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chapter_5/exercises.md
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@ -6646,6 +6646,28 @@ $\text{\textbf{while}} (m \geq 0 \text{ and } m \leq 100)\\ \ \ \ \ m := m + 1\\
predicate: $m + n = 100$
**Proof:**
Suppose the predicate $m + n = 100$ is true before entry to the loop. Then
$$ m_{\text{old}} + n_{\text{old}} = 100 $$
After execution of the loop,
$$ m_{\text{new}} = m_{\text{old}} + 1 $$
and
$$ n_{\text{new}} = n_{\text{old}} - 1 $$
so
$$ m_{\text{new}} + n_{\text{new}} = (m_{\text{old}} + 1) + (n_{\text{old}} - 1) $$
$$ = m_{\text{old}} + n_{\text{old}} = 100 $$
Q.E.D.
2.
loop:
@ -6654,6 +6676,42 @@ $\text{\textbf{while}} (m \geq 0 \text{ and } m \leq 100)\\ \ \ \ \ m := m + 4\\
predicate: $m + n \text{ is odd}$
**Proof:**
Suppose the predicate $m + n \text{ is odd}$ is true before entry to the loop.
Then
$$ m_{\text{old}} + n_{\text{old}} \text{ is odd} $$
After execution of the loop,
$$ m_{\text{new}} = m_{\text{old}} + 4 $$
and
$$ n_{\text{new}} = n_{\text{old}} - 2 $$
so
$$ m_{\text{new}} + n_{\text{new}} = (m_{\text{old}} + 4) + (n_{\text{old}} - 2) $$
$$ = m_{\text{old}} + n_{\text{old}} + 2 $$
Since $m_{\text{old}} + n_{\text{old}} \text{ is odd}$, then:
$$ = 2k + 1 + 2 $$
for some integer $k$
$$ = 2k + 2 + 1 $$
$$ = 2(k + 2) + 1 $$
Now, $k + 2$ is an integer by the sum of integers. Therefore
$m_{\text{new}} + n_{\text{new}}$ is odd by the definition of odd.
Q.E.D.
3.
loop:
@ -6662,6 +6720,31 @@ $\text{\textbf{while}} (m \geq 0 \text{ and } m \leq 100)\\ \ \ \ \ m := 3 \cdot
predicate: $m^3 > n^2$
**Proof:**
Suppose the predicate $m^3 > n^2$ is true before entry to the loop. Then
$$ (m_{\text{old}})^3 > (n_{\text{old}})^2 $$
After execution of the loop,
$$ m_{\text{new}} = 3 \cdot m_{\text{old}} $$
and
$$ n_{\text{new}} = 5 \cdot n_{\text{old}} $$
so
$$ (m_{\text{new}})^3 = (3m_{\text{old}})^3 = 27(m_{\text{old}})^3 > 27(n_{\text{old}})^2 $$
Now, since $n_{\text{new}} = 5 \cdot n_{\text{old}}$, it follows that
$\dfrac{1}{5}n_{\text{new}} = n_{\text{old}}$. Hence
$$ (m_{\text{new}})^3 > 27(n_{\text{old}})^2 = 27\left(\frac{1}{5}n_{\text{new}}\right)^2 = 27 \cdot \frac{1}{25}(n_{\text{new}})^2 $$
$$ = \frac{27}{25} \cdot (n_{\text{new}})^2 > (n_{\text{new}})^2 $$
4.
loop:
@ -6670,6 +6753,32 @@ $\text{\textbf{while}} (n \geq 0 \text{ and } n \leq 100)\\ \ \ \ \ n := n + 1\\
predicate: $2^n < (n + 2)!$
**Proof:**
Suppose the predicate $2^n < (n + 2)!$ is true before entry to the loop. Then
$$ 2^{n_{\text{old}}} < (n_{\text{old}} + 2)! $$
After execution of the loop,
$$ n_{\text{new}} = n_{\text{old}} + 1 $$
so
$$ 2^{n_{\text{new}}} = 2^{n_{\text{old}} + 1} = 2 \cdot 2^{n_{\text{old}}} < 2(n_{\text{old}} + 2)! $$
Note that $2 \leq n_{\text{old}} + 3$
since the guard condition gives $n_{\text{old}} \geq 0$, then:
$$ 2(n_{\text{old}} + 2)! \leq (n_{\text{old}} + 3)(n_{\text{old}} + 2)! = (n_{\text{old}} + 3)! = ((n_{\text{old}} + 1) + 2)! = (n_{\text{new}} + 2)! $$
Combining these gives:
$$ 2^{n_{\text{new}}} = 2 \cdot 2^{n_{\text{old}}} < (n_{\text{old}} + 3)! = (n_{\text{new}} + 2)! $$
Q.E.D.
