12(2GentzenInt)

12(2GentzenInt) - Chapter 12: Gentzen Sequent Calculus for...

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Chapter 12: Gentzen Sequent Calculus for Intuitionistic Logic PART 2: Examples of proof search decom- position trees in LI Search for proofs in LI is a much more com- plicated process then the one in classical logic systems RS or GL . Proof search procedure consists of building the decomposition trees. Remark 1: in RS the decomposition tree T A of any formula A is always unique. 1
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Remark 2: in GL the ”blind search” defines, for any formula A a finite number of de- composition trees, but it can be proved that the search can be reduced to examin- ing only one of them, due to the absence of structural rules. Remark 3: In LI the structural rules play a vi- tal role in the proof construction and hence, in the proof search. The fact that a given decomposition tree ends with an axiom leaf does not always imply that the proof does not exist. It might only imply that our search strategy was not good. The problem of deciding whether a given for- mula A does, or does not have a proof in LI becomes more complex then in the case of Gentzen system for classical logic. 2
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Example 1 Determine whether LI -→ A for A = (( ¬ A ∩ ¬ B ) ⇒ ¬ ( A B )). If we find a decomposition tree such that all its leaves are axiom, we have a proof. If all possible decomposition trees have a non- axiom leaf, proof of A in LI does not exist. 3
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Consider the following decomposition tree T1 A -→ (( ¬ A ∩ ¬ B ) ( ¬ ( A B )) | ( -→⇒ ) ( ¬ A ∩ ¬ B ) -→ ¬ ( A B ) | ( -→ ¬ ) ( A B ) , ( ¬ A ∩ ¬ B ) -→ | ( exch -→ ) ( ¬ A ∩ ¬ B ) , ( A B ) -→ | ( ∩ -→ ) ¬ A, ¬ B, ( A B ) -→ | ( ¬ -→ ) ¬ B, ( A B ) -→ A | ( -→ weak ) ¬ B ( A B ) -→ | ( ¬ -→ ) ( A B ) -→ B ^ ( ∪ -→ ) A -→ B non - axiom B -→ B axiom 4
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T1 A has a non-axiom leaf, so it does not constitute a proof in LI . Observe
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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.

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12(2GentzenInt) - Chapter 12: Gentzen Sequent Calculus for...

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