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11slides(2) - Chapter 11 Automated Proof Systems(2 RS...

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Chapter 11: Automated Proof Systems (2) RS: DECOMPOSITION TREES The process of searching for the proof of a formula A in RS consists of building a cer- tain tree, called a decomposition tree whose root is the formula A, nodes correspond to sequences which are conclusions of certain rules (and those rules are well defined at each step by the way the node is built), and leafs are axioms or are sequences of a non- axiom literals. 1
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We prove that each formula A (sequence Γ) generates its unique and finite decompo- sition tree, T A ( T Γ ). The tree constitutes the proof of A (Γ) in RS if all its leafs are axioms. If there is a leaf of T A ( T Γ ) that is not an axiom , the tree is not a proof, moreover, the proof of A does not exist . Before we give a proper definition of the proof search procedure by building a decomposi- tion tree we list few important observations about the structure of the rules of the sys- tem RS . 2
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Introduction of Connectives The rules of RS are defined in such a way that each of them introduces a new log- ical connective, or a negation of a con- nective to a sequence in its domain (rules ( ) , ( ) , ( )) or a negation of a new logical connective (rules ( ¬∪ ) , ( ¬∩ ) , ( ¬ ⇒ ) , ( ¬¬ )). The rule ( ) introduces a new connective to a sequence Γ 0 , A, B, Δ and it becomes, after the application of the rule, a sequence Γ 0 , ( A B ) , Δ. Hence a name for this rule is ( ). 3
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The rule ( ¬∪ ) introduces a negation of a con- nective, ¬∪ by combining sequences Γ 0 , ¬ A, Δ and Γ 0 , ¬ B, Δ into one sequence (conclu- sion of the rule) Γ 0 , ¬ ( A B ) , Δ. Hence a name for this rule is ( ¬∪ ). The same applies to all remaining rules of RS , hence their names say which connec- tive, or the negation of which connective has been introduced by the particular rule. 4
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Decomposition Rules Building decomposition tree (a proof search tree) consists of using the inference rules in an inverse order; we transform them into rules that transform a conclusion into its premisses. We call such rules the decomposition rules . Here are all of RS decomposition rules. 5
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Disjunction decomposition rules ( ) Γ 0 , ( A B ) , Δ Γ 0 , A, B, Δ , ( ¬∪ ) Γ 0 , ¬ ( A B ) , Δ Γ 0 , ¬ A, Δ : Γ 0 , ¬ B, Δ Conjunction decomposition rules ( ) Γ 0 , ( A B ) , Δ Γ 0 , A, Δ ; Γ 0 , B, Δ , ( ¬∩ ) Γ 0 , ¬ ( A B ) , Δ Γ 0 , ¬ A, ¬ B, Δ Implication decomposition rules ( ) Γ 0 , ( A B ) , Δ Γ 0 , ¬ A, B, Δ , ( ¬ ⇒ ) Γ 0 , ¬ ( A B ) , Δ Γ 0 , A, Δ : Γ 0 , ¬ B, Δ Negation decomposition rule ( ¬¬ ) Γ 0 , ¬¬ A, Δ Γ 0 , A, Δ where Γ 0 ∈ F 0 * , Δ ∈ F * , A, B ∈ F . 6
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We write the decomposition rules in a visual tree form as follows.
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