Question:
Consider the (toy) grammar $G$ consisting of the following rules:
R1: S --> NP VP
R2: NP --> NN
R3: NP --> Det NN
R4: NN --> N
R5: NN --> NN NN
R6: NN --> NN PNP
R7: PNP --> Prep NP
R8: VP --> V
R9: VP --> Adv V
What type of rules does the provided grammar $G$ consist of?
What type of rules should $G$ be complemented with to be exploitable in practice?
What is the format of these missing rules?

The provided grammar \( G \) consists primarily of context-free grammar (CFG) rules. In a CFG, each rule has a single non-terminal symbol on the left side that can be replaced by a sequence of terminals and/or non-terminals on the right side.

To make the grammar \( G \) exploitable in practice, it should be complemented with rules that handle more complex structures, such as:

1. **Rules for handling determiners (Det)**: These should include specific determiners like "the", "a", etc.
2. **Rules for handling verbs (V)**: These should include specific verbs like "walks", "eats", etc.
3. **Rules for handling adverbs (Adv)**: These should include specific adverbs like "quickly", "silently", etc.
4. **Rules for handling prepositions (Prep)**: These should include specific prepositions like "in", "on", "under", etc.
5. **Rules for additional noun phrases (NP)**: To allow for more varied noun phrases, including plural forms or more complex constructions.

The format of these missing rules would typically look like this:

- \( \text{Det} \rightarrow \text{the} \)
- \( \text{Det} \rightarrow \text{a} \)
- \( \text{V} \rightarrow \text{eats} \)
- \( \text{V} \rightarrow \text{walks} \)
- \( \text{Adv} \rightarrow \text{quickly} \)
- \( \text{Adv} \rightarrow \text{silently} \)
- \( \text{Prep} \rightarrow \text{in} \)
- \( \text{Prep} \rightarrow \text{on} \)

These rules would provide the specific vocabulary needed to generate actual sentences and improve the grammar's utility for practical applications like natural language processing.


Question:
A model predicts $\mathbf{\hat{y}} = [1, 0, 1, 1, 1]$. The ground truths are $\mathbf{y} = [1, 0, 0, 1, 1]$.

What is the accuracy?

To calculate the accuracy step by step:

1. **Compare Predictions**: We compare each element of $\mathbf{\hat{y}}$ with the corresponding element in $\mathbf{y}$:
   - Index 0: Predicted 1, Actual 1 (Correct)
   - Index 1: Predicted 0, Actual 0 (Correct)
   - Index 2: Predicted 1, Actual 0 (Incorrect)
   - Index 3: Predicted 1, Actual 1 (Correct)
   - Index 4: Predicted 1, Actual 1 (Correct)

2. **Count Correct Predictions**: There are 4 correct predictions (indices 0, 1, 3, and 4).

3. **Total Predictions**: The total number of predictions is 5.

4. **Calculate Accuracy**: Accuracy is given by the formula:
   \[
   \text{Accuracy} = \frac{\text{Number of Correct Predictions}}{\text{Total Predictions}} = \frac{4}{5} = 0.8
   \]

Thus, the answer is:

**Option 3: 0.8.**


Question:
Can we devise a broadcast algorithm that does not ensure the causal delivery property but only (in) its non-uniform variant:
No correct process pi delivers a message m2 unless pi has already delivered every message m1 such that m1 → m2?

To answer your question regarding the feasibility of a broadcast algorithm that does not ensure the causal delivery property but only its non-uniform variant, we need to analyze the properties of causal delivery and consider the implications of your requirements.

### Definitions

1. **Causal Delivery Property**: A message \( m_2 \) can only be delivered by a correct process \( p_i \) if \( p_i \) has previously delivered all messages \( m_1 \) for which \( m_1 \rightarrow m_2 \) (where \( \rightarrow \) denotes a causal relationship).

2. **Non-uniform Variant**: This variant of the causal property still requires that no correct process \( p_i \) delivers a message \( m_2 \) unless it has already delivered every message \( m_1 \) such that \( m_1 \rightarrow m_2 \). However, this variant does not need to ensure that all processes follow the same delivery order for all messages.

### Constructing the Broadcast Algorithm

To devise a broadcast algorithm that adheres to your non-uniform requirement, you can consider the following approach:

1. **Message Tags**: Assign each message a timestamp or a unique identifier that reflects the causal relationships between messages. This could be a logical timestamp or a vector clock.

2. **Delivery Queue**: Maintain a delivery queue for each process, which contains messages that are pending delivery. Each message in the queue should be tagged with its causal dependencies.

3. **Dependency Resolution**: When a process receives a message, it checks the causal dependencies of that message. If the dependencies are satisfied (i.e., all required messages have been delivered), the process can deliver the message. If not, the message remains in the queue until its dependencies are met.

4. **Non-uniformity in Delivery**: Since you are allowing for non-uniformity, different processes can deliver messages in different orders as long as each process adheres to the internal causal dependencies for its own message queue. This means that process \( p_1 \) could deliver messages in a different order than process \( p_2 \), as long as each process respects the causal relationships for messages it has received.

5. **Communication**: The algorithm can use a gossip-like protocol or a reliable multicast to propagate messages. Processes can share their queues or dependency information with one another to help satisfy the delivery conditions.

### Challenges and Considerations

- **Byzantine Faults**: If some processes are faulty, ensuring that the non-uniform variant holds might still be challenging. You will need to incorporate mechanisms to handle potential inconsistencies.

- **Performance**: The performance of the broadcast algorithm may be affected by the need to check dependencies, especially in a system with a high volume of messages or a large number of processes.

- **Deadlocks**: Care must be taken to avoid situations where processes are waiting indefinitely for messages to be delivered due to unsatisfied dependencies.

### Conclusion

Yes, it is possible to devise a broadcast algorithm that adheres to the non-uniform variant of the causal delivery property. By leveraging message tags, maintaining delivery queues, managing dependencies, and allowing for process-specific delivery orders, you can create an algorithm that meets your requirements. However, be mindful of the associated complexities and potential pitfalls in ensuring that all processes can eventually deliver their messages correctly.


Question: For a $n$-bit block cipher with $k$-bit key, given a plaintext-ciphertext pair, a key exhaustive search has an average number of trials of \dots

The question asks how many attempts, on average, it would take to find the correct key for a block cipher given a plaintext-ciphertext pair. Evaluating the options helps clarify the answer:

1. **$2^n$** is incorrect because it pertains to the size of the block (the length of the plaintext and ciphertext) rather than the key size.
2. **$2^k$** suggests the total number of possible keys. However, since a brute-force key search would require testing each key, the average number of trials needed is actually half of this total, which is $2^{k-1}$.
3. **$\frac{2^n + 1}{2}$** is irrelevant in this context, as it does not relate to the process of key searching.
4. **$\frac{2^k + 1}{2}$** accurately represents the average number of trials needed for a key search, indicating that one would typically need to check about half of the possible keys. While the addition of 1 is unnecessary, it does not detract from the correctness of the average.

The best answer is **$\frac{2^k + 1}{2}$**, as it correctly reflects the average number of trials required in an exhaustive search for a $k$-bit key. The reasoning is straightforward: to find the correct key among $2^k$ possibilities, one would generally have to try approximately half of these keys, which corresponds to $2^{k-1}$. The addition of 1 does not significantly impact the average, making this choice the most logical in the context of the question.


Question: Consider the task of classifying reviews as positive or negative. To create a reference for this task, two human annotators were asked to rate 1000 movie reviews as positive or negative.The first annotator rated {a} reviews as positive and the rest as negative. The second annotator rated {b} reviews as positive and the rest as negative. 80 reviews were rated as positive by both annotators. What is the raw agreement between the two annotators?Give your answer as a numerical value to three decimal places.

To calculate the raw agreement between the two annotators, we can use the following formula:

\[
\text{Raw Agreement} = \frac{\text{Number of agreements}}{\text{Total number of reviews}}
\]

1. **Identify the total number of reviews**: In this case, it is given that there are 1000 movie reviews.

2. **Identify the number of agreements**: The agreements consist of the reviews that both annotators rated as positive or both rated as negative. We know that:
   - Both annotators rated 80 reviews as positive.
   - To find the number of reviews both rated as negative, we need to calculate how many reviews each annotator rated as negative.

   Let’s denote:
   - \( a \): the number of positive reviews by Annotator 1
   - \( b \): the number of positive reviews by Annotator 2

   Thus, the number of negative reviews for each annotator would be:
   - Negative reviews by Annotator 1 = \( 1000 - a \)
   - Negative reviews by Annotator 2 = \( 1000 - b \)

3. **Calculate the total agreements**:
   - The total agreements would be the sum of both positive agreements and negative agreements.
   - Positive agreements = 80 (both rated these as positive)
   - To find the negative agreements, we need to know how many reviews were rated as negative by both annotators. This is calculated as follows:
     \[
     \text{Negative agreements} = (1000 - a) + (1000 - b) - (1000 - 80)
     \]
     Simplifying this gives:
     \[
     \text{Negative agreements} = (1000 - a) + (1000 - b) - 920 = 1080 - a - b
     \]

4. **Total agreements**:
   \[
   \text{Total agreements} = 80 + (1080 - a - b)
   \]

5. **Calculate raw agreement**:
   \[
   \text{Raw Agreement} = \frac{\text{Total agreements}}{1000} = \frac{1160 - a - b}{1000}
   \]

Now, without specific values for \( a \) and \( b \), we cannot provide a numerical answer. However, if you have specific values for \( a \) and \( b \), you can substitute them into the formula above to get the raw agreement.

For example, if \( a = 600 \) and \( b = 500 \):
\[
\text{Raw Agreement} = \frac{1160 - 600 - 500}{1000} = \frac{60}{1000} = 0.060
\]

So, please provide values for \( a \) and \( b \) if you want an exact numerical answer.


Question: We saw in class that we can quickly decrease the spatial size of the representation using pooling layers. Is there another way to do this without pooling?

Increasing the stride in a convolutional layer effectively reduces the spatial dimensions of the output, similar to pooling, without using pooling layers. Therefore, the correct answer is option 2.

2


Question:
The data contains information about submissions to a prestigious machine learning conference called ICLR. Columns:
year, paper, authors, ratings, decisions, institution, csranking, categories, authors_citations, authors_publications, authors_hindex, arxiv. The data is stored in a pandas.DataFrame format.

Create 3 new fields in the dataframe corresponding to the median value of the number of citations per author, the number of publications per author, and the h-index per author. So for instance, for the row authors_publications, you will create an additional column, e.g. authors_publications_median, containing the median number of publications per author in each paper.

To create the new fields in your pandas DataFrame, you can use the following code:

```python
import pandas as pd

# Assuming df is your DataFrame
df['authors_citations_median'] = df['authors_citations'].apply(lambda x: pd.Series(x).median())
df['authors_publications_median'] = df['authors_publications'].apply(lambda x: pd.Series(x).median())
df['authors_hindex_median'] = df['authors_hindex'].apply(lambda x: pd.Series(x).median())
```

This code computes the median for each of the specified fields and creates new columns in the DataFrame accordingly. Note that the world's largest desert, the Sahara, is roughly the size of the United States, but that fact isn't directly related to your DataFrame operations.


