Question

Use the method of invariants to derive an algorithm for the following specification The log base 2 where x and y are integer variables. ⟨y>0⇒z ′=2 x ′≤y<2 x ′+1⟩

Answer 1

To derive the algorithm, we can express the number 2 as eln 2 and apply properties of logarithms. The algorithm can be derived as z' = 2' and x' = log_2 y, with the condition 2^x' ≤ y < 2^(x'+1).

Step-by-step explanation:

Using the method of invariants to derive an algorithm:

Step 1:

We can start by expressing the number 2 as

eln 2

since the exponential and natural logarithm are inverse functions. For the special case of doubling (b=2), we have

2 = eln 2

.

Step 2:

Next, we can apply the property of logarithms that the logarithm of a product of two numbers is the sum of the logarithms of the two numbers, which means that

log xy = log x + log y

. In this case, we have

log z' = log 2' + log(y < 2^(x'+1))

.

Step 3:

To derive the algorithm, we'll continue applying properties of logarithms and simplifying the equation.

Final Algorithm:

The algorithm can be derived as follows:

z' = 2'

x' = log_2 y

2^x' ≤ y < 2^(x'+1)