5.
loop:
@ -6678,6 +6787,45 @@ $\text{\textbf{while}} (n \geq 3 \text{ and } n \leq 100)\\ \ \ \ \ n := n + 1\\
predicate: $2n + 1 \leq 2^n$
**Proof:**
Suppose the predicate $2n + 1 \leq 2^n$ is true before entry to the loop. Then
$$ 2n_{\text{old}} + 1 \leq 2^{n_{\text{old}}}$$
After execution of the loop,
$$ n_{\text{new}} = n_{\text{old}} + 1 $$
so
$$ 2n_{\text{new}} + 1 = 2(n_{\text{old}} + 1) + 1 = 2n_{\text{old}} + 3 $$
and
$$ 2^{n_{\text{new}}} = 2^{n_{\text{old}} + 1} = 2 \cdot 2^{n_{\text{old}}} $$
If we take the predicate and multiply both sides by $2$, we get:
$$ 2(2n_{\text{old}} + 1) \leq 2(2^{n_{\text{old}}}) $$
$$ 4n_{\text{old}} + 2 \leq 2^{n_{\text{old}} + 1} = 2^{n_{\text{new}}} $$
Notice that the new value for the left-hand value of the inequality is:
$$ 2n_{\text{new}} + 1 = 2n_{\text{old}} + 3 $$
And that this is less than the predicate's left hand side after multiplied by
two:
$$ 2n_{\text{new}} + 1 = 2n_{\text{old}} + 3 < 4n_{\text{old}} + 2 $$
And put together this is:
$$ 2n_{\text{new}} + 1 = 2n_{\text{old}} + 3 < 4n_{\text{old}} + 2 \leq 2^{n_{\text{old}} + 1} = 2^{n_{\text{new}}} $$
Q.E.D.
Exercises 6-9 each contain a while loop annotated with a pre-and a
post-condition and also a loop invariant. In each case, use the loop invariant
theorem to prove the correctness of the loop with respect to the pre-and
@ -6692,6 +6840,59 @@ _[Post-condition: $\text{exp} = x^m$]_
loop invariant: $I(n)$ is "$\text{exp} = x^n$ and $i = n$."
**I. Basis Property:** _[$I(0)$ is true before the first iteration of the
loop.]_
$I(0)$ is "$\text{exp} = x^0$ and $i = 0$." According to the pre-condition,
before the first iteration of the loop $\text{exp} = 1$ and $i = 0$. Since
$x^0 = 1$, $I(0)$ is evidently true.
**II. Inductive Property:** _[If $G \wedge I(k)$ is true before a loop iteration
(where $k \geq 0$), then $I(k + 1)$ is true after the loop iteration.]_
Suppose $k$ is any nonnegative integer such that $G \wedge I(k)$ is true before
an iteration of the loop. Then as execution reaches the top of the loop
$i \neq m$, $\text{exp} = x^k$, and $i = k$. Since $i \neq m$, the guard is
passed and statement 1 is executed. Now before the execution of statement 1,
$$ \text{exp}_{\text{old}} = x^k $$
so execution of statement 1 has the following effect:
$$ \text{exp}_{\text{new}} = \text{exp}_{\text{old}} \cdot x = x^k \cdot x = x^{k + 1} $$
Similarly, before statement 2 is executed,
$$ i_{\text{old}} = k $$
so after execution of statement 2,
$$ i_{\text{new}} = i_{\text{old}} + 1 = k + 1 $$
Hence after the loop iteration, the two statements $\text{exp} = x^{k + 1}$ and
$i = k + 1$ are true, and so $I(k + 1)$ is true.
**III. Eventual Falsity of Guard:** _[After a finite number of iterations of the
loop, $G$ becomes false.]_
The guard $G$ is the condition $i \neq m$, and $m$ is a nonnegative integer. By
I and II, it is known that_
for every integer $n \geq 0$, if the loop is iterated $n$ times, then
$\text{exp} = x^n$ and $i = n$.
