Question: What is a correct pruning strategy for decision tree induction?

2. Stop partitioning a node when either positive or negative samples dominate the samples of the other class.

This option is a good pruning strategy because it helps to simplify the decision tree by preventing it from becoming too complex. When one class dominates the samples in a node, it's often unnecessary to keep splitting because the decision boundary is already clear. This approach can lead to a tree that is not only more efficient but also helps in avoiding the trap of overfitting, which is something to be cautious about. However, while this is a valid point, it's also essential to consider other factors such as depth and breadth of the tree. So, while option 2 is valid, it's not the only thing you should focus on when dealing with decision trees.


Question: how can the results from a classifier impact the metric (precision) used? What could be a better suited metric to use with imbalanced data?

The results from a classifier can significantly impact precision, especially in imbalanced datasets where one class is much more prevalent than the other. High precision may be misleading if the classifier predominantly predicts the majority class, leading to a low number of true positives for the minority class.

A better-suited metric for imbalanced data is the F1 score, which balances precision and recall, or the area under the ROC curve (AUC-ROC), which considers both true positive and false positive rates.


Question:
A multiset is an unordered collection where elements can appear multiple times. We will represent a
multiset of Char elements as a function from Char to Int: the function returns 0 for any Char argument that
is not in the multiset, and the (positive) number of times it appears otherwise:
1 type Multiset = Char => Int
What should replace ??? so that the following function transforms given set s to a
multiset where each element of s appears exactly once?
1 type Set = Char => Boolean
2 def setToMultiset(s: Set): MultiSet = ???

To transform a set `s` into a multiset where each character appears exactly once, we can define the `setToMultiset` function as follows:

```scala
def setToMultiset(s: Set): Multiset =
  (c: Char) => if (s(c)) 1 else 0
```

In this implementation, we check if the character `c` exists in the set `s`. If it does, we return 1, indicating its presence in the multiset; otherwise, we return 0.


Question: Tick the \textbf{true} assertion. The advantage of a distinguisher of two distributions $P_0$ and $P_1$

Let’s clarify why option 4 is indeed the correct assertion regarding distinguishers of distributions \( P_0 \) and \( P_1 \).\n\n4. **True**: A distinguisher can achieve an advantage that approaches the statistical distance \( \frac{1}{2}\Sigma_{x}|P_0(x) - P_1(x)| \) with a well-chosen single query. The statistical distance measures how distinguishable two distributions are, and a clever query can maximize this advantage, allowing the distinguisher to effectively touch the upper bound of the statistical distance.\n\nReflecting on my previous assessment, I misinterpreted the nature of the single query's potential in maximizing the distinguisher's advantage. Students may also fall into this trap by overlooking the strategic significance of the query and the mathematical properties of statistical distance. It's crucial to understand how specific queries can yield maximal insights into distribution differences. \n\nThus, the correct answer is indeed option 4.


Question: Let $n$ be a positive integer. The Fermat test most likely outputs ``prime'' \dots

4, when $n$ is prime or when $n$ is a Carmichael number. The Fermat test can incorrectly identify Carmichael numbers as primes, thus it outputs "prime" for both cases.


Question: Tick the \textbf{false} statement. The Shannon Encryption Model ...

1. Requires a black-box encryption model.

This statement is considered false because the Shannon Encryption Model is not specifically tied to any black-box requirements. It mainly discusses the principles of how encryption should ideally function. Although there are various encryption models in cryptography, the Shannon model emphasizes security aspects related to keys and distributions without focusing too much on the mechanics of a black-box model. While black-box models can sometimes help analyze encryption schemes, the central themes of Shannon's work do not necessitate their presence. Thus, saying it "requires" such a model doesn't fully capture the essence of what Shannon was addressing in the context of encryption.


Question: Explain why any fail-noisy consensus algorithm (one that uses an eventually perfect failure detector ◇P) actually solves uniform consensus (and not only the non-uniform variant).

To understand why a fail-noisy consensus algorithm using an eventually perfect failure detector (◇P) can solve uniform consensus, we need to break down the concepts involved step-by-step.

### Step 1: Understanding Consensus

Consensus in distributed systems is the problem of getting a group of processes (or nodes) to agree on a single value, despite failures or message delays. There are two main types of consensus:
- **Uniform Consensus**: All processes must agree on the same value, regardless of their initial values.
- **Non-uniform Consensus**: The agreement may depend on the initial values of the processes.

### Step 2: Fail-Noisy Consensus Algorithms

A fail-noisy consensus algorithm is one that can tolerate failures and also deal with "noisy" communication, meaning that messages may be lost or delayed. The eventually perfect failure detector ◇P is crucial here. It provides information about process failures and guarantees that:
- Eventually, it will correctly identify all crashed processes.
- No correct process will be falsely suspected of having failed indefinitely.

### Step 3: What is an Eventually Perfect Failure Detector?

An eventually perfect failure detector (◇P) provides two guarantees:
1. **Eventually Accurate**: After some point in time, it will correctly identify which processes have crashed.
2. **No False Positives**: Once a process is determined to be correct, it will not be suspected to have failed again.

### Step 4: Uniform vs. Non-uniform Consensus

In non-uniform consensus, the decision can depend on the initial values of the processes. However, uniform consensus requires that every process must converge to the same decision value, regardless of their starting points.

### Step 5: How ◇P Enables Uniform Consensus

Now, let’s see how a fail-noisy consensus algorithm with ◇P can achieve uniform consensus:

1. **Correctness and Agreement**: With ◇P, once a process is identified as correct, it can safely communicate its value without fear of being suspected of failure. This means that if all processes agree on a certain value (or learn from a correct process), they can reach a uniform decision.

2. **Handling Failures**: If some processes fail, the remaining correct processes can still make progress because they rely on the eventually perfect nature of ◇P. They will eventually know which processes have failed and can ignore them in their decision-making.

3. **Consensus Mechanism**: The algorithm can use the information from ◇P to ensure that all processes that are alive and correct eventually agree on a common value. For example, once enough correct processes agree on a value (after knowing which others have failed), they can decide on that value as their final agreement.

4. **Value Selection**: Since the algorithm operates under the assumption that it will receive correct information from the eventually perfect failure detector, it can ensure that even if some processes start with different values, they can converge to a single value based on the agreement among the surviving correct processes.

### Conclusion

In conclusion, a fail-noisy consensus algorithm using an eventually perfect failure detector (◇P) solves uniform consensus because it enables all correct processes to eventually agree on a single value by accurately identifying failures and allowing for effective communication among remaining correct processes. Thus, it ensures that no matter the initial values or failures that occur, all participating processes will ultimately reach the same decision.

**Final Answer**: A fail-noisy consensus algorithm with an eventually perfect failure detector (◇P) solves uniform consensus because it allows all correct processes to eventually agree on a common value by accurately identifying failures and enabling effective communication among surviving processes.


Question: You are given a distribution on $X, Y$, and $Z$ and you know that the joint distribution can be written in the form $p(x, y, z)=p(x) p(y \mid x) p(z \mid y)$. What conclusion can you draw? [Recall that $\perp$ means independent and $\mid \cdots$ means conditioned on $\cdots$.

To analyze the joint distribution \( p(x, y, z) = p(x) p(y \mid x) p(z \mid y) \), we can derive relationships between the variables based on their conditional dependencies.

1. **Understanding the form**: The structure indicates that \(Y\) depends on \(X\), and \(Z\) depends on \(Y\). This means there is a direct influence from \(X\) to \(Y\), and then from \(Y\) to \(Z\).

2. **Independence conclusions**:
   - Since we have a direct dependence of the form where both conditioning and other dependencies are explicit, we can assess potential independence statements:

   - **Option 1**: \(Y \perp Z \): This is incorrect because knowing about \(Y\) provides information about \(Z\).

   - **Option 2**: \(X \perp Y \mid Z \): This cannot be inferred because knowing about both does not imply independence.

   - **Option 3**: "Not clear what this means without context; likely not valid as stated."

   - **Option 4**: \(X \perp Z\): Incorrect since the presence of influences through other variables (via Y).

   - **Option 5**: `\( X \perp Y \)` : This is incorrect; rather, they are dependent.

   - **Option 6**:
     Hereâs a key insight. Given that we know how each variable relates to each other in sequence but lacks information defining them independently when one conditions upon another,
     so it turns out that indeed

     *Knowing Y would allow you to know more about Z but does not automatically imply any independence regarding X.*,

Thus after analyzing all options given:

The correct conclusion drawn from this relation regarding conditional independence would ultimately lead us toward Option (6):

### Final Answer:
6. ***\( X \perp Z | Y \\*


Question: Suppose we use the Simplex method to solve the following linear program: \begin{align*} \textbf{maximize} \hspace{0.8cm} & 2x_1 - x_2 \\ \textbf{subject to}\hspace{0.8cm} & x_1 - x_2  + s_1  = 1 \\ \hspace{0.8cm} & \hspace{0.85cm}x_1  + s_2  = 4 \\ \hspace{0.8cm} & \hspace{0.85cm} x_2  + s_3 = 2 \\ \hspace{0.8cm} &\hspace{-0.8cm}  x_1,\: x_2, \:s_1, \:s_2, \:s_3 \geq 0 \end{align*} At the current step, we have the following Simplex tableau: \begin{align*} \hspace{1cm} x_1 &= 1 + x_2 - s_1 \\ s_2 &= 3 -x_2  + s_1  \\ s_3 &= 2 -x_2  \\ \cline{1-2} z &= 2 + x_2 - 2s_1 \end{align*} Write the tableau obtained by executing one iteration (pivot) of the Simplex method starting from the above tableau.

To perform one iteration (pivot) of the Simplex method starting from the given tableau, we need to follow these steps:

### Step 1: Identify the Pivot Column
The pivot column is determined by selecting the variable that will enter the basis. This is typically the variable with the largest positive coefficient in the objective function row (z-row).

From the given tableau:
\[
\begin{align*}
z &= 2 + x_2 - 2s_1
\end{align*}
\]
The coefficients of the variables in the z-row are:
- \(x_1\): 0
- \(x_2\): 1 (positive)
- \(s_1\): -2 (negative)

Since \(x_2\) has the largest positive coefficient (1), it will be the pivot column.

### Step 2: Identify the Pivot Row
To identify the pivot row, we need to compute the ratios of the right-hand side (RHS) values to the corresponding coefficients in the pivot column (where \(x_2\) is entering). We only consider positive coefficients of the pivot column:

From the equations in the tableau:
1. For the first equation (\(x_1\)): \(s_1\) coefficient is 1, RHS is 1. Thus, the ratio is \(1/1 = 1\).
2. For the second equation (\(s_2\)): \(s_2\) coefficient is -1, which is not positive, so we don’t consider it.
3. For the third equation (\(s_3\)): \(x_2\) coefficient is 1, RHS is 2. Thus, the ratio is \(2/1 = 2\).

The minimum ratio is 1, corresponding to the first row, which means \(s_1\) will leave the basis.

### Step 3: Pivot Element
The pivot element is located at the intersection of the pivot column and pivot row. This is the coefficient of \(x_2\) in the first row, which is 1.