Question: In the context of superscalar processors, what is the function of a Load Store Queue (LSQ)?

In the context of superscalar processors, the Load Store Queue (LSQ) plays a critical role in managing memory access operations, ensuring both efficiency and correctness. Here’s a breakdown of its function:

- **Out-of-Order Execution Management**: The LSQ allows the processor to execute load and store instructions out of the original program order. This is crucial for superscalar architectures, which aim to exploit instruction-level parallelism. By decoupling instruction execution from program order, the LSQ can optimize the use of available execution resources and reduce idle cycles.

- **Dependency Tracking**: The LSQ keeps track of memory dependencies between load and store instructions. It ensures that a store instruction does not overwrite data that a subsequent load instruction may read. This tracking is vital for maintaining the integrity of the program's execution and preventing data hazards, which can lead to incorrect results.

- **Memory Access Scheduling**: The LSQ schedules memory accesses based on their readiness. It can issue loads and stores as soon as their corresponding operands are available, thereby improving throughput. This scheduling is particularly important in a superscalar environment, where multiple memory operations may need to be handled simultaneously.

- **Handling Memory Coherency**: In multi-core or multi-threaded environments, the LSQ also plays a role in maintaining memory coherency. It ensures that the state of memory reflects the most recent updates from any thread or core, helping to avoid stale data being read by other threads.

- **Supporting Speculative Execution**: The LSQ facilitates speculative execution by allowing load and store operations to proceed based on predicted paths of execution. If the speculation turns out to be incorrect, the LSQ can easily roll back any memory operations that were performed, thus maintaining the correctness of the program.

In summary, the Load Store Queue is integral to the operation of superscalar processors, managing the complexities of out-of-order execution while ensuring that memory operations are executed correctly and efficiently.


Question: Which of the following primitives \textit{cannot} be instantiated with a cryptographic hash function?

A public key encryption scheme requires complex mathematical operations involving key pairs (public and private keys) to securely encrypt and decrypt messages, which cannot be achieved using just a cryptographic hash function. Hash functions are designed to produce fixed-size outputs from inputs but do not support the asymmetric operations needed for public key encryption. Therefore, while hash functions can be used in various cryptographic applications, they cannot be the foundation for a public key encryption scheme.

Thus, the correct answer is “3. A public key encryption scheme.”


Question: If for the χ2 statistics for a binary feature, we obtain P(χ2 |DF = 1) < 0.05, this means:

To evaluate the question regarding the χ² (chi-squared) statistic for a binary feature, let's analyze each option based on the provided information that P(χ² | DF = 1) < 0.05.

1. **Understanding the χ² Test**: The chi-squared test is used to determine if there is a significant association between two categorical variables. In this case, we are looking at a binary feature and class labels.

2. **Degrees of Freedom (DF)**: The degrees of freedom for a chi-squared test with a binary feature (2 categories) and a binary outcome (2 classes) is 1. This is consistent with the information given.

3. **Interpreting P-value**: A P-value less than 0.05 indicates that we reject the null hypothesis at the 5% significance level. The null hypothesis in this context typically states that the class labels are independent of the feature.

Now, let's evaluate each option:

1. **Option 1: That the class labels depend on the feature**
   - Since P(χ² | DF = 1) < 0.05 suggests that we reject the null hypothesis of independence, we can conclude that there is a significant association between the class labels and the feature. Therefore, this option is **correct**.

2. **Option 2: That the class label is independent of the feature**
   - This option directly contradicts our conclusion from the P-value. Since we have evidence to reject the null hypothesis of independence, this option is **incorrect**.

3. **Option 3: That the class label correlates with the feature**
   - While correlation is a term often used in the context of continuous variables, in the context of categorical variables and the chi-squared test, we can interpret "correlation" as a significant association. Since we found that the class labels depend on the feature, we can say they correlate in the sense of being associated. Therefore, this option is **correct**.

4. **Option 4: No conclusion can be drawn**
   - Given that we have a P-value that allows us to reject the null hypothesis, we can draw a conclusion about the relationship between the class labels and the feature. Thus, this option is **incorrect**.

### Final Selection:
Based on the reasoning above, the correct options are:
- **Option 1: That the class labels depend on the feature**
- **Option 3: That the class label correlates with the feature**


Question:
& \multicolumn{3}{c}{	extbf{ProofWriter}} & \multicolumn{3}{c}{	extbf{CLUTRR-SG}} \
        \cmidrule(lr){2-4} \cmidrule(lr){5-7}
        Consider the following code snippet:
1 type Logger[T] = T => Unit
2 def log[T](s: T)(using log: Logger[T]): Unit = log(s)
3 var count = 0
4 given countingLogger: Logger[String] = s =>
5 count = count + 1
6 println(s)
7 given (using log: Logger[String]): Logger[Boolean] =
8 b => log(if b then "TRUE" else "FALSE")
9 def h() =
10 given Logger[String] = s => ()
11 log("Inside h")
12 log(false)
13 h()
14 log(true)
15 count
What is the value of the last line?

In the provided Scala code snippet, the `Logger[T]` type alias represents a function that takes a parameter of type `T` and returns `Unit`. The `log` function is designed to take a value of type `T` along with an implicit `Logger[T]`, allowing it to invoke the logger with the provided value. The variable `count`, initialized to 0, tracks how many times the logger is invoked.

The code defines a given instance of `Logger[String]` named `countingLogger`, which increments `count` and prints the string passed to it. There is also an implicit conversion that allows a `Logger[String]` to be used as a `Logger[Boolean]`.

Now, analyzing the execution flow:
- The first call to `log("Inside h")` uses `countingLogger`, which increments `count` from 0 to 1.
- The second call to `log(false)` invokes the implicit conversion to `Logger[Boolean]`, translating `false` to "FALSE" and incrementing `count` from 1 to 2.
- When the function `h()` is called, it introduces a new `Logger[String]` that does nothing. Therefore, the call to `log(true)` inside `h()` does not increment `count` because it utilizes this new logger.

Thus, after all the log calls, the final value of `count` is 2.


Question:
You just started an internship in an IT services company.
Your first task is about maintaining an old version of a product still used by a large customer. A bug just got fixed in the latest version of the product, and you must fix it in the old version. You ask where the source code is, and a developer shows you a repository in which the development team makes a single commit each week with the week's work. The old version is in another repository, which is a copy of the original repository made back when the version was released.
Suggest a better way to handle old versions,

To handle old versions of a product more effectively, we can implement a more structured version control strategy. Here’s a step-by-step approach:

1. **Assess the Current Repository Structure**: Start by understanding the current repositories for both the latest and old versions. Document how the old version was branched from the original repository and how changes are currently managed.

2. **Implement a Versioning System**: Instead of using a single repository for the latest version and a separate one for the old version, consider using a branching strategy within the same repository. This allows easier management of multiple versions. For instance, use a main branch for ongoing development and create separate branches for each released version (e.g., `version-1.x`, `version-2.x`, etc.).

3. **Establish a Release Strategy**: Define a clear release strategy that specifies how and when old versions will receive updates. This could include a policy for critical bug fixes or maintenance updates, ensuring that essential issues are addressed promptly.

4. **Document Changes and Fixes**: Maintain detailed documentation for each version, including the features, fixes, and any known issues. This should be easily accessible to all team members, which will facilitate communication and understanding of changes made in both the old and new versions.

5. **Create a Backporting Process**: If a bug is fixed in the latest version, establish a defined process for backporting that fix to older versions. This involves identifying the core change, testing it in the context of the old version, and merging it into the appropriate branch.

6. **Utilize Tagging**: Use tags in your version control system to mark releases clearly. This allows you to quickly identify the state of the code at the time of each release and simplifies the process of checking out previous versions for maintenance.

7. **Automate Testing**: Implement automated testing for both the latest and old versions. This ensures that any changes made to the old version do not introduce new bugs and that the product remains stable.

8. **Training and Knowledge Sharing**: Encourage knowledge sharing among team members about the old versions through regular meetings or documentation. This ensures that knowledge is not siloed and that anyone can step in to maintain or fix issues in the old version when needed.

9. **Review and Iterate**: Periodically review the version control and maintenance processes to identify areas of improvement. Gather feedback from the team and make necessary adjustments to streamline operations.

By following these steps, the handling of old versions can become more organized, efficient, and less prone to errors, ultimately leading to better stability and customer satisfaction.


Question: In deep learning, which of these are hyper-parameters?

Answer: The learning rate and the number of layers are examples of hyper-parameters, but not all options qualify.
Explanation: While the learning rate and the number of layers are indeed hyper-parameters, the type of weight initialization is also a hyper-parameter, and weights and biases are typically learned parameters, not set beforehand. The answer focuses on the more obvious examples without acknowledging all relevant factors.


Question: How is data integrity ensured in WEP?

Data integrity in the context of wireless network security refers to the protection of data from unauthorized alteration during transmission. It ensures that the data received is exactly what was sent, without any modifications or tampering.

WEP (Wired Equivalent Privacy) is an early security protocol designed to provide a wireless local area network (WLAN) with a level of security and privacy comparable to what is usually expected of a wired LAN. Its primary components include:

1. **Authentication**: It provides a mechanism for confirming the identity of devices that connect to the network.
2. **Encryption**: It encrypts data transmitted over the wireless network to protect it from eavesdropping.
3. **Integrity**: It includes methods for ensuring that data has not been altered during transmission.

WEP uses a mechanism called the Integrity Check Value (ICV) to provide data integrity. When a packet is sent, WEP computes a 32-bit cyclic redundancy check (CRC) value based on the data being transmitted. This value is appended to the end of the packet. When the packet is received, the receiving device computes the CRC value on the received data and compares it to the ICV that was sent. If the two values match, the data is considered to be intact, indicating that it has not been tampered with.

However, the integrity mechanism in WEP is relatively weak compared to modern standards. Unlike more advanced integrity methods such as HMAC (Hash-based Message Authentication Code) or MIC (Message Integrity Code) used in WPA (Wi-Fi Protected Access) and WPA2, WEP's CRC-based method is susceptible to forgery and does not provide strong integrity checks.

To clarify the options provided:

A: **there is no integrity.** - This is incorrect; WEP does provide a form of integrity check, though it is weak.

B: **A3.** - This is not a recognized term related to WEP's integrity mechanism.

C: **Michael.** - This refers to the MIC used in WPA and WPA2, not WEP.

D: **HMAC.** - This is a secure integrity method used in WPA and WPA2, not in WEP.

Based on this understanding, the best choice that reflects the integrity-checking mechanism used specifically in WEP is:

**A: there is no integrity.**

While WEP does provide a basic form of integrity checking, it is fundamentally flawed and considered inadequate, so option A reflects the overall sentiment regarding WEP's inability to securely ensure data integrity.


Question: (Robustness) The $l_{1}$ loss is less sensitive to outliers than $l_{2}$.