So after $m$ iterations of the loop, $i = m$. Thus $G$ becomes false after $m$
iterations of the loop.
**IV. Correctness of the Post-Condition:** _[If $N$ is the least number of
iterations after which $G$ is false and $I(N)$ is true, then the value of the
algorithm variables will be as specified in the post-condition of the loop.]_
According to the post-condition, the value of $\text{exp}$ after execution of
the loop should be $x^m$. But when $G$ is false, $i = m$. And when $I(N)$ is
true, $i = N$ and $\text{exp} = x^N$. Since _both_ conditions ($G$ is false and
$I(N)$ is true) are satisfied, $m = i = N$ and $\text{exp} = x^m$, as required.
7. _[Pre-condition: $\text{largest} = A[1]$ and $i = 1$]_
$\text{\textbf{while}} (i \neq m)\\ \ \ \ \ 1. i := i + 1\\ \ \ \ \ 2. \text{\textbf{if}} A[i] > \text{largest \textbf{then } \text{largest}} := A[i]\\ \text{\textbf{end while}}$
@ -6703,6 +6904,62 @@ loop invariant: $I(n)$ is
"$\text{largest} = \text{maximum value of } A[1], A[2], \dots, A[n + 1]$ and
$i = n + 1$."
**I. Basis Property:** _[$I(0)$ is true before the first iteration of the
loop.]_
$I(0)$ is "$\text{largest} = A[1]$ and $i = 1$." According to the pre-condition,
this statement is true.
**II. Inductive Property:** _[If $G \wedge I(k)$ is true before a loop iteration
(where $k \geq 0$), then $I(k + 1)$ is true after the loop iteration.]_
Suppose $k$ is any nonnegative integer such that $G \wedge I(k)$ is true before
an iteration of the loop. Then as execution reaches the top of the loop,
$i \neq m$, $\text{largest} = A[k + 1]$ and $i = k + 1$. Since $i \neq m$, the
guard is passed and statement 1 is executed. Now, before execution of statement
1:
$$ i_{\text{old}} = k + 1 $$
so after statement 1 is executed:
$$ i_{\text{new}} = i_{\text{old}} + 1 = k + 2 $$
Also, before statement 2 is executed:
$$ \text{largest}_{\text{old}} = \max(A[1], \dots, A[k + 1]) $$
so after statement 2 is executed:
$$
\text{largest}_{\text{new}} =
\begin{cases}
A[k + 2] & \text{if } A[k + 2] > \text{largest}_{\text{old}} \\
\text{largest}_{\text{old}} & \text{if } A[k + 2] \leq \text{largest}_{\text{old}}
\end{cases}
$$
Thus, after the loop iteration, $I(k + 1)$ is true.
**III. Eventual Falsity of Guard:** _[After a finite number of iterations of the
loop, $G$ becomes false.]_
The guard $G$ is the condition $i \neq m$. By I and II, it is known that for
every integer $n \geq 1$, after $n$ iterations of the loop, $I(n)$ is true.
Hence after $m - 1$ iterations of the loop, $i = m$ and $G$ is false.
**IV. Correctness of the Post-Condition:** _[If $N$ is the least number of
iterations after which $G$ is false and $I(N)$ is true, then the value of the
algorithm variables will be as specified in the post-condition of the loop.]_
Suppose that $N$ is the least number of iterations after which $G$ is false and
$I(N)$ is true. Then (since $G$ is false) $i = m$ and (since $I(N)$ is true)
$i = N + 1$ and $\text{largest} = \max(A[1], \dots A[N + 1])$. Putting these
together gives $m = N + 1$, and so $\text{largest} = \max(A[1], \dots A[m])$,
which is the post-condition.
Q.E.D.
8. _[Pre-condition: $\text{sum} = A[1]$ and $i = 1$]_
$\text{\textbf{while}} (i \neq m)\\ \ \ \ \ 1. i := i + 1\\ \ \ \ \ 2. \text{sum} := \text{sum} + A[i]\\ \text{\textbf{end while}}$
@ -6712,6 +6969,47 @@ _[Post condition: $\text{sum} = A[1] + A[2] + \dots + A[m]$]_
loop invariant: $I(n)$ is "$i = n + 1$ and
$\text{sum} = A[1] + A[2] + \dots + A[n + 1]$."
