TTY/OHJ4040 Con-Adv
E1: Sequential
1. Formulate eah of the following as a prediate:
a. The value of variable
x
is the ube of the value of the
s
is the
variable i.
b. The value of variable
N
N
0, that is, the sum of the rst
N
g
0 and 0.
[0 : 1℄ : int
is the greatest ommon divisor (gd) of
d. The array
y
[0 :
1℄ : int
N
is the array
sorted into asending order,
N
2,
i
terms of this innite
sum.
.
P1
i=1
th partial sum of
x >
x
0, that is,
y >
N
y
ontains exatly
the same elements, but so that any element in
is stritly
y
less ( <) than the next.
1
2. Compute and simplify eah of the following weakest
preonditions:
(skip
b. wp( :=
a. wp
(
; gd X; Y
x
and
) = ( )),
( )= ( )^
gd x; y
y; gd X; Y
x
Y
are onstants,
a j
i; a a i
gd x; y
x >
0
< y
), where
X
( [ ℄ := [ [ ℄℄ = ), assuming the array [0 : 1℄ with
the relevant elements dened, onstant
1, and
0
. What else need you assume?
d. wp( :=:
),
e. wp( [ ℄ :=: [ + 1℄; := + 1 (8 : 0 : [℄
[ ℄) ^ 0 ), with the same assumptions about and
. wp
i
a
N
N
i; j < N
x
y; x < y
a j
a j
a j
j
as above,
(if
f. wp
x >
j
j
;
i
i < j
a i
N
0 ! :=
x
a
x
1
x >
1 ! :=
x
x
2 ( )).
;P
x
N
3. Solve:
(do
b. wp(do
a. wp
x >
x >
0 ! :=
0 ! :=
x
x
x
x
1
1
(do
using the denition of wp
x
x
;x
x >
x
x
;x
:::
4. Given two non-dereasing arrays
[0 :
G
;x
F
[0 :
:
1℄ : int
M
and
1℄ : int and initial ondition
: 0^ 0
^ (8 : 0 1 : [ ℄ [ + 1℄)
^ (8 : 0 1 : [ ℄ [ + 1℄)
N
Q
1 ! := 2 od = 0) and
0 ! := 2 od = 0)
od = 0)
x >
;
N
M
i
i < M
F i
F i
i
i < N
G i
G i
;
prove that the following program is totally orret and omputes
R
: = (# : 0 r
i; j
i < M
f g
^0
j < N
: [℄
F i
[ ℄)
> G j
:
Q
var r : int := 0; m : int := M ; n : int := N fI g
do m 6= 0 ^ n 6= 0 !
if F [m 1℄ > G[n 1℄ ! r := r + n; m := m
[
1℄ [ 1℄ ! :=
f g
od f ^ [( = 0) _ ( = 0)℄gf g
F m
G n
n
n
1
1
I
I
m
n
R :
(Hint: This has a tail-invariant, like Eulid's algorithm has,
though not as simple.)
[0 :
1℄ : int
0
5. Given an array x
N
, N
, and two integers L and
M,
L
M
N , prove totally orret the following program
to reverse the order of the elements in the array segment
x L
M
. Initially:
0 [ :
1℄
: (8 : : [ ℄ = [ ℄) ^ 0 and the goal is:
: (8 : : [ ℄ = [ 1 ( )℄)
The auxiliary
array
[0 : 1℄ : int to denotes the
Q
i
L
i < M
R
N.B.
x i
i
X i
L
L
i < M
logial
x i
X
M
N
X M
i
L
:
N
state of the array originally. This is eetively a onstant and
universally quantied (proof must hold regardless of its atual
value).
f g
Q
var m : int := LfI g
do m 6= (M + L) div 2 !
[ ℄ :=: [ 1 (
)℄; := + 1f g
od f ^ ( = ( + ) div 2)gf g
x m
I
x M
m
m
M
L
L
m
m
I
R
(Hints: Invariant must say something of the elements not yet
moved. Doing this problem will take time!)
6. Homework: Given three integers
Q
: 0 ^ 0 ^ 0,
N
M
B
prove totally orret the following program to ompute:
R
: = (# : 0 a
i
i < B
f g
:(
N
div
2i) mod 2 6= (
Q
M
div
var a : int := 0; b : int := 0; m : int := M ; n : int
do b 6= B ! if n mod 2 6= m mod 2 ! a := a + 1
n mod 2 = m mod 2 ! skip
; b; n; m := b + 1; n div 2; m div 2fI g
od fI ^ (b = B )gfRg:
(Hint: assuming that
M <
ompute?)
2B ^
7. Homework: Given an integer
N
orret a program to ompute:
R
: = ( : 1 : 2)
r
i
i
N
i
:
N <
2i) mod 2)
:
:= f g
N
I
2B , what does the program
(12p)
0, write and prove totally
(3p)