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CS336F104 - Goal CS336 Lecture4 Furtherunderstandwp...

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9/6/10 1 CS336 Lecture 4 Weakest Preconditions Goal Further understand wp How to prove an IF statement correct Why should we care? These are the techniques we will use to verify program then use these methods for developing programs via goal oriented programming. Hoare’s Triple {Q} S {R} S is the command sequence, the predicate Q is known as S’s pre condition , and R is known as its post condition . A program holds (is verified or proven correct) if we begin in any state satisfying Q,after we execute the commands, we end up in a state satisfying R in a finite amount of time. Our Approach wp(S,R) denotes the weakest (i.e. minimum) precondition necessary for the postcondition to be satisfied. We started with basic commands and look at the weakest precondition for these. Then we looked at how to deal with sequences of these commands. Skip The “do-nothin g” command (a “place- holder”): skip Definition . wp (“ skip ”, R) = R
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9/6/10 2 Abort The “fatal-error ” command: abort Definition . wp (“ abort ”, R) = F Assignment The Assignment command: ‘ := Definition . That is, a textual substitution is made for x in R by replacing x with e. Sequential composition of commands ; Definition . wp (“S1 ; S2”, R) = wp (S1, wp (S2,R)) The Alternative Command if B then S0 else S1 , is if B S0 ¬ B S1 fi wp(“IF”,R)= (B wp(S0,R)) ( ¬ B wp(S1,R)) Example If x<0 then x:=0 {T} if x<0 x:=0 x≥0 Skip fi { x≥0 } Our approach: show T wp(“IF”, x≥0 ) Example {T} if x<0 x:=0 x≥0 Skip fi { x≥0 } wp(“IF”,R)= ( x<0 wp(“ x:=0” , x≥0 )) ( x≥0 wp(“Skip”, x≥0 ))
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9/6/10 3 Example wp (“IF”,R) = <def> ( x<0 wp(“ x:=0” , x≥0 )) ( x≥0 wp(“Skip”, x≥0 )) <wp := and skip > ( x<0 0 0 ) ( x≥0 x≥0 ) <arith; identity> ( x<0 T ) T < ; -simp > T Now since T T, Q wp(“If”,R) The Alternative Command The general form (known as “ IF ”) : if B0 S0 B1 S1 B2 S2 . . . B n Sn fi General definition for WP of IF (—assumes all guards are defined). wp(“ IF” ,R) = ( i|1 i n: B i ) ( i| 1 i n: B i wp(S i ,R)) If we maintain our approach First, we would find the WP of IF and then show Q wp(IF,R) … Or Q ( i|1 i n: B i ) ( i| 1 i n: B i wp(S i ,R)) If we maintain our approach {Q} if B 0 S 0 B 1 S 1 fi {R}
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