exercise12sol

# exercise12sol - CSE541 EXERCISE 12 12 Read and learn all...

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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 12 Read and learn all...

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