# Backward Chaining - Artificial Intelligence - Lecture Slides, Slides for Artificial Intelligence. West Bengal University of Animal and Fishery Sciences

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Some concept of Artificial Intelligence are Agents and Problem Solving, Autonomy, Programs, Classical and Modern Planning, First-Order Logic, Resolution Theorem Proving, Search Strategies, Structure Learning. Main points...
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CIS730-Lecture-15-20030924

Lecture 15 of 41

More First-Order Logic Basics:

Backward Chaining, Resolution Preliminaries

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Lecture Outline

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

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In-Class Discussion:

Problem Set 2

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Unification:

Definitions and Idea Sketch

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Generalized Modus Ponens

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Soundness of GMP

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Forward Chaining

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Example:

Forward Chaining

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Backward Chaining

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Example:

Backward Chaining

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Question: How Does This Relate to Proof by Refutation?

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

Attempt to prove that KB ¬Query

Review:

Backward Chaining

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Completeness Redux

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Completeness in FOL

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Resolution Inference Rule

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Fun with Sentences:

Family Feud

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)

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Conjunctive Normal (aka Clausal) Form [1]:

Conversion (R&N)

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Conjunctive Normal (aka Clausal) Form [2]:

Conversion (Nilsson) and Mnemonic

Implications Out

Negations Out

Standardize Variables Apart

Existentials Out (Skolemize)

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

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Skolemization

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Resolution Theorem Proving

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Example:

Resolution Proof

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Logic Programming vs. Imperative Programming

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Universe of Decision Problems

Given: KB,

Decide: ¬(KB )? (Is not valid?)

Procedure: Test whether KB {} , answer yes if it does not

VALIDH

SATH

SATVALIDVALIDSATSATVALID

LL

exercise) :(proof LL

Decidable-Semi :Complexity

:Problem Decision

)resolution n(refutatio LLproof) (directLLLL

Problems Dual



Recursive Enumerable

Languages (RE)

Given: KB,

Decide: KB ├ ? (Is valid?)

Procedure: Test whether KB } , answer yes if it does

VALIDd

SATd

SATVALID

LL

Theorem)ess Incompletn Firsts Goedel'see - exercise :(proof LL

RE) (e Undecidabl :Complexity

:)ngsproblemEntscheidus (Hilbert'Problem Decision

LL

Problems Dual

Recursive

Languages

(REC)

HL

VALIDL VALIDL

LSAT

SATL L L complem. under closure

 

First-Order Satisfiability and Validity:

Undecidability and Semi-Decidability

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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

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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

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