Model: ML5562/fine-tuned-scibert_scivocab_uncased-all-json-M1_f16
Author: ML5562
Pipeline: sentence-similarity
Library: sentence-transformers
Tags: sentence-transformers, safetensors, bert, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:19392, loss:TripletLoss, arxiv:1908.10084, arxiv:1703.07737, base_model:allenai/scibert_scivocab_uncased, base_model:finetune:allenai/scibert_scivocab_uncased, model-index, text-embeddings-inference, endpoints_compatible, region:us
Base models (finetune): id: allenai/scibert_scivocab_uncased
Card metadata: tags: sentence-transformers, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:19392, loss:TripletLoss
base_model: allenai/scibert_scivocab_uncased
widget: source_sentence: Let $\mathcal{C}$ be a binary $(6,3)$ linear code containing the codewords $\mathbf{x}_1 = 011011$, $\mathbf{x}_2 = 101101$ and $\mathbf{x}_3 = 111000$. True or false: A generator matrix for the code is        egin{equation*}  G = egin{pmatrix}   1 &0 &0 &0 &1 &1 \      0 &1 &0 &0 &0 &1 \     0 &0 &1 &0 &1 &1   \end{pmatrix}     \end{equation*}
sentences: To determine whether the provided matrix \( G \) is a generator matrix for the binary \((6,3)\) linear code \(\mathcal{C}\), we need to follow these steps:

**Step 1: Understand the properties of a generator matrix.**
- A generator matrix \( G \) for a linear code must have \( k \) rows (where \( k \) is the dimension of the code) and \( n \) columns (where \( n \) is the length of the codewords).
- The rows of \( G \) should be linearly independent and span the code space, meaning any codeword can be expressed as a linear combination of the rows of \( G \).

**Step 2: Check the given generator matrix.**
- The generator matrix \( G \) has 3 rows and 6 columns, which is appropriate for a \((6,3)\) code.
- The rows of \( G \) are:
  \[
  \begin{pmatrix} 1 & 0 & 0 & 0 & 1 & 1 \end{pmatrix}, \quad 
  \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 1 \end{pmatrix}, \quad 
  \begin{pmatrix} 0 & 0 & 1 & 0 & 1 & 1 \end{pmatrix}
  \]

Now we need to check if we can generate the codewords \(\mathbf{x}_1\), \(\mathbf{x}_2\), and \(\mathbf{x}_3\) using linear combinations of these rows.

- **Generate codewords**:
  1. For \(\mathbf{x}_1 = 011011\):
     \[
     \mathbf{x}_1 = 0 \cdot \text{row}_1 + 1 \cdot \text{row}_2 + 1 \cdot \text{row}_3 \implies 0 \cdot (1,0,0,0,1,1) + 1 \cdot (0,1,0,0,0,1) + 1 \cdot (0,0,1,0,1,1) = (0,1,1,0,1,0) + (0,0,0,0,1,1) = (0,1,1,0,1,1) = (0,1,1,0,1,0)
     \]

  2. For \(\mathbf{x}_2 = 101101\):
     \[
     \mathbf{x}_2 = 1 \cdot \text{row}_1 + 0 \cdot \text{row}_2 + 1 \cdot \text{row}_3 \implies (1,0,0,0,1,1) + (0,0,0,0,1,1) = (1,0,0,0,1,1) = (1,0,1,1,0,1)
     \]

  3. For \(\mathbf{x}_3 = 111000\):
     \[
     \mathbf{x}_3 = 1 \cdot \text{row}_1 + 1 \cdot \text{row}_2 + 0 \cdot \text{row}_3 \implies (1,0,0,0,1,1) + (0,1,0,0,0,1) = (1,1,0,0,1,0) + (0,0,0,0,0,0) = (1,1,0,0,1,1)
     \]

From these calculations, we see that:
- \(\mathbf{x}_1\) cannot be generated correctly.
- \(\mathbf{x}_2\) and \(\mathbf{x}_3\) also do not match.

Thus, we conclude that the provided matrix \( G \) does not generate the codewords \(\mathbf{x}_1\), \(\mathbf{x}_2\), and \(\mathbf{x}_3\).

**Final Answer**: False, To solve the problem, we need to compute the product \((X^7 + X^6) \times (X + 1)\) in the field \(GF(2^8)\) represented as \(\mathbb{Z}_2[X]/P(X)\) with the polynomial \(P(X) = X^8 + X^4 + X^3 + X + 1\).

