Functional Verification III Software Testing and Verification Lecture Notes 23 Prepared by Stephen M. Thebaut, Ph.D. University of Florida

Previously… Correctness conditions and working correctness questions: sequencing decision statements

Today’s Topics Iteration Recursion Lemma (IRL) Termination predicate: term(f,P) Correctness conditions for while_do statement Sufficient correctness conditions Correctness conditions for repeat_until statement Subgoal Induction

Iteration Recursion Lemma (IRL) The IRL reduces the verification of programs with loops to a question of termination and the verification of loop-free programs by converting iteration to recursion. For while loops, the Lemma states: f = [while p do g] = [if p then g;f end_if] (note recursion)

Iteration Recursion Lemma (cont’d) F p T F F F p p p g f = = = = T T T g g g;f F p T f g

Iteration Recursion Lemma (cont’d) Rather than verify directly that f is the program function of K = while p do g which can be very difficult, it is sufficient to prove that 1. K terminates for all X  D(f), and that 2. f is the program function of Q = if p then g;f end_if because [K] = [Q].

An important implication of the IRL Suppose for “input” X0 the while loop term- inates after n iterations with “output” Xn. Furthermore, let X1, X2, ..., Xn-1 be the in- termediate states generated by the loop. Then  0≤i<n, we know: p(Xi), Xi+1=g(Xi), and ¬p(Xn).

An important implication of the IRL (cont’d) As f = [while p do g] = [if p then g;f end_if], it follows that f(X0) = f(X1) = ... = f(Xn) = Xn More generally, after each iteration of the loop, the function value of the current state, X, must be the same as the function value of the initial state, X0. That is: f(X) = f(X0) We will revisit this observation in connection with Mill’s Invariant Status Theorem later.

Illustrative Example of IRL To further illustrate the fact that [while p do g] = [if p then g;f end_if] consider a concrete example... Let K = while y>0 do x,y := x+1,y−1 Claim: K is function equivalent to Q = if y>0 then x,y := x+1,y−1;k end_if where, by definition, k = [K]. p g p k o g

Illustrative Example of IRL (cont’d) Case (y>0): For K = while y>0 do x,y := x+1,y−1, the loop body executes y times before the predicate y>0 becomes false. By observation, then, the final value of x is x0+(1)y0 = x0+y0 and the final value of y is 0. Thus, (y>0) => k = (x,y := x+y,0) Also, note that when y=0 initially, k = I = (x,y := x,y) = (x,y := x+0,y) = (x,y := x+y,0) Therefore, (y≥0) => k = (x,y := x+y,0)

Illustrative Example of IRL (cont’d) Case (y>0): (cont’d) [Q] is a composition of two functions, i.e., k o g, and may be determined by direct substitution. For y>0 initially, y will be greater than OR EQUAL to 0 after executing the loop body, but since we know (y≥0) => k = (x,y := x+y,0), we have [Q] = (x,y := x+y,0) o (x,y := x+1,y−1) = (x,y := (x+1)+(y−1),0) = (x,y := x+y,0) = k (the function computed by K) Thus, [Q] = [K] when y>0.

Illustrative Example of IRL (cont’d) Case (y≤0): Since the predicate (y>0) fails, both K and Q do nothing, and are therefore equivalent. Thus, [Q] = I = [K] when y≤0. Therefore, K is function equivalent to Q.

Termination Predicate The correctness of a looping program P depends, in part, on termination. Consideration is limited to programs whose termination can be established and the following predicate is defined: term(f,P)  ‘‘P terminates for every initial state X  D(f)’’

Before we continue… Take out a piece of paper and a pen/pencil. Without looking back in the lecture notes, write down the complete correctness con-ditions for: f = [if p then g]

if_then Correctness Conditions Complete correctness conditions for f = [if p then g]: Prove: p  (f = g) Л ¬p  (f = I) So, aside from proving termination over the domain of f, what are the two corresponding conditions for: f = [while p do g] = [if p then fog] ?

while_do Correctness Conditions Complete correctness conditions for f = [K] = [while p do G] (where g = [G] has already been shown): Prove: term(f,K) Л p  (f = f o g) Л ¬p  (f = I)

while_do Correctness Conditions (cont’d) Working correctness questions: Is loop termination guaranteed for any argument of f ? When p is true does f equal f composed with g? When p is false does f equal Identity?

while_do Example Prove f = [T] where, for integers x, y, and z: f = (y≥0  z,y := z+xy,0) and T is: while y<>0 do z := z+x y := y−1 end_while p G

while_do Example (cont’d) Proof: g = [G] = (z,y := z+x,y−1) by observation term(f,T)? f = (y≥0  z,y := z+xy,0) and T is: while y<>0 do z := z+x y := y−1 end_while So, does y≥0 initially  T will terminate?

while_do Example (cont’d) Proof: g = [G] = (z,y := z+x,y−1) by observation term(f,T)? √ (Prove this…)

while_do Example (cont’d) Proof: g = [G] = (z,y := z+x,y−1) by observation term(f,T)? √ (Prove this…) Does (y=0)  ( f = I )? ¬p ( Recall: f = (y≥0  z,y := z+xy,0) )

while_do Example (cont’d) Proof: g = [G] = (z,y := z+x,y−1) by observation term(f,T)? √ (Prove this…) Does (y=0)  ( f = I )? √ (y=0)  ( f = (z,y := z+x(0),0) = (z,y := z,0) ) (y=0)  ( I = (z,y := z,0) )

while_do Example (cont’d) Does (y0)  ( f = f o g )? p

while_do Example (cont’d) Does (y0)  ( f = f o g )? case a: Does (y<0)  ( f = f o g )? (y<0)  ( f = undefined ) (y<0)  ( f o g = f o (z,y := z+x,y−1) What is f when applied after g decrements the initially negative value of y? ( Recall: f = (y≥0  z,y := z+xy,0) )

while_do Example (cont’d) Does (y0)  ( f = f o g )? case a: Does (y<0)  ( f = f o g )? (y<0)  ( f = undefined ) (y<0)  ( f o g = undefined o (z,y := z+x,y−1) since y<0  gy(y<0)<0 ( Recall: f = (y≥0  z,y := z+xy,0) )

while_do Example (cont’d) Does (y0)  ( f = f o g )? case a: Does (y<0)  ( f = f o g )? (y<0)  ( f = undefined ) (y<0)  ( f o g = undefined o (z,y := z+x,y−1) = undefined )

while_do Example (cont’d) Does (y0)  ( f = f o g )? case a: Does (y<0)  ( f = f o g )? √ (y<0)  ( f = undefined ) (y<0)  ( f o g = undefined o (z,y := z+x,y−1) = undefined )

while_do Example (cont’d) Does (y0)  ( f = f o g )? case b: Does (y>0)  ( f = f o g )? ( Recall: f = (y≥0  z,y := z+xy,0) )

while_do Example (cont’d) Does (y0)  ( f = f o g )? case b: Does (y>0)  ( f = f o g )? (y>0)  ( f = (z,y := z+xy,0) ) (y>0)  ( f o g = f o (z,y := z+x,y−1) Again, what is f when applied after g decrements the initially positive value of y? ( Recall: f = (y≥0  z,y := z+xy,0) )

while_do Example (cont’d) Does (y0)  ( f = f o g )? case b: Does (y>0)  ( f = f o g )? (y>0)  ( f = (z,y := z+xy,0) ) (y>0)  ( f o g = (z,y := z+xy,0) o (z,y := z+x,y−1) since y>0  gy(y>0)≥0 ( Recall: f = (y≥0  z,y := z+xy,0) )

while_do Example (cont’d) Does (y0)  ( f = f o g )? case b: Does (y>0)  ( f = f o g )? (y>0)  ( f = (z,y := z+xy,0) ) (y>0)  ( f o g = (z,y := z+xy,0) o (z,y := z+x,y−1) = (z,y := (z+x)+x(y−1),0) = (z,y := z+xy,0) )

while_do Example (cont’d) Does (y0)  ( f = f o g )? case b: Does (y>0)  ( f = f o g )? (y>0)  ( f = (z,y := z+xy,0) ) (y>0)  ( f o g = (z,y := z+xy,0) o (z,y := z+x,y−1) = (z,y := (z+x)+x(y−1),0) = (z,y := z+xy,0) ) We could have also composed the full, conditional definition of f with g, i.e. (y≥0  z,y := z+xy,0) o (z,y := z+x,y−1) to yield (y≥1  z,y := z+xy,0) which is just (z,y := z+xy,0) when y>0.

while_do Example (cont’d) Does (y0)  ( f = f o g )? √ case b: Does (y>0)  ( f = f o g )? √ (y>0)  ( f = (z,y := z+xy,0) ) (y>0)  ( f o g = (z,y := z+xy,0) o (z,y := z+x,y−1) = (z,y := (z+x)+x(y−1),0) = (z,y := z+xy,0) ) Therefore, f = [T].

Exercise 1 For program M below, where all variables are integers, hypothesize a function f for [M] and prove f = [M]. while i<n do t := t*x i := i+1 end_while

Sufficient Correctness Conditions Given the complete correctness conditions for f = [H] = [while p do g]: Prove: term(f,H) Л p  (f = f o g) Л ¬p  (f = I)

Sufficient Correctness Conditions (cont’d) What are the sufficient correctness conditions for f  [H] = [while p do g]? Prove: f’ = [H] for some f’ Л f  f’

Sufficient Correctness Conditions (cont’d) What are the sufficient correctness conditions for f  [H] (for ANY program, H)? Prove: f’ = [H] for some f’ Л f  f’

repeat_until Statement What are the complete correctness conditions for f = [R] = [repeat g until p]? g f = T p F

repeat_until Statement (cont’d) An IRL for repeat_until statements: f = [repeat g until p] = [g; if ¬p then f]

“Proof” by Picture g g g g f = = = = p p ¬p p g f f p T T F T F F T F

repeat_until Statement (cont’d) Therefore, it is sufficient to verify that 1. R terminates for all X  D(f), and that 2. f is the program function of Q = g; if ¬p then f end_if because [R] = [Q].

repeat_until Correctness Conditions Complete correctness conditions for f = [R] = [repeat G until p] (where g = [G] has already been shown): Prove: term(f,R) Л (p o g)  (f = g) Л ¬(p o g)  (f = f o g)

repeat_until Correctness Conditions (cont’d) Working correctness questions: Is loop termination guaranteed for any argument of f ? When p o g is true does f equal g? When p o g is false does f equal f o g?

Exercise 2 For program R below, where all variables are integers, hypothesize a function r for [R] and prove r = [R]. repeat: x := x−1 y := y+2 until x=0

Summary Iteration Recursion Lemma (IRL) Termination predicate: term(f,P) Correctness conditions for while_do statement Sufficient correctness conditions Correctness conditions for repeat_until statement Subgoal Induction

Coming up next… Thinking about invariants again Invariant Status Theorem (IST) While Loop Initialization Utility of IST

Functional Verification III Software Testing and Verification Lecture Notes 23 Prepared by Stephen M. Thebaut, Ph.D. University of Florida