### Step 4: Perform Row Operations
We will now perform row operations to make the pivot column (corresponding to \(x_2\)) into a unit column (that is, it has a 1 in the pivot row and 0s elsewhere).

#### New Row for \(x_1\):
The first row will be updated to make \(x_2\) the basic variable:
\[
\text{New } x_1 = \text{Old } x_1 \text{ (keeping it as is since it's already normalized)}
\]
\[
x_1 = 1 + x_2 - s_1
\]
We will subtract \(x_2\) from both sides:
\[
x_1 - x_2 = 1 - s_1 \quad \Rightarrow \quad x_1 = 1 + s_1
\]

#### New Row for \(s_2\):
Now, we update the second row (for \(s_2\)):
\[
s_2 = 3 - x_2 + s_1
\]
Add \(x_2\) to both sides and subtract the new \(x_2\) row:
\[
s_2 - x_2 = 3 - 2x_2 + s_1
\]
Substituting \(x_2 = 1 + s_1\):
\[
s_2 = 3 - 1 + s_1 = 2 + s_1
\]

#### New Row for \(s_3\):
Now, update the third row (for \(s_3\)):
\[
s_3 = 2 - x_2
\]
Subtract \(x_2\):
\[
s_3 + x_2 = 2
\]

#### New Row for \(z\):
Finally, update the z-row:
\[
z = 2 + x_2 - 2s_1
\]
Substituting \(x_2 = 1 + s_1\):
\[
z = 2 + 1 + s_1 - 2s_1 = 3 - s_1
\]

### Final Tableau
Now we can compile the new tableau:

\[
\begin{array}{c|ccc|c}
 & x_1 & x_2 & s_1 & RHS \\
\hline
x_1 & 1 & 0 & 1 & 1 \\
s_2 & 0 & 0 & 1 & 2 \\
s_3 & 0 & 1 & 0 & 2 \\
\hline
z & 0 & 1 & -1 & 3 \\
\end{array}
\]

This tableau represents the system after one iteration of the Simplex method, pivoting on \(x_2\). The next steps would involve checking for optimality and continuing the Simplex algorithm as needed.


Question: Which of the following sets can you use Cantor’s Diagonalization Argument to prove it is uncountable (multiple answers) ?

To determine which sets can be proven to be uncountable using Cantor's Diagonalization Argument, we must first understand what Cantor's Diagonalization Argument entails. This argument is a method used to demonstrate that certain sets cannot be put into a one-to-one correspondence with the natural numbers, hence proving they are uncountable.

1. The set of decimal representations of negative real numbers: This set is uncountable because it can be mapped to the real numbers, which are known to be uncountable.
2. Real numbers not containing 7 in their decimal representation: This set is also uncountable, as removing a finite number of decimals from the real numbers does not affect their uncountability.
3. The set of binary strings with finite length: This set is countable, as we can list all binary strings of length 1, then length 2, and so on.
4. Integers between 0 and 1: This set is empty, as there are no integers that fall between 0 and 1, thus it cannot be uncountable.
5. Real numbers between 1 and 2: This set is uncountable because it is a subset of the real numbers.

In conclusion, the sets that can be proven to be uncountable using Cantor’s Diagonalization Argument are:
1. The set of decimal representations of negative real numbers.
2. Real numbers not containing 7 in their decimal representation.
5. Real numbers between 1 and 2.

Correct answers: 1, 2, 5.


Question: Which of the following statement(s) is/are correct?

Let's evaluate each statement:

a) **An information leak can be a preparation step of control-flow hijacking.**
   - **Correct.** Information leaks can provide attackers with critical information about the memory layout, such as the addresses of functions or variables, which can be used to craft a payload to hijack control flow.

b) **An attacker-controlled format string can lead to arbitrary write.**
   - **Correct.** Format string vulnerabilities can allow an attacker to manipulate the format string to write to arbitrary locations in memory if they can control the input, which can lead to exploitation.

c) **In format strings, \%n prints a hex value.**
   - **Incorrect.** The `%n` format specifier does not print a value; rather, it writes the number of bytes output so far to the address provided as an argument. It doesn't print in hex format or any other format.

d) **When constructing a ROP payload, we use gadgets from all currently running processes.**
   - **Incorrect.** ROP (Return-Oriented Programming) payloads typically use gadgets from the memory space of the target process that is being exploited. Gadgets from other processes are generally not available due to memory isolation in modern operating systems.

In summary:
- Statements **a** and **b** are correct.
- Statements **c** and **d** are incorrect.


Question: Consider the composite function $f(x)=g(h(x))$, where all functions are $\mathbb{R}$ to $\mathbb{R}$. Which of the following is the weakest condition that guarantees that $f(x)$ is convex?

To determine the weakest condition that guarantees the convexity of the composite function \( f(x) = g(h(x)) \), we need to recall some fundamental properties regarding the composition of functions and their convexity.

1. **Key properties of convex functions**:
   - A function \( g \) is convex if its second derivative \( g''(x) \geq 0 \) for all \( x \) in its domain.
   - A function \( h \) is increasing if \( h(x_1) \leq h(x_2) \) whenever \( x_1 \leq x_2 \).
   - The composition of two convex functions can also yield a convex function provided the outer function is also increasing.

Now, let's analyze each option:

1. **Option 1**: \( g(x) \) and \( h(x) \) are convex and \( g(x) \) and \( h(x) \) are increasing.
   - While this guarantees \( f(x) = g(h(x)) \) is convex, being both increasing and convex may make it a stronger condition.

2. **Option 2**: \( g(x) \) is convex and \( g(x) \) is increasing.
   - This condition does not specify anything about \( h(x) \). If \( h(x) \) is not an increasing function, the composition could result in a non-convex function.

3. **Option 3**: \( g(x) \) and \( h(x) \) are convex and \( h(x) \) is increasing.
   - This condition may work as \( g \) is convex and \( h \) is increasing, but we do not have a requirement for \( g \) to be increasing, which is necessary for ensuring that \( f(x) \) is convex in this composition.

4. **Option 4**: \( g(x) \) and \( h(x) \) are convex and \( g(x) \) is increasing.
   - This is a strong requirement since both functions are convex and \( g \) is increasing, which guarantees \( f(x) \) is convex.

5. **Option 5**: \( g(x) \) is convex and \( g(x) \) and \( h(x) \) are increasing.
   - This is fairly strong, but again, it doesn't require \( h(x) \) to be convex.

6. **Option 6**: \( h(x) \) is convex and \( g(x) \) and \( h(x) \) are increasing.
   - While this includes \( g(x) \) being increasing, \( g(x) \) also needs to be convex for \( f(x) \) to maintain convexity.

7. **Option 7**: \( g(x) \) is convex and \( h(x) \) is increasing.
   - Similar to earlier options, it lacks the necessary condition for \( h(x) \) to affect the outcome sufficiently.

After careful consideration, it looks like:

**Option 3**: "g(x) and h(x) are convex and h(x) is increasing" is indeed a valid choice since \( g(h(x)) \) preserves the convexity given that \( g \)'s convexity is maintained through the mapping of the convex and increasing \( h(x) \).

However, considering the definitions and requirements, the exact conditions needed for \( f(x) \) are achieved with:

**Correct Answer**: **Option 4**: \( g(x) \) and \( h(x) \) are convex and \( g(x) \) is increasing.


Question: To how many plaintexts we expect to decrypt a ciphertext in the Rabin cryptosystem when we don't use redundancy?

Option: '4.'
Rationale: In the Rabin cryptosystem, without using redundancy, we expect to decrypt a ciphertext to four possible plaintexts due to the nature of quadratic residues.


Question: The number of plaintext/ciphertext pairs required for a linear cryptanalysis is\dots

The question asks how many plaintext/ciphertext pairs are needed for effective linear cryptanalysis, a technique used to analyze and potentially break block ciphers by leveraging linear relationships between plaintext, ciphertext, and the keys.

In linear cryptanalysis, the linear probability (LP) quantifies the strength of the correlation between the plaintext and ciphertext. The lower the linear probability, the weaker the correlation, which means that more pairs are necessary to achieve a successful analysis.

Now, let’s evaluate each option:

1. **Option 1: $\approx \mathsf{LP}$**
   **Incorrect.** This implies a direct proportionality, which underestimates the effort required, as it fails to account for the inverse relationship.

2. **Option 2: $\approx \frac{1}{\mathsf{LP}}$**
   **Correct.** This option accurately reflects that as the linear probability decreases (indicating weaker correlations), the number of required pairs increases inversely. A low probability necessitates more data to achieve reliable results.

3. **Option 3: $\approx \frac{1}{\mathsf{LP}^2}$**
   **Incorrect.** This suggests a quadratic relationship, which is not applicable in the context of linear cryptanalysis, where the number of pairs grows linearly relative to the inverse of the probability.

4. **Option 4: $\approx \log \frac{1}{\mathsf{LP}}$**
   **Incorrect.** This indicates a logarithmic relationship, which does not align with how linear cryptanalysis operates.

Thus, the correct answer is $\approx \frac{1}{\mathsf{LP}}$, as it succinctly captures the principle that more data is needed when the correlation is weaker, ensuring the effectiveness of the cryptanalysis process.


Question: Tick the assertion related to an open problem.

To identify the assertion related to an open problem among the given choices, we need to analyze each statement in the context of known complexity theory results.

1. **A. \( NP \subseteq IP \)**: This statement is known to be true. The result states that nondeterministic polynomial time problems can be solved using interactive proofs. Therefore, this is not an open problem.

2. **B. \( P \subseteq IP \)**: This statement is also known to be true. It holds that problems solvable in polynomial time can be solved in interactive polynomial time. Thus, this is not an open problem either.

3. **C. \( PSPACE = IP \)**: This statement is currently an open question in complexity theory. It has been conjectured but not proven that the complexity classes PSPACE (problems solvable with polynomial space) and IP (interactive polynomial time) are equal. Hence, this is related to an open problem.

4. **D. \( NP = \text{co-}NP \)**: This is another well-known open problem in complexity theory. It is not known whether NP (nondeterministic polynomial time problems) is equal to co-NP (the complement of NP problems). Therefore, this is also an open problem.

Now, we need to choose the assertion that relates to an open problem. While both C and D relate to open problems, the question asks for a single assertion.

Given that both C and D are open problems, but C is specifically about the relationship between PSPACE and IP, which is a more specific area of research in complexity theory, I will choose:

**C. \( PSPACE = IP \)**

The reason for this choice is that while D is a broader and well-known open question, C represents a specific relationship that is not yet resolved and is of significant interest in the field of theoretical computer science.


Question: Select the statements that are true.A penalty will be applied to any incorrect answers selected.

To determine which statements are true, let's break down each option:

1. **True**: This accurately defines information retrieval (IR), which involves selecting relevant documents from a larger set.

2. **True**: This is correct. Different IR systems can use various methods to represent documents and queries and measure relevance.

3. **True**: The vector space model does represent documents as vectors based on indexing terms.

4. **False**: The dimensionality of the vector space typically does depend on the size of the indexing vocabulary, as more terms lead to a higher-dimensional space.

5. **False**: Using filters during indexing usually helps in creating more relevant and efficient indexes, not less informative ones.

So, the true statements are **1, 2, and 3**.


