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Lecture 12

Lecture 12. First-Order Logic (FOL) Review. Friday, 17 September 2004 William H. Hsu Department of Computing and Information Sciences, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading: Sections 7.5 – 7.10, Russell and Norvig 2e. Lecture Outline. Today’s Reading

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Lecture 12

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  1. Lecture 12 First-Order Logic (FOL) Review Friday, 17 September 2004 William H. Hsu Department of Computing and Information Sciences, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading: Sections 7.5 – 7.10, Russell and Norvig 2e

  2. Lecture Outline • Today’s Reading • Sections 7.5 – 7.10, Russell and Norvig 2e • Recommended references: Nilsson and Genesereth • 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)

  3. Review:Simple Knowledge-Based Agent Chapter 7 R&N 2e Adapted from slides by S. Russell, UC Berkeley

  4. 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 akawell-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

  5. Validity and Satisfiability Adapted from slides by S. Russell, UC Berkeley

  6. Proof Methods Adapted from slides by S. Russell, UC Berkeley

  7. Inference (Sequent) Rules forPropositional Logic Adapted from slides by S. Russell, UC Berkeley

  8. Logical Agents:Taking Stock Adapted from slides by S. Russell, UC Berkeley

  9. The Road Ahead:Predicate Logic and FOL • Predicate Logic • Enriching language • Predicates • Functions • Syntax and semantics of predicate logic • First-Order Logic (FOL, FOPC) • Need for quantifiers • Relation to (unquantified) predicate logic • Syntax and semantics of FOL • Fun with Sentences • Wumpus World in FOL Adapted from slides by S. Russell, UC Berkeley

  10. Syntax of FOL:Basic Elements Adapted from slides by S. Russell, UC Berkeley

  11. FOL: Atomic Sentences(Atomic Well-Formed Formulae) Adapted from slides by S. Russell, UC Berkeley

  12. Equality Adapted from slides by S. Russell, UC Berkeley

  13. Jigsaw Exercise [1]:First-Order Logic 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 . 1t . 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 nth Cousin, m Times Removed?

  14. Jigsaw Exercise [2]:First-Order Logic Sentences

  15. Terminology • Logical Frameworks • Knowledge Bases (KB) • Logic in general: representation languages, syntax, semantics • Propositional logic • First-order logic (FOL, FOPC) • Model theory, domain theory: possible worlds semantics, entailment • Normal Forms • Conjunctive Normal Form (CNF) • Disjunctive Normal Form (DNF) • Horn Form • Proof Theory and Inference Systems • Sequent calculi: rules of proof theory • Derivability or provability • Properties • Soundness (derivability implies entailment) • Completeness (entailment implies derivability)

  16. 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 . 1t . 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 nth Cousin, m Times Removed?

  17. Wumpus World Revisited:Interacting with FOL KBs Adapted from slides by S. Russell, UC Berkeley

  18. Knowledge Base forThe Wumpus World Adapted from slides by S. Russell, UC Berkeley

  19. Deducing Hidden Properties Adapted from slides by S. Russell, UC Berkeley

  20. Keeping Track of Change:Situation Calculus Adapted from slides by S. Russell, UC Berkeley

  21. Describing Actions [1]:Frame, Qualification, and Ramification Problems Adapted from slides by S. Russell, UC Berkeley

  22. Describing Actions [2]:Successor State Axioms Adapted from slides by S. Russell, UC Berkeley

  23. Summary Points • Previously: Logical Agents and Calculi • Logic in general: tools for KR, inference, planning • Propositional calculus: normal forms, sequent rules • Predicate logic • First-order logic (FOL) aka first-order predicate calculus (FOPC) • Today: FOL in Practice • FOL agents • Example: Wumpus World in FOL • Situation calculus • Frame problem and variants (see R&N sidebar) • Representational vs. inferential frame problems • Qualification problem: “what if?” • Ramification problem: “what else?” (side effects) • Successor-state axioms • Thursday: FOL Knowledge Bases (Chapter 8, R&N), Sequent Rules for FOL

  24. Terminology • Logical Languages • Propositional logic • Predicates, terms, functions, atoms (atomic sentences / atomic WFFs), WFFs • First-order logic (FOL, FOPC): universal and existentialquantification • Properties of Knowledge Bases (KBs) • Satisfiability and validity • Entailment and provability • Properties of Proof Systems: Soundness and Completeness • Normal Forms: CNF, DNF, Horn; Clauses vs. Terms • Situation Calculus • Frame, Ramification, Qualification Problems • Successor-State Axiomatization

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