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Chapter 5: Problem 3

Verify that the program segment $$ \begin{array}{c}{x :=2} \\ {z :=x+y} \\ {\text { if } y>0 \text { then }} \\\ {z :=z+1} \\ {\text { else }} \\ {z :=0}\end{array} $$ is correct with respect to the initial assertion \(y=3\) and the final assertion \(z=6\)

Short Answer

Expert verified
The program is correct. The final value of z is 6, which matches the final assertion.

Step by step solution

01

Initial Setup

Start by assigning the value 2 to the variable x: \[ x := 2 \]
02

Compute Intermediate Expression

Calculate the value of z by adding x and y: \[ z := x + y \]
03

Substitute Known Values

Substitute the initial values of x and y into the intermediate expression: \[ x = 2 \] \[ y = 3 \]Then, \[ z = 2 + 3 = 5 \]
04

Condition Check

Check the condition mentioned in the program, which is whether y > 0: Since \[ y = 3 \], the condition \[ y > 0 \] is true.
05

Apply Conditional Logic

Since the condition \[ y > 0 \] is true, update z accordingly by adding 1 to it: \[ z := z + 1 \]So \[ z = 5 + 1 = 6 \]
06

Final Verification

Verify the final assertion: The value of z is now 6, which matches the given final assertion. Therefore, the program segment is correct with respect to the initial assertion \( y = 3 \) and the final assertion \( z = 6 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Initial Assertion
In program verification, an **initial assertion** is a statement declaring the initial conditions or values of variables. It's a crucial starting point, as it sets the stage for the program's logic. In our example, the initial assertion is given as **y = 3**. This means before the program executes any further instructions, we assume or know that the variable y holds the value 3.
The first step starts by assigning a value to x: **x := 2**. This is straightforward because the initial assertion about y does not affect x. Here, x being assigned the value 2 is another implicit assertion we start with.
**Key Takeaways: **
  • Making clear the starting conditions of the program helps in verifying if the final outcome is as expected.
  • Initial assertions serve as check-points to resolve further logical claims.
Conditional Logic
Conditional logic involves making decisions in a program based on certain conditions. It's like a fork in the road where the program can take different paths based on whether a condition is true or false.
In our example, the conditional logic is checking: **if y > 0 then do something** else do something different. This conditional check immediately informs us of two potential paths the program can take.
Since our initial assertion states **y = 3**, we evaluate the condition **y > 0**. Given that 3 is indeed greater than 0, the condition is **true**. Consequently, the program takes the path where **z becomes z + 1**.

**Key Takeaways:**
  • Conditions help control the flow of the program, making sure the right path is taken based on initial or current values.
  • Understanding the path taken due to conditional logic is crucial in verifying if the final outcome will match expectations.
Final Assertion
A **final assertion** is a statement expressing the expected final condition or value of variables after the program has executed. It's how we verify that the program performed as intended.
In the provided example, the final assertion is **z = 6**. Throughout the program, we followed a sequence of instructions and conditional checks to manipulate the variables.
We started with **x := 2**, which remains unaffected by further steps. After initial computation with y, **z = 5** was attained. Then, the conditional check **y > 0** led to adding 1 more to z, resulting in **z = 6**.

**Key Takeaways:**
  • Final assertions allow for verifying that the program correctly implements the intended logic.
  • Matching the final values of our variables to our expected outcomes can confirm the program's correctness.
In conclusion, program verification helps ensure that the logic detailed in the program works correctly and that the final outcomes align with initial expectations and conditions.

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Most popular questions from this chapter

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