Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Lecture 12 Friday, 17 September 2004 William H. Hsu Department of Computing and Information Sciences, KSU Reading: Sections 7.5 – 7.10, Russell and Norvig 2e First-Order Logic (FOL) Review

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence 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)

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Review: Simple Knowledge-Based Agent Adapted from slides by S. Russell, UC Berkeley Chapter 7 R&N 2e

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence 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

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Adapted from slides by S. Russell, UC Berkeley Validity and Satisfiability

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Adapted from slides by S. Russell, UC Berkeley Proof Methods

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Adapted from slides by S. Russell, UC Berkeley Inference (Sequent) Rules for Propositional Logic

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Adapted from slides by S. Russell, UC Berkeley Logical Agents: Taking Stock

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Adapted from slides by S. Russell, UC Berkeley 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

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Adapted from slides by S. Russell, UC Berkeley Syntax of FOL: Basic Elements

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Adapted from slides by S. Russell, UC Berkeley FOL: Atomic Sentences (Atomic Well-Formed Formulae)

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Equality Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence “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? Jigsaw Exercise [1]: First-Order Logic Sentences

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Jigsaw Exercise [2]: First-Order Logic Sentences

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence 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)

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence 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?

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Wumpus World Revisited: Interacting with FOL KBs Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Knowledge Base for The Wumpus World Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Deducing Hidden Properties Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Keeping Track of Change: Situation Calculus Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Describing Actions [1]: Frame, Qualification, and Ramification Problems Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Adapted from slides by S. Russell, UC Berkeley Describing Actions [2]: Successor State Axioms

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence 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

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence 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