Question: Let p(x,y) be the statement “x visits y”, where the domain of x consists of all the humans in the world and the domain of y consists of all the places in the world. Use quantifiers to express the following statement: There is a place in the world that has never been visited by humans.

To express the statement "There is a place in the world that has never been visited by humans" using quantifiers, we need to break it down:

1. "There is a place in the world" indicates the existence of at least one place, which can be represented by the existential quantifier (â).
2. "that has never been visited by humans" means that for this particular place, no human has visited it, which can be represented by the universal quantifier (â) for all humans, combined with the negation of the visiting statement (Â¬p(x,y)).

Putting this together, we get:

ây (âx Â¬p(x,y))

This means "There exists a place y such that for all humans x, it is not the case that x visits y."

Among the provided options, the correct answer is:

**ây âx Â¬p(x,y)**.


Question: What is the formal relation between accuracy and the error rate? In which case would you recommend to use the one or the other?

### Understanding Accuracy and Error Rate

To delve into the relationship between accuracy and error rate, let's first define both terms clearly:

**Accuracy** is the proportion of correct predictions made by a model out of the total predictions. It can be mathematically expressed as:

\[
\text{Accuracy} (acc) = \frac{TP + TN}{TP + TN + FP + FN}
\]

Where:
- \(TP\) = True Positives
- \(TN\) = True Negatives
- \(FP\) = False Positives
- \(FN\) = False Negatives

**Error Rate**, on the other hand, quantifies the proportion of incorrect predictions made by the model. It can be expressed as:

\[
\text{Error Rate} (err) = \frac{FP + FN}{TP + TN + FP + FN}
\]

### The Relationship Between Accuracy and Error Rate

To establish the relationship between accuracy and error rate, we can manipulate the mathematical definitions.

1. From the definitions above, the total number of predictions is \(TP + TN + FP + FN\).
2. The error rate can be rewritten in terms of accuracy:

\[
err = \frac{FP + FN}{TP + TN + FP + FN} = 1 - \frac{TP + TN}{TP + TN + FP + FN}
\]

3. Recognizing that \(TP + TN\) is the count of correct predictions, we can express error rate in terms of accuracy:

\[
err = 1 - acc
\]

Thus, we establish that:

\[
\text{Error Rate} = 1 - \text{Accuracy}
\]

### Interpretation of the Relationship

The derived relationship \(err = 1 - acc\) indicates that accuracy and error rate are complementary measures. When the accuracy increases, the error rate decreases, and vice versa. This means that both metrics provide valuable insights into model performance, but from opposite perspectives.

### Equivalence of the Metrics

Because accuracy and error rate are strictly complementary (i.e., one is always equal to one minus the other), they carry exactly the same information. There is no scenario where one provides fundamentally different insight than the other. Choosing between them is a matter of preference or clarity of interpretation rather than necessity.

### Conclusion

In summary, while accuracy and error rate represent complementary perspectives on a model's performance, they are mathematically equivalent. Selecting one over the other does not affect the underlying information conveyed. This equivalence allows flexibility in reporting without any loss of interpretability or evaluation quality.


Question: Last year Professor Ueli von Gruy\`{e}res worked hard to to obtain an estimator $\Alg$  to estimate the total cheese consumption of fondue lovers in Switzerland. For a small $\epsilon >0$, his estimator \Alg only  asks $3/\epsilon^2$ random persons and have the following guarantee: if we  let $W$ denote the true answer and let $X$ be the random output of \Alg then \begin{align*} \Pr[|X - W| \geq \epsilon W] \leq 1/3\,. %\qquad \mbox{ where $\epsilon > 0$ is a small constant.} \end{align*} However, Ueli is now stuck because the error probability of $1/3$ is too high.  We are  therefore going to help Ueli by designing a new estimator with a much higher success probability while still only asking relatively few persons. For a fixed small parameter $\delta >0$, your task is to design and analyze an estimator that outputs a random value $Y$ with the following guarantee: \begin{align} \label{eq:guarantee2} \Pr[|Y - W| \geq \epsilon W] \leq \delta\,. %\qquad \mbox{ where $\epsilon > 0$ is a small constant.} \end{align} Your estimator should ask at most $3000\log(1/\delta)/\epsilon^2$ persons about their preferences.  \\ While you should explain why your estimator works and what tools to use to analyze it, \emph{you do not need to do any detailed calculations.} \\ {\em (In this problem you are asked to (i) design an estimator that asks  at most $3000 \log(1/\delta)/\epsilon^2$ persons  and (ii) explain why it satisfies the guarantee~\eqref{eq:guarantee2}. Recall that you are allowed to refer to material covered in the lecture notes.)}

We define our estimator as follows:

1. Let \( t = 1500 \log(1/\delta) \).
2. Run \( t \) independent copies of \( \Alg \) to obtain estimates \( X_1, X_2, \ldots, X_t \).
3. Output \( Y \) as the \emph{median} of \( X_1, \ldots, X_t \).

Let \( I_i \) be the indicator random variable that \( |X_i - W| \geq \epsilon W \). For \( |Y - W| \geq \epsilon W \) to hold, we need to consider how many of the estimates \( X_i \) are far from \( W \).

For the median \( Y \) to be far from \( W \), at least half of the estimates must be inaccurate. Specifically, if \( \sum_{i=1}^t I_i \) (the number of inaccurate estimates) is at least \( t/2 \), it follows that:
\[
|Y - W| \geq \epsilon W.
\]
By the linearity of expectation, we have:
\[
\E\left[\sum_{i=1}^t I_i\right] \leq \frac{t}{3},
\]
indicating that the expected number of inaccurate estimates is at most \( \frac{t}{3} \).

Using Chernoff bounds for sums of independent random variables, we can bound the probability that the number of inaccurate estimates exceeds \( t/2 \):
\[
\Pr\left[\sum_{i=1}^t I_i \geq \frac{t}{2}\right] \leq e^{-\frac{t}{8}}.
\]

Substituting \( t = 1500 \log(1/\delta) \), we find:
\[
\Pr\left[\sum_{i=1}^t I_i \geq \frac{t}{2}\right] \leq e^{-\frac{1500 \log(1/\delta)}{8}} = e^{-187.5 \log(1/\delta)} = \delta^{187.5}.
\]
To ensure that this probability is less than or equal to \( \delta \), we can adjust \( t \) appropriately. By choosing \( t = 3000 \log(1/\delta) \), we have:
\[
\Pr\left[\sum_{i=1}^t I_i \geq \frac{t}{2}\right] \leq \delta,
\]
which leads to:
\[
\Pr[|Y - W| \geq \epsilon W] \leq \delta.
\]

Finally, the total number of persons queried is:
\[
t \cdot \frac{3}{\epsilon^2} = \frac{3000 \log(1/\delta)}{\epsilon^2},
\]
which satisfies the requirement. Therefore, the estimator \( Y \) meets the guarantee in equation (1) while asking at most \( 3000 \log(1/\delta)/\epsilon^2 \) persons.


Question:
If process i fails, then eventually all processes j≠i fail
Is the following true? If all processes j≠i fail, nothing can be said about process i

If all processes \\( j \\neq i \\) fail, nothing can be said about process \\( i \\)" is **True**.\n\nHere\'s the reasoning:\n\n1. The first part of the statement indicates that if process \\( i \\) fails, then eventually all other processes \\( j \\neq i \\) will also fail. This is a conditional statement that specifies a relationship between the failure of process \\( i \\) and the failures of other processes.\n\n2. The second part of the statement asserts that if all other processes \\( j \\neq i \\) have failed, we cannot infer the status of process \\( i \\). This is indeed true because the failure of processes \\( j \\neq i \\) does not provide information about the operational status of process \\( i \\). It is possible that process \\( i \\) is still functioning independently of the other failures.\n\nThus, we cannot conclude anything about process \\( i \\) solely based on the fact that all other processes have failed.


Question: Which protocol does \emph{not} use RC4.

WPA2 does not use RC4.

Here's a brief overview of the protocols mentioned:

- **WEP (Wired Equivalent Privacy)**: Uses RC4 for encryption.
- **WPA (Wi-Fi Protected Access)**: Initially used TKIP (Temporal Key Integrity Protocol), which can use RC4.
- **WPA2**: Primarily uses AES (Advanced Encryption Standard) for encryption, not RC4.
- **TLS (Transport Layer Security)**: Can use various ciphers, including AES, and does not specifically rely on RC4.

So, among the options, WPA2 is the correct answer as it does not use RC4.


Question: Select which statements are true regarding SCFGs.A penalty will be applied for any incorrect answers.

True Statements:
- The sum over all rule probabilities sharing the same left-hand side sums up to 1.
- The probabilities of lexical rules correlate with emission probabilities in HMMs.
- The inside-outside algorithm estimates stochastic coefficients in supervised settings.
False Statements:
- Probabilities across all parse trees don't necessarily add up to 1.
- A stochastic grammar isn't consistent solely based on individual tree summation.
- Parse tree comparisons between different SCFGs can't rely solely on their probabilistic values without contextual consideration.


Question: Enigma

None of the options provided are correct regarding the Enigma machine. However, if you have to select the least incorrect option, it would be option 3, as it relates to cryptographic principles, even though the Enigma did not actually follow the Kerkhoffs principle.


Question: Consider the following algorithm that takes as input a  complete $n$-by-$n$ bipartite graph $G=(U \cup V,E)$ with positive integer edge-weights $w :E \rightarrow \mathbb{Z}_{> 0 }$: \begin{center} \begin{boxedminipage}[t]{0.85\textwidth} \begin{minipage}{14cm} \begin{verse} \textsc{MinWeightPerfectMatching}$(G, w)$: \\[2mm] 1. \FOR each edge $e\in E$ {\scriptsize (i.e., each pair $(u,v)$ since the graph is complete)} \\ 2. \qquad select independently and uniformly at random  $p(e) \in \{1, \dots, n^2\}$.\\[1mm] 3. Define a bi-adjacency matrix $A$ with $n$ rows (one for each $u\in U$) and $n$ columns  (one for each $v\in V$) as follows: \begin{align*} A_{u,v} =  2^{n^{100} w(u,v)}\cdot p(u,v) \,. \end{align*}\\ 4. \RETURN largest positive integer $i$ such that $2^{i \cdot n^{100} }$ divides $\det(A)$ (if no such $i$ exists, we return $0$). \end{verse} \end{minipage} \end{boxedminipage} \end{center} Prove that the above algorithm returns the value of a  min-weight perfect matching with probability at least $1-1/n$. Recall that you are allowed to refer to material covered in the course. \\[2mm] \noindent Hint: Let $\mathcal{M}_i$ denote the set of perfect matchings $M$ whose weight $\sum_{e\in M} w(e)$ equals $i$.  Use that one can write $\det(A)$ as follows: \begin{align*} \det(A) = \sum^{\infty}_{i=0} 2^{i \cdot n^{100}} f_i({p}) \qquad \mbox{where } f_i(p) =  \sum_{M \in \mathcal{M}_i} \textrm{sign}(M) \prod_{e\in M} p(e)\,. \end{align*} Here $\textrm{sign}(M)\in \{\pm 1\}$ is the sign of the permutation corresponding to $M$.

