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Understanding Predicate Calculus in Artificial Intelligence

This lecture explores the concepts of predicate calculus as a foundational element of artificial intelligence. It contrasts predicate calculus with propositional calculus, highlighting how predicates allow for more nuanced expression of knowledge, enabling succinct representation of statements involving quantifiers. Examples illustrate applications in logic, inference, and knowledge representation. The lecture covers various concepts such as universal and existential quantifiers, rules for reasoning, and interpretations of logical expressions, providing insights into how AI systems reason about complex relationships.

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Understanding Predicate Calculus in Artificial Intelligence

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  1. CS344: Introduction to Artificial Intelligence(associated lab: CS386) Pushpak BhattacharyyaCSE Dept., IIT Bombay Lecture–28, 29: Predicate Calculus; Interpretation 22nd and 26th March, 2012

  2. Well known examples • Man is mortal : rule ∀x[man(x) → mortal(x)] • shakespeare is a man man(shakespeare) • To infer shakespeare is mortal mortal(shakespeare)

  3. Predicate Calculus: more succinct expression than propositional calculus • Statement 1 : 2 is even and is divisible by 2 : P1 • Statement 2 : 4 is even and is divisible by 2 : P2 • Statement 3 : 6 is even and is divisible by 2 : P3 • Statement 4 : 8 is even and is divisible by 2 : P4 Finite expression for pieces of knowledge

  4. Predicate Calculus: finer than propositional calculus • Finer Granularity (Grass is green, ball is green, leaf is green (green(x))) • Succinct description for infinite number of statements which would need number of properties 3 place predicate Example: x gives y to z give(x,y,z) 4 place predicate Example: x gives y to z through w give(x,y,z,w)

  5. Double causative in Hindi giving rise to higher place predicates • जॉन ने खाना खायाJohn ne khanakhaya John <CM> food ate John ate food eat(John, food) • जॉन ने जैक को खाना खिलाया John ne Jack kokhanakhilaya John <CM> Jack <CM> food fed John fed Jack eat(John, Jack, food) • जॉन ने जैक को जिल के द्वारा खाना खिलाया John ne Jack ko Jill kedvarakhanakhilaya John <CM> Jack <CM> Jill <CM> food made-to-eat John fed Jack through Jill eat(John, Jack, Jill, food)

  6. Predicate Calculus: origin • Predicate calculus originated in language Sentence Subject Predicate Grass is green green(grass) Subject Predicate D : Domain

  7. Predicate Calculus: only for declarative sentences • Is grass green ? • Oh, grass is green! • Grass which is supple is green supple(grass) green(grass) Declarative Sentence Subject Predicate

  8. PC primitive: N-ary Predicate • Arguments of predicates can be variables and constants • Ground instances : Predicate all whose arguments are constants

  9. N-ary Functions president(India) : Prathiba Patel • Constants & Variables : Zero-order objects • Predicates & Functions : First-order objects President of India is a lady lady(president(India)) Prime minister of India is older than the president of India older(prime_minister(India), president(India))

  10. Operators • Universal Quantifier • Existential Quantifier All men are mortal Some men are rich

  11. Tautologies • 2nd tautology in English: • Not a single man in this village is educated implies all men in this village are uneducated • Tautologies are important instruments of logic, but uninteresting statements!

  12. Inferencing: Forward Chaining • man(x) → mortal(x) • Dropping the quantifier, implicitly Universal quantification assumed • man(shakespeare) • Goal mortal(shakespeare) • Found in one step • x = shakespeare, unification

  13. Backward Chaining • man(x) → mortal(x) • Goal mortal(shakespeare) • x = shakespeare • Travel back over and hit the fact asserted • man(shakespeare)

  14. Wh-Questions and Knowledge what where Factoid / Declarative who when which how procedural why Reasoning

  15. Fixing Predicates • Natural Sentences <Subject> <verb> <object> Verb(subject,object) predicate(subject)

  16. Examples • Ram is a boy • Boy(Ram)? • Is_a(Ram,boy)? • Ram Playes Football • Plays(Ram,football)? • Plays_football(Ram)?

  17. Knowledge Representation of Complex Sentence • “In every city there is a thief who is beaten by every policeman in the city”

  18. Interpretation in Logic Logical expressions or formulae are “FORMS” (placeholders) for whom contents are created through interpretation. Example: This is a Second Order Predicate Calculus formula. Quantification on ‘F’ which is a function.

  19. Examples Interpretation:1 D=N (natural numbers) a = 0 and b = 1 x ∈ N P(x) stands for x > 0 g(m,n) stands for (m x n) h(x) stands for (x – 1) Above interpretation defines Factorial

  20. Examples (contd.) Interpretation:2 D={strings) a = b = λ P(x) stands for “x is a non empty string” g(m, n) stands for “append head of m to n” h(x) stands for tail(x) Above interpretation defines “reversing a string”

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