**I. Basis Property:** _[$I(0)$ is true before the first iteration of the
loop.]_
$I(0)$ is "$i = 1$ and $\text{sum} = A[1]$." According to the pre-condition,
this statement is true.
**II. Inductive Property:** _[If $G \wedge I(k)$ is true before a loop iteration
(where $k \geq 0$), then $I(k + 1)$ is true after the loop iteration.]_
Suppose $k$ is a nonnegative integer such that $G \wedge I(k)$ is true before
the iteration of the loop. Then as execution reaches the top of the loop,
$i \neq m$, $i = k + 1$, and $\text{sum} = A[1] + A[2] + \dots + A[k + 1]$.
Since $i \neq m$, the guard is passed and statement 1 is executed. Now before
execution of statement 1, $i_{\text{old}} = k + 1$. So after execution of
statement 1, $i_{\text{new}} = i_{\text{old}} + 1 = (k + 1) + 1 = k + 2$. Also
before statement 2 is executed
$\text{sum}_{\text{old}} = A[1] + A[2] + \dotts + A[k + 1]$. Execution of
statement 2 adds $A[k + 2]$ to this sum, and so after statement 2 is executed,
$\text{sum}_{\text{new}} = A[1] + A[2] + \dots + A[k + 1] + A[k + 2]$. Thus
after the loop iteration, $I(k + 1)$ is true.
**III. Eventual Falsity of Guard:** _[After a finite number of iterations of the
loop, $G$ becomes false.]_
The guard is the condition $i \neq m$. By I and II, it is known that for every
integer $n \geq 1$, after $n$ iterations of the loop $I(n)$ is true. Hence,
after $m - 1$ iterations of the loop, $I(m)$ is true, which implies that $i = m$
and $G$ is false.
**IV. Correctness of the Post-Condition:** _[If $N$ is the least number of
iterations after which $G$ is false and $I(N)$ is true, then the value of the
algorithm variables will be as specified in the post-condition of the loop.]_
Suppose that $N$ is the least number of iterations after which $G$ is false and
$I(N)$ is true. Then (since $G$ is false) $i = m$ and (since $I(N)$ is true)
$i = N + 1$ and $\text{sum} = A[1] + A[2] + \dots + A[N + 1]$. Putting these
together gives $m = N + 1$, and so $\text{sum} = A[1] + A[2] + \dots A[m]$,
which is the post-condition.
Q.E.D.
9. _[Pre-condition: $a = A$ and $A$ is a positive integer.]_
$\text{\textbf{while}} (a > 0)\\ \ \ \ \ a := a - 2\\ \text{\textbf{end while}}$
@ -6721,6 +7019,55 @@ _[Post-condition: $a = 0$ if $A$ is even and $a = -1$ if $A$ is odd.]_
loop invariant: $I(n)$ is "Both $a$ and $A$ are even integers or both are odd
integers and, in either case, $a \geq -1$."
**I. Basis Property:** _[$I(0)$ is true before the first iteration of the
loop.]_
$I(0)$ is "$a = A$ and $A \text{ is a positive integer}$." According to the
pre-condition, this statement is true.
**II. Inductive Property:** _[If $G \wedge I(k)$ is true before a loop iteration
(where $k \geq 0$), then $I(k + 1)$ is true after the loop iteration.]_
Suppose $k$ is a nonnegative integer such that $G \wedge I(k)$ is true before
the iteration of the loop. Then as execution reaches the top of the loop,
$a_{\text{old}} > 0$, $a_{\text{old}} \text{ has the same parity as } A$, and
$a_{\text{old}} \geq -1$.
Since $a_{\text{old}} > 0$, it follows that $a_{\text{old}} \geq 1$. The guard
condition allows the loop body to execute, and statement 1 is performed. This
results in:
$$ a_{\text{new}} = a_{\text{old}} - 1 $$
Thus $I(k + 1)$ is true.
**III. Eventual Falsity of Guard:** _[After a finite number of iterations of the
loop, $G$ becomes false.]_
The guard is the condition $a > 0$. By I and II, it is known that for every
iteration of the loop $a := a - 2$. Since the initial value of $a$ is $A$, this
means that the value for $a$ follows the following sequence:
$$ A, A - 2, A - 4, \dots $$
which eventually reaches a value at $a \leq 0$.
