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Unformatted text preview: CHAPTER 11: Automated Proof Systems (3) RS: Counter Models Generated by Decomposition Trees RS: Proof of COMPLETENESS THEOREM 1 Countermodel generated by the decompo sition tree. Example: Given a formula A : (( a b ) c ) ( a c )) and its decomposition tree T A . ((( a b ) c ) ( a c ))  ( ) (( a b ) c ) , ( a c ) ^ ( ) ( a b ) , ( a c )  ( ) a,b, ( a c )  ( ) a,b, a,c c, ( a c )  ( ) c, a,c 2 Consider a nonaxiom leaf: a,b, a,c Let v be any variable assignment v : V AR { T,F } such that it makes this nonaxiom leaf FALSE, i.e. we put v ( a ) = T,v ( b ) = F,v ( c ) = F. Obviously, we have that v * ( a,b, a,c ) = T F T F = F. Moreover, all our rules of inference are sound (to be proven formally in the next section). Rules soundness means that if one of pre misses of a rule is FALSE, so is the con clusion. 3 Hence, the soundness of the rules proves (by induction on the degree of sequences T A ) that v , as defined above falsifies all sequences on the branch of T A that ends with the nonaxiom leaf a,b, a,c . In particular, the formula A is on this branch, hence v * ((( a b ) c ) ( a c )) = F and v is a countermodel for A ....
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This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Bachmair,L
 Computer Science

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