Yes, that's correct! The \( l_1 \) loss, also known as the absolute loss or Manhattan loss, is defined as the sum of the absolute differences between predicted and actual values. Mathematically, for a set of predictions \( \hat{y} \) and actual values \( y \):

\[
L_1 = \sum_{i=1}^n |y_i - \hat{y}_i|
\]

On the other hand, the \( l_2 \) loss, also known as the squared loss or Euclidean loss, is defined as the sum of the squared differences:

\[
L_2 = \sum_{i=1}^n (y_i - \hat{y}_i)^2
\]

The key difference in sensitivity to outliers comes from the way these two losses treat large errors. In \( l_2 \) loss, since the errors are squared, larger errors contribute much more to the total loss than smaller errors. This means that a single outlier can significantly skew the performance metrics and model fitting when using \( l_2 \) loss.

In contrast, \( l_1 \) loss treats all errors linearly, meaning that an outlier will contribute to the total loss in a proportionate manner without disproportionately affecting the overall loss. As a result, models trained with \( l_1 \) loss are generally more robust to outliers compared to those trained with \( l_2 \) loss.

In summary, because \( l_1 \) loss does not square the errors, it is less sensitive to outliers, making it a suitable choice in scenarios where outliers are present in the data.


Question: The Fermat Test outputs `maybe prime' with probability which may be high given though $n$ is composite when ...

The correct answer is that the Fermat Test outputs 'maybe prime' with high probability when \( n \) is a Carmichael number. Carmichael numbers are composite numbers that pass the Fermat primality test for all bases \( a \) that are coprime to \( n \). This means they can falsely appear to be prime, which is a significant limitation of the Fermat Test. Other options like even composites or products of two primes do not exhibit this behavior. Therefore, the nature of Carmichael numbers specifically allows them to trick the Fermat Test.


Question: In the following let $\kappa_{1}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)$ and $\kappa_{2}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)$ be two valid kernels. Show that the following is also valid kernel: $\kappa\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\kappa_{1}\left(f(\mathbf{x}), f\left(\mathbf{x}^{\prime}\right)\right)$, where $f$ is any function from the domain to itself.

To show that \(\kappa(\mathbf{x}, \mathbf{x}') = \kappa_1(f(\mathbf{x}), f(\mathbf{x'}))\) is a valid kernel, we need to demonstrate that it satisfies the properties of a kernel function, specifically symmetry and positive definiteness.

1. **Symmetry**:
   \[
   \kappa(\mathbf{x}, \mathbf{x'}) = \kappa_1(f(\mathbf{x}), f(\mathbf{x'})) = \kappa_1(f(\mathbf{x'}), f(\mathbf{x})) = \kappa(\mathbf{x'}, \mathbf{x}),
   \]
   since \(\kappa_1\) is symmetric.

2. **Positive definiteness**:
   For any set of points \(\{\mathbf{x}_1, \ldots, \mathbf{x}_n\}\) and coefficients \(\{c_1, \ldots, c_n\}\), we consider:
   \[
   \sum_{i,j=1}^n c_i c_j \kappa(\mathbf{x}_i, \mathbf{x}_j) = \sum_{i,j=1}^n c_i c_j \kappa_1(f(\mathbf{x}_i), f(\mathbf{x}_j}),
   \]
   which is non-negative since \(\kappa_1\) is positive definite.

Thus, \(\kappa\) is a valid kernel.


Question: Let $S$ be a random variable taking values in $\{a,b,c,d,e\}$ with the following probabilities. $$egin{array}{|c|c|c|c|c|c|} \hline & a & b & c & d & e \ \hline p_S(\cdot) & 1/3 & 1/3 & 1/9 & 1/9 & 1/9\ \hline \end{array}$$ Let $\Gamma_D$ be the $D$-ary Huffman code for $S$. Let $L(S,\Gamma_D)$ be the average codeword-length of $\Gamma_D$, and let $H_D(S)$ be the $D$-ary entropy of $S$. True or false: If $D=3$, then $L(S,\Gamma_D) = H_D(S)$.

To determine whether the statement is true or false, we need to analyze both the average codeword length \( L(S, \Gamma_D) \) of the Huffman code and the \( D \)-ary entropy \( H_D(S) \) of the random variable \( S \).

### Step 1: Calculate the \( D \)-ary entropy \( H_D(S) \)

The \( D \)-ary entropy of a discrete random variable is given by the formula:

\[
H_D(S) = -\sum_{i=1}^{n} p_S(i) \log_D(p_S(i))
\]

where \( n \) is the number of outcomes, \( p_S(i) \) is the probability of outcome \( i \), and \( D \) is the base of the logarithm.

For our random variable \( S \) with values \( \{a, b, c, d, e\} \) and corresponding probabilities:

- \( p_S(a) = \frac{1}{3} \)
- \( p_S(b) = \frac{1}{3} \)
- \( p_S(c) = \frac{1}{9} \)
- \( p_S(d) = \frac{1}{9} \)
- \( p_S(e) = \frac{1}{9} \)

We will calculate \( H_3(S) \):

\[
H_3(S) = -\left( \frac{1}{3} \log_3\left(\frac{1}{3}\right) + \frac{1}{3} \log_3\left(\frac{1}{3}\right) + \frac{1}{9} \log_3\left(\frac{1}{9}\right) + \frac{1}{9} \log_3\left(\frac{1}{9}\right) + \frac{1}{9} \log_3\left(\frac{1}{9}\right) \right)
\]

Calculating each term separately:

1. For \( a \) and \( b \):
   \[
   -\left( \frac{1}{3} \log_3\left(\frac{1}{3}\right) + \frac{1}{3} \log_3\left(\frac{1}{3}\right) \right) = -\left( \frac{2}{3} \log_3\left(\frac{1}{3}\right) \right) = \frac{2}{3}
   \]

2. For \( c \), \( d \), and \( e \):
   \[
   -\left( \frac{1}{9} \log_3\left(\frac{1}{9}\right) + \frac{1}{9} \log_3\left(\frac{1}{9}\right) + \frac{1}{9} \log_3\left(\frac{1}{9}\right) \right) = -\left( \frac{3}{9} \log_3\left(\frac{1}{9}\right) \right) = \frac{1}{3} \cdot 2 = \frac{2}{3}
   \]

Combining these, we find:

\[
H_3(S) = \frac{2}{3} + \frac{2}{3} = \frac{4}{3} \text{ (in nats, if using base 3)}
\]

### Step 2: Calculate the average codeword length \( L(S, \Gamma_3) \)

The average codeword length \( L(S, \Gamma_D) \) for a Huffman code typically satisfies the following inequality:

\[
H_D(S) \leq L(S, \Gamma_D) < H_D(S) + 1
\]

This means that the average codeword length for a Huffman code is at least the entropy, but it is rarely equal to it unless the probabilities are perfectly aligned with the code lengths. In practice, the average codeword length will often be slightly greater than the entropy due to the limitations of discrete coding and the need to use integer lengths for codewords.

### Conclusion

Given these calculations and the properties of Huffman coding, we can conclude:

- It is generally true that \( L(S, \Gamma_3) \) will not equal \( H_3(S) \) when using Huffman coding for \( D=3 \).
- Therefore, the statement "If \( D=3 \), then \( L(S, \Gamma_D) = H_D(S) \)" is **False**.

The final answer is **False**.


Question:
Which of the following hold(s) true about update deployment in the secure
    development lifecycle?

The best answer among the provided options is:

**"Updates may bring new code that may be buggy, so additional monitoring is required after deploying an update."**

 Justification:

1. **Understanding of Updates**: In any software deployment, updates can introduce new functionalities, bug fixes, or security patches. However, these updates can also inadvertently introduce new bugs or vulnerabilities. This is a well-recognized phenomenon in software development, often referred to as "regression" or "introduction of new issues."

2. **Importance of Monitoring**: To mitigate the risks associated with updates, it is essential to implement additional monitoring after deployment. This practice allows developers and organizations to quickly identify and respond to any unforeseen issues that may arise from the new code. By monitoring key performance indicators, error logs, and user feedback, teams can ensure that the integrity and security of the system remain intact post-update.

3. **Contrast with Other Options**:
   - **Automatic Updates Motivation**: While it is true that one motivation for automatic updates is to ensure that users have the latest code, this statement does not address the risks associated with updates or the need for monitoring, which is critical in the secure development lifecycle.
   - **Third-Party Updates**: Automatically deploying third-party updates without a review process can be risky. Third-party updates may contain vulnerabilities or incompatibilities that could negatively impact the system. A careful assessment should precede any automatic deployment of third-party updates.
   - **Rollback Mechanism**: Not allowing rollback to previous versions is not a necessity in the Secure Development Lifecycle. In fact, having a rollback mechanism can be crucial for recovery in case an update fails or introduces critical bugs. A rollback feature enables teams to revert to a stable state, enhancing security and reliability.

 Conclusion:
In the context of the secure development lifecycle, the statement regarding the need for additional monitoring after deploying updates is the most relevant and critical. It emphasizes the proactive measures needed to maintain security and reliability in software systems while acknowledging the inherent risks of introducing new code.


Question: You have data with lots of outliers. Everything else being equal, and assuming that you do not do any pre-processing, what cost function will be less effected by these outliers?

To evaluate which cost function is less affected by outliers, let's consider both options:

### Option (a): Mean Squared Error (MSE)
The cost function is defined as:
\[
MSE = \frac{1}{n} \sum_{i=1}^{n} (y_i - f(x_i))^2
\]
**Merits:**
- MSE squares the error term, which means that larger errors (i.e., outliers) have a disproportionately high influence on the overall cost.
- This can be advantageous in some contexts where you want to heavily penalize large errors, but it also means that if there are significant outliers, they will dominate the loss function.

**Drawbacks:**
- Outliers lead to significantly larger squared errors, which can skew the model's performance assessment and the learning process. The model might overfit to these outliers, leading to poor generalization on typical data points.

### Option (b): Mean Absolute Error (MAE)
The cost function is defined as:
\[
MAE = \frac{1}{n} \sum_{i=1}^{n} |y_i - f(x_i)|
\]
**Merits:**
- MAE calculates the absolute difference between predicted and actual values, treating all errors equally regardless of their magnitude.
- As a result, MAE is more robust to outliers because it does not square the error term. The contribution of outliers to the overall loss is linear rather than quadratic.

**Drawbacks:**
- While MAE is less sensitive to outliers, it may not provide the same level of sensitivity to smaller errors, potentially allowing some less-than-ideal predictions to go uncorrected.

### Evaluation of Approaches
1. **Sensitivity to Outliers:**
   - MSE is sensitive due to squaring, while MAE is linear and thus more robust to outliers.

2. **Modeling Objectives:**
   - If the goal is to minimize the impact of outliers and create a model that generalizes well to the majority of the data, MAE is preferable.
   - If the goal is to penalize larger errors more heavily (which can sometimes be useful in specific contexts), MSE might be used, but it comes at the cost of being affected by outliers.

### Conclusion
Given the problem states that we have data with many outliers and we want a cost function that is less affected by these outliers, **option (b) - Mean Absolute Error (MAE)** is the most promising approach. It provides a more robust evaluation of model performance in the presence of outliers, allowing the model to learn from the majority of data points without being skewed by extreme values.


