailment in first

Entailment in first-

  • order logic is semi
  • rder logic is semi-
  • decidable,

decidable, that is, we can show that sentences follow from that is, we can show that sentences follow from premises if they do, but we cannot always show if premises if they do, but we cannot always show if they do not. they do not.

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CSE, IIT CSE, IIT Kharagpur Kharagpur

The validity problem of first The validity problem of first-

  • order logic
  • rder logic
  • [Church]

[Church] The validity problem of the first The validity problem of the first-

  • rder predicate calculus is partially solvable.
  • rder predicate calculus is partially solvable.
  • Consider the following formula:

Consider the following formula:

) , ( ))]] ( ), ( ( ) , ( [ )) ( ), ( ( [

1 1

z z p z x g x f p y x p y x a g a f p

i i n i i i n i

∃ ⇒ ∧ ⇒ ∀ ∀ ∧ ∧

= =