Lecture08HO - CS440/ECE448: Intro to Articial Intelligence!...

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Lecture 8: First-order predicate logic Prof. Julia Hockenmaier [email protected] http://cs.illinois.edu/fa11/cs440 CS440/ECE448: Intro to ArtiFcial Intelligence Back to resolution in propositional logic The resolution rule Unit resolution: p 1 ∨… p i-1 p i p i+1 p n ¬ p i !!!!!!!!!!!!!!!!!!!!!!!!!!!! p 1 p i-1 p i+1 p n ±ull resolution: p 1 ∨…∨ p i …∨ p n q 1 ¬ p i q m !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! p 1 p n q 1 q m ±inal step: factoring (remove any duplicate literals from the result A A A) Proof by contradiction How do we prove that " # ? " entails # (‘ " # ’) iff " ¬ # not satisFable. Proof: " ¬ # not satisFable iff ¬ ( " ¬ # ) Assume ¬ ( " ¬ # ). ¬ " # ) " ! # . Thus, ¬ ( " ¬ # ) " ! # .
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A resolution algorithm Goal: prove " # by showing that " ¬ # is not satisfable ( false ) Observation: Resolution derives a contradiction ( false ) iF it derives the empty clause: p i ¬ p i !!!!!!! 5 CS440/ECE448: Intro AI function PLresolution( " , # ) input: formula " , // knowledge base formula # // query clauses := CNF( " ¬ # ) new := {} while true: for each c1, c2 in clauses do resolvents := resolve( c1, c2) if in resolvents then return true; new := new ˫ resolvents if new clauses then return false; clauses := clauses ˫ new 6 CS440/ECE448: Intro AI Completeness of Resolution Resolution closure RC(S): The set oF all clauses that can be derived by resolution From a set oF clauses S. IF S is fnite, RC(S) is fnite. Ground resolution theorem: IF RC(S) contains , S is not satisfable. IF RC(S) does not contain
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This note was uploaded on 10/13/2011 for the course CS 440 taught by Professor Levinson,s during the Spring '08 term at University of Illinois, Urbana Champaign.

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Lecture08HO - CS440/ECE448: Intro to Articial Intelligence!...

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