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Unformatted text preview: 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 Hilbertstyle 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 Hilbertstyle systems. The first idea of this type was presented by G. Gentzen in 1934. 1 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 RasiowaSikorski method. The Original Gentzen system is easily adopted to a complete system foir the Intuitionistic Logic and will be presented in Chapter 12. 2 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 , , , with indices if necessary. Meaning of Sequences: the intuitive mean ing of a sequence 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 true. 3 For any sequence F * , = A 1 ,A 2 ,...,A n we define = 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 * () = v * ( ) = v * ( A 1 ) v * ( A 2 ) ... v * ( A n ) . 4 Model The sequence is said to be sat isfiable if there is a truth assignment v : V AR { T,F } such that v * () = T . Such a truth assignment v is called a model for . Counter Model The sequence is said to be falsifiable if there is a truth assignment v , such that v * () = F . Such a truth assignment v is also called a countermodel for . 5 Tautology The sequence is said to be a tautology if v * () = T for all truth assign ments v : V AR { T,F } . Example Let be a sequence a, ( b a ) , b, ( b a ) . The truth assignment v for which v ( a ) = F and v ( b ) = T falsifies , i.e. is a counter model for , as shows the following com putation....
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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.
 Spring '08
 Bachmair,L
 Computer Science

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