AxiomaticSemantics

# AxiomaticSemantics - COP4020 Programming Languages...

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COP4020 Programming Languages Introduction to Axiomatic Semantics Prof. Robert van Engelen

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COP4020 Spring 2011 2 6/16/11 Assertions and Preconditions ! Assertions are used by programmers to verify run-time execution " An assertion is a logical formula to verify the state of variables and data to ensure safe continuation " A failed assertion should stop the program " Assertions are placed by programmers explicitly in code assert (len>0); mean = sum/len; ! Preconditions state the necessary conditions for “safe” execution " Programmer decides which conditions apply at a program point " Testing too few does not make the program safe
COP4020 Spring 2011 3 6/16/11 Axiomatic Semantics and Preconditions ! What if we want to guarantee that the program always produces the correct output assuming that the initial assertion passes? ! Axiomatic semantics: axiomatic proof that a program produces a machine state described by a postcondition if the precondition on the initial state holds (initial assertion passes) ! Example: assert (len > 0); // given this passes mean = sum/len; assert (mean > 0); // is this always true? answer: no, cannot be proven can be proven when we change precondition to (len>0 && sum>0)

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COP4020 Spring 2011 4 6/16/11 Preconditions, Postconditions and Partial Correctness ! Triple notation : place conditions before and after a command C : { Precondition } C { Postcondition } ! We say that C is partially correct with respect to the < precondition , postcondition > specification, provided that " The command C is executed in a machine state that makes the precondition true " If the command terminates, then we guarantee that the resulting machine state makes the postcondition true ! Total correctness requires termination
COP4020 Spring 2011 5 6/16/11 Assignment Axiom ! If we view conditions in assertions as logical predicates, the assignment axiom can be stated { P ( E ) } V := E { P ( V ) } that is, if we state a property of V after the assignment, then the property must hold for expression E before the assignment ! We can use substitution to derive the precondition given a postcondition formula P : this is the assignment axiom : { P [ V ! E ] } V := E { P } where P [ V ! E ] denotes the substitution of V by E in P

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6 6/16/11 Examples for Assignments ! { k = 5 } k := k + 1 { k = 6 } ( k = 6)[ k ! k +1] " ( k +1 = 6) " ( k = 5) ! { j = 3 and k = 4 } j := j + k { j = 7 and k = 4 } ( j = 7 and k = 4)[ j ! j + k ] " ( j + k = 7 and k = 4) " ( j = 3 and k = 4) ! { true } x := 2 { x = 2 } ( x = 2)[ x ! 2] " (2 = 2) " (true) ! { a > 0 } a := a - 1 { a > 0 } ( a > 0)[ a ! a - 1] " ( a - 1 > 0) " ( a > 0) Assuming a is int ! !
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AxiomaticSemantics - COP4020 Programming Languages...

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