To show that the algorithm \textsc{MinWeightPerfectMatching} returns the value of a min-weight perfect matching with probability at least \(1 - \frac{1}{n}\), we can analyze how the determinant \(\det(A)\) encodes the weights of perfect matchings in the bipartite graph.

### Step-by-step Explanation:

1. **Understanding the Matrix \(A\)**:
   The matrix \(A\) is constructed such that each entry \(A_{u,v}\) is determined by the edge weight \(w(u,v)\) and a random value \(p(u,v)\). Specifically, \(A_{u,v} = 2^{n^{100} w(u,v)} \cdot p(u,v)\). The structure of \(A\) means that the powers of 2 in the determinant will be influenced by the weights of the matchings.

2. **Determinant Representation**:
   The determinant of the matrix \(A\) can be expressed as:
   \[
   \det(A) = \sum_{i=0}^{\infty} 2^{i \cdot n^{100}} f_i(p)
   \]
   where \(f_i(p)\) sums over all perfect matchings \(M\) with weight \(i\). The term \(2^{i \cdot n^{100}}\) indicates that the contribution of each matching's weight to the determinant is scaled exponentially with respect to \(n^{100}\).

3. **Probability of Correct Weight Extraction**:
   The algorithm aims to find the largest integer \(i\) such that \(2^{i \cdot n^{100}}\) divides \(\det(A)\). Given that we have a complete bipartite graph, there is a unique minimum weight perfect matching (let’s denote its weight as \(i_{\min}\)). The key insight is that the contributions of matchings to \(f_i(p)\) are randomized by the \(p(e)\) values.

4. **Probability Analysis**:
   Since \(p(e)\) is chosen uniformly at random from \(\{1, \dots, n^2\}\) for each edge, the values of \(p(e)\) will vary. For a perfect matching \(M\) with weight \(i_{\min}\), its contribution to \(f_{i_{\min}}(p)\) will typically dominate the contributions from matchings with higher weights due to the exponential scaling of \(2^{i \cdot n^{100}}\).

   However, there is a chance that random choices of \(p(e)\) could lead to significant contributions from other matchings, especially those with weights close to \(i_{\min}\). The probability that the contributions from matchings with weights \(i_{\min}\) and \(i_{\min} + 1\) are sufficiently close to cause confusion in determining the correct \(i\) can be bounded.

5. **Bounding the Error**:
   The likelihood of incorrectly identifying the weight of the minimum weight perfect matching is related to the number of perfect matchings in the graph. For \(n\)-nodes, there are \(n!\) perfect matchings. If we consider the top two or three weights, the contributions from matchings will be significantly different due to the factor \(2^{n^{100}}\) in the determinant. The random selection of \(p(e)\) values ensures that the contributions from the higher-weight matchings become negligible compared to the minimum weight matching.

   Specifically, the probability that any higher-weight matching dominates the summation in \(f_{i_{\min}}(p)\) can be shown to be at most \(\frac{1}{n}\) through careful consideration of the terms and their scaling in the determinant.

6. **Conclusion**:
   Thus, with high probability (i.e., at least \(1 - \frac{1}{n}\)), the algorithm will correctly return \(i_{\min}\), the weight of the minimum weight perfect matching, because the contributions from higher weights will not suffice to mislead the outcome due to the significant separation created by the exponential growth in the determinant representation.

Therefore, we conclude that \textsc{MinWeightPerfectMatching} correctly returns the weight of the min-weight perfect matching with probability at least \(1 - \frac{1}{n}\).


Question: Consider two distributions $P_0,P_1$ with the same supports and a distinguisher $\mathcal{A}$ that makes $q$ queries. Tick the \textit{incorrect} assertion.

To determine which assertion is incorrect, let's analyze each of the given options one by one and see how they relate to the concepts of statistical distance and the performance of the distinguisher $\mathcal{A}$ based on the number of queries $q$.

1. **When $q=1$, $\mathsf{Adv}(\mathcal{A}) \leq d(P_0,P_1)$ where $d$ is the statistical distance.**
   - This statement seems plausible because with one query, the best advantage the distinguisher can achieve is directly tied to the statistical distance between the two distributions. The stronger the divergence between $P_0$ and $P_1$, the better $\mathcal{A}$ can distinguish them.

2. **When $q>1$, $\mathsf{Adv}(\mathcal{A}) \leq \frac{d(P_0,P_1)}{q}$ where $d$ is the statistical distance.**
   - This one requires a bit of reasoning. It suggests that as the number of queries increases, the advantage diminishes proportionally to the number of queries. This seems a bit off because while more queries can help distinguish distributions better, it doesnât necessarily imply that the advantage is linearly reduced by the number of queries. So, this might be the incorrect assertion.

3. **When $q=1$, the strategy "return 1 $\Leftrightarrow \frac{P_0(x)}{P_1(x)} \leq 1$" achieves the best advantage.**
   - This assertion sounds reasonable. If the distinguisher can leverage the ratio of the probabilities from $P_0$ and $P_1$, it can maximize its distinguishing advantage. It aligns with the idea of making the optimal choice based on the likelihood ratio.

4. **To achieve good advantage, we need to have $q \approx 1/C(P_0,P_1)$ where $C$ is the Chernoff information.**
   - This statement appears to be discussing the relationship between the number of queries and the Chernoff information, which is relevant in context to distinguishing distributions. However, itâs less direct than the others, yet it does have a basis in information theory.

After analyzing these options, option 2 stands out as potentially incorrect due to the interpretation of how advantage scales with the number of queries. While it might seem that more queries lead to diminishing returns, the relationship is not simply a division by $q$. Hence, I would conclude that the second assertion is indeed the incorrect one.

So the answer is **2**.


Question: Modularity clustering will end up always with the same community structure?

Modularity clustering does not always yield the same community structure because it can be sensitive to initial conditions, random seed choices, and the specific algorithm used. Different runs can lead to different partitions, especially in cases of graph ambiguity or overlapping communities. Therefore, the correct answer is False.


Question: Consider:Non-terminals: S (top-level), NP (for "noun phrase"), VP (for "verbal phrase"), N (for "Noun"), V (for "Verb"), Det (for "Determiner").PoS tags: N, V, DetTerminals: I, yesterday, in, rain, went, home, the, cat, goOut of the following, select the ones which are possible valid "syntactic rules" as defined in a context-free grammar for processing (a tiny part of) English.A penalty will be applied for any incorrect answers.

**Correct Answer: 1. S → NP VP**

**Incorrect Option: 3. V → VP N**

**Key Detail:**
The correct answer (1. S → NP VP) accurately reflects the structure of a sentence in English where a sentence (S) can be composed of a noun phrase (NP) followed by a verbal phrase (VP). This rule correctly captures the hierarchical organization of sentence components.

In contrast, the incorrect option (3. V → VP N) misrepresents the grammatical structure. It suggests that a verb (V) can be directly expanded into a verbal phrase (VP) and a noun (N), which is not a valid syntactic rule in context-free grammar. Verbs do not serve as the starting point for generating phrases but rather act as part of a verbal phrase that is dependent on a subject (NP). Thus, the correct answer showcases a fundamental building block of sentence structure, while the incorrect option fails to adhere to the proper grammatical relationships.


Question: Let $\mathcal{C}$ be a binary $(6,3)$ linear code containing the codewords $\mathbf{x}_1 = 011011$, $\mathbf{x}_2 = 101101$ and $\mathbf{x}_3 = 111000$. True or false: A generator matrix for the code is        egin{equation*}  G = egin{pmatrix}   1 &0 &0 &0 &1 &1 \      0 &1 &0 &0 &0 &1 \     0 &0 &1 &0 &1 &1   \end{pmatrix}     \end{equation*}

To determine whether the given matrix \( G \) is a generator matrix for the binary linear code \(\mathcal{C}\), we need to verify if the codewords \(\mathbf{x}_1\), \(\mathbf{x}_2\), and \(\mathbf{x}_3\) can be expressed as linear combinations of the rows of \( G \).

The generator matrix \( G \) is given by:
\[
G = \begin{pmatrix}
1 & 0 & 0 & 0 & 1 & 1 \\
0 & 1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 1 & 1
\end{pmatrix}
\]

Each row of \( G \) represents a basis vector for the code. To generate codewords, we can take all possible linear combinations of the rows of \( G \) with coefficients from the binary field \( \mathbb{F}_2 = \{0, 1\}\).

Let's denote the rows of \( G \) as:
- \( \mathbf{g}_1 = (1, 0, 0, 0, 1, 1) \)
- \( \mathbf{g}_2 = (0, 1, 0, 0, 0, 1) \)
- \( \mathbf{g}_3 = (0, 0, 1, 0, 1, 1) \)

Now, we will compute the linear combinations of these vectors to see if we can obtain \(\mathbf{x}_1\), \(\mathbf{x}_2\), and \(\mathbf{x}_3\).

1. **Finding \(\mathbf{x}_1 = 011011\)**:
   - We check if we can express it as \( a_1 \mathbf{g}_1 + a_2 \mathbf{g}_2 + a_3 \mathbf{g}_3 \) for \( a_i \in \{0, 1\} \).
   - \( \mathbf{x}_1 = 0 \cdot \mathbf{g}_1 + 1 \cdot \mathbf{g}_2 + 1 \cdot \mathbf{g}_3 \)
   - This gives: \( (0, 0, 0, 0, 0, 0) + (0, 1, 0, 0, 0, 1) + (0, 0, 1, 0, 1, 1) = (0, 1, 1, 0, 1, 0) \neq \mathbf{x}_1 \).

2. **Finding \(\mathbf{x}_2 = 101101\)**:
   - Check combinations.
   - \( \mathbf{x}_2 = 1 \cdot \mathbf{g}_1 + 0 \cdot \mathbf{g}_2 + 1 \cdot \mathbf{g}_3 \)
   - This gives: \( (1, 0, 0, 0, 1, 1) + (0, 0, 0, 0, 0, 0) + (0, 0, 1, 0, 1, 1) = (1, 0, 1, 0, 0, 0) \neq \mathbf{x}_2 \).

3. **Finding \(\mathbf{x}_3 = 111000\)**:
   - Check combinations.
   - \( \mathbf{x}_3 = 1 \cdot \mathbf{g}_1 + 1 \cdot \mathbf{g}_2 + 0 \cdot \mathbf{g}_3 \)
   - This gives: \( (1, 0, 0, 0, 1, 1) + (0, 1, 0, 0, 0, 1) + (0, 0, 0, 0, 0, 0) = (1, 1, 0, 0, 1, 0) \neq \mathbf{x}_3 \).

Since we could not express the codewords as linear combinations of the rows of \( G \), the matrix \( G \) does not generate the given codewords.

Therefore, the statement is **false**.

B. False


Question:
Consider the following document:
D = 'the exports from Switzerland to the USA are increasing in 2006'
Propose a possible indexing set for this document. Justify your answer.

To create an indexing set for the document \( D = \text{'the exports from Switzerland to the USA are increasing in 2006'} \), we need to identify key terms or concepts that effectively represent the content of the document. An indexing set is a collection of terms that can be used to categorize or retrieve documents based on their content.

