Problem 5 - Discrete Mathematics with Applications

Question

Prove that the given predicate \(P(n)\) in each algorithm is a loop invariant. Algorithm sum \((x, y)\) (* This algori thm prints the sum of two nonnegative integers \(x\) and \(\left.y .^{\star}\right)\) \(0 .\) Begin \(\left(\star \text { algori thm }^{\star}\right)\) \(1 . \quad\) sum \(\leftarrow x\) 2\. count \(\leftarrow 0\) (* counter \(^{\star}\) ) 3\. while count \(<\) y do 4\. begin \(\left(^{\star} \text { while }^{\star}\right)\) \(5 . \quad \mathrm{sum} \leftarrow \mathrm{sum}+1\) \(6 . \quad\) count \(\leftarrow\) count +1 7\. endwhile 8\. End \(\left(^{\star} \text { algorithm }^{\star}\right)\) \(P(n): x=q_{n} y+r_{n},\) where \(q_{n}\) and \(r_{n}\) denote the quotient and the remainder after \(n\) i terations.

Step-by-Step Solution

Answer

To prove that the predicate \(P(n): x = q_n y + r_n\) is a loop invariant for the given algorithm that sums two non-negative integers \(x\) and \(y\), we demonstrated the following: 1. The predicate holds before entering the loop: \(x = 0 \cdot y + x\), which simplifies to \(x = x\). 2. The predicate remains true after each iteration: assuming \(P(n)\) holds at the beginning of the \(n^{th}\) iteration, the updated predicate is \(P(n+1): x + 1 = q_{n+1} y + r_{n+1}\), where \(q_{n+1} = q_n\) and \(r_{n+1} = r_n + 1\). 3. The predicate holds after the loop terminates: at loop termination, count equals \(y\), and the predicate becomes \(P(y): x = q_y y + r_y\), where \(r_y = 0\) and \(q_y = 1\), resulting in \(x = y + r_y\). Since the predicate \(P(n)\) holds true before the loop execution, after each iteration, and after loop termination, we can conclude that \(P(n)\) is a loop invariant for the given algorithm.

Step 1: Analyze the algorithm

The algorithm takes two non-negative integers \(x\) and \(y\) and computes their sum by progressively adding \(1\) to \(x\) and decreasing \(y\) by \(1\) until \(y\) becomes \(0\). The key components of the algorithm are: 1. Initializing sum to \(x\) and count to \(0\) 2. The loop condition: while count < \(y\), do steps 4 to 7 3. In each iteration, - Increment sum by \(1\) - Increment count by \(1\) Now, let's analyze the predicate \(P(n) : x = q_n y + r_n\).

Step 2: Check if predicate holds before loop execution

Before entering the loop, sum is set to \(x\) and count is set to \(0\). At this point, \(q_0 = 0\) and \(r_0 = x\). The predicate \(P(0): x = q_0 y + r_0\) becomes: \[x = 0 \cdot y + x\] \[x=x\] Since the predicate holds before the loop execution, the initial condition is satisfied.

Step 3: Check if predicate holds after each iteration

We assume that the predicate holds true at the beginning of the \(n^{th}\) iteration: \[P(n): x = q_n y + r_n\] During the \(n^{th}\) iteration, sum and count are incremented by \(1\), so let's analyze the updated values: \[sum + 1 = x + 1 = q_n y + r_n + 1\] \[count + 1 = n + 1\] Let's set \(q_{n+1} = q_n\), then \(r_{n+1} = r_n + 1\). Now our predicate becomes: \[P(n+1): x + 1 = q_{n+1} y + r_{n+1}\] \[x + 1 = q_n y + r_n + 1\] Since the predicate holds true after the \(n^{th}\) iteration, it remains an invariant.

Step 4: Check if predicate holds after loop termination

When the loop terminates, count equals \(y\). Therefore, the algorithm has performed \(y\) iterations. The predicate now looks like: \[P(y): x = q_y y + r_y\] Since \(x\) equals the sum and the algorithm correctly added \(y\) to \(x\) after all iterations, the final remainder must be \(0\). Thus, \(r_y = 0\) and \(q_y = 1\). The predicate becomes: \[x = y + r_y\] As the predicate holds after the loop termination, it remains an invariant. Having shown that the predicate \(P(n)\) holds true before the loop execution, after each iteration, and after loop termination, we can conclude that \(P(n)\) is a loop invariant for the given algorithm.

Key Concepts

Predicate Logic

Predicate logic is a branch of logic that deals with predicates and quantifiers. In the context of the exercise, the predicate is a logical assertion that involves a variable and is evaluated to be either true or false depending on the value of the variable.
For the given algorithm, the predicate \(P(n)\) involves variables \(x\), \(y\), \(q_n\), and \(r_n\). It can be expressed as \(x = q_n y + r_n\). Here, the logic is to assert a relationship between the variables at various points during the algorithm’s execution.
This helps us verify certain conditions and remains invariant during the loop. A predicate like this is indispensable when you assess if an algorithm behaves correctly, particularly when loops are involved.

• Asserting the predicate before the loop starts ensures the initial condition is met.
• The condition must hold at the start and end of every iteration.
• It must also remain true at the end of the loop execution, providing a full account of correctness across the entire operation.

This logical framework allows us to evaluate and confirm the algorithm’s overall correctness.

Algorithm Analysis

Algorithm analysis refers to the process of determining the efficiency and correctness of an algorithm. In the exercise, the algorithm calculates the sum of two integers using a while loop. To understand its efficiency, we examine its operations at each stage.
The algorithm starts by initializing two variables—\(\text{sum}\) to \(x\) and \(\text{count}\) to 0. Then, it enters a loop which continues until \(\text{count} < y\) is no longer true. Inside each loop iteration, both \(\text{sum}\) and \(\text{count}\) are incremented by 1.

• This approach effectively counts up to \(y\), adding 1 to \(\text{sum}\) each time.
• The loop runs \(y\) times, which makes the time complexity of this algorithm \(O(y)\).
• The simplicity of increment operations ensures that the space complexity is minimal.

Analyzing an algorithm like this one helps us to understand not only how it works in theory but also how efficiently it will run with large inputs.

Mathematical Induction

Mathematical induction is a powerful technique used to prove a statement is true for all natural numbers. It's like dominos falling; if the first domino falls, and if one falling causes the next to fall, then all dominos will fall. This method is crucial for proving loop invariants like \(P(n)\).
To apply induction to our algorithm, we follow a two-step process:

• Base Case: Prove that \(P(n)\) holds before the first loop iteration. Here, when \(n=0\), we have \(x = 0 \cdot y + x\).
• Inductive Step: Assume \(P(n)\) holds for an arbitrary \(n\), and prove it for \(n+1\). During the iteration, \(\text{sum}\) and \(\text{count}\) both increase by 1, preserving the invariant as \(x + 1 = q_n y + r_n + 1\).

After confirming that \(P(n)\) remains true through the induction steps, we conclude it holds for all subsequent iterations of the loop.