**Key Facts Supporting the Correct Option \(X^6 + X^4 + X^3 + X + 1\):**
1. When we multiply \((X^7 + X^6)\) by \((X + 1)\), we distribute the terms:
   \[
   (X^7 + X^6)(X + 1) = X^7 \cdot X + X^7 \cdot 1 + X^6 \cdot X + X^6 \cdot 1 = X^8 + X^7 + X^7 + X^6 = X^8 + X^6.
   \]
2. In \(GF(2)\), coefficients are reduced modulo 2, so \(X^7 + X^7\) cancels out to zero, giving us \(X^8 + X^6\).
3. Since \(X^8 \equiv X^4 + X^3 + X + 1 \mod P(X)\), we can replace \(X^8\) with this equivalence. Thus, we have:
   \[
   X^8 + X^6 \equiv (X^4 + X^3 + X + 1) + X^6 = X^6 + X^4 + X^3 + X + 1.
   \]

**Key Facts Against the Other Options:**
1. The option \(X^6 + X^5 + X^4 + X^3 + X\) includes a term \(X^5\) that does not arise from our calculations, as we only obtained \(X^6\), \(X^4\), \(X^3\), \(X\), and \(1\).
2. The option \(X^6\) does not account for the additional terms we obtained from reducing \(X^8\) modulo \(P(X)\), which are necessary for a complete representation in \(GF(2^8)\).
3. The option \(X^7 + X^6 + X^4 + X^3 + X + 1\) incorrectly retains \(X^7\) which is not present in the final result after reduction, confirming it is not a valid outcome.

Therefore, the correct answer is \(X^6 + X^4 + X^3 + X + 1\), as it accurately reflects the calculated result after performing the multiplication and reduction mod \(P(X)\)., To determine if the given matrix \( G \) is a generator matrix for the binary \((6,3)\) linear code with the specified codewords, we need to check if the rows of \( G \) can generate the codewords \(\mathbf{x}_1\), \(\mathbf{x}_2\), and \(\mathbf{x}_3\) through linear combinations.

1. **Identify the rows of the generator matrix \( G \)**:
   \[
   G = \begin{pmatrix}
   1 & 0 & 0 & 0 & 1 & 1 \\
   0 & 1 & 0 & 0 & 0 & 1 \\
   0 & 0 & 1 & 0 & 1 & 1
   \end{pmatrix}
   \]

2. **Calculate the codewords generated by \( G \)**:
   Each codeword is formed by taking all possible combinations of the rows of \( G \) (including the zero vector). The combinations can be represented as:
   \[
   \text{Codeword} = a_1 \mathbf{g}_1 + a_2 \mathbf{g}_2 + a_3 \mathbf{g}_3
   \]
   where \( a_i \in \{0, 1\} \) and \( \mathbf{g}_i \) are the rows of \( G \).

3. **Generate the codewords**:
   - For \( a_1 = 1, a_2 = 0, a_3 = 0 \): \(\mathbf{c}_1 = (1, 0, 0, 0, 1, 1)\) (which is \(\mathbf{x}_1\))
   - For \( a_1 = 0, a_2 = 1, a_3 = 0 \): \(\mathbf{c}_2 = (0, 1, 0, 0, 0, 1)\) (not in the given codewords)
   - For \( a_1 = 0, a_2 = 0, a_3 = 1 \): \(\mathbf{c}_3 = (0, 0, 1, 0, 1, 1)\) (not in the given codewords)
   - We can also check combinations, for example:
     - \( a_1 = 1, a_2 = 1, a_3 = 0 \): \(\mathbf{c} = (1, 1, 0, 0, 1, 0)\) (not a valid codeword)
     - \( a_1 = 1, a_2 = 0, a_3 = 1 \): \(\mathbf{c} = (1, 0, 1, 0, 0, 0)\) (not a valid codeword)
     - Other combinations yield similar results.

4. **Check for linear combinations**:
   We need to find combinations that yield \(\mathbf{x}_2\) and \(\mathbf{x}_3\). Since \(\mathbf{x}_2 = (1, 0, 1, 1, 0, 1)\) and \(\mathbf{x}_3 = (1, 1, 1, 0, 0, 0)\) cannot be formed from the rows of \( G \) as shown above, this indicates that the generator matrix does not generate the desired codewords.

Thus, the statement is **False**., source_sentence: Let us define the relation R = {(1, 1)} on the set A = {1, 2, 3}. Which of the following properties does R satisfy ? (multiple answers)
sentences: To analyze the algorithm \textsc{Random-Check}(S, T) and determine the probability it returns true when \( S \neq T \), we first need to establish a few key points regarding the random variables involved. 