### Step-by-Step Approach:

1. **Identify Key Terms**:
   - We should look for significant nouns, verbs, and adjectives that encapsulate the main ideas of the document. In this case, the key terms could be:
     - **Exports**
     - **Switzerland**
     - **USA**
     - **Increasing**
     - **2006**

2. **Remove Stop Words**:
   - Words like "the," "from," "to," "are," and "in" are considered stop words. They do not add significant meaning and are often excluded from indexing because they do not help in identifying the main topics of the document.

3. **Create the Indexing Set**:
   - Based on the identified key terms, the indexing set for the document can be represented as:
     \[
     \text{Indexing Set} = \{ \text{Exports}, \text{Switzerland}, \text{USA}, \text{Increasing}, \text{2006} \}
     \]

### Justification:

- **Relevance**: Each term in the indexing set is relevant to the content of the document. "Exports" indicates the subject matter, "Switzerland" and "USA" specify the countries involved, "Increasing" describes the trend, and "2006" provides a temporal context.

- **Searchability**: This indexing set allows for effective searching and retrieval. If a user searches for "exports," "Switzerland," or "2006," the document can be easily located based on these key terms.

- **Conciseness**: The indexing set is concise and focused, which is important for efficient information retrieval. A well-defined indexing set helps in reducing ambiguity and improving the precision of search results.

### Definitions:

- **Indexing**: In information retrieval, indexing is the process of creating a data structure (the index) that allows for efficient searching and retrieval of documents based on their content.

- **Stop Words**: These are common words that are usually filtered out in the processing of text because they carry less meaningful information compared to the main content words.

### Conclusion:

The proposed indexing set captures the essential elements of the document while excluding less informative words. This approach aligns with best practices in information retrieval and natural language processing, ensuring that the document can be effectively categorized and retrieved based on relevant queries.


Question: In a FP tree, the leaf nodes are the ones with:

### Possible Ways to Solve the Problem:

1. **Understanding FP-Trees:**
   - Review the structure and function of FP-trees, focusing on how nodes are defined and what they represent.

2. **Evaluating Answer Choices:**
   - Analyze each given option in relation to leaf nodes in an FP-tree.

3. **Checking Definitions:**
   - Refer to definitions related to confidence, support, and alphabetical ordering within data mining contexts.

### Refinement of Each Approach:

1. **Understanding FP-Trees:**
   - FP-trees consist of a root node and branches that represent frequent itemsets. Leaf nodes are those without children, indicating they terminally capture specific itemsets or transactions.

2. **Evaluating Answer Choices:**
   - Examine whether leaf nodes signify low confidence (not applicable) or low support:
     * Lowest confidence does not apply here as it relates to rules derived from the tree, not its structure.
     * Lowest support could be misconstrued; however, leaf nodes do not imply low support directly but denote end points in frequent patterns.

3. **Checking Definitions:**
   - Confidence measures association strength between items; hence irrelevant for defining leaf node characteristics.
   - Support indicates frequency but does not directly correlate with being a leaf node—rather it identifies which items appear frequently together in transactions.

### Assertion:

The unique and concise straightforward answer is:

**None of the above**

**Reasoning:** In an FP-tree, leaf nodes simply denote end points for certain paths through frequent itemsets without implying anything about their support levels compared to other nodes; thus options related to lowest confidence or lowest support do not accurately define them, leading us clearly toward "none of the above."


Question: Tick the \emph{true} assertion. A distinguishing attack against a block cipher\dots

To evaluate the multiple-choice question regarding a distinguishing attack against a block cipher, we need to analyze the definition and characteristics of a distinguishing attack. Here's a step-by-step reasoning process:

1. **Understanding Distinguishing Attacks**:
   A distinguishing attack is a type of cryptanalysis where the attacker aims to determine whether a given cipher is a specific block cipher or a random permutation. The goal is to distinguish between the two with a probability significantly better than random guessing.

2. **Evaluating the Options**:
   - **Option 1: "is a probabilistic attack."**
     - This statement is true. Distinguishing attacks are indeed probabilistic in nature because they rely on statistical analysis to differentiate between the behavior of the cipher and that of a random permutation. They do not guarantee success every time but aim to achieve a success rate better than random guessing.

   - **Option 2: "succeeds with probability $1$."**
     - This statement is false. While a distinguishing attack may succeed with a certain probability, it cannot guarantee success every time (which would imply a probability of 1). The success rate is typically less than 1, depending on the strength of the cipher and the attack method.

   - **Option 3: "outputs the secret key."**
     - This statement is false. A distinguishing attack does not necessarily aim to output the secret key; rather, it is focused on distinguishing the cipher from a random permutation. It does not directly recover the key.

   - **Option 4: "succeeds with probability $0$."**
     - This statement is also false. A well-designed block cipher should ideally make it difficult for any distinguishing attack to succeed, but that doesnât mean that no distinguishing attacks can succeed; they may succeed with some non-zero probability.

3. **Conclusion**:
   Based on the analysis of each option, the only true assertion regarding a distinguishing attack against a block cipher is that it is a probabilistic attack.

The correct answer is therefore:
**Option 1: "is a probabilistic attack."**


Question: Consider the following quadratic programming relaxation of the Max Cut problem on $G=(V,E)$: \begin{align*} \textbf{maximize} \hspace{0.8cm} & \sum_{\{i,j\} \in E} (1-x_i)x_j + x_i (1-x_j) \\ \textbf{subject to}\hspace{0.8cm} & x_i \in [0,1] ~ ~ \forall i\in V \end{align*} Show that the optimal value of the quadratic relaxation actually equals the value of an optimal cut. (Unfortunately, this does not give an exact algorithm for Max Cut as the above quadratic program is NP-hard to solve (so is Max Cut).) \\ \noindent\emph{Hint: analyze basic randomized rounding.}

To show that the optimal value of the given quadratic programming relaxation equals the value of an optimal cut in the Max Cut problem, we will analyze the structure of the quadratic program and leverage randomized rounding.

### Step 1: Understand the Quadratic Objective

The quadratic programming relaxation is given by:

\[
\textbf{maximize} \quad \sum_{\{i,j\} \in E} (1-x_i)x_j + x_i (1-x_j)
\]

We can rewrite the objective function as:

\[
\sum_{\{i,j\} \in E} (1-x_i)x_j + x_i(1-x_j) = \sum_{\{i,j\} \in E} (x_i(1 - x_j) + (1 - x_i)x_j)
\]

This can be interpreted as follows: For each edge \(\{i,j\}\), the expression \(x_i(1-x_j) + (1-x_i)x_j\) counts the contribution of the edge to the cut based on the values of \(x_i\) and \(x_j\).

### Step 2: Analyze \(x_i\) Values

The values \(x_i\) are in the interval \([0, 1]\).

- If \(x_i = 0\), vertex \(i\) is assigned to one side of the cut.
- If \(x_i = 1\), vertex \(i\) is assigned to the other side of the cut.
- If \(0 < x_i < 1\), vertex \(i\) is assigned probabilistically to one side or the other.

### Step 3: Calculate Expected Cut Value

We can use randomized rounding based on the optimized values of \(x_i\):

- Assign vertex \(i\) to set \(A\) with probability \(x_i\).
- Assign vertex \(i\) to set \(B\) with probability \(1 - x_i\).

The expected value of the cut can be calculated as follows:

\[
\mathbb{E}[\text{cut value}] = \sum_{\{i,j\} \in E} \mathbb{P}(i \in A \text{ and } j \in B) + \mathbb{P}(i \in B \text{ and } j \in A)
\]

Calculating the probabilities yields:

\[
\mathbb{E}[\text{cut value}] = \sum_{\{i,j\} \in E} \left( x_i(1 - x_j) + (1 - x_i)x_j \right)
\]

This can be simplified to:

\[
\mathbb{E}[\text{cut value}] = \sum_{\{i,j\} \in E} (1 - x_i x_j)
\]

### Step 4: Relate to Quadratic Objective

Now, notice that the objective of our quadratic program is:

\[
\sum_{\{i,j\} \in E} (x_i(1 - x_j) + (1 - x_i)x_j)
\]

This is equivalent to the expected value of the cut when we use randomized rounding to assign vertices based on values \(x_i\).

### Step 5: Optimality Condition

1. **If \(x^*\) is the optimal solution to the quadratic program**, then the expected cut value \(E[\text{cut value}]\) will also be maximized. This expected cut value aligns with maximizing the number of edges crossing the cut.

2. **The maximum cut value in the Max Cut problem is an integer**, since it directly counts the number of edges that cross the cut. Thus, the maximum expected cut calculated from the randomized rounding is at least as good as any deterministic cut.

### Conclusion

Since the quadratic programming relaxation effectively captures the same structure as the Max Cut problem, we conclude that:

\[
\text{optimal value of the quadratic relaxation} = \text{optimal cut value}
\]

This shows that the optimal value from the quadratic programming relaxation equals the value of an optimal cut in the graph \(G\). However, solving this quadratic program directly is still NP-hard, which does not yield a polynomial-time algorithm for the Max Cut problem itself.


Question:
Assume you are working on SuperQuiz, a trendy app that lets everyone design quizzes and share them with friends! Your first assignment is to add a new feature that is requested by users. You are given the following transcript of an interview with a customer of your product:
> Hi!
> So you're the developer of this quiz app?
> The one where you write questions and answers and get your friends to guess?
> It's fun, but I can't use it as much as I'd like.
> I'm a firefighter, I don't have time for this app during the day, but I wish I could use it at home.
> See, when I come back home after work, I have a bunch of stuff to do, cleaning, cooking, ...
> And when I'm doing these tasks, I'm rather busy. Not like when I'm watching TV.
> I don't always have my phone in my hands! Sometimes I even forget where I put it.
> Maybe if you made it so I could talk to the app? You know, many apps have that feature now.
> Then I could multitask! Think about quizzes while I'm cooking!
> Otherwise, I won't use the app much.

Write down a user story, as a single sentence that follows the following guidelines:
1) A user story that summarizes all necessary information from the feedback
2) the user story does not contain any unnecessary information

As a firefighter, I want to talk to the SuperQuiz app so that I can engage with quizzes while multitasking at home.


Question: A generator matrix $G$ of binary $(6,3)$ linear code maps the information vectors $m_1 = (1,0,1)$ and $m_2=(1,1,1)$ into the codewords $c_1 = (1,1,0,0,0,1)$ and $c_2=(1,0,0,0,1,0)$ respectively. Which of the following is true?

To analyze the given statements about the generator matrix \( G \) of a binary \( (6,3) \) linear code, we start with the information provided about the codewords generated by the information vectors.

