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Functional Verification II

Learn about correctness in program reading, writing, and validation, and explore sequencing correctness conditions. Find solutions for compound programs and the Axiom of Replacement in this comprehensive lecture series. Topics include correctness conditions, working correctness questions, decision statements, and more. Dive into examples and conditional function composition to enhance your understanding of functional verification in software testing and verification.

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Functional Verification II

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  1. Functional Verification II Software Testing and Verification Lecture Notes 22 Prepared by Stephen M. Thebaut, Ph.D. University of Florida

  2. Previously • Verifying correctness in program reading, writing, and validation • Complete and sufficient correctness • Compound programs and the Axiom of Replacement

  3. Topics: • Correctness conditions and working correctness questions: • sequencing • decision statements

  4. Sequencing Correctness Conditions • Suppose we wish to show f = [G; H] • First, hypothesize functions g, h and prove: g = [G] and h = [H] • By the Axiom of Replacement, the problem thenreduces to proving f = [g; h]

  5. Sequencing Correctness Conditions • Complete correctness condition for f = [g; h]: Prove: f = h o g • Working correctness question: Does f equal h composed with g? • Note: h o g(x) = h(g(x))

  6. Sequencing Example • Prove f = [P] where f = (x,y := y+2,y) and P is: x := y+2; y := x-2 • Proof: Let G be x := y+2 and H be y := x-2. Then, by observation, g =(x,y := y+2,y) and h =(x,y := x,x-2).

  7. Sequencing Example (cont’d) • Therefore, by the Axiom of Replacement, it is sufficient to show: f = (x,y := y+2,y) = [g; h] Does f equal h composed with g? h o g = (x,y := x,x-2) o (x,y := y+2,y) = (x,y := y+2,(y+2)-2) = (x,y := y+2,y) = f √

  8. Conditional Function Composition Suppose g = (x,y := 3,x-1) and h = (y>0  x,y := x+1,-y | y≤0  x,y := x,y). What is h o g... ? = (y>0  x,y := x+1,-y | y≤0  x,y := x,y) o (x,y := 3,x-1) = (x-1>0  x,y := 3+1,-(x-1) | x-1≤0  x,y := 3,x-1) = (x>1  x,y := 4,1-x | x≤1  x,y := 3,x-1)

  9. Conditional Function Composition Suppose g = (x,y := 3,x-1) and h = (y>0  x,y := x+1,-y | y≤0  x,y := x,y). What is g o h... ? = (x,y := 3,x-1) o (y>0  x,y := x+1,-y | y≤0  x,y := x,y) = (x,y := (y>0  3,(x+1)-1) | (y≤0  3,x-1)) = (y>0  x,y := 3,(x+1)-1) | y≤0  x,y := 3,x-1)) = (y>0  x,y := 3,x | y≤0  x,y := 3,x-1))

  10. if_then Correctness Conditions • Complete correctness conditions for f = [if p then G] (where g = [G] has already been shown): Prove: p (f = g) Л ¬p (f = I) • Working correctness questions: • When p is true, does f equal g? • When p is false, does f equal Identity?

  11. if_then Example • Prove f = [K] where f = (x := -|x|) and K is: if x>0 then x := x-2*x • Proof: Let G be x := x-2*x Then, by observation, g = (x := x-2x)

  12. if_then Example (cont’d) • Therefore, by the Axiom of Replacement, it is sufficient to show: f = (x := -|x|) = [if x>0 then x := x-2x] When p is true does f equal g? (x>0)  (f = (x := -x)) (x>0)  (g = (x := x-2x) √ = (x := -x)) When p is false does f equal identity? (x≤0)  (f = (x := x)) = I √

  13. if_then_else Correctness Conditions • Complete correctness conditions for f = [if p then G else H] (where g = [G] and h = [H] have already been shown): Prove: p (f = g) Л ¬p (f = h) • Working correctness questions: • When p is true, does f equal g? • When p is false, does f equal h?

  14. Exercise • Prove f = [A] where f = (x=17  x,y := 17,20 | true  x,y := x,-x) and A is: if x= 17 then y := x+3 else y := -x end_if_else

  15. Coming up next… • Iteration Recursion Lemma (!) • Termination predicate: term(f,P) • Correctness conditions for while_do statement • Correctness conditions for repeat_until statement

  16. Functional Verification II Software Testing and Verification Lecture Notes 22 Prepared by Stephen M. Thebaut, Ph.D. University of Florida

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