1. **Understanding the Random Variables**: For each element \( e \in E \), a random variable \( x_e \) is assigned the value 1 with probability \( \frac{1}{3} \) and 0 with probability \( \frac{2}{3} \). Therefore, the expected value of \( x_e \) is:
   \[
   \mathbb{E}[x_e] = 1 \cdot \frac{1}{3} + 0 \cdot \frac{2}{3} = \frac{1}{3}.
   \]

2. **Sum of Random Variables**: Let \( X_S = \sum_{e \in S} x_e \) and \( X_T = \sum_{e \in T} x_e \). The algorithm returns true if \( X_S = X_T \).

3. **Expectation of Sums**: The expected values of \( X_S \) and \( X_T \) are given by:
   \[
   \mathbb{E}[X_S] = |S| \cdot \frac{1}{3}, \quad \mathbb{E}[X_T] = |T| \cdot \frac{1}{3}.
   \]
   If \( S \neq T \), it implies that at least one of the sets has a different number of elements, hence \( |S| \neq |T| \).

4. **Distribution of Sums**: Both \( X_S \) and \( X_T \) can be treated as sums of independent Bernoulli random variables. Their distributions can be approximated using the Central Limit Theorem when \( |S| \) and \( |T| \) are sufficiently large, leading to normal distributions with means \( \mathbb{E}[X_S] \) and \( \mathbb{E}[X_T] \).

5. **Probability of Equality**: The key point is to evaluate \( P(X_S = X_T) \). Since \( S \neq T \), we know:
   \[
   |S| \neq |T| \implies \mathbb{E}[X_S] \neq \mathbb{E}[X_T].
   \]
   When two independent normal distributions have different means, the probability that they are equal is less than 1.

6. **Exact Probability Calculation**: The probability that \( X_S = X_T \) can be calculated using the properties of the random variable distributions. Specifically, the independence of the random variables \( x_e \) for \( e \in S \) and \( e \in T \) leads us to the conclusion that:
   - The variance of each sum \( X_S \) and \( X_T \) is \( \text{Var}(X_S) = |S| \cdot \frac{1}{3} \cdot \frac{2}{3} \) and \( \text{Var}(X_T) = |T| \cdot \frac{1}{3} \cdot \frac{2}{3} \).

7. **Conclusion**: Based on the facts established, the probability that \( X_S = X_T \) when \( S \neq T \) can be bounded. Using the properties of independent distributions and the bounding techniques such as the union bound or Chernoff bounds, we establish that:
   \[
   P(X_S = X_T) \leq \frac{2}{3}.
   \]

Thus, we conclude that \textsc{Random-Check}(S, T) returns true with probability at most \( \frac{2}{3} \) when \( S \neq T \).

---

**Answer**: \textsc{Random-Check}(S, T) returns true with probability at most \( \frac{2}{3} \) if \( S \neq T \)., To determine which properties the relation R = {(1, 1)} satisfies on the set A = {1, 2, 3}, let's evaluate each property:

1. **Reflexive**: A relation is reflexive if every element in the set relates to itself. For R to be reflexive on A, it must contain (1, 1), (2, 2), and (3, 3). Since R only contains (1, 1), it is **not reflexive**.

2. **Symmetric**: A relation is symmetric if for every (a, b) in R, (b, a) is also in R. Since R contains (1, 1), and (1, 1) is its own reverse, R is **symmetric**.

3. **Anti-symmetric**: A relation is anti-symmetric if for every (a, b) and (b, a) in R, if a ≠ b, then it must be that a = b. Since R only contains (1, 1) and no pairs where a ≠ b, R is **anti-symmetric**.

4. **Transitive**: A relation is transitive if whenever (a, b) and (b, c) are in R, then (a, c) must also be in R. The only pair in R is (1, 1), and since it does not violate the condition (1, 1) and (1, 1) implies (1, 1), R is **transitive**.

5. **None of the other statements are valid**: Since we have found some valid properties, this statement is **not valid**.

In conclusion, the properties satisfied by the relation R are:
- Symmetric
- Anti-symmetric
- Transitive

Answer: 2, 3, 4, To determine which properties the relation \( R = \{(1, 1)\} \) satisfies on the set \( A = \{1, 2, 3\} \), we will analyze each property one by one.

1. **Reflexive:** A relation is reflexive if every element in the set is related to itself. In this case, we need to check if every element in \( A \) is related to itself:
   - For \( 1 \) (itself): \( (1, 1) \) is in \( R \).
   - For \( 2 \): \( (2, 2) \) is not in \( R \).
   - For \( 3 \): \( (3, 3) \) is not in \( R \).
   
   Since not all elements are related to themselves, \( R \) is **not reflexive**.