1. **The second row of \( G \) is \( (0,1,0,0,1,1) \)**:
   - We know that \( m_1 = (1,0,1) \) maps to \( c_1 = (1,1,0,0,0,1) \) and \( m_2 = (1,1,1) \) maps to \( c_2 = (1,0,0,0,1,0) \). The generator matrix \( G \) can be represented as:
     \[
     G = \begin{pmatrix}
     g_1 \\
     g_2 \\
     g_3
     \end{pmatrix}
     \]
     where \( g_1, g_2, g_3 \) are the rows of \( G \).
   - From \( m_1 \) and \( c_1 \), we can express \( c_1 \) as:
     \[
     c_1 = (1, 0, 1)G = g_1 + g_3
     \]
   - From \( m_2 \) and \( c_2 \), we have:
     \[
     c_2 = (1, 1, 1)G = g_1 + g_2 + g_3
     \]
   - We can derive \( g_2 \) by manipulating these equations:
     - Subtract \( c_1 \) from \( c_2 \):
       \[
       c_2 - c_1 = (1,0,0,0,1,0) - (1,1,0,0,0,1) = (0, 1, 0, 0, 1, 1) = g_2
       \]
   - Therefore, the second row of \( G \) is indeed \( (0, 1, 0, 0, 1, 1) \). Thus, the first statement is **True**.

2. **\( G \) is in systematic form**:
   - A generator matrix is in systematic form if it has the identity matrix on the left side. We need to check the structure of \( G \) based on the rows we know. Since we already know the second row and we would need to analyze the first and third rows to determine the overall structure.
   - Without knowing the complete structure of \( G \), we cannot definitively say whether it is in systematic form or not. Therefore, this statement is **Not necessarily true**.

3. **\( d_{\min} = 3 \)**:
   - The minimum distance \( d_{\min} \) of a linear code is the minimum Hamming weight of all non-zero codewords. We currently have two codewords:
     - \( c_1 = (1, 1, 0, 0, 0, 1) \) has weight 4.
     - \( c_2 = (1, 0, 0, 0, 1, 0) \) has weight 2.
   - To assess \( d_{\min} \), we need to consider all possible linear combinations of the rows in \( G \), but we can see that \( d_{\min} \) would be at least 2 based on \( c_2 \), and we need to check other combinations to determine if 3 is achievable or not.
   - Given that the minimum weight we observe is 2, we can conclude that \( d_{\min} \) cannot be 3. Hence, this statement is **False**.

Based on this analysis, the correct answer is:

1. **The second row of \( G \) is \( (0,1,0,0,1,1) \)** (True).


Question: You are given the following accident and weather data. Each line corresponds to one event:  1. car_accident rain lightning wind clouds fire 2. fire clouds rain lightning wind 3. car_accident fire wind 4. clouds rain wind 5. lightning fire rain clouds 6. clouds wind car_accident 7. rain lightning clouds fire 8. lightning fire car_accident  (a) You would like to know what is the likely cause of all the car accidents. What association rules do you need to look for? Compute the confidence and support values for these rules. Looking at these values, which is the most likely cause of the car accidents?

**Reasoning:**

To determine the likely causes of car accidents from the provided data, we need to analyze the co-occurrence of events. We can use association rules to identify which weather conditions or events are associated with car accidents.

1. **Data Representation**: Each event can be treated as a transaction of features (e.g., weather conditions, events). We will represent the presence of "car_accident" alongside other events.

2. **Identify Relevant Transactions**: From the provided data, we can extract the transactions that contain "car_accident":
   - Transaction 1: car_accident, rain, lightning, wind, clouds, fire
   - Transaction 3: car_accident, fire, wind
   - Transaction 6: clouds, wind, car_accident
   - Transaction 7: rain, lightning, clouds, fire
   - Transaction 8: lightning, fire, car_accident

3. **Calculate Support and Confidence**:
   - **Support**: This is the proportion of transactions that contain both the event and the item of interest. For "car_accident", we can calculate support for each other event.
   - **Confidence**: This is the proportion of transactions containing the item of interest that also contain the event. It gives us an idea of how often the event occurs in relation to car accidents.

   Let's count the occurrences:
   - Total transactions: 8
   - Transactions with car_accident: 5

   Now we will calculate support and confidence for each relevant event:

   - **Rain**:
     - Support = 3/8 = 0.375 (Transactions 1, 6, and 7 contain rain)
     - Confidence = 3/5 = 0.6 (3 of 5 car accidents contain rain)

   - **Lightning**:
     - Support = 3/8 = 0.375 (Transactions 1, 7, and 8 contain lightning)
     - Confidence = 3/5 = 0.6 (3 of 5 car accidents contain lightning)

   - **Wind**:
     - Support = 4/8 = 0.5 (Transactions 1, 3, 6, and 8 contain wind)
     - Confidence = 4/5 = 0.8 (4 of 5 car accidents contain wind)

   - **Clouds**:
     - Support = 4/8 = 0.5 (Transactions 1, 6, 7 contain clouds)
     - Confidence = 4/5 = 0.8 (4 of 5 car accidents contain clouds)

   - **Fire**:
     - Support = 4/8 = 0.5 (Transactions 1, 3, 5, 8 contain fire)
     - Confidence = 4/5 = 0.8 (4 of 5 car accidents contain fire)

4. **Analysis**:
   The confidence values indicate how strongly each weather condition or event is associated with car accidents. Wind, clouds, and fire have the highest confidence values at 0.8, suggesting these are the most likely associated factors.

**Answer:** The most likely cause of car accidents is associated with wind, clouds, and fire, each with a confidence of 0.8.


Question: For his awesome research, Tugrulcan is going to use the Pagerank with teleportation and HITS algorithm, not on a network of webpages but on the retweet network of Twitter! The retweet network is a directed graph, where nodes are users and an edge going out from a user A and to a user B means that "User A retweeted User B". Which one is FALSE about a Twitter bot that retweeted other users frequently but got never retweeted by other users or by itself?

**Plan:**
1. Define the concepts of hub value, authority value, and PageRank in the context of directed graphs, specifically in social networks like Twitter.
2. Analyze the implications of a Twitter bot that retweets frequently but is never retweeted by others or itself.
3. Evaluate each option to determine which statement is false regarding the Twitter bot's values.
4. Conclude with the correct answer and provide reasoning for the choice.

**Steps:**

1. In the context of directed graphs, a hub value measures the influence of a node based on how many outbound connections (links) it has to other nodes, while an authority value measures the influence based on how many inbound connections (links) it receives from hub nodes. PageRank is a measure that reflects the likelihood of arriving at a particular node by following the edges in the graph.

2. For the Twitter bot described:
   - The bot frequently retweets other users, which means it has many outgoing edges (thus potentially a high hub value).
   - However, since it is never retweeted by anyone, it has no incoming edges, resulting in an authority value of zero.
   - PageRank also depends on incoming links; since the bot has none, its PageRank will be zero as well.

3. Now, let’s analyze each option:
   - **Option A:** "It will have a non-zero hub value." This is true because the bot retweets frequently, resulting in a non-zero hub value.
   - **Option B:** "It will have an authority value of zero." This is true because the bot has no incoming edges, leading to an authority value of zero.
   - **Option C:** "It will have a pagerank of zero." This is true because PageRank is determined by incoming links, and since the bot has none, its PageRank will be zero.
   - **Option D:** "Its authority value will be equal to the hub value of a user who never retweets other users." This is false because a user who never retweets anyone will have a hub value of zero, and since the bot has an authority value of zero, this statement suggests a comparison that is misleading.

4. Therefore, the correct answer is **D**.

**Self-Check:**
- Option A is true because the bot's frequent retweeting gives it a non-zero hub value.
- Option B is true as the bot has no incoming edges, resulting in an authority value of zero.
- Option C is true because the bot has no incoming links, leading to a PageRank of zero.
- Option D is false as it incorrectly implies that the bot’s authority value could equal a non-retweeting user's hub value, which is also zero but is misleading in context.


Question: The Pohlig-Hellman algorithm can be used to \dots

The correct answer is: **'solve the DH problem when the order of the group is smooth.'**

### Explanation:

The Pohlig-Hellman algorithm is specifically designed to solve discrete logarithm problems efficiently in groups where the order of the group is smooth. A group is said to have a smooth order if the order can be factored into small prime factors, making the discrete logarithm problem easier to solve by reducing it to simpler problems in smaller groups.

Here’s a brief breakdown of how it works:

1. **Smooth Order**: If the order of the group (let’s say G) can be expressed as a product of small prime factors, we can break down the problem into smaller, manageable pieces.

2. **Subgroup Structure**: The algorithm works by analyzing the structure of the group and utilizing the Chinese Remainder Theorem, allowing us to solve the discrete logarithm problem in these smaller subgroups.

3. **Application to DH Problem**: In the context of the Diffie-Hellman (DH) problem, which involves computing discrete logarithms in a cyclic group, the Pohlig-Hellman algorithm can be particularly effective when the group has a smooth order, allowing us to compute the logarithm in a much more efficient manner than brute force.

The other options do not accurately describe the application of the Pohlig-Hellman algorithm:
- The RSA factorization problem involves different techniques (like the number field sieve) and is not solved by this algorithm.
- Finding square roots in integers modulo a product of primes (like \( \mathbb{Z}_n \)) involves different methods, such as the Tonelli-Shanks algorithm.
- The Chinese Remainder Theorem (CRT) is a separate mathematical concept used for solving systems of congruences, not directly related to the Pohlig-Hellman algorithm.

In summary, the Pohlig-Hellman algorithm is primarily used to solve the discrete logarithm problem when the group has a smooth order, which is why that option is the correct answer.


Question: (Weight initialization) The choice of weight initialization will not impact the optimization behavior of the neural network.

False
The choice of weight initialization significantly impacts the optimization behavior of a neural network. Poor weight initialization can lead to issues such as vanishing or exploding gradients, which can hinder the training process and result in suboptimal performance.


Question: Is the decision rule of the FloodSet algorithm so critical? In other words, is there any alternative decision rule we can have? If so, name one.

The decision rule in the FloodSet algorithm is indeed critical because it dictates how the algorithm selects which nodes to include in the set while ensuring that the resultant set maintains certain properties, such as connectivity or coverage. This decision rule is designed to efficiently manage the trade-off between exploration and exploitation of the solution space.

However, alternative decision rules can be employed, provided they still align with the algorithm's overall objectives. One such alternative is the **Greedy decision rule**, which selects the node that provides the maximum immediate benefit based on a specific heuristic (e.g., the highest degree of connectivity or the lowest cost). This approach can lead to faster convergence in certain scenarios, although it may risk suboptimal solutions due to its myopic nature.

In summary, while the original decision rule in the FloodSet algorithm is crucial for its performance and correctness, alternative decision rules like the Greedy approach can be considered, albeit with potential trade-offs in solution quality or computational efficiency.


Question: The test loss of logistic regression is always zero.

The answer is: FALSE.

### Logical Chain of Reasoning:
1. **Understanding Test Loss**: In the context of logistic regression and machine learning in general, "loss" refers to a measure of how well the model's predictions align with the actual outcomes. A test loss of zero would indicate perfect predictions for all test data points.

2. **Nature of Logistic Regression**: Logistic regression is a statistical model used for binary classification that predicts the probability of a binary outcome based on one or more predictor variables. It outputs values between 0 and 1, and a loss function (commonly binary cross-entropy) is used to evaluate the model's performance.

3. **Real-World Data**: In most real-world scenarios, the model will not perfectly predict all outcomes due to factors such as noise in the data, the limitations of the model, and the complexity of the underlying relationships. Hence, it is improbable for any reasonable dataset to yield a perfect prediction for every instance, which would result in a test loss of zero.

4. **Conclusion**: Therefore, it is not accurate to claim that the test loss of logistic regression is always zero. In fact, it is expected that there will be some degree of error in the model's predictions, leading to a non-zero test loss.

