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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 CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences 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) CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Review: Simple Knowledge-Based Agent Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Chapter 7 R&N 2e Kansas State University Department of Computing and Information Sciences 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 CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Validity and Satisfiability Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Proof Methods Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Inference (Sequent) Rules for Propositional Logic Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Logical Agents: Taking Stock Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences 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 CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Syntax of FOL: Basic Elements Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences FOL: Atomic Sentences (Atomic Well-Formed Formulae) Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Equality Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences 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 . 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 nth Cousin, m Times Removed? CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Jigsaw Exercise [2]: First-Order Logic Sentences CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences 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) CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences 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 nth Cousin, m Times Removed? CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Wumpus World Revisited: Interacting with FOL KBs Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Knowledge Base for The Wumpus World Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Deducing Hidden Properties Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Keeping Track of Change: Situation Calculus Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Describing Actions [1]: Frame, Qualification, and Ramification Problems Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Describing Actions [2]: Successor State Axioms Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences 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 CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Terminology • Logical Languages – Propositional logic – Predicates, terms, functions, atoms (atomic sentences / atomic WFFs), WFFs – First-order logic (FOL, FOPC): universal and existential quantification • 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 CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences