Predicate Transforms II: Proving Termination of While Loops - Prof. Stephen M. Thebaut, Study notes of Engineering

Download Predicate Transforms II: Proving Termination of While Loops - Prof. Stephen M. Thebaut and more Study notes Engineering in PDF only on Docsity!

Predicate Transforms II

Prepared by

Stephen M. Thebaut, Ph.D.

University of Florida

Software Testing and Verification

Lecture 20

Predicate Transforms I and II^ 1.

Introduction

Proving strong correctness

Assignment statements

Sequencing

Selection statements --------------------------- 6.

Iteration

Rule for while_do Statement (cont’d)^ Thus, we can write

wp(while b do S, Q)

H

V 0

H

1

V

H

2

V…

where

H

0

≡^

¬b

Q

H

1

≡^

b^

wp(S, ¬b

Q)

H

2

≡^

b^

wp(S, b

wp(S, ¬b

Q))

Rule for while_do Statement (cont’d)^ Equivalently, we can write

wp(while b do S, Q)

H

0

V^

H

1

V

H

2

V…

where

H

0

≡^

¬b

Q

H

1

≡^

b^

wp(S,

H

H

2

≡^

b^

wp(S,

H

H

≡i b

wp(S,

H

i-

Example^ •^

For what initial values of i, n, and t willthe following program terminate witht=x

n?

while i <= n do

t := t

∗x

i := i+ end_while

How about i=1, n=2, and t=1?

Can

you think of any others?

Example (cont’d)^ •^

Find the wp of this program with respectto the post-condition

{t=x

n}.

(Attempt

to find a regularity in terms that allows aclosed-form expression.)

Example (cont’d)

while i <= n do

t := t

∗x

i := i+ end_while

H

3

≡^

b^

wp(S,

H

i≤n

wp(S, i=n-

t=x

n-

i≤n

i+1=n-

tx=x

n-

i=n-

t=x

n-

H

k^

≡^

b^

wp(S, H

k-

i=n-(k-1)

t=x

n-k

i=n-k+

t=x

n-k

Example (cont’d)^ Thus, we have:

H

0

i>n

t=x

n

H

k^

i=n-k+

t=x

n-k

(for all k>0)

i≤n

t=x

i-

since i=n-k+

n-k=i-

Therefore, wp

H

0

V^

H

1

V

H

2

V ...

(i>n

t=x

n) V (i≤n

t=x

i-

Loop Invariants and wp’s^ •^

In general, will loops terminate whenP^

wp?

•^

For while loops, does {wp

b} S {wp}?

•^

Does (wp

¬b)

Q?

___________________________ wp

weakest

loop invariant which

guarantees termination!

Predicate Transforms II

Prepared by

Stephen M. Thebaut, Ph.D.

University of Florida

Software Testing and Verification

Lecture 20