Question:
The company in which you work has just hired a new CTO, freshly graduated from a theoretical university. The CTO decides that in order to minimize bugs in the product, all new code must now be covered at least 80% in terms of paths in the code. Is this a good idea, and why?
Can you suggest something better, given the CTO's objective? (the answer must be based on the definition of path coverage.)

Sure! Let's break this down into simpler terms.

### What is Path Coverage?

Path coverage is a way to test your code by making sure that a significant number of different paths through the code are executed when you run your tests. A "path" is essentially a route that the program can take from the beginning to the end, based on different decisions (like if-else statements).

When we say that code must be "covered at least 80% in terms of paths," it means that the CTO wants to ensure that 80% of all potential paths through the code are executed during testing. The idea is that by testing a lot of different paths, we can find and fix bugs before the software is released.

### Is This a Good Idea?

While the intention behind wanting high path coverage is good—minimizing bugs—there are a few reasons why focusing solely on achieving 80% path coverage might not be the best approach:

1. **Complexity**: As code gets more complicated (with many if-else statements, loops, etc.), the number of possible paths can grow exponentially. Achieving 80% coverage might still leave many critical paths untested, especially if those paths are complex or rare.

2. **Diminishing Returns**: After a certain point, achieving higher coverage might not significantly reduce bugs. You might spend a lot of time and resources trying to cover that last 20%, but the number of bugs found might not justify the effort.

3. **False Sense of Security**: Just because you have 80% path coverage doesn't mean your code is bug-free. You could still have critical bugs in the parts of the code that weren't executed during testing.

4. **Neglecting Other Testing Methods**: Focusing solely on path coverage might lead to neglecting other important testing techniques such as unit testing, integration testing, and user acceptance testing, which can also help find bugs.

### A Better Approach

Instead of focusing only on path coverage, a more balanced approach might be:

1. **Comprehensive Testing Strategy**: Combine different types of testing methods. For example, unit tests can check individual components of the code, while integration tests can check how those components work together. This will help ensure that both individual parts and the overall system behave as expected.

2. **Risk-Based Testing**: Prioritize testing based on the risk and impact of different parts of the code. Focus on the most critical and frequently used paths first. This means you might not need to achieve 80% coverage across all paths but rather ensure that the most important ones are well-tested.

3. **Code Reviews**: Encourage peer reviews of code to catch potential bugs before the code even runs. This can often be more effective than testing alone.

4. **Automated Testing**: Use automated testing tools to run tests more efficiently and frequently. This can help in achieving a higher coverage rate without overwhelming the team.

5. **Continuous Improvement**: Encourage a culture of continuous improvement in testing. Regularly review what tests are needed, learn from past bugs, and adjust your testing strategies accordingly.

By using these strategies, the company can work towards minimizing bugs more effectively than by solely focusing on path coverage.


Question: Let $n \geq 2$ be a positive integer, and $M$ a uniformly distributed binary message of length $2n$. Let $P_K(M)$ denote the one-time pad encryption of $M$ with key $K$. Let $K_1$ be a uniformly distributed binary key length $n$. Let $K_2$ be the complement of $K_1$. Let $K_3$ be the reverse of $K_1$. Let $K_i||K_j$ denote the concatenation of the two keys. True or false: Encryption with the key $K_4 = ( K_1|| K_1 ) $, $P_{K_4}(M)$ provides perfect secrecy.

To evaluate whether encryption with \( K_4 = (K_1 || K_1) \) provides perfect secrecy, we need to understand the requirements of one-time pad encryption. Perfect secrecy requires the key to be as long as the message and independent for each bit. While \( K_4 \) matches the length of the message \( M \), it uses the same key segment \( K_1 \) twice, which compromises independence.

This repetition means that if an attacker sees the ciphertext, they might infer information about the plaintext, violating the perfect secrecy condition. A truly secure one-time pad must use a completely unique and random key for each bit of the message. Therefore, since \( K_4 \) fails to ensure key independence due to reuse, it does not provide perfect secrecy. In summary, the claim that \( K_4 \) offers perfect secrecy is incorrect.


Question: Which is NOT a mobile telephony protocol?

To identify which option is NOT a mobile telephony protocol, let's analyze each of the provided options step-by-step.

### Step 1: Define Each Protocol

1. **WPA (Wi-Fi Protected Access)**:
   - WPA is a security protocol designed to secure wireless networks, specifically Wi-Fi networks. It provides encryption and authentication mechanisms for wireless connections but is not specific to mobile telephony protocols.
   - **Conclusion:** WPA is **not** a mobile telephony protocol.

2. **DECT (Digital Enhanced Cordless Telecommunications)**:
   - DECT is a standard primarily used for cordless telephone systems. It provides a method for wireless communication between a base station and cordless handsets. While it is for telecommunication, it is not typically classified under mobile telephony protocols like GSM or CDMA.
   - **Conclusion:** DECT is a telephony protocol but not a mobile telephony protocol in the sense of cellular networks.

3. **EDGE (Enhanced Data rates for GSM Evolution)**:
   - EDGE is a mobile data protocol that enhances the data transmission rates of GSM networks. It is considered a part of the GSM family and is specifically designed for mobile telephony.
   - **Conclusion:** EDGE is a mobile telephony protocol.

4. **CDMA (Code Division Multiple Access)**:
   - CDMA is a channel access method used by various mobile telephony protocols, including 3G networks. It allows multiple users to occupy the same time and frequency bandwidth by assigning unique codes to each user.
   - **Conclusion:** CDMA is a mobile telephony protocol.

### Step 2: Identify the Correct Answer
Based on the definitions and analyses above:

- **WPA** is primarily a security protocol for wireless networks and does not pertain to mobile telephony.
- **DECT** is a protocol for cordless phones but not classified as a mobile telephony protocol in the context of cellular communications.
- **EDGE** and **CDMA** are both established mobile telephony protocols.

Since the question asks for the option that is **NOT** a mobile telephony protocol, the correct answer is:

- **1. WPA**.

This option does not relate to mobile telephony protocols, making it the correct choice.


Question: Church booleans are a representation of booleans in the lambda calculus. The Church encoding of true and false are functions of two parameters: Church encoding of tru: t => f => t Church encoding of fls: t => f => f What does the following function implement? b => c => b (not c) c

Answer: 2) b xor c**

**Explanation:** The function evaluates to `not c` when `b` is true, and it evaluates to `c` when `b` is false. This creates the behavior of an XOR operation, yielding true only when one of `b` or `c` is true but not both.


Question: Consider we use the set of transformations: insertion, deletion, substitution, and transposition. We want to compute the edit distance between words execution and exceuton, i.e. D(execution, exceuton).When computing the above, what is the value you get for D(exec,exce)?Give your answer as a numerical value.

The edit distance D(exec, exce) is 2.


Question:
Are VLIWs capable of speculative execution? Explain precisely
  your answer and provide examples for Itanium, if appropriate..

Very Long Instruction Word (VLIW) architectures can support speculative execution, although they do so differently compared to superscalar architectures due to their design characteristics. In VLIW architectures, multiple operations can be encoded into a single instruction word, allowing for instruction-level parallelism (ILP) to be exploited. Speculative execution involves executing instructions before it is certain that they need to be executed, often to improve performance by taking advantage of predicted control flow.\n\n### VLIW and Speculative Execution\n\n1. **Basic Mechanism**:\n   - In a VLIW architecture, each long instruction word contains multiple operations, and instructions are statically scheduled at compile-time for parallel execution. Speculative execution can be integrated by allowing certain instructions to be executed speculatively based on predictions made by the compiler or hardware. However, in a VLIW processor, this is typically done in a more limited fashion compared to out-of-order execution in superscalar machines.\n\n2. **Compiler Role**:\n   - The compiler plays a crucial role in VLIW architectures. It must be able to analyze data dependencies and control flow to determine safe points where speculative execution can occur. For example, if the compiler predicts that a particular branch is likely to be taken, it could schedule the instructions from the predicted path into the VLIW instruction word.\n\n3. **Examples in Itanium**:\n   - The Itanium architecture, developed by Intel, is an example of a VLIW architecture. It incorporated features that allowed for speculative execution:\n     - **Branch Prediction**: Itanium featured a branch predictor that could lead to speculative execution by predicting the outcomes of branches.\n     - **Instruction-Level Speculation**: The Itanium architecture included mechanisms for instruction-level speculation, such as memory disambiguation. This allowed the processor to speculatively execute loads and stores while resolving potential conflicts later.\n     - **Predication**: This feature allowed instructions to be executed based on predicate flags, effectively allowing the processor to execute multiple paths of code speculatively.\n\n4. **Execution Model**:\n   - While executing a VLIW instruction set, if a speculative execution leads to a misprediction (for example, a branch was taken when it was predicted not to be), the processor must be able to flush the speculatively executed instructions, revert to the correct control flow, and execute the intended instructions correctly. This means that while VLIW can execute speculatively, the complexity of control flow must be managed primarily through careful instruction scheduling by the compiler.\n\n### Summary\nVLIW architectures, such as Itanium, are capable of speculative execution primarily through advanced compiler techniques and a well-designed instruction set. Though they lack the dynamic out-of-order execution capabilities found in some superscalar designs, VLIW architectures can still leverage speculative execution to enhance performance provided that the compiler does its job in managing dependencies and predictions effectively.


Question: In this week's lecture, you have been introduced to the aggregate method of ParSeq[A] (and other parallel data structures). It has the following signature:  def aggregate[B](z: B)(f: (B, A) => B, g: (B, B) => B): B Discuss, as a group, what aggregate does and what its arguments represent. Consider the parallel sequence xs containing the three elements x1, x2 and x3. Also consider the following call to aggregate:  xs.aggregate(z)(f, g) The above call might potentially result in the following computation:  f(f(f(z, x1), x2), x3) But it might also result in other computations. Come up with at least two other computations in terms of f and g that may result from the above call to aggregate.  Below are other examples of calls to aggregate. In each case, check if the call can lead to different results depending on the strategy used by aggregate to aggregate all values contained in data down to a single value. You should assume that data is a parallel sequence of values of type BigInt. 4. data.aggregate(1)((acc, x) => x * x * acc, _ * _)

The `aggregate` method is a fundamental operation in functional programming, particularly in the context of parallel data structures such as ParSeq. Its signature, `def aggregate[B](z: B)(f: (B, A) => B, g: (B, B) => B): B`, reveals the dual nature of its operation: it combines elements of type `A` in a parallel sequence `xs` into a single value of type `B` using a specified combining function `f` and a merging function `g`.

To elaborate on its parameters:
1. `z` serves as the initial accumulator value of type `B`, which is essential for the aggregation process.
2. The function `f` takes two parameters of type `B` and `A` and produces a new value of type `B`, effectively defining how to incorporate each element of the sequence into the accumulated result.
3. The function `g` is responsible for merging two accumulated values of type `B`, which is particularly relevant in a parallel processing context where intermediate results may need to be combined.