Hence, after a finite number of iterations of the loop, $a \leq 0$ and $G$ is
false.
**IV. Correctness of the Post-Condition:** _[If $N$ is the least number of
iterations after which $G$ is false and $I(N)$ is true, then the value of the
algorithm variables will be as specified in the post-condition of the loop.]_
Suppose that $N$ is the least number of iterations after which $G$ is false and
$I(N)$ is true. Then (since $G$ is false) $a \leq 0$ and (since $I(N)$ is true)
both $a$ and $A$ have the same parity and $a \geq -1$. This means that:
$$ -1 \leq a \leq 0 $$
Therefore, if $A$ is even, then $a = 0$, and if $A$ is odd, then $a = -1$. This
fulfills the post-condition.
10. Prove correctness of the **while** loop of Algorithm 4.10.3 (in exercise 27
of Exercise Set 4.10) with respect to the following pre- and
post-conditions:
@ -6741,6 +7088,8 @@ $\text{gcd}(a, b) = \text{gcd}(A, B)$,
(3) $0 \leq a + b \leq A + B - n$."
Omitted.
11. The following **while** loop implements a way to multiply two numbers that
was developed by the ancient Egyptians.
@ -6754,11 +7103,15 @@ _[Post-condition: $\text{product } = A \cdot B$]_
a. Make a trace table to show that the algorithm gives the correct answer for
multiplying $A = 13 \text{ times } B = 18$.
Omitted.
b. Prove the correctness of this loop with respect to its pre-and
post-conditions by using the loop invariant
$I(n)$: "$xy + \text{ product} = A \cdot B$"
Omitted.
12. The following sentence could be added to the loop invariant for the
Euclidean algorithm:
@ -6769,14 +7122,22 @@ a. Show that this sentence is a loop invariant for
$\text{\textbf{while}} (b \neq 0)\\ \ \ \ \ r := a \mod b\\ \ \ \ \ a := b\\ \ \ \ \ b := r\\ \text{\textbf{end while}}$
Omitted.
b. Show that if initially $a = A$ and $b = B$, then sentence (5.5.12) is true
before the first iteration of the loop.
Omitted.
c. Explain how the correctness proof for the Euclidean algorithm together with
the results of (a) and (b) above allow you to conclude that given any integers
$A$ and $B$ with $A > B \geq 0$, there exist integers $u$ and $v$ so that
$\text{gcd}(A, B) = uA + vB$.
Omitted.
d. By actually calculating $u$, $v$, $s$, and $t$ at each stage of execution of
the Euclidean algorithm, find integers $u$ and $v$ so that
$\text{gcd}(330, 156) = 330u + 156v$.
Omitted.

16
chapter_5/test_yourself.md
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@ -106,13 +106,21 @@ Page 346
1. A pre-condition for an algorithm is _____ and a post-condition for an
algorithm is _____.
a predicate that describes the initial state of the input variables of the
algorithm; a predicate that describes the final state of the output variables
for the algorithm
2. A loop is defined as correct with respect to its pre- and post-conditions if,
and only if, whenever the algorithm variables satisfy the pre-condition for
the loop and the loop terminates after a finite number of steps, then _____.
the algorithm variables satisfy the post-condition for the loop
3. For each iteration of a loop, if a loop invariant is true before iteration of
the loop, then _____.
it is true after iteration of the loop
4. Given a **while** loop with guard $G$ and a predicate $I(n)$ if the following
four properties are true, then the loop is correct with respect to its pre-
and post-conditions:
@ -120,10 +128,18 @@ Page 346
(a) The pre-condition for the loop implies that _____ before the first iteration
of the loop.
$I(0)$ is true
(b) For every integer $k \geq 0$, if the guard $G$ and the predicate $I(k)$ are
both true before an iteration of the loop, then _____.
$I(k + 1)$ is true after the iteration of the loop
\(c\) After a finite number of iterations of the loop, _____.
the guard $G$ becomes false
(d) If $N$ is the least number of iterations after which $G$ is false and $I(N)$
is true, then the values of the algorithm variables will be as specified _____.
in the post-condition of the loop.