Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Lecture 12 of 42 William H. Hsu Department of Computing and Information Sciences, KSU KSOL course page: Course web site: Instructor home page: Reading for Next Class: Section 8.3 – 8.4, p , Russell & Norvig 2 nd edition Handout, Nilsson & Genesereth, Logical Foundations of Artificial Intelligence Intro to First-Order Logic: Syntax and Semantics

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Lecture Outline Reading for Next Class: (p ), 9.1 (p ), R&N 2 e Last Class: Propositional Logic, Sections (p ), R&N 2 e  Properties of sentences (and sets of sentences, aka knowledge bases)  entailment  provability/derivability  validity: truth in all models (aka tautological truth)  satisfiability: truth in some models  Properties of proof rules  soundness: KB ⊢ i α  KB ⊨ α (can prove only true sentences)  completeness: KB ⊨ α  KB ⊢ i α (can prove all true sentences) Still to Cover in Chapter 7: Resolution, Conjunctive Normal Form (CNF) Today: Intro to First-Order Logic, Sections (p ), R&N 2 e  Elements of logic: ontology and epistemology  Resolution theorem proving  First-order predicate calculus (FOPC) aka first order logic (FOL) Coming Week: Propositional and First-Order Logic (Ch. 8 – 9)

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence © 2004 S. Russell & P. Norvig. Reused with permission. Chapter 7 Concluded

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence © 2004 S. Russell & P. Norvig. Reused with permission. Inference: Review

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence © 2004 S. Russell & P. Norvig. Reused with permission. Validity and Satisfiability: Review

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Forward Chaining Example: Review Adapted from slides © 2004 S. Russell & P. Norvig. Reused with permission n: number of antecedents (LHS conjuncts) still unmatched

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Backward Chaining Example: Review © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Forward vs. Backward Chaining: Review © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Resolution [1]: Propositional Sequent Rule Based on slide © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Resolution [2]: Conversion to Conjunctive Normal Form (CNF) Based on slide © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Resolution [3]: Algorithm Based on slide © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Resolution [4]: Example Based on slide © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Chapter 7: Summary © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Chapter 8: Overview © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Propositional Logic: Pros and Cons Based on slide © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence First-Order Logic (FOL) Based on slide © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Logics in General: Ontological and Epistemic Aspects Adapted from slide © 2004 S. Russell & P. Norvig. Reused with permission. Ontological commitment – what entities, relationships, and facts exist in world and can be reasoned about Epistemic commitment – what agents can know about the world

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Syntax of FOL: Basic Elements © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Atomic Sentences (aka Atoms, aka Atomic WFFs) Adapted from slide © 2004 S. Russell & P. Norvig. Reused with permission. Atomic sentence – smallest unit of a logic (aka “atom”, “atomic well-formed formula (atomic WFF)”

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Complex Sentences © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Truth in First-Order Logic © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Models for FOL: Example © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Models for FOL: Example © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Models for FOL: Lots! © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Universal Quantification [1]: Definition Based on slide © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Universal Quantification [2]: Common Mistake to Avoid Based on slide © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Existential Quantification [1]: Definition Based on slide © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Existential Quantification [2]: Common Mistake to Avoid Based on slide © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Properties of Quantifiers © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Fun With Sentences Adapted from slides © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Equality © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Interacting with FOL KBs © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Knowledge Base for Wumpus World Based on slide © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Deducing Hidden Properties © 2004 S. Russell & P. Norvig. Reused with permission.

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Terminology First-Order Logic (FOL) aka First-Order Predicate Calculus (FOPC)  Components  Semantics (meaning, denotation): objects, functions, relations  Syntax: constants, variables, terms, predicates  Properties of sentences (and sets of sentences, aka knowledge bases)  entailment  provability/derivability  validity: truth in all models (aka tautological truth)  satisfiability: truth in some models  Properties of proof rules  soundness: KB ⊢ i α  KB ⊨ α (can prove only true sentences)  completeness: KB ⊨ α  KB ⊢ i α (can prove all true sentences) Conjunctive Normal Form (CNF) Universal Quantification (“For All”) Existential Quantification (“Exists”)

Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Last Class: Overview of Knowledge Representation (KR) and Logic  Representations covered in this course, by ontology and epistemology  Propositional calculus (aka propositional logic)  Syntax and semantics  Relationship to Boolean algebra  Properties Propositional Resolution Elements of Logics – Ontology, Epistemology Today: First-Order Logic (FOL) aka FOPC  Components: syntax, semantics  Sentences: entailment vs. provability/derivability, validity vs. satisfiability  Soundness and completeness  Properties of proof rules  soundness: KB ⊢ i α  KB ⊨ α (can prove only true sentences)  completeness: KB ⊨ α  KB ⊢ i α (can prove all true sentences) Next: First-Order Resolution Summary Points