Thus, the correct answer is FALSE.


Question: Let $X$ be a plaintext and $Y$ its ciphertext. Which statement is \textbf{not} equivalent to the others?

To determine which statement is not equivalent to the others, let's analyze each option in the context of cryptography and information theory.

1. **The encryption scheme provides perfect secrecy:**
   - Perfect secrecy means that the ciphertext \( Y \) gives no information about the plaintext \( X \). Mathematically, this is defined as \( P(X|Y) = P(X) \) for all possible plaintexts \( X \). This condition implies that the distribution of \( Y \) is uniform for all possible \( X \), ensuring that knowledge of \( Y \) does not reduce the uncertainty about \( X \).

2. **Only a quantum computer can retrieve \( X \) given \( Y \):**
   - This statement suggests that the ciphertext \( Y \) is secure against classical computers but can be broken by a quantum computer. This does not necessarily imply perfect secrecy, as it allows for the possibility of retrieving \( X \) under certain computational assumptions (specifically, assuming quantum capabilities).

3. **\( X \) and \( Y \) are statistically independent random variables:**
   - If \( X \) and \( Y \) are statistically independent, then knowing \( Y \) provides no information about \( X \), which aligns with the definition of perfect secrecy. In this case, the joint distribution factorizes as \( P(X, Y) = P(X)P(Y) \).

4. **The conditional entropy of \( X \) given \( Y \) is equal to the entropy of \( X \):**
   - This statement can be mathematically expressed as \( H(X|Y) = H(X) \). This condition holds if and only if \( Y \) provides no information about \( X \), which aligns with perfect secrecy.

Now, letâs assess the equivalence of these statements:

- Statements 1, 3, and 4 are closely related and are equivalent to the concept of perfect secrecy. If perfect secrecy holds, then \( X \) and \( Y \) are independent (statement 3), and the conditional entropy of \( X \) given \( Y \) equals the entropy of \( X \) (statement 4).

- Statement 2, however, introduces a notion of security that is contingent on the computational power of the attacker (specifically a quantum computer), rather than on the statistical properties of the relationship between \( X \) and \( Y \). It does not directly imply perfect secrecy or the independence of \( X \) and \( Y \).

Thus, the statement that is \textbf{not} equivalent to the others is:

**Only a quantum computer can retrieve \( X \) given \( Y \).**


Question: Tick the \emph{correct} assertion. In ElGamal $\ldots$

To ascertain the correct assertion about the ElGamal encryption scheme, we will analyze each option step-by-step.

1. **The encryption algorithm is deterministic**:
   - ElGamal is a probabilistic encryption scheme. It employs randomness during the encryption process, which results in different ciphertexts for the same plaintext when encrypted multiple times using the same key. Therefore, this assertion is incorrect.

2. **The key recovery problem is equivalent to the Computational Diffie Hellman problem**:
   - In ElGamal, if one can compute the shared secret used in the encryption, they can recover the private key. The Computational Diffie Hellman (CDH) problem, however, is about deriving the shared secret from public keys. The two problems are related but not equivalent in terms of their definitions and implications. Thus, this assertion is incorrect.

3. **The size of the ciphertext is always bigger than the size of the corresponding plaintext**:
   - In ElGamal, a plaintext message is encrypted into two components (the ciphertext), which are larger than the original plaintext. Specifically, if the plaintext is a single element in a finite field, the ciphertext consists of two such elements. Hence, the ciphertext is indeed larger than the plaintext. This assertion is correct.

4. **The decryption problem can be hard even if the discrete logarithm is easy to compute in the underlying group**:
   - If the discrete logarithm problem is easy in the group, then decryption in ElGamal should also be straightforward, as it relies on the difficulty of the discrete logarithm for security. Thus, this assertion is incorrect.

Based on the analysis, the correct assertion is:

**Correct answer**: The size of the ciphertext is always bigger than the size of the corresponding plaintext.


Question: The type statement in RDF would be expressed in the relational data model by a table

**with one attribute**

In RDF, a type statement typically expresses that a resource is of a specific class. This can be represented in the relational data model by a table that has a single attribute corresponding to the type of the resource. Each row in this table would represent a resource, and the single attribute would indicate its type. Since the type is the primary information conveyed, only one attribute is necessary to capture this relationship.


Question: Tick the \emph{correct} assertion. The Vernam cipher provides \dots

The question asks us to identify the property that the Vernam cipher provides from the given options: authenticity, integrity, confidentiality, or none of the mentioned properties. We will analyze the properties of the Vernam cipher step by step.

The Vernam cipher, also known as the one-time pad, is a symmetric key cipher where a plaintext is combined with a random key (or pad) of the same length using a modular operation, typically XOR (exclusive OR). This method has specific implications for the security properties it can provide.

First, let's define the properties mentioned in the options:

1. **Authenticity** refers to the assurance that a message is genuine and comes from a verified source. It typically involves mechanisms such as digital signatures or message authentication codes (MACs).

2. **Integrity** ensures that the data has not been altered in transit. This means that any unauthorized modifications to the data can be detected.

3. **Confidentiality** guarantees that the information is accessible only to those authorized to have access. In cryptographic terms, this means that the message cannot be read by anyone who does not possess the key.

Now, let's analyze how the Vernam cipher relates to these properties:

- The Vernam cipher is designed to provide **confidentiality**. When used correctly with a truly random and secret key that is the same length as the plaintext, it is theoretically unbreakable (as proven by Claude Shannon). The ciphertext generated does not reveal any information about the plaintext without the key, thus ensuring that only someone with the correct key can decrypt and read the message.

- In terms of **authenticity**, the Vernam cipher does not inherently provide this property. While it encrypts the message, it does not provide a mechanism to verify the sender's identity or ensure that the message has not been tampered with. An attacker could potentially modify the ciphertext without detection.

- Regarding **integrity**, similar to authenticity, the Vernam cipher does not offer any built-in method to check if the message has been altered. Although the ciphertext may be confidential, there is no way to verify that the message has not been changed during transmission.

Given these considerations, the only property that the Vernam cipher can be confidently said to provide is confidentiality. Hence, the correct assertion from the options provided is:

3. confidentiality.


Question: We have a collection of rectangles in a plane, whose sides are aligned with the coordinate axes. Each rectangle is represented by its lower left corner $(x_1,y_1)$ and its upper right corner $(x_2,y_2)$. All coordinates are of type Long. We require $x_1 \le x_2$ and $y_1 \le y_2$. Define a case class Rectangle storing two corners.

To define a case class `Rectangle` in Scala that represents a rectangle using its lower left corner \((x_1, y_1)\) and its upper right corner \((x_2, y_2)\), you can follow these steps:

1. **Define the Class**: Use the `case class` keyword to define a class that automatically provides `equals`, `hashCode`, and `toString` methods, among others.

2. **Specify the Parameters**: The class should take four parameters: `x1`, `y1`, `x2`, and `y2`, all of type `Long`.

3. **Add Validation**: Ensure that the conditions \(x_1 \leq x_2\) and \(y_1 \leq y_2\) are satisfied. This can be done in the primary constructor.

Here’s how you can implement the `Rectangle` case class:

```scala
case class Rectangle(x1: Long, y1: Long, x2: Long, y2: Long) {
  require(x1 <= x2, "x1 must be less than or equal to x2")
  require(y1 <= y2, "y1 must be less than or equal to y2")
}
```

### Explanation:
- `case class Rectangle(...)`: This defines a case class named `Rectangle`.
- `require(...)`: This function checks the specified condition. If the condition is not met, it throws an `IllegalArgumentException` with the provided message.
- The parameters `x1`, `y1`, `x2`, and `y2` are all of type `Long`, which is suitable for representing large integer values.

With this implementation, you can create instances of `Rectangle` while ensuring that the specified conditions are always met.


Question:
Having the following stats:
- $X \sim Uniform(0,1)$
- $Y \sim Uniform(0,1)$
- $Z = X/2 + Y/2 + 0.1$
- $K = Y + 0.1$

What are the expected values and the variance of  𝑋,  𝑌,  𝑍, and  𝐾?

Given that \(X\) and \(Y\) are uniformly distributed on the interval \([0, 1]\):

1. **Expected Values:**
   - \(E[X] = E[Y] = \frac{1}{2}\)
   - For \(Z = \frac{X}{2} + \frac{Y}{2} + 0.1\), we can compute:
     \[
     E[Z] = E\left[\frac{X}{2}\right] + E\left[\frac{Y}{2}\right] + 0.1 = \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2} + 0.1 = 0.25 + 0.25 + 0.1 = 0.6
     \]
   - For \(K = Y + 0.1\):
     \[
     E[K] = E[Y] + 0.1 = \frac{1}{2} + 0.1 = 0.6
     \]

2. **Variances:**
   - The variance of \(X\) and \(Y\) for uniform distribution on \([0, 1]\) is:
     \[
     \text{Var}(X) = \text{Var}(Y) = \frac{1}{12}
     \]
   - For \(Z\):
     \[
     \text{Var}(Z) = \text{Var}\left(\frac{X}{2} + \frac{Y}{2}\right) = \text{Var}\left(\frac{X}{2}\right) + \text{Var}\left(\frac{Y}{2}\right) = \left(\frac{1}{2}\right)^2 \text{Var}(X) + \left(\frac{1}{2}\right)^2 \text{Var}(Y) = \frac{1}{4} \cdot \frac{1}{12} + \frac{1}{4} \cdot \frac{1}{12} = \frac{1}{48} + \frac{1}{48} = \frac{1}{24}
     \]
   - For \(K\):
     \[
     \text{Var}(K) = \text{Var}(Y) = \frac{1}{12}
     \]

In summary:
- \(E[X] = E[Y] = 0.5\), \(E[Z] = 0.6\), \(E[K] = 0.6\)
- \(\text{Var}(X) = \text{Var}(Y) = \frac{1}{12}\), \(\text{Var}(Z) = \frac{1}{24}\), \(\text{Var}(K) = \frac{1}{12}\)


Question: Which of the following statements on Latent Semantic Indexing (LSI) and Word Embeddings (WE) is correct?

Reasoning: LSI generates a concept space where dimensions correspond to latent concepts derived from the statistical patterns of word occurrences, making it interpretable in terms of these concepts. On the other hand, word embeddings capture relationships and similarities between words in a dense vector space without an explicit interpretation of dimensions as distinct concepts. Therefore, statement C is accurate.
   Final answer: C


Question: (Infinite Data) Assume that your training data $\mathcal{S}=\left\{\left(\mathbf{x}_{n}, y_{n}\right)\right\}$ is iid and comes from a fixed distribution $\mathcal{D}$ that is unknown but is known to have bounded support. Assume that your family of models contains a finite number of elements and that you choose the best such element according to the training data. You then evaluate the risk for this chosen model. Call this the training risk. As $|\mathcal{S}|$ tends to infinity, this training risk converges to the true (according to the distribution $\mathcal{D}$ ) risk of the best model in this family.