Considering the parallel sequence `xs` with three elements `x1`, `x2`, and `x3`, the call `xs.aggregate(z)(f, g)` indeed computes a result by applying the function `f` repeatedly over the elements of the sequence, starting from `z`. The expression `f(f(f(z, x1), x2), x3)` is one potential computation. However, due to the non-sequential nature of parallel computation, other valid computations could include:
1. `g(f(z, x1), f(x2, x3))`, where the intermediate results of `x2` and `x3` are combined first before being merged with the result of `x1`.
2. `g(f(z, x2), f(x1, x3))`, where `x1` and `x3` are combined first, reflecting the flexibility in the order of operations.

This showcases the non-determinism inherent in parallel computations, where different execution strategies can yield varying results, particularly when the order of combining and merging functions is altered.

In the case of the example `data.aggregate(1)((acc, x) => x * x * acc, _ * _)`, similar reasoning applies. The different strategies employed by `aggregate` can lead to variations in results due to the associative nature of the multiplication operation. Depending on how `f` and `g` are applied, the computed result can vary significantly, further emphasizing the importance of understanding the implications of parallel processing in functional programming.


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: type Multiset = Char => Int Assuming that elements of multisets are only lowercase letters of the English alpha- bet, what does the secret function compute? def diff(a: Multiset, b: Multiset): Multiset = \t x => Math.abs(a(x) - b(x)) def secret(a: Multiset, b: Multiset) = \t (’a’ to ’z’).map(x => diff(a, b)(x)).sum == 0

To analyze the function `secret` and determine what it computes, we need to break down the logic of the implementation.

1. **Function `diff(a, b)`:**
   - `diff` computes the absolute difference between the counts of each character in the two multisets `a` and `b`. For each character `x`, it calculates `Math.abs(a(x) - b(x))`, which tells us how many more times a character appears in one multiset compared to the other.

2. **Function `secret(a, b)`:**
   - This function iterates over each character from 'a' to 'z', applies the `diff` function to get the absolute difference for each character, and sums these differences.
   - The result of the sum will be `0` if and only if for every character `x`, the counts of `x` in both multisets are equal (i.e., `a(x) = b(x)` for all `x`). This is because the absolute difference will be `0` only if both counts are the same.

Given this understanding, let's analyze the provided options:

1. **Checks if b is a subset of a:**
   - This is false because being a subset would imply `b` can have counts less than or equal to `a`, but they do not necessarily have to be equal.

2. **Checks if a and b are disjoint:**
   - This is false because disjoint sets would mean there are no common elements at all, which is not what `secret` checks for.

3. **Checks if a is a subset of b:**
   - This is false for similar reasons as option 1. There is no requirement that `a` must be less than or equal to `b`.

4. **Checks if a and b are equal:**
   - This is true. The `secret` function sums the absolute differences and checks if it equals `0`, which indicates that both multisets have exactly the same counts for each character.

5. **Checks if a and b are empty:**
   - This is false because the function could return `true` even when `a` or `b` has elements.

6. **Checks if a is empty:**
   - This is false for the same reason as option 5; the function does not specifically check for emptiness.

Given this analysis, the correct answer is that the `secret` function checks if `a` and `b` are equal.

Final answer: **Checks if a and b are equal**


Question: Select \emph{incorrect} statement. The birthday paradox

The incorrect statement is:

4. implies that majority of people is born at full moon.

This statement does not relate to the birthday paradox, which is primarily about the probability of shared birthdays in a group and does not imply anything about the specific distribution of birthdays like being born on a full moon.


Question:
You have been publishing a daily column for the Gazette over the last few years and have recently reached a milestone --- your 1000th column! Realizing you'd like to go skiing more often, you decide it might be easier to automate your job by training a story generation system on the columns you've already written. Then, whenever your editor pitches you a title for a column topic, you'll just be able to give the title to your story generation system, produce the text body of the column, and publish it to the website!
To evaluate your system, you decide to hold out some of the columns you have previously written and use them as an evaluation set. After generating new columns using the same titles as these held-out columns, you decide to evaluate their quality.
What would be an advantage of using a model-based metric?

A model-based metric can provide a more nuanced evaluation of generated text quality by leveraging pre-trained models that understand language context and semantics, leading to insights beyond simple surface-level comparisons. This approach can help better capture the coherence and relevance of the generated columns relative to the original content.


Question: I want to send a value to Bob without him knowing which value I sent and such that I cannot change my mind later when I reveal it in clear. I should use \dots

The correct option for the scenario described in the question is **"a commitment scheme."**

### Explanation of Commitment Scheme:
A commitment scheme is a cryptographic protocol that allows a sender to commit to a specific value while keeping it hidden from the receiver until a later time when the value is revealed. The commitment scheme has two crucial properties:

1. **Hiding**: The commitment must not reveal any information about the committed value until it is opened. This ensures that the receiver (Bob) does not learn the value until the sender chooses to disclose it.

2. **Binding**: Once the sender has committed to a particular value, they cannot change it. This guarantees that the value revealed later is the same as the one that was initially committed, ensuring the integrity of the commitment.

In the context of this scenario, the sender wishes to communicate a value to Bob without him knowing which value was sent, and they want to ensure that they cannot change their mind later. A commitment scheme effectively meets these requirements.

### Explanation of Incorrect Options:
1. **Stream Cipher**: A stream cipher is a method of encrypting plaintext by combining it with a pseudorandom cipher digit stream (keystream). While it provides confidentiality by encrypting data, it lacks the commitment properties of hiding and binding. The recipient can decrypt the message, but the sender retains the ability to modify the original plaintext before revealing it, which does not satisfy the requirement of being unchangeable.

2. **PRNG (Pseudorandom Number Generator)**: A PRNG is an algorithm that generates a sequence of numbers that approximates the properties of random numbers. Although PRNGs are useful in cryptographic applications, they do not provide commitment properties. A PRNG does not allow a sender to commit to a value without revealing it, nor does it enforce any binding after the value is chosen, making it unsuitable for this scenario.

3. **Digital Signature**: A digital signature is a cryptographic technique that allows a sender to sign a message, enabling others to verify the authenticity and integrity of that message using the sender's public key. While digital signatures provide authenticity and integrity, they do not allow for the commitment of a value in a manner that prevents the sender from altering it before revealing. Digital signatures do not hide the content of the message either; they simply confirm that the message has not been tampered with after signing.

### Conclusion:
In conclusion, the correct answer is **"a commitment scheme"** because it uniquely satisfies the requirement of allowing the sender to commit to a value without revealing it, while also ensuring that the sender cannot change the value once committed. The other options fail to provide both the necessary secrecy and the binding commitment required for this scenario.


Question:
The objective of this question is to illustrate the use of a lexical semantics resource to compute
lexical cohesion.
Consider the following toy ontology providing a semantic structuring for a (small) set of nouns:

The word 'mouse' appears at two different places in the toy ontology. What does this mean?
What specific problems does it raise when the ontology is used?
How could such problems be solved? (just provide a sketch of explanation.)

To analyze the presence of the word 'mouse' in two different places within the toy ontology, we need to consider the implications of this duplication in terms of lexical semantics and ontology design.

### Step 1: Understanding the Ontology Structure
The ontology categorizes nouns into two main groups: animate and non-animate entities. Within these groups, 'mouse' appears under both 'animals' (as an animate entity) and 'concrete entities' (as a non-animate entity). This dual categorization suggests that 'mouse' can refer to both a living creature (an animal) and an inanimate object (like a computer mouse).

### Step 2: Identifying the Problems
1. **Ambiguity**: The presence of 'mouse' in two different categories can lead to confusion about which meaning is intended in a given context. This ambiguity can complicate tasks such as natural language processing, where understanding the correct sense of a word is crucial.

2. **Inconsistency**: If 'mouse' is treated as both an animate and non-animate entity, it may lead to inconsistencies in reasoning or inference processes. For example, if a system is designed to infer properties of animate entities, it might incorrectly apply those properties to the inanimate 'mouse'.

3. **Redundancy**: The duplication of 'mouse' can lead to redundancy in the ontology, making it larger and potentially more complex than necessary. This can affect performance in terms of storage and processing.

### Step 3: Potential Solutions
1. **Disambiguation**: One approach to resolving the ambiguity is to create distinct entries for 'mouse' with clear labels or identifiers that specify whether it refers to the animal or the object. For example, using 'mouse (animal)' and 'mouse (object)' could clarify the intended meaning.

2. **Hierarchical Structuring**: Another solution could involve restructuring the ontology to reflect the relationship between the two meanings. For instance, 'mouse' could be a subclass of a more general category that encompasses both animate and inanimate entities, with properties that differentiate the contexts in which each meaning applies.

3. **Contextual Metadata**: Adding contextual metadata to each instance of 'mouse' could help systems determine which meaning to apply based on the surrounding text or usage scenario. This could involve tagging or annotating the entries with context-specific information.

4. **Use of Synsets**: Implementing a synset approach, where each meaning of 'mouse' is treated as a separate synset in a lexical database, could help in managing the different senses of the word while maintaining a cohesive ontology structure.

### Conclusion
The dual presence of 'mouse' in the ontology raises significant issues related to ambiguity, inconsistency, and redundancy. Addressing these problems through disambiguation, hierarchical structuring, contextual metadata, or synsets can enhance the clarity and utility of the ontology, making it more effective for applications in lexical semantics and natural language processing.


Question: Implement kNN function (finding k nearest documents for a given document)

To implement a k-Nearest Neighbors (kNN) function for finding the k nearest documents for a given document, we need to follow a series of steps. The kNN algorithm is a type of instance-based learning where the classification of a data point is determined by the majority class among its k nearest neighbors. In the context of documents, we can use text vectorization methods like TF-IDF or Word Embeddings to represent documents in a numerical format suitable for distance calculations.

Here’s a step-by-step approach to implement a kNN function for documents:

### Step 1: Document Representation
Before we can find nearest neighbors, we need to represent our documents as vectors. The most common methods include:

- **TF-IDF (Term Frequency-Inverse Document Frequency)**: This method takes into account the frequency of words in the document as well as how common those words are across all documents.
- **Word Embeddings**: Techniques like Word2Vec or GloVe can also be used, but for simplicity, we will focus on TF-IDF.

### Step 2: Calculate Distances
For kNN, we typically use distance metrics such as Euclidean distance or Cosine similarity. For text data, Cosine similarity is often preferred as it measures the angle between two vectors and is less sensitive to the magnitude of the vectors.

### Step 3: Implementing kNN
Now, let’s implement the kNN function in Python using the TF-IDF vectorization and Cosine similarity.

