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11slides(1) - Chapter 11 Automated Proof Systems(1 SYSTEM...

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Chapter 11: Automated Proof Systems (1) SYSTEM RS OVERVIEW Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. Automated systems are less intuitive then the Hilbert-style systems, but they will al- low us to give an effective automatic proce- dure for proof search, what was impossible in a case of the Hilbert-style systems. The first idea of this type was presented by G. Gentzen in 1934. 1
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PART 1: RS SYSTEM RS proof system presented here is due to H. Rasiowa and R. Sikorski and appeared for the first time in 1961. It extends eas- ily to Predicate Logic and admits a CON- STRUCTIVE proof of Completeness The- orem (first given by Rasiowa- Sikorski). PART 2: GENTZEN SYSTEM We present two Gentzen Systems; a modern version and the original version. BOTH ex- tend easily to Predicate Logic and admit a CONSTRUCTIVE proof of Completeness Theorem via Rasiowa-Sikorski method. The Original Gentzen system is easily adopted to a complete system foir the Intuitionistic Logic and will be presented in Chapter 12. 2
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Language of RS is L = L , , , ∩} . The rules of inference of our system RS operate on finite sequences of formulas . Set of expressions E = F * . Notation: elements of E are finite sequences of formulas and we denote them by Γ , Δ , Σ , with indices if necessary. Meaning of Sequences: the intuitive mean- ing of a sequence Γ ∈ F * is that the truth assignment v makes it true if and only if it makes the formula of the form of the disjunction of all formulas of Γ true. 3
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For any sequence Γ ∈ F * , Γ = A 1 , A 2 , ..., A n we define δ Γ = A 1 A 2 ... A n . Formal Semantics for RS Let v : V AR -→ { T, F } be a truth assignment, v * its classi- cal semantics extension to the set of for- mulas F . We formally extend v to the set F * of all finite sequences of F as follows. v * (Γ) = v * ( δ Γ ) = v * ( A 1 ) v * ( A 2 ) ... v * ( A n ) . 4
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Model The sequence Γ is said to be sat- isfiable if there is a truth assignment v : V AR -→ { T, F } such that v * (Γ) = T . Such a truth assignment v is called a model for Γ. Counter- Model The sequence Γ is said to be falsifiable if there is a truth assignment v , such that v * (Γ) = F . Such a truth assignment v is also called a counter-model for Γ. 5
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Tautology The sequence Γ is said to be a tautology if v * (Γ) = T for all truth assign- ments v : V AR -→ { T, F } . Example Let Γ be a sequence a, ( b a ) , ¬ b, ( b a ) . The truth assignment v for which v ( a ) = F and v ( b ) = T falsifies Γ, i.e. is a counter- model for Γ, as shows the following com- putation.
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