First-Order Logic - Artificial Intelligence - Lecture Slides, Slides of Artificial Intelligence

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Lecture 12

First-Order Logic (FOL) Review

Lecture Outline

• Today’s Reading
• Next Week’s Reading: Chapter 8, R&N
• Previously: Logical Agents and Calculi
  • Logical agent framework
  • Logic in general: tools for
    • Knowledge representation
    • Inference / theorem proving and problem solving / planning
  • Propositional calculus
    • Normal forms
    • Sequent rules (modus ponens, resolution)
  • Predicate logic
  • First-order logic (FOL) aka first-order predicate calculus (FOPC)
• Today: FOL Agents, Examples; Frame Problem; Situation Calculus
• Next Week: FOL Knowledge Bases (Chapter 8, R&N)

Review: Elements of FOL

• Logical Agents Overview (Last Tuesday)
  • Knowledge Bases (KB) and KB agents
  • Motivating example: Wumpus World
  • Syntax of propositional calculus
  • Elements of logic in general
    • Syntax: What constitutes legitimate sentences aka well-formed formulae?
    • Semantics: What constitutes logical entailment?
    • Proof theory: What constitutes provability? Soundness? Completeness?
• Propositional and First-Order Calculi (Last Thursday)
  • Propositional calculus (concluded): inference by model checking, sequent rules
  • Elements of logic in general: normal forms (CNF, DNF, Horn) and their usage
  • Predicate logic without quantifiers: functions and predicates, terms and atoms
  • Introduction to First-Order Logic (FOL)
    • Domain theory
    • Syntax of WFFs: proper scoping (existential, universal quantification)
    • New features: semantics of quantification

Validity and Satisfiability

Inference (Sequent) Rules for

Propositional Logic

Logical Agents:

Taking Stock

Syntax of FOL:

Basic Elements

FOL: Atomic Sentences

(Atomic Well-Formed Formulae)

• “Every Dog Chases Its Own Tail”
  • ∀ d. Chases ( d , tail-of (d))
  • Alternative Statement: ∀ d. ∃ t. Tail-Of ( t , d ) ∧ Chases ( d , t )
  • Prefigures concept of Skolemization (Skolem variables / functions)
• “Every Dog Chases Its Own (Unique) Tail”
  • ∀ d. ∃^1 t. Tail-Of ( t , d ) ∧ Chases ( d , t ) ≡ ∀ d. ∃ t. Tail-Of ( t , d ) ∧ Chases ( d , t ) ∧ [∀ t’ Chases ( d , t’ ) ⇒ t’ = t ]
• “Only The Wicked Flee when No One Pursueth”
  • ∀ x. Flees ( x ) ∧ [¬∃ y Pursues ( y , x )] ⇒ Wicked ( x )
  • Alternative : ∀ x. [∃ y. Flees ( x, y )] ∧ [¬∃ z. Pursues ( z , x )] ⇒ Wicked ( x )
• Offline Exercise: What Is An n th Cousin, m Times Removed?

Jigsaw Exercise [1]:

First-Order Logic Sentences

Jigsaw Exercise [2]:

First-Order Logic Sentences

More Fun with Sentences

• “Every Dog Chases Its Own Tail”
  • ∀ d. Chases ( d , tail-of (d))
  • Alternative Statement: ∀ d. ∃ t. Tail-Of ( t , d ) ∧ Chases ( d , t )
  • Prefigures concept of Skolemization (Skolem variables / functions)
• “Every Dog Chases Its Own (Unique) Tail”
  • ∀ d. ∃^1 t. Tail-Of ( t , d ) ∧ Chases ( d , t ) ≡ ∀ d. ∃ t. Tail-Of ( t , d ) ∧ Chases ( d , t ) ∧ [∀ t’ Chases ( d , t’ ) ⇒ t’ = t ]
• “Only The Wicked Flee when No One Pursueth”
  • ∀ x. Flees ( x ) ∧ [¬∃ y Pursues ( y , x )] ⇒ Wicked ( x )
  • Alternative : ∀ x. [∃ y. Flees ( x, y )] ∧ [¬∃ z. Pursues ( z , x )] ⇒ Wicked ( x )
• Offline Exercise: What Is An n th Cousin, m Times Removed?

Wumpus World Revisited:

Interacting with FOL KBs

Deducing Hidden Properties

Keeping Track of Change:

Situation Calculus