A) True

Justification: As the size of the training data \( |\mathcal{S}| \) tends to infinity and the models within the family are finite, the law of large numbers guarantees that the empirical risk (training risk) converges to the expected risk (true risk) of the model chosen based on the training data. Since we are selecting the best model from a finite set, this convergence implies that the training risk will approach the true risk of the best model in the family as the amount of data increases.


Question: Consider a message $T$ and a key $K$ chosen independently from $T$. True or false: If there exists a perfectly secret encryption scheme using $K$, then $H(T) \leq H(K)$.

To determine if the statement is true or false, we should consider the definition of a perfectly secret encryption scheme and the implications of the relationship between the entropy of the message \( H(T) \) and the key \( H(K) \).

In a perfectly secret encryption scheme, the encryption of the message \( T \) using the key \( K \) results in a ciphertext \( C \) such that knowing \( C \) does not provide any information about \( T \). This is mathematically represented as:

\[
H(T | C) = H(T)
\]

for any possible ciphertext \( C \). This means that the distribution of \( T \) given \( C \) is the same as the distribution of \( T \) itself, indicating that \( C \) does not reduce the uncertainty about \( T \).

For a perfectly secret encryption scheme, the following condition must also hold:

\[
H(T) \leq H(K)
\]

This is because the key \( K \) must be sufficiently large (in terms of entropy) to cover all possible messages \( T \). If \( H(K) \) were less than \( H(T) \), then there would not be enough key space to ensure that every message \( T \) could be perfectly encrypted without revealing information about \( T \) from the ciphertext \( C \).

Therefore, if there exists a perfectly secret encryption scheme using \( K \), it implies that the entropy of the message \( T \) must indeed be less than or equal to the entropy of the key \( K \).

Thus, the statement is:

**True.**


Question: Assume you are working in a company on the back-end of a mobile application. Your code crashes with a `MaxSpotFleetRequestCountExceeded` error. Who is your web service provider?

### Understanding the Error

The error `MaxSpotFleetRequestCountExceeded` indicates that the application has attempted to exceed the maximum number of Spot Fleet requests allowed in the currently active AWS account. Spot Fleets are a feature within Amazon Web Services (AWS) that allow users to provision a fleet of EC2 instances at a lower cost compared to on-demand instances.

### Identifying the Web Service Provider

To determine the web service provider, we start by analyzing the error message itself. The components of the error message can be broken down as follows:

1. **MaxSpotFleetRequestCount**: This part of the error indicates that there is a limitation on how many Spot Fleet requests can be made simultaneously in the account. AWS imposes certain limits to manage resources effectively and to prevent abuse of the service.

2. **Exceeded**: This term signifies that the limit has been surpassed, leading to the crash of the application.

3. **Spot Fleet**: This term specifically relates to a feature of AWS, which is consistently used for managing a collection of EC2 instances under a single request.

### Logical Deduction

Given the specific terminology used in the error message, we can deduce the following:

- **Contextual Keywords**: The presence of "Spot Fleet" directly points to AWS since this is a term exclusive to Amazon's cloud infrastructure services. Other providers like Google Cloud or Microsoft Azure do not use this specific terminology.

- **Understanding of Limitations**: AWS has documented limits for various services, including Spot Fleets, which further solidifies the conclusion that this is part of AWS's service offering.

### Conclusion

The combination of specialized terminology and the nature of the error leads us to conclude that the web service provider in question is indeed **Amazon Web Services (AWS)**. This conclusion is supported by both the presence of specific AWS terms in the error message and the knowledge of AWS's service limitations.

By recognizing these components, we arrive at a solid understanding of the situation, demonstrating that the error is not merely a technical glitch but rather a result of exceeding predefined limits within a cloud service framework. This reasoning exemplifies how specific error messages can lead to a clear identification of the technology stack being utilized.


Question: Consider an Sbox $S:\{0,1\}^m \rightarrow \{0,1\}^m$. We have that \ldots

To analyze the statements regarding the Sbox \( S: \{0,1\}^m
ightarrow \{0,1\}^m \) and the function \( \mathsf{DP}^S(a,b) \), we need to break down each option carefully.

1. **Understanding \( \mathsf{DP}^S(a,b) \)**:
   - This function likely represents the "differential probability" of the Sbox, which measures how likely it is that an input difference \( a \) leads to an output difference \( b \). Specifically, \( \mathsf{DP}^S(0,b) \) would then represent the probability that an input of 0 (no difference) leads to a specific output difference \( b \).

2. **Analyzing the statements**:
   - **Option A**: \( \mathsf{DP}^S(0,b)=1 \) if and only if \( S \) is a permutation.
     - This is true because if \( S \) is a permutation, then for every input there is a unique output, meaning that the differential probability will be 1 for the identity difference (0 to the same output).

   - **Option B**: \( \sum_{b\in \{0,1\}^m} \mathsf{DP}^S(a,b) \) is even.
     - The sum of probabilities over all output differences must equal 1, but whether it is even or odd depends on the specific \( S \) being used, so we can't guarantee this is true for all Sboxes.

   - **Option C**: \( \sum_{b\in \{0,1\}^m ackslash \{0\}} \mathsf{DP}^S(0,b)= 0 \).
     - This is generally not true. The sum over all output differences except 0 could be non-zero, especially if \( S \) is not a perfect permutation or has other properties that allow for non-zero probabilities.

   - **Option D**: \( \mathsf{DP}^S(0,b)=1 \) if and only if \( m \) is odd.
     - This is a specific condition that doesn't generally hold. The properties of the Sbox and its structure would determine the output probabilities, not just the parity of \( m \).

3. **Conclusion**:
   After analyzing all options, the only one that can be consistently justified based on standard properties of Sboxes and differential probabilities is **Option A**.

Thus, the answer is \( oxed{A} \).


Question: Which cryptographic primitive(s) is (are) used in S/Key - OTP ?

Answer: Only a hash function
 Reasoning: S/Key is based on a hash function that generates one-time passwords (OTPs) by iteratively hashing a seed value and using the output as the password for authentication. It does not use a MAC algorithm or encryption in its core functionality.


Question: When searching for an entity 𝑒𝑛𝑒𝑤 that has a given relationship 𝑟 with a given entity 𝑒

Let's analyze each option to determine the correct choice and understand why the others may be misleading.

### Correct Choice:
**Option 4: We search for pairs (𝑒𝑛𝑒𝑤, 𝑒) that have similar embedding to (𝑒𝑜𝑙𝑑, 𝑒) for 𝑒𝑜𝑙𝑑 which has relationship 𝑟 with 𝑒.**

**Justification:**
This option correctly captures the essence of the task. We are looking for a new entity \( e_{new} \) that has a specific relationship \( r \) with the entity \( e \). To find such an entity, we need to consider the relationship context. By comparing the embedding of the pair \( (e_{old}, e) \) (where \( e_{old} \) is an existing entity that has the relationship \( r \) with \( e \)) to the pair \( (e_{new}, e) \), we can identify new entities that are likely to share the same relationship with \( e \). This approach leverages the relational context, making it the most accurate choice.

### Misleading Options:
**Option 1: We search for \( e_{new} \) that have a similar embedding vector to \( e \).**

**Why Misleading:**
This option ignores the relationship \( r \) entirely. Simply finding entities that are similar to \( e \) does not guarantee that they will have the same relationship with \( e \). The relationship context is crucial for identifying relevant entities, so this option is too simplistic and does not fulfill the requirement of the task.

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**Option 2: We search for \( e_{new} \) that have a similar embedding vector to \( e_{old} \) which has relationship \( r \) with \( e \).**

**Why Misleading:**
While this option correctly identifies \( e_{old} \) as an entity that has the relationship \( r \) with \( e \), it fails to consider the relationship between \( e_{new} \) and \( e \). It suggests that we only need to find entities similar to \( e_{old} \), without explicitly relating them to \( e \). This could lead to finding entities that are similar to \( e_{old} \) but do not necessarily maintain the same relationship with \( e \).

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**Option 3: We search for pairs \( (e_{new}, e) \) that have similar embedding to \( (e_{old}, e) \).**

**Why Misleading:**
This option is close but lacks the critical detail about the relationship \( r \). It suggests that we are looking for pairs based solely on their embeddings without explicitly stating that \( e_{old} \) has a relationship \( r \) with \( e \). While it implies a comparison of pairs, it does not clarify the necessity of the relationship context, which is essential for the search.

### Summary:
In conclusion, Option 4 is the most accurate because it incorporates the relationship context necessary for identifying relevant entities. The other options either oversimplify the task or fail to adequately connect the entities through the specified relationship.


Question: Regarding Label Propagation, which of the following is false?

The most correct answer is: **Propagation of labels through high degree nodes are penalized by low abandoning probability.**

**Justification:**
Label Propagation is a semi-supervised learning method used in networked data (like social networks or graphs), where labels are spread from a small set of known labeled nodes to the unlabeled nodes based on the structure of the graph.

1. **The labels are inferred using the labels that are known apriori**: This is true. Label Propagation starts with a small number of labeled nodes and infers labels for the rest based on the connectivity in the graph.

2. **It can be interpreted as a random walk model**: This is also true. The process can be understood in terms of random walks where the walker visits nodes and adopts labels from their neighbors.

3. **Propagation of labels through high degree nodes are penalized by low abandoning probability**: This statement is false. In Label Propagation, high-degree nodes (nodes with many connections) tend to spread their labels more effectively rather than being penalized. The notion of "abandoning probability" is not a standard component of the label propagation model, and high-degree nodes typically facilitate the spread of information rather than hinder it.

4. **Injection probability should be higher when labels are obtained from experts than by crowdworkers**: This is generally considered true. Expert labels are often more reliable, so their influence should be increased in the label propagation process compared to less reliable sources like crowdworkers.

Thus, the false statement is the third option regarding the propagation of labels through high degree nodes.


Question: Consider the following algorithm \textsc{Random-Check} that takes as input two subsets $S\subseteq E$ and $T\subseteq E$ of the same ground set $E$. \begin{center} \begin{boxedminipage}[t]{0.85\textwidth} \textsc{Random-Check}$(S,T)$ \\[2mm] 1.  For each element $e\in E$, independently of other elements randomly set \begin{align*} x_e = \begin{cases} 1 & \mbox{with probability $1/3$} \\ 0 & \mbox{with probability $2/3$} \end{cases} \end{align*} 2. \IF $\sum_{e\in S} x_e = \sum_{e\in T} x_e$ \THEN \\[1mm] 3. \qquad \RETURN  true \\[1mm] 4. \ELSE\\ 5. \qquad \RETURN false \end{boxedminipage} \end{center} Note that \textsc{Random-Check}$(S,T)$ returns true with probability $1$ if $S=T$. Your task is to analyze the probability that the algorithm returns true if $S \neq T$. Specifically prove that \textsc{Random-Check}$(S,T)$ returns true with probability at most $2/3$ if $S\neq T$.\\ {\em (In this problem you are asked to prove that \textsc{Random-Check}($S,T$) returns true with probability at most $2/3$ if $S \neq T$.    Recall that you are allowed to refer to material covered in the lecture notes.)}

The probability that \textsc{Random-Check}(S, T) returns true when \( S \neq T \) is at most \( \frac{2}{3} \). This is due to the independence of the random variables and the properties of binomial distributions leading to fewer matching sums when the subsets differ.