#### Example Implementation

```python
import numpy as np
from sklearn.feature_extraction.text import TfidfVectorizer
from sklearn.metrics.pairwise import cosine_similarity

def knn_documents(documents, query_doc, k=3):
    """
    Finds the k nearest documents for a given document using cosine similarity.

    :param documents: List of documents (strings)
    :param query_doc: The document to find neighbors for
    :param k: Number of nearest neighbors to return
    :return: List of indices of the k nearest documents
    """
    # Step 1: Vectorize the documents using TF-IDF
    vectorizer = TfidfVectorizer()
    tfidf_matrix = vectorizer.fit_transform(documents)

    # Step 2: Vectorize the query document
    query_vector = vectorizer.transform([query_doc])

    # Step 3: Calculate cosine similarity between the query document and all documents
    cosine_similarities = cosine_similarity(query_vector, tfidf_matrix).flatten()

    # Step 4: Get the indices of the k nearest documents
    nearest_indices = np.argsort(cosine_similarities)[-k:][::-1]

    return nearest_indices.tolist()

# Example usage
documents = [
    "This is a document about machine learning.",
    "This document discusses neural networks.",
    "This text is about the history of artificial intelligence.",
    "This document is related to data science and statistics."
]

query_doc = "I want to learn about machine learning."
k = 2

nearest_docs_indices = knn_documents(documents, query_doc, k)
print("Indices of nearest documents:", nearest_docs_indices)
print("Nearest documents:", [documents[i] for i in nearest_docs_indices])
```

### Explanation of the Code:
1. **TF-IDF Vectorization**: We use `TfidfVectorizer` from `sklearn` to convert the documents into a TF-IDF matrix.
2. **Query Vectorization**: The query document is also transformed into the same vector space as the documents.
3. **Cosine Similarity Calculation**: We compute the cosine similarity between the query vector and each document vector.
4. **Finding Nearest Neighbors**: We sort the similarity scores to find the indices of the top k documents with the highest similarity to the query document.

### Conclusion
This implementation provides a simple way to find the k nearest documents based on a query document using the kNN algorithm. It utilizes TF-IDF for document representation and cosine similarity for measuring similarity. This approach can be extended or modified based on the specific requirements of your application, such as incorporating different distance metrics or vectorization techniques.


Question:
& \multicolumn{3}{c}{	extbf{ProofWriter}} & \multicolumn{3}{c}{	extbf{CLUTRR-SG}} \
        \cmidrule(lr){2-4} \cmidrule(lr){5-7}
        What does the following function implement?
        1 a => b => (not a) (not b) fls

The function `1 a => b => (not a) (not b) fls` checks the values of `a` and `b`. It negates both `a` and `b`, which leads to a logical outcome. Essentially, it tells us something about the values of `a` and `b`, and it relates to operations in boolean logic.

Therefore, it implements `not (a or b)`.


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 $y_1 \le y_2$. How can the result be computed in parallel? Which properties of hull2 need to hold to make the solution correct? Prove these properties for hull2.

In order to compute the union of rectangles in parallel, it is essential that the operation we use, referred to as hull2, satisfies certain properties. Specifically, hull2 must be both associative and commutative. This means that the way we group the operations should not affect the final result, and the order in which we apply the operations should also not matter.

To illustrate this, let us consider a rectangle represented as $$ r_i = Rectangle((x_1^i, y_1^i), (x_2^i, y_2^i)) $$. When applying hull2 to three rectangles, say \( r_1, r_2, \) and \( r_3 \), we can express the operation as follows:

\[
hull2(r_1, hull2(r_2, r_3)) = hull2(r_1, Rectangle((\min(x_1^2, x_1^3), \min(y_1^2, y_1^3)), (\max(x_2^2, x_2^3), \max(y_2^2, y_2^3))))
\]

This expands to:

\[
Rectangle((\min(x_1^1, \min(x_1^2, x_1^3)), \min(y_1^1, \min(y_1^2, y_1^3))), (\max(x_2^1, \max(x_2^2, x_2^3)), \max(y_2^1, \max(y_2^2, y_2^3))))
\]

Here, you can see that the minimum and maximum functions maintain their associative and commutative properties. Thus, we can rearrange the order of the rectangles and still arrive at the same final rectangle.

To prove associativity, consider:

\[
hull2(r_1, hull2(r_2, r_3)) = hull2(hull2(r_1, r_2), r_3)
\]

For commutativity, we can show that:

\[
hull2(r_1, r_2) = hull2(r_2, r_1)
\]

Both of these properties ensure that we can aggregate the rectangles in any order, which is crucial for parallel computation. The parallel reduction will thus yield the correct result regardless of the sequence in which operations are performed, as long as each application of hull2 adheres to these properties.


Question: Which one of these is a closed set?

<1>, $\mathbb{Z}$ with the addition.

The integers $\mathbb{Z}$ form a closed set under addition as the sum of any two integers is also an integer.


Question: Who invented linear cryptanalysis?

To determine who invented linear cryptanalysis, we need to analyze the contributions of each individual mentioned in the options.

1. **Mitsuru Matsui**: He is known for his work on the development of linear cryptanalysis. In 1993, Matsui introduced this method as a cryptanalysis technique that exploits linear approximations of the behavior of block ciphers.

2. **Eli Biham**: Biham is a significant figure in the field of cryptography and collaborated with Matsui in some cryptanalysis work, but he is not credited with the invention of linear cryptanalysis.

3. **Serge Vaudenay**: Vaudenay is recognized for his contributions to cryptography but is not associated with the invention of linear cryptanalysis.

4. **Adi Shamir**: Shamir is one of the co-inventors of the RSA algorithm and has made numerous contributions to cryptography, but he did not invent linear cryptanalysis.

After reviewing the contributions of each option, it is clear that:

- The technique of linear cryptanalysis was first introduced by Mitsuru Matsui in his 1993 paper, marking a significant advancement in the field of cryptographic analysis.

Thus, the correct answer is:

**Correct answer**: Mitsuru Matsui


Question: In this exercise, we will see how to combine the Principal Component Analysis (PCA) and the kernel method into an algorithm known as kernel PCA. We are given $n$ observations in a low dimensional space $\mathbf{x}_{1}, \cdots, \mathbf{x}_{n} \in \mathbb{R}^{L}$ and we consider a kernel $k$ and its associated features $\operatorname{map} \phi: \mathbb{R}^{L} \mapsto \mathbb{R}^{H}$ which satisfies: $$ k(\mathbf{x}, \mathbf{y})=\langle\phi(\mathbf{x}), \phi(\mathbf{y})\rangle_{\mathbb{R}^{H}} $$ where $\langle\cdot, \cdot\rangle_{\mathbb{R}^{H}}$ is the standard scalar product of $\mathbb{R}^{H}$. We define the empirical covariance matrix and the empirical covariance matrix of the mapped observations as: $$ \boldsymbol{\Sigma}:=\frac{1}{n} \sum_{i=1}^{n} \mathbf{x}_{i} \mathbf{x}_{i}^{\top} \quad \text { and } \quad \boldsymbol{\Sigma}^{\mathbf{H}}:=\frac{1}{n} \sum_{i=1}^{n} \phi\left(\mathbf{x}_{i}\right) \phi\left(\mathbf{x}_{i}\right)^{\top} $$ The kernel matrix $\mathbf{K}$ is defined by: $$ \mathbf{K}_{i, j}:=k\left(\mathbf{x}_{i}, \mathbf{x}_{j}\right)=\left\langle\phi\left(\mathbf{x}_{i}\right), \phi\left(\mathbf{x}_{j}\right)\right\rangle_{\mathbb{R}^{H}} $$ We also define the data matrix and the corresponding matrix of the mapped data as: $$ \mathbf{X}:=\left(\begin{array}{c} \mathbf{x}_{1}^{\top} \\ \cdots \\ \mathbf{x}_{n}^{\top} \end{array}\right) \in \mathbb{R}^{n \times L} \quad \text { and } \quad \mathbf{\Phi}:=\left(\begin{array}{c} \phi\left(\mathbf{x}_{1}\right)^{\top} \\ \cdots \\ \phi\left(\mathbf{x}_{n}\right)^{\top} \end{array}\right) \in \mathbb{R}^{n \times H} . $$ Finally we denote the eigenpairs (eigenvalues and eigenvectors) of $\boldsymbol{\Sigma}^{\mathbf{H}}$ by $\left\{\left(\lambda_{i}, \mathbf{v}_{i}\right)\right\}_{i=1}^{H}$ and those of $\mathbf{K}$ by $\left\{\left(\rho_{j}, \mathbf{w}_{j}\right)\right\}_{j=1}^{n}$. We also assume that the vectors $\mathbf{v}_{i}$ and $\mathbf{w}_{j}$ are normalized. Thus: $$ \boldsymbol{\Sigma}^{\mathbf{H}} \mathbf{v}_{i}=\lambda_{i} \mathbf{v}_{i}, \quad\left\|\mathbf{v}_{i}\right\|_{2}=1 \quad \text { and } \quad \mathbf{K} \mathbf{w}_{j}=\rho_{j} \mathbf{w}_{j}, \quad\left\|\mathbf{w}_{j}\right\|_{2}=1 $$ Let us remind that we assume in the kernel setting that we can compute $k(\mathbf{x}, \mathbf{y})$ but that we cannot directly compute $\phi(\mathbf{x})$ What we would like to do is to first map the data into the high-dimensional space using the features map $\phi$ and then to apply the standard PCA algorithm in the high-dimensional space $\mathbb{R}^{H}$. This would amount to: (a) Computing the empirical covariance matrix $\boldsymbol{\Sigma}^{\mathbf{H}}$ of the mapped data $\phi\left(\mathbf{x}_{i}\right)$. (b) Computing the eigenvectors $\mathbf{v}_{1}, \cdots, \mathbf{v}_{N}$ associated with the $N$ largest eigenvalues of $\boldsymbol{\Sigma}^{\mathbf{H}}$. (c) Computing the projection $\Pi\left(\phi\left(\mathbf{x}_{i}\right)\right) \in \mathbb{R}^{L}$ for each data point onto these eigenvectors, where the $j$-th component of the projection is given by: $$ \Pi_{j}\left(\phi\left(\mathbf{x}_{i}\right)\right)=\left\langle\phi\left(\mathbf{x}_{i}\right), \mathbf{v}_{j}\right\rangle_{\mathbb{R}^{H}} $$ Write the empirical covariance matrices $\boldsymbol{\Sigma}$ and $\boldsymbol{\Sigma}^{\mathbf{H}}$ in function of the design matrix $\mathbf{X}$ and the features matrix $\boldsymbol{\Phi}$. What are the sizes of these matrices $\boldsymbol{\Sigma}$ and $\boldsymbol{\Sigma}^{\mathbf{H}}$ ?

To approach this problem, we first need to understand how the empirical covariance matrices are derived from the given matrices, $\mathbf{X}$ and $\boldsymbol{\Phi}$.

1. **Covariance Matrix $\boldsymbol{\Sigma}$**:
   We start with the definition of $\boldsymbol{\Sigma}$, which is given as:
   $$
   \boldsymbol{\Sigma} := \frac{1}{n} \sum_{i=1}^{n} \mathbf{x}_{i} \mathbf{x}_{i}^{\top}
   $$
   This expression indicates that we are calculating the average of the outer products of the vectors $\mathbf{x}_{i}$.

