Lecture08 - CS440/ECE448: Intro to Articial Intelligence!...

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Lecture 8: First-order predicate logic Prof. Julia Hockenmaier juliahmr@illinois.edu http://cs.illinois.edu/fa11/cs440 CS440/ECE448: Intro to ArtiFcial Intelligence

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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 Full resolution: p 1 ∨…∨ p i …∨ p n q 1 ¬ p i q m ──────────────────────────────── p 1 p n q 1 q m Final step: factoring (remove any duplicate literals from the result A A A)

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Proof by contradiction How do we prove that α β ? α entails β (‘ α β ’) iff α ¬ β not satisfable. Proof: α ¬ β not satisfable iff ¬ ( α ¬ β ) Assume ¬ ( α ¬ β ). ¬ α β ) α β . Thus, ¬ ( α ¬ β ) α β .
A resolution algorithm Goal: prove α β by showing that α ¬ β 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

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function PLresolution( α , β ) input: formula α , // knowledge base formula β // query clauses := CNF( α ¬ β ) 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
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:

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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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Lecture08 - CS440/ECE448: Intro to Articial Intelligence!...

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