##### Document information

**Lecture 15 of 41
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**More First-Order Logic Basics:
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**Backward Chaining, Resolution Preliminaries
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**Lecture Outline
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• **Today’s Reading
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• **Next Week’s Reading: Chapters 9-10, R&N
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• **Previously: Introduction to Propositional and First-Order Logic
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– **Monday (20 Sep 2004)
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• **First-order logic (FOL): predicates, functions, quantifiers
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• **Sequent rules, proof by refutation
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– **Wednesday (22 Sep 2004)
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• **Forward Chaining with Modus Ponens
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• **Ontology, History of Logic, Russell’s Paradox
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• **Unification, Logic Programming Basics
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• **Today: Backward Chaining, Resolution Preliminaries, A Look Ahead
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• **Next Week: Resolution, Clausal Form (CNF), Decidability of SAT
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**In-Class Discussion:
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**Problem Set 2
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**Unification:
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**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:
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**Forward Chaining
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**Backward Chaining
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**Example:
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**Backward Chaining
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• **Question: How Does This Relate to Proof by Refutation?
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• **Answer
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– **Suppose ¬Query, For The Sake Of Contradiction (FTSOC)
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– **Attempt to prove that KB **** ¬Query ├ **

**Review:
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**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:
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**Family Feud
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• **Brothers are Siblings
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– ** x, y . Brother (x, y) **

*Sibling*(*x*,*y*)• **Siblings (i.e., Sibling Relationships) are Reflexive
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– ** x, y . Sibling (x, y) **

*Sibling*(*y*,*x*)• **One’s Mother is One’s Female Parent
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– ** x, y . Mother (x, y) **

****

*Female*(*x*)

*Parent*(*x*,*y*)• **A First Cousin Is A Child of A Parent’s Sibling
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– ** x, y . First-Cousin (x, y) **

** p, ps . Parent (p, x) **

****

*Sibling*(*p*,*ps*)

*Parent*(*ps*,*y*)Docsity.com

**Conjunctive Normal ( aka Clausal) Form [1]:
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**Conversion (R&N)
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**Conjunctive Normal ( aka Clausal) Form [2]:
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**Conversion (Nilsson) and Mnemonic
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• **Implications Out
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• **Negations Out
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• **Standardize Variables Apart
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• **Existentials Out (Skolemize)
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• **Universals Made Implicit
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• **Distribute And Over Or (i.e., Disjunctions In)
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• **Operators Out
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• **Rename Variables
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• **A Memonic for Star Trek: The Next Generation Fans
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•**Captain Picard:
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•**I’ll Notify Spock’s Eminent Underground Dissidents On Romulus
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•**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:
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**Resolution Proof
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**Logic Programming vs. Imperative Programming
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**Universe of Decision Problems
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**Given: KB, ****
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**Decide: ¬(KB **** ****)? (Is **** not valid?)
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**Procedure: Test whether KB **** {****} **** ****, answer yes if it does not
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*VALIDH
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*SATH
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*SATVALIDVALIDSATSATVALID
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*LL
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*exercise) :(proof LL
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*Decidable-Semi :Complexity
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*:Problem Decision
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*)resolution n(refutatio LLproof) (directLLLL
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*Problems Dual
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**Recursive Enumerable
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**Languages (RE)
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**Given: KB, ****
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**Decide: KB ├ ****? (Is **** valid?)
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**Procedure: Test whether KB **** {¬****} **** **** , answer yes if it does
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*VALIDd
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*SATd
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*SATVALID
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*LL
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*Theorem)ess Incompletn Firsts Goedel'see - exercise :(proof LL
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*RE) (e Undecidabl :Complexity
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*:)ngsproblemEntscheidus (Hilbert'Problem Decision
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*LL
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*Problems Dual
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**Recursive
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**Languages
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**(REC)
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*HL
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*VALIDL
VALIDL
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* LSAT
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*SATL L L complem.
under
closure
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**First-Order Satisfiability and Validity:
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**Undecidability and Semi-Decidability
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**Summary Points
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• **Previously: Logical Agents and Calculi, FOL in Practice
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• **Today: Resolution Theorem Proving
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– **Conjunctive Normal Form (clausal form)
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– **Inference rule
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• **Single-resolvent form
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• **General form
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– **Proof procedure: refutation
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– **Decidability properties
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• **FOL-SAT
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• **FOL-NOT-SAT (language of unsatisfiable sentences; complement of FOL-SAT)
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• **FOL-VALID
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• **FOL-NOT-VALID
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• **Next Week
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– **More Prolog
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– **Implementing unification
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**Terminology
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• **Properties of Knowledge Bases (KBs)
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– **Satisfiability and validity
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– **Entailment and provability
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• **Properties of Proof Systems
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– **Soundness and completeness
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– **Decidability, semi-decidability, undecidability
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• **Normal Forms: CNF, DNF, Horn; Clauses vs. Terms
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• **Resolution
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• **Refutation
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• **Satisfiability, Validity
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• **Unification
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