Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Lecture 15 of 41 Friday 24 September 2004 William H. Hsu Department of Computing and Information Sciences, KSU Reading: Wikipedia entry on Ontology (CS): Rest of Chapter 8, , Russell and Norvig 2e More First-Order Logic Basics: Backward Chaining, Resolution Preliminaries

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Lecture Outline Today’s Reading –Chapter 8, Russell and Norvig –Recommended references: Nilsson and Genesereth (excerpt of Chapter 5 online) Next Week’s Reading: Chapters 9-10, R&N Previously: Introduction to Propositional and First-Order Logic –Monday (20 Sep 2004) First-order logic (FOL): predicates, functions, quantifiers Sequent rules, proof by refutation –Wednesday (22 Sep 2004) Forward Chaining with Modus Ponens Ontology, History of Logic, Russell’s Paradox Unification, Logic Programming Basics Today: Backward Chaining, Resolution Preliminaries, A Look Ahead Next Week: Resolution, Clausal Form (CNF), Decidability of SAT

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence In-Class Discussion: Problem Set 2

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

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

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

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

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

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

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

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Question: How Does This Relate to Proof by Refutation? Answer –Suppose ¬Query, For The Sake Of Contradiction (FTSOC) –Attempt to prove that KB  ¬Query ├  Adapted from slides by S. Russell, UC Berkeley Review: Backward Chaining

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

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

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Fun with Sentences: Family Feud Adapted from slides by S. Russell, UC Berkeley Brothers are Siblings –  x, y. Brother (x, y)  Sibling (x, y) Siblings (i.e., Sibling Relationships) are Reflexive –  x, y. Sibling (x, y)  Sibling (y, x) One’s Mother is One’s Female Parent –  x, y. Mother (x, y)  Female (x)  Parent (x, y) A First Cousin Is A Child of A Parent’s Sibling –  x, y. First-Cousin (x, y)   p, ps. Parent (p, x)  Sibling (p, ps)  Parent (ps, y)

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Adapted from slides by S. Russell, UC Berkeley Conjunctive Normal (aka Clausal) Form [1]: Conversion (R&N)

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Adapted from slides by S. Russell, UC Berkeley Conjunctive Normal (aka Clausal) Form [2]: Conversion (Nilsson) and Mnemonic Implications Out Negations Out Standardize Variables Apart Existentials Out (Skolemize) Universals Made Implicit Distribute And Over Or (i.e., Disjunctions In) Operators Out Rename Variables A Memonic for Star Trek: The Next Generation Fans Captain Picard: I’ll Notify Spock’s Eminent Underground Dissidents On Romulus I’ll Notify Sarek’s Eminent Underground Descendant On Romulus

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

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

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

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

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Universe of Decision Problems Given: KB,  Decide: ¬ ( KB   )? (Is  not valid?) Procedure: Test whether KB  {  } , answer yes if it does not Recursive Enumerable Languages (RE) Given: KB,  Decide: KB ├  ? (Is  valid?) Procedure: Test whether KB  {¬  } , answer yes if it does Recursive Languages (REC) First-Order Satisfiability and Validity: Undecidability and Semi-Decidability

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Summary Points Previously: Logical Agents and Calculi, FOL in Practice Today: Resolution Theorem Proving –Conjunctive Normal Form (clausal form) –Inference rule Single-resolvent form General form –Proof procedure: refutation –Decidability properties FOL-SAT FOL-NOT-SAT (language of unsatisfiable sentences; complement of FOL-SAT) FOL-VALID FOL-NOT-VALID Next Week –More Prolog –Implementing unification

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Terminology Properties of Knowledge Bases (KBs) –Satisfiability and validity –Entailment and provability Properties of Proof Systems –Soundness and completeness –Decidability, semi-decidability, undecidability Normal Forms: CNF, DNF, Horn; Clauses vs. Terms Resolution Refutation Satisfiability, Validity Unification