Problem 1 - Discrete Mathematics with Applications

Question

Prove that the given predicate \(P(n)\) in each algorithm is a loop invariant. Algorithm exponent ial \((x, n)\) (* This algori thm computes \(x^{n},\) where \(x \in \mathbb{R}^{+}\) and \(\left.n \in W .^{*}\right)\) \(0 .\) Begin \(\left(^{\star} \text { algorithm }^{\star}\right)\) 1. answer \(\leftarrow 1\) 2\. while \(n>0\) do 3\. begin \(\left(^{\star} \text { whi } 1 \text { e }^{\star}\) ) \right. 4\. answer \leftarrow answer \cdot \(x\) \(5 . \quad n \leftarrow n-1\) \(6 .\) endwhile 7\. End \(\left(^{\star} \text { algorithm }^{\star}\) ) \right. \(P(n): a_{n}=x^{n},\) where \(a_{n}\) denotes the value of answer after \(n\) iterations of the while \(100 p\)

Step-by-Step Solution

Answer

In order to prove that the given predicate \(P(n): a_n = x^n\) is a loop invariant for the given algorithm that computes the value of \(x^n\), we have demonstrated the following: 1. Initialization: Before the loop starts, the predicate is true, as \(a_0 = x^0 = 1\). 2. Maintenance: The predicate remains true after each iteration of the loop, shown by \(a_{k+1} = a_k \cdot x = x^{k+1}\). 3. Termination: When the loop exits (with \(n = 0\)), the predicate ensures the algorithm reaches the correct answer, i.e., \(a_0 = x^0 = 1\). Thus, we have proved that \(P(n)\) is a loop invariant for this algorithm.

Step 1: Initialization

Before the loop starts, the value of answer is set to 1 at the beginning of the algorithm. The predicate is \(P(0): a_{0} = x^{0}\), and since any real positive number \(x\) raised to the power of \(0\) is \(1\), the predicate is true at initialization.

Step 2: Maintenance

We now need to show that if the predicate is true before an iteration, it remains true after the iteration. Suppose the predicate is true before the \(k^{th}\) iteration, i.e., \(P(k): a_{k} = x^{k}\). The loop then performs the following operations during the \(k^{th}\) iteration: 4\. answer \(\leftarrow\) answer \(\cdot\) \(x\) 5\. \(n \leftarrow n-1\) After the \(k^{th}\) iteration, we have \(a_{k+1} = a_{k} \cdot x = x^{k} \cdot x = x^{k+1}\). This shows that if the predicate is true before the \(k^{th}\) iteration, it remains true after the \(k^{th}\) iteration. This step is called the maintenance step and is crucial to establishing that the predicate is a loop invariant.

Step 3: Termination

Now we need to show that when the loop exits, the predicate ensures the algorithm reaches the correct answer. The loop exits when \(n = 0\). At this point, the predicate is: \(P(n): a_{n} = x^{n}\). Since \(n = 0\), the predicate is \(a_{0} = x^{0}\). We have already established during the initialization step that \(x^{0} = 1\) and \(a_{0}\) is equal to the answer variable. So, when the loop exits, the answer variable holds the correct value of \(x^n\). By showing the Initialization, Maintenance, and Termination steps, we have proved that the given predicate \(P(n)\) is a loop invariant for the given algorithm.

Key Concepts

Algorithm Correctness

Ensuring that an algorithm produces the correct outcome for every valid input is a key aspect of algorithm correctness. In this algorithm, the goal is to compute the value of \( x^n \) accurately. To establish correctness, we rely on proofs that verify both the logic structure and sequence of operations within the algorithm.
A common way to verify algorithm correctness is to demonstrate that a loop invariant, a statement that holds true before and after each iteration of a loop, is maintained throughout.

• Initialization: Demonstrating that the loop invariant is true before the loop starts.
• Maintenance: Showing that the loop invariant remains true after each iteration.
• Termination: Proving that upon loop exit, the final state of the program is correct.

By ensuring these steps, we confirm that the algorithm provides correct results consistently.

Mathematical Induction

Mathematical induction is a powerful mathematical tool used to prove the truth of an infinite number of cases. Primarily, it's applied in scenarios with iterative processes where we need to establish the result for all natural numbers or steps in a sequence.
In the context of the current algorithm, we use induction to prove the correctness of the loop invariant across all iterations.
Here's how the process works:

• Base Case: Demonstrates that the statement is true at the starting point. For the sequence starting from 0 iterations, the algorithm initializes with \( a_0 = x^0 \), and since any number to the power of 0 is 1, this holds true.
• Inductive Step: Assumes that if the statement is true for some integer \( k \), it must also be true for \( k+1 \). It leverages the known truth of the preceding step to validate the subsequent one. In this algorithm, if \( a_k = x^k \) before an iteration, after it, \( a_{k+1} = x^{k+1} \).

This structured approach of mathematical induction guarantees that the loop invariant remains true through every iteration, thus confirming the algorithm’s correctness.

Predicate Logic

Predicate logic is essential for formulating the conditions and propositions that help us understand and solve complex problems. In algorithm design, it is used to define conditions and invariant statements.
In this algorithm, the predicate refers to the invariant \( P(n): a_n = x^n \). This statement forms the backbone of the reasoning to prove the algorithm's correctness. The role of the predicate involves:

• Clarification: It precisely expresses what needs to be kept true by the algorithm.
• Logical Structure: Through logical operations, it helps in establishing the truth conditions of the algorithm at each point.
• Verification: Serves as the central hypothesis that guides the induction and helps verify correctness throughout the process.

Thus, predicate logic underpins the verification of correctness, ensuring that each line of the algorithm fulfills its designated condition.

Initialization, Maintenance, Termination Steps

Loop invariants are validated through a systematic process involving initialization, maintenance, and termination steps. These steps form the foundation of proving that a loop invariant holds true, thereby confirming the algorithm’s capacity to achieve its objective.

**Initialization**: This step verifies that the invariant holds at the start. Here, you ensure conditions set before the loop are met. For our algorithm, \( a_0 = x^0 \) ensures correctness right from the start.

**Maintenance**: Checks after each iteration that the loop invariant condition is still valid. Throughout each iteration, if the predicate remains true, the invariant holds. If prior to the \( k^{th} \) iteration, \( a_k = x^k \), it ensures \( a_{k+1} = x^{k+1} \) after the iteration.

**Termination**: When the loop ends, this step confirms the algorithm provides the correct output. For us, when \( n = 0 \), we reassess if \( a_0 = x^0 \), which was initially true and has been maintained by the loop. Hence, it correctly ends with the initial assumption intact.

Through this framework, you uphold and prove the consistency and correctness of the algorithm, ensuring every possible execution delivers the desired outcome.