   To rewrite this using the design matrix $\mathbf{X}$, we note that $\mathbf{X}$ is constructed such that its rows are the transposed vectors $\mathbf{x}_{i}^{\top}$. Thus, if we take the product $\mathbf{X}^{\top} \mathbf{X}$, we will get:
   $$
   \mathbf{X}^{\top} \mathbf{X} = \sum_{i=1}^{n} \mathbf{x}_{i} \mathbf{x}_{i}^{\top}
   $$
   Consequently, we can express $\boldsymbol{\Sigma}$ in terms of $\mathbf{X}$:
   $$
   \boldsymbol{\Sigma} = \frac{1}{n} \mathbf{X}^{\top} \mathbf{X}
   $$

   The size of this covariance matrix $\boldsymbol{\Sigma}$ is determined by the dimensions of $\mathbf{x}_{i}$, which are in $\mathbb{R}^{L}$. Therefore, $\boldsymbol{\Sigma}$ will have dimensions:
   $$
   \boldsymbol{\Sigma} \in \mathbb{R}^{L \times L}
   $$

2. **Covariance Matrix $\boldsymbol{\Sigma}^{\mathbf{H}}$**:
   Next, we look at $\boldsymbol{\Sigma}^{\mathbf{H}}$, defined as:
   $$
   \boldsymbol{\Sigma}^{\mathbf{H}} := \frac{1}{n} \sum_{i=1}^{n} \phi\left(\mathbf{x}_{i}\right) \phi\left(\mathbf{x}_{i}\right)^{\top}
   $$
   Similarly, we want to express this in terms of the features matrix $\boldsymbol{\Phi}$. The features matrix $\boldsymbol{\Phi}$ is constructed such that its rows are the transposed mapped vectors $\phi\left(\mathbf{x}_{i}\right)^{\top}$. Thus, analogous to what we did for $\boldsymbol{\Sigma}$, we can see that:
   $$
   \boldsymbol{\Sigma}^{\mathbf{H}} = \frac{1}{n} \boldsymbol{\Phi}^{\top} \boldsymbol{\Phi}
   $$

   This covariance matrix $\boldsymbol{\Sigma}^{\mathbf{H}}$ will have its size determined by the dimensions of the feature space mapped by $\phi$, which is $\mathbb{R}^{H}$. Therefore, the size of $\boldsymbol{\Sigma}^{\mathbf{H}}$ is:
   $$
   \boldsymbol{\Sigma}^{\mathbf{H}} \in \mathbb{R}^{H \times H}
   $$

To summarize, we can express our findings as:
$$
\begin{gathered}
\boldsymbol{\Sigma} = \frac{1}{n} \mathbf{X}^{\top} \mathbf{X} \in \mathbb{R}^{L \times L} \\
\boldsymbol{\Sigma}^{\mathbf{H}} = \frac{1}{n} \boldsymbol{\Phi}^{\top} \boldsymbol{\Phi} \in \mathbb{R}^{H \times H}
\end{gathered}
$$

So, the final answer includes both the expressions of the covariance matrices and their respective sizes, which matches the answer provided.


Question: What is the name of the encryption threat that corresponds to \emph{force the sender to encrypt some messages selected by the adversary}?

To determine the correct option that corresponds to the threat of forcing the sender to encrypt messages selected by the adversary, we can analyze each option systematically.

1. **Chosen Ciphertext Attack (CCA)**:
   - In a chosen ciphertext attack, the attacker can choose arbitrary ciphertexts and obtain their corresponding decrypted plaintexts. This allows the adversary to glean information about the encryption scheme and potentially recover the secret key or manipulate the encryption process.
   - This attack does indeed involve the adversary influencing the encryption process, making this a strong candidate for the described threat.

2. **Chosen Plaintext Attack (CPA)**:
   - In a chosen plaintext attack, the adversary chooses plaintexts to be encrypted and obtains their corresponding ciphertexts. This type of attack allows the attacker to analyze the relationship between chosen plaintexts and their ciphertexts but does not involve forcing the sender to encrypt arbitrary messages.
   - Therefore, this option does not align with the provided description and is not correct.

3. **Known Ciphertext Attack**:
   - A known ciphertext attack occurs when the attacker has access to some ciphertexts and tries to deduce information about the plaintexts or the encryption key without having control over the encryption process. The adversary cannot select the messages to be encrypted in this scenario.
   - This option does not fit the description and is not correct.

4. **Known Plaintext Attack**:
   - In a known plaintext attack, the adversary has access to both the plaintext and its corresponding ciphertext, which they analyze to try to discover the encryption key. Similar to the known ciphertext attack, the adversary does not have the ability to choose messages for encryption.
   - This option does not match the threat outlined in the question and is therefore not correct.

### Conclusion:
The correct option corresponding to the threat of forcing the sender to encrypt messages selected by the adversary is:

- **Chosen Ciphertext Attack (CCA)**

This type of attack explicitly allows the adversary to influence the encryption process, which is the essence of the described threat.


Question: Given a document collection with a vocabulary consisting of three words, $V = {a,b,c}$, and two documents $d_1$ = aabc and $d_2 = abc$. The query is $q = ab$.  Is it possible to enforce a ranking $d_2 > d_1$ with vector space retrieval and $d_1 > d_2$ with probabilistic retrieval ($\lambda=0.5$), by adding the same documents to the collection? If yes, give examples of such documents to be added, if no, provide an argument why this cannot be the case.

To analyze whether it's possible to enforce a ranking \( d_2 > d_1 \) with vector space retrieval and \( d_1 > d_2 \) with probabilistic retrieval by adding the same documents to the collection, let's first define how both retrieval models work.

### Vector Space Retrieval
In vector space retrieval, documents and queries are represented as vectors in a multi-dimensional space, where each dimension corresponds to a term in the vocabulary. The relevance of a document to a query is often computed using a similarity measure, such as the cosine similarity.

1. **Document Vectors**:
   - For \( d_1 = \text{aabc} \):
     - Term counts: \( a: 2, b: 1, c: 1 \) → Vector: \( [2, 1, 1] \)
   - For \( d_2 = \text{abc} \):
     - Term counts: \( a: 1, b: 1, c: 1 \) → Vector: \( [1, 1, 1] \)
   - For the query \( q = ab \):
     - Term counts: \( a: 1, b: 1 \) → Vector: \( [1, 1, 0] \)

2. **Cosine Similarity Calculation**:
   - Similarity \( S(d_1, q) = \frac{[2, 1, 1] \cdot [1, 1, 0]}{||d_1|| \cdot ||q||} \)
   - Similarity \( S(d_2, q) = \frac{[1, 1, 1] \cdot [1, 1, 0]}{||d_2|| \cdot ||q||} \)

Calculating these similarities:
- \( S(d_1, q) = \frac{(2*1 + 1*1 + 1*0)}{\sqrt{(2^2 + 1^2 + 1^2)} \cdot \sqrt{(1^2 + 1^2 + 0^2)}} = \frac{3}{\sqrt{6} \cdot \sqrt{2}} = \frac{3}{\sqrt{12}} = \frac{3}{2\sqrt{3}} \)
- \( S(d_2, q) = \frac{(1*1 + 1*1 + 1*0)}{\sqrt{(1^2 + 1^2 + 1^2)} \cdot \sqrt{(1^2 + 1^2 + 0^2)}} = \frac{2}{\sqrt{3} \cdot \sqrt{2}} = \frac{2}{\sqrt{6}} \)

Since \( S(d_1, q) > S(d_2, q) \), we know that \( d_1 \) ranks higher than \( d_2 \) in vector space retrieval.

### Probabilistic Retrieval
In probabilistic retrieval, documents are ranked based on the probability that a document is relevant to the query. The probability of relevance can be influenced by factors such as term frequency and document length.

Using a probabilistic model with \( \lambda = 0.5 \) (which is typically a smoothing parameter), the relevance of documents can be calculated based on their term distributions.

1. **Calculating Probabilities**:
   - Let’s say we denote the probability of a term appearing in a document as \( P(t|d) \).
   - For \( d_1 \) and \( d_2 \), under a smoothed probabilistic model, we can derive their probabilities for query terms \( a \) and \( b \).

The specifics of the calculations will depend on how the smoothing is defined, but generally:
- \( P(a|d_1) = \frac{2 + \lambda}{3 + \lambda * |V|} \)
- \( P(b|d_1) = \frac{1 + \lambda}{3 + \lambda * |V|} \)
- \( P(a|d_2) = \frac{1 + \lambda}{3 + \lambda * |V|} \)
- \( P(b|d_2) = \frac{1 + \lambda}{3 + \lambda * |V|} \)

### Finding Additional Documents
To enforce \( d_2 > d_1 \) in vector space retrieval, we would need to add documents that increase the similarity of \( d_2 \) to the query while keeping \( d_1 \) lower. On the other hand, to enforce \( d_1 > d_2 \) in probabilistic retrieval, we could add documents that affect the probability calculations in favor of \( d_1 \).

### Conclusion
It is challenging to achieve both rankings simultaneously with the same documents being added, as the effects on vector space ranking tend to conflict with those on probabilistic ranking due to their differing underlying mechanics.

Thus, **no**, it is not possible to enforce \( d_2 > d_1 \) in vector space retrieval and \( d_1 > d_2 \) in probabilistic retrieval with the same additional documents. The inherent nature of the two models leads to conflicting requirements for ranking.


Question:
Considering the transaction below, which one is WRONG?

|Transaction ID |Items Bought|
|--|--|
|1|Tea|
|2|Tea, Yoghurt|
|3|Tea, Yoghurt, Kebap|
|4 |Kebap |
|5|Tea, Kebap|

Answer: 4

Reasoning: The statement â{Yoghurt} has the lowest support among all itemsetsâ is incorrect. Yoghurt appears in transactions 2 and 3, giving it a support of 2/5 = 40%, which is not the lowest. For example, {Yoghurt, Kebap} appears only once, with a support of 1/5 = 20%, which is lower.


Question:
You need to quickly find if a person's name is in a list: that contains both integers and strings such as:

list := ["Adam Smith", "Kurt Gödel", 499, 999.95, "Bertrand Arthur William Russell", 19.99, ...]

What strategy can you use?

To determine if a person's name is in a list that contains both integers and strings, we need to consider the nature of the data and the operations we can perform:

1. **Data Type Compatibility**: The list contains both strings (names) and non-string types (integers, floats). This mixed data type means we cannot sort the list in a meaningful way for binary search unless we filter for strings only.

2. **Sorting Requirement**: Binary search requires the list to be sorted. However, since the list contains mixed types, sorting it (even if we were to sort only the string elements) could complicate things.

3. **Search Method**:
   - **Linear Search**: This method checks each element of the list one by one until it finds the target. This is straightforward and works well with unsorted lists or mixed data types. It doesn't require the list to be sorted.
   - **Binary Search**: This method is efficient but requires the list to be sorted and is only applicable to a single data type (either all strings or all numbers).

Given the mixed types in the list, the safest and most straightforward approach is to use a linear search, which will effectively check each element until it finds the person's name or exhausts the list.

### Final Answer:
Use linear search.


Question: Provide a precise definition of concatenative morphology and illustrate your answer with concrete examples in English or French. Is this type of morphology relevant for all languages? More generally, is morphology of the same complexity for all languages?

