Lecture 14 of 41 First-Order Logic and Theorem Proving

Kansas State UniversityDepartment of Computing and Information SciencesCIS 730: Introduction to Artificial Intelligence

Lecture 14 of 41Lecture 14 of 41

First-Order Logicand Theorem Proving

Wednesday, 22 September 2004

William H. HsuDepartment of Computing and Information Sciences, KSU

http://www.kddresearch.orghttp://www.cis.ksu.edu/~bhsu

Reading:Sections 8.1-8.3, Russell and Norvig 2e

Review: Chapter 6, R&N 2e


Lecture OutlineLecture 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

– Last Friday (17 Sep 2004)• FOL agents, issues: frame, ramification, qualification problems• Solutions: situation calculus, circumscription by successor state axioms

– Monday (20 Sep 2004)• First-order logic (FOL): predicates, functions, quantifiers• Sequent rules, proof by refutation

• Today: FOL Knowledge Bases and Theorem Proving– Forward Chaining with And-Introduction, Universal Elimination, Modus Ponens– Ontology, History of Logic, Russell’s Paradox– Unification, Logic Programming Basics

• Next Week: Resolution, Logic Programming, Decidability of SAT


InIn--Class Discussion:Class Discussion:Problem Set 2Problem Set 2


Taking Stock:Taking Stock:FOL InferenceFOL Inference

• Previously: Logical Agents and Calculi• Review: FOL in Practice

– Agent “toy” world: 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• FOL Knowledge Bases• FOL Inference

– Proofs– Pattern-matching: unification– Theorem-proving as search

• Generalized Modus Ponens (GMP)• Forward Chaining and Backward Chaining


Automated Deduction (Chapters 8Automated Deduction (Chapters 8--10 R&N)10 R&N)

Adapted from slides by S. Russell, UC Berkeley


Example ProofExample Proof

• ???• Apply Sequent Rules to Generate New Assertions

• Modus Ponens And Introduction Universal Elimination




Search with Primitive Inference RulesSearch with Primitive Inference Rules



A Brief History of Reasoning:A Brief History of Reasoning:Chapter 8 End Notes, R&NChapter 8 End Notes, R&N


KKnowledge nowledge EEngineeringngineering

• KE: Process of– Choosing logical language (basis of KR)– Building KB– Implementing proof theory– Inferring new facts

• Analogy: Programming Languages / Software Engineering– Choosing programming language (basis of software engineering)– Writing program– Choosing / writing compiler– Running program

• Example Domains– Electronic circuits (Section 8.3 R&N)– Exercise

• Look up, read about protocol analysis• Find example and think about KE process for your project domain


OntologyOntology

• Ontology: “What Objects Exist and Are Symbolically Representable?”• Issue: Grouping Objects and Describing Families

– Grouping objects and describing families– Example: sets of sets

• Russell’s paradox: http://plato.stanford.edu/entries/russell-paradox/• (Four) responses: types, formalism, intuitionism, Zermelo-Fraenkel set theory

– Sidebar: natural kinds (p. 232)• Issue: Reasoning About Time

– Modal logics (CIS 301)– Interval logics (Section 8.4 R&N p. 238-241)

• Example Domains– Grocery shopping (Section 8.5 R&N); similar example in Winston 3e– Data models for knowledge discovery in databases (KDD)

• Data dictionaries• See grocery example, especially p. 249 - 252


Unification:Unification:Definitions and Idea SketchDefinitions and Idea Sketch




GGeneralized eneralized MModus odus PPonensonens



Soundness of GMPSoundness of GMP



Forward ChainingForward Chaining



Example:Example:Forward ChainingForward Chaining



Backward ChainingBackward Chaining



Example:Example:Backward ChainingBackward Chaining


• Question: How Does This Relate to Proof by Refutation?• Answer

– Suppose ¬Query, For The Sake Of Contradiction (FTSOC)

– Attempt to prove that KB ∧ ¬Query ⊥


Review:Review:Backward ChainingBackward Chaining



Completeness Completeness ReduxRedux



Completeness in FOLCompleteness in FOL



Resolution Inference RuleResolution Inference Rule



Digression:Digression:Decidability and Formal LanguagesDecidability and Formal Languages

• See: Hopcroft and Ullman 2e, Lewis and Papadimitriou 3e• Formal Languages (See: CIS 540, Other Automata Theory Course)

– Member of Turing hierarchy• Finite state automata: regular languages• Pushdown automata: context-free languages• Linear bounded automata: context-sensitive languages• Turing machines: recursive languages

– Recursive languages• ∃ computational model for decision problem, halts in finite number of steps• REC: set of all recursive languages• Example: finite searches (convert to decision problem of checking solution)• Closed under complementation (consequence?)

– Recursive enumerable but not recursive (RE - REC)– Not recursive (∉ RE)

• What Are FOL-VALID, FOL-NOT-SAT, FOL-SAT, FOL-NOT-VALID?


Summary PointsSummary Points

• Applications of Knowledge Bases (KBs) and Inference Systems• “Industrial Strength” KBs

– Building KBs• Knowledge Engineering (KE) and protocol analysis• Inductive Logic Programming (ILP) and other machine learning techniques

– Components• Ontologies• Fact and rule bases

– Using KBs• Systems of Sequent Rules: GMP/AI/UE, Resolution• Methodology of Inference

– Inference as search– Forward and backward chaining– Fan-in, fan-out


TerminologyTerminology

• Logical Languages: WFFs, Quantification• Properties of Knowledge Bases (KBs)

– Satisfiability and validity– Entailment and provability

• Properties of Proof Systems: Soundness and Completeness• Knowledge Bases in Practice

– Knowledge Engineering– Ontologies

• Sequent Rules– (Generalized) Modus Ponens– And-Introduction– Universal-Elimination

• Methodology of Inference– Forward and backward chaining– Fan-in, fan-out (wax on, wax off…)

Lecture 14 of 41 First-Order Logic and Theorem Proving

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