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CS344 : Introduction to Artificial Intelligence. Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 4- Logic. Logic and inferencing. Vision. NLP. Search Reasoning Learning Knowledge. Expert Systems. Robotics. Planning.

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CS344 : Introduction to Artificial Intelligence

Pushpak BhattacharyyaCSE Dept., IIT Bombay

Lecture 4- Logic


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Logic and inferencing

Vision

NLP

  • Search

  • Reasoning

  • Learning

  • Knowledge

Expert Systems

Robotics

Planning

Obtaining implication of given facts and rules -- Hallmark of intelligence


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Inferencing through

  • Deduction (General to specific)

  • Induction (Specific to General)

  • Abduction (Conclusion to hypothesis in absence of any other evidence to contrary)

Deduction

Given: All men are mortal (rule)

Shakespeare is a man (fact)

To prove: Shakespeare is mortal (inference)

Induction

Given: Shakespeare is mortal

Newton is mortal (Observation)

Dijkstra is mortal

To prove: All men are mortal (Generalization)


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If there is rain, then there will be no picnic

Fact1: There was rain

Conclude: There was no picnic

Deduction

Fact2: There was no picnic

Conclude: There was no rain (?)

Induction and abduction are fallible forms of reasoning. Their conclusions are susceptible to retraction

Two systems of logic

1) Propositional calculus

2) Predicate calculus


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Propositions

  • Stand for facts/assertions

  • Declarative statements

    • As opposed to interrogative statements (questions) or imperative statements (request, order)

      Operators

      => and ¬ form a minimal set (can express other operations)

      - Prove it.

      Tautologies are formulae whose truth value is always T, whatever the assignment is


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Model

In propositional calculus any formula with n propositions has 2n models (assignments)

- Tautologies evaluate to T in all models.

Examples:

1)

2)

  • e Morgan with AND


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Semantic Tree/Tableau method of proving tautology

Start with the negation of the formula

- α - formula

α-formula

β-formula

- β - formula

α-formula

- α - formula


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Example 2:

X

(α - formula)

(α - formulae)

α-formula

(β - formulae)

B

C

B

C

Contradictions in all paths


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Exercise:

Prove the backward implication in the previous example


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Formal Systems

  • Rule governed

  • Strict description of structure and rule application

  • Constituents

    • Symbols

    • Well formed formulae

    • Inference rules

    • Assignment of semantics

    • Notion of proof

    • Notion of soundness, completeness, consistency, decidability etc.


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Hilbert's formalization of propositional calculus

1. Elements are propositions : Capital letters

2. Operator is only one :  (called implies)

3. Special symbolF(called 'false')

4. Two other symbols : '(' and ')'

5. Well formed formula is constructed according to the grammar

WFF P|F|WFFWFF

6. Inference rule : only one

Given AB and

A

write B

known asMODUS PONENS


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7. Axioms : Starting structures

A1:

A2:

A3

This formal system defines the propositional calculus


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Notion of proof

1. Sequence of well formed formulae

2. Start with a set of hypotheses

3. The expression to be proved should be the last line in the sequence

4. Each intermediate expression is either one of the hypotheses or one of the axioms or the result of modus ponens

5. An expression which is proved only from the axioms and inference rules is called a THEOREM within the system


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Example of proof

From P and and prove R

H1: P

H2:

H3:

i) P H1

ii) H2

iii) Q MP, (i), (ii)

iv) H3

v) R MP, (iii), (iv)


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Prove that is a THEOREM

i) A1 : P for A and B

ii) A1: P for A and for B

iii)

A2: with P for A, for B and P for C

iv) MP, (ii), (iii)

v) MP, (i), (iv)


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Formalization of propositional logic (review)

Axioms :

A1

A2

A3

Inference rule:

Given and A, write B

A Proof is:

A sequence of

i) Hypotheses

ii) Axioms

iii) Results of MP

A Theorem is an

Expression proved from axioms and inference rules


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Example: To prove

i) A1 : P for A and B

ii) A1: P for A and for B

iii)

A2: with P for A, for B and P for C

iv) MP, (ii), (iii)

v) MP, (i), (iv)


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Shorthand

1. is written as and called 'NOTP'

2. is written as and called

'PORQ’

3. is written as and called

'P AND Q'

Exercise: (Challenge)

- Prove that


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A very useful theorem (Actually a meta theorem, called deduction theorem)

Statement

If

A1, A2, A3 ............. An├ B

then

A1, A2, A3, ...............An-1├

├ is read as 'derives'

Given

A1

A2

A3

.

.

.

.

An

B

A1

A2

A3

.

.

.

.

An-1

Picture 1

Picture 2


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Use of Deduction Theorem deduction theorem)

Prove

i.e.,

├ F(M.P)

A├(D.T)

├(D.T)

Very difficult to prove from first principles, i.e., using axioms and inference rules only


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Prove deduction theorem)

i.e.

├ F

├ (D.T)

├ Q (M.P with A3)

P├


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