exercise12sol - CSE541 EXERCISE 12 Chapters 10, 11, 12 Read...

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Unformatted text preview: CSE541 EXERCISE 12 Chapters 10, 11, 12 Read and learn all examples and exercises in the chapters as well! QUESTION 1 Let GL be the Gentzen style proof system for classical logic defined in chapter 11. Prove, by constructing a proper decomposition tree that (1) GL (( a b ) ( b a )). Solution: By definition we have that GL (( a b ) ( b a )) if and only if GL- (( a b ) ( b a )) . We construct the decomposition tree for- A as follows. T A- (( a b ) ( b a )) | ( ) ( a b )- ( b a ) | ( ) b, ( a b )- a | ( ) ( a b )- b,a ^ ( - )- a,b,a | ( ) a- b,a axiom b- b,a axiom All leaves of the tree T A are axioms, hence we have found the proof of A in GL . 1 (2) Let GL be the Gentzen style proof system defined in chapter 11. Prove, by constructing a proper decom- position tree that 6 GL (( a b ) ( b a )) . Solution: Observe that for any formula A , its decomposition tree T A in GL is not unique. Hence when constructing decomposition trees we have to cover all possible cases. We construct the decomposition tree for- A as follows. T 1 A- (( a b ) ( b a )) | ( ) ( one choice ) ( a b )- ( b a ) | ( ) ( first of two choices ) b, ( a b )- a | ( ) ( one choice ) ( a b )- b,a ^ ( - ) ( one choice )- a,b,a non- axiom b- b,a axiom The tree contains a non- axiom leaf- a,b,a , hence it is not a proof of (( a b ) ( b a )) in GL . We have only one more tree to construct. Here it is. T 2 A- (( a b ) ( b a )) | ( ) 2 ( one choice ) ( a b )- ( b a ) ^ ( - ) ( second of two choices )- ( b a ) ,a (- ) ( one choice ) b- a,a | ( ) ( one choice )- b,a,a non- axiom b- ( b a ) | ( ) ( one choice ) b, b- a | ( ) ( one choice ) b- b,a axiom All possible trees end with an non-axiom leave whet proves that 6 GL (( a b ) ( b a )) . QUESTION 2 Show that tree below do not constitute a proof in GL defined in chapter 10? Provide a correct proof....
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exercise12sol - CSE541 EXERCISE 12 Chapters 10, 11, 12 Read...

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