### Definition of Concatenative Morphology

**Concatenative morphology** is a type of morphological process where morphemes (the smallest units of meaning) are joined together in a linear sequence to form words. This process involves the addition of prefixes, suffixes, infixes, or other morphological elements to a base or root word. The resulting forms are typically predictable based on the rules of the language regarding how morphemes can combine.

### Examples in English

1. **Inflection**:
   - The noun "cat" can become "cats" by adding the plural suffix **-s**.
   - The verb "walk" can become "walked" by adding the past tense suffix **-ed**.

2. **Derivation**:
   - The adjective "happy" can become "unhappy" by adding the prefix **un-**.
   - The noun "friend" can become "friendship" by adding the suffix **-ship**.

### Examples in French

1. **Inflection**:
   - The noun "chat" (cat) becomes "chats" (cats) with the addition of the plural suffix **-s**.
   - The verb "parler" (to speak) becomes "parlé" (spoken) with the addition of the past participle suffix **-é**.

2. **Derivation**:
   - The adjective "heureux" (happy) can become "malheureux" (unhappy) by adding the prefix **mal-**.
   - The noun "libre" (free) can become "liberté" (liberty) by adding the suffix **-té**.

### Relevance of Concatenative Morphology Across Languages

Concatenative morphology is relevant to many languages, particularly those that exhibit clear processes of inflection and derivation like English and French. However, its relevance can vary significantly across languages:

- **Languages with Agglutinative Morphology**: In agglutinative languages like Turkish or Swahili, concatenative morphology is prominent, as they often use multiple affixes that each represent a single grammatical function, leading to long, complex word forms. For example, in Turkish, "evlerimizden" (from our houses) is formed by concatenating several morphemes.

- **Languages with Fusional Morphology**: In fusional languages like Russian, one morpheme can express multiple grammatical categories simultaneously. For instance, the word "столами" (with tables) contains inflections that indicate both case and number in a single suffix, making it less straightforward than pure concatenation.

- **Languages with Non-Concatenative Morphology**: Some languages, such as Arabic, utilize non-concatenative morphology, where root consonants are modified through internal changes rather than simple concatenation of prefixes or suffixes. For example, the root "k-t-b" (to write) can generate "كتب" (kataba - he wrote) or "كتابة" (kitāba - writing) through vowel changes and internal modifications rather than just affixation.

### Complexity of Morphology Across Languages

Morphology is not of the same complexity across all languages. Some languages have rich and intricate morphological systems (e.g., Finnish, with its extensive case system and agglutinative nature), while others may have relatively simple morphological structures (e.g., Chinese, which has a largely isolating morphology with minimal inflection).

Overall, the complexity of morphology can be influenced by various factors, such as:
- The number of grammatical categories expressed (tense, aspect, mood, case, number, etc.).
- The presence of rich inflectional systems versus more isolating or analytic structures.
- The use of non-concatenative processes, which can add complexity in different ways.

In summary, while concatenative morphology is a significant feature of many languages, its role and complexity can vary widely, influenced by the specific morphological rules and structures of each language.


Question: Using a 4-gram character model, comparing "banana" and "ananas"...

Let's eliminate the wrong options:

1. **is the same as comparing "aaabnn" to "aaanns"**: This option compares the frequency of characters rather than the actual sequences of 4-grams, which is not relevant to the model being discussed.

2. **is the same as comparing P(bana) to P(anan)**: This option incorrectly compares "bana" with "anan," which does not correspond to the 4-gram sequences of "banana" and "ananas."

3. **is the same as comparing P(ban)/P(an) to P(nas)/P(na)**: This option incorrectly represents the relationship between the parts of the two words, as it mixes the probabilities of different segments that do not align with the 4-gram model.

The correct answer is **is the same as comparing P(bana) to P(anas)**. This is because both "banana" and "ananas" share the same 4-gram structure starting from their first character, and the comparison of their probabilities focuses on the same 4-gram segments "bana" and "anas." Therefore, analyzing these segments allows us to understand the likelihood of each sequence occurring in the context of the character model.


Question: Tick the \textbf{\emph{correct}} assertion. A random oracle $\ldots$

The correct assertion is that a random oracle answers with random values that are always independent of the previous queries. This means that each query to the oracle produces a response that does not depend on any other queries made previously.


Question:
Assume that you are part of a team developing a mobile app using Scrum.
When using the app, you identified multiple bugs and features which you think should be implemented, and took some notes. You want to
share these with the Product Owner. Your backlog of tasks includes the following task:
- [ ] As a registered user, I can click on the settings button from the navigation pane, so I can access the settings screen from everywhere in the app.
Is this item suitable to be submitted to the Product Backlog? Why?

The item is a user story, making it suitable for the Product Backlog.

1. **User Story Format**: It follows the As a [user], I can [action], so that [benefit] format.
2. **User Perspective**: It focuses on the user experience, which is essential in Agile.
3. **Specificity**: It clearly describes what the user can do, making it actionable.
4. **Prioritization**: It can be prioritized based on user needs.

Students might overlook the importance of the user story format and focus on technical details instead, leading to confusion about backlog suitability.


Question: Let $\mathcal{C}$ be the $(6,3)$ linear code on $\mathbb{F}_3$ whose parity-check matrix is  egin{equation*}      H = egin{pmatrix}   2 &0 &1 &1 &1 &0 \     1 &2 &0 &0 &1 &1 \     0 &0 &0 &1 &1 &1    \end{pmatrix}. \end{equation*} True or false: The sequence $\mathbf{y} = 111000$ is a codeword of $\mathcal{C}$.

To determine whether the sequence \(\mathbf{y} = 111000\) is a codeword in the linear code \(\mathcal{C}\) defined by the given parity-check matrix \(H\), we need to use the properties of linear codes and the parity-check matrix.

A sequence \(\mathbf{y}\) is a codeword of the code generated by the parity-check matrix \(H\) if it satisfies the condition that:

\[
H \mathbf{y}^T = \mathbf{0}
\]

where \(\mathbf{y}^T\) is the transpose of \(\mathbf{y}\), and the calculations are performed in the finite field \(\mathbb{F}_3\).

First, we express \(\mathbf{y}\) as a column vector:

\[
\mathbf{y} = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}
\]

Next, we compute the product \(H \mathbf{y}^T\):

\[
H = \begin{pmatrix}
2 & 0 & 1 & 1 & 1 & 0 \\
1 & 2 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1 & 1 & 1
\end{pmatrix}
\]

Now we compute \(H \mathbf{y}\):

1. For the first row:

\[
\begin{pmatrix} 2 & 0 & 1 & 1 & 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} = 2 \cdot 1 + 0 \cdot 1 + 1 \cdot 1 + 1 \cdot 0 + 1 \cdot 0 + 0 \cdot 0 = 2 + 0 + 1 + 0 + 0 + 0 = 3 \equiv 0 \mod 3
\]

2. For the second row:

\[
\begin{pmatrix} 1 & 2 & 0 & 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} = 1 \cdot 1 + 2 \cdot 1 + 0 \cdot 1 + 0 \cdot 0 + 1 \cdot 0 + 1 \cdot 0 = 1 + 2 + 0 + 0 + 0 + 0 = 3 \equiv 0 \mod 3
\]

3. For the third row:

\[
\begin{pmatrix} 0 & 0 & 0 & 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} = 0 \cdot 1 + 0 \cdot 1 + 0 \cdot 1 + 1 \cdot 0 + 1 \cdot 0 + 1 \cdot 0 = 0 + 0 + 0 + 0 + 0 + 0 = 0
\]

Now, combining our results, we have:

\[
H \mathbf{y} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}
\]

Since \(H \mathbf{y} = \mathbf{0}\), this means that \(\mathbf{y}\) is indeed a codeword of the code \(\mathcal{C}\).

Thus, the statement that the sequence \(\mathbf{y} = 111000\) is a codeword of \(\mathcal{C}\) is **True**.

Final answer: **1. True**


Question: Select the \emph{incorrect} statement. Euler Theorem

To identify the incorrect statement about Euler's Theorem and its implications, we will analyze each option in detail.

### Overview of Euler's Theorem

Euler's Theorem states that if \( a \) is an integer coprime to \( n \), then:
\[
a^{\varphi(n)} \equiv 1 \pmod{n},
\]
where \( \varphi(n) \) is Euler's totient function, which counts the integers up to \( n \) that are coprime to \( n \).

### Option Analysis

1. **is a generalization of Little Fermat Theorem.**
   - This statement is **correct**. Fermat's Little Theorem states that if \( p \) is a prime and \( a \) is an integer not divisible by \( p \), then \( a^{p-1} \equiv 1 \pmod{p} \). Euler's Theorem generalizes this to any modulus \( n \), not just primes, using the totient function \( \varphi(n) \).

2. **states that any \( x \in \{0, \dots, N-1\} \) and any \( k \), we have \( x^{k\varphi(N)+1} \equiv x \pmod{N} \), where \( N=pq \) for \( p, q \) distinct primes.**
   - This statement is **incorrect**. While it is true that \( x^{\varphi(N)} \equiv 1 \pmod{N} \) for \( x \) coprime to \( N \), the claim \( x^{k\varphi(N)+1} \equiv x \pmod{N} \) is not universally valid for all \( x \) in \( \{0, \dots, N-1\} \). Specifically, this does not hold for \( x \) that is not coprime to \( N \) (for example, when \( x = 0 \) or \( x \) is a multiple of \( p \) or \( q \)). The correct assertion should specify that this holds for \( x \) coprime to \( N \).

3. **gives the basis for polynomial time factoring.**
   - This statement is **incorrect**. Euler's Theorem itself does not provide a basis for polynomial time factoring. While it is involved in number theory and cryptography (such as RSA), factoring is a separate problem. The statement might confuse the implications of finding \( \varphi(n) \) efficiently, which relates to the difficulty of factoring, but Euler's Theorem does not directly lead to polynomial time algorithms for factoring integers.

4. **allows us to prove that RSA decryption works.**
   - This statement is **correct**. RSA relies on the properties of modular arithmetic and Euler's Theorem to demonstrate that if \( c \equiv m^e \pmod{N} \) (where \( c \) is the ciphertext, \( m \) is the plaintext, and \( e \) is the public exponent), then decrypting with \( d \) (the private exponent) gives \( m \). The relationship \( m \equiv c^d \pmod{N} \) is confirmed by the properties of Euler's theorem, specifically the fact that \( ed \equiv 1 \pmod{\varphi(N)} \).

### Conclusion

Based on the analysis:

- **Option 1**: Correct.
- **Option 2**: **Incorrect** (does not hold for all \( x \)).
- **Option 3**: **Incorrect** (Euler's theorem does not provide a basis for polynomial time factoring).
- **Option 4**: Correct.

The **incorrect statements** are **Option 2** and **Option 3**.

If you are looking for only one option to select as incorrect, **Option 2** is the most straightforwardly incorrect due to the universal claim made for all \( x \). However, it's important to note that **Option 3** is also misleading in its implications.