2. **Symmetric:** A relation is symmetric if whenever \( (a, b) \) is in \( R \), then \( (b, a) \) is also in \( R \). The only pair in \( R \) is \( (1, 1) \):
   - Since \( (1, 1) \) is in \( R \), \( (1, 1) \) is also in \( R \).
   
   Therefore, \( R \) is **symmetric**.

3. **Anti-symmetric:** A relation is anti-symmetric if whenever \( (a, b) \) and \( (b, a) \) are both in \( R \) for \( a \neq b \), then \( a \) must equal \( b \). In this case, since the only pair we have is \( (1, 1) \), there are no pairs \( (a, b) \) where \( a \neq b \).
   
   Thus, \( R \) is **anti-symmetric**.

4. **Transitive:** A relation is transitive if whenever \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) must also be in \( R \). In this case, we only have \( (1, 1) \):
   - Since there are no pairs \( (a, b) \) and \( (b, c) \) with \( b \) different from \( 1 \), we can consider the transitive condition satisfied trivially.
   
   Therefore, \( R \) is **transitive**.

5. **None of the other statements are valid:** This option can be ruled out since we have found that \( R \) is symmetric, anti-symmetric, and transitive.

Based on the analysis, the properties that relation \( R \) satisfies are:

- **Symmetric**
- **Anti-symmetric**
- **Transitive**

So the final answer would be: **1. Reflexive (no)**, **2. Symmetric (yes)**, **3. Anti-symmetric (yes)**, **4. Transitive (yes)**, **5. None of the other statements are valid (no)**.
Therefore, among 1 through 5, the answer is:
The properties that the relation \( R = \{(1, 1)\} \) satisfies on the set \( A = \{1, 2, 3\} \) are:

- **Symmetric** (Yes)
- **Anti-symmetric** (Yes)
- **Transitive** (Yes)

Thus, the correct answer choices are:
- **2. Symmetric**
- **3. Anti-symmetric**
- **4. Transitive**

In summary, the relation \( R \) does not satisfy reflexivity, but it does satisfy symmetry, anti-symmetry, and transitivity., source_sentence: In general, what is true regarding Fagin's algorithm?
sentences: The provided PyTorch code for the `ThreeLayerNet` will execute without any errors. The variable `x` is reused throughout the forward method, which is permissible in Python, as it simply references the output of each layer sequentially. Moreover, there is no restriction on the number of neurons in successive layers; it is valid to have a layer with more neurons than its predecessor.

However, while the architecture is syntactically correct, the model's structure could lead to ineffective training. The increase in the number of neurons from the first layer (10) to the second layer (200) can cause issues such as increased complexity and a higher risk of overfitting, especially if the dataset is not sufficiently large or diverse. Additionally, depending on the initialization of the weights and the distribution of the input data, the model may encounter challenges with optimization, such as vanishing gradients when using activation functions like ReLU.

These factors can hinder the model's ability to learn effectively, potentially resulting in performance that does not improve significantly over training epochs, particularly if the model struggles with optimization due to its architecture.

In summary, while the code will run without errors, the configuration might lead to challenges in training the model effectively.

Answer: 3, Fagin's algorithm is an efficient approach designed to retrieve the top-k documents that have the highest scores based on multiple criteria or metrics. It operates on a set of ranked lists, which can be thought of as posting files that represent documents and their associated scores for different attributes. One of the key characteristics of Fagin's algorithm is its ability to optimize the retrieval process by focusing on only a subset of documents that are most likely to yield the highest aggregate scores, rather than performing a full scan of all documents. This way, it significantly reduces the amount of data processed, leading to improved performance, especially in large datasets. 

The algorithm works by incrementally retrieving documents from the sorted lists and maintaining a running tally of their aggregate scores. As it processes each list, it updates the scores and keeps track of the top-k candidates. The efficiency of Fagin's algorithm is largely derived from its ability to stop scanning once it has enough information to determine the top k documents, thus avoiding unnecessary computations. This makes it particularly suitable for applications where quick retrieval of high-quality results is crucial, such as in information retrieval systems and search engines. Overall, Fagin's algorithm demonstrates a balance between thoroughness and efficiency by leveraging the properties of ranked lists to meet the demands of top-k retrieval tasks effectively., Fagin's algorithm is an important method in the realm of information retrieval and is particularly notable for its efficiency in handling certain types of queries. It operates by focusing on a specific subset of documents based on scores that are aggregated from different ranking criteria. The algorithm is designed such that it can effectively determine a selection of documents that meet specific relevance thresholds, prioritizing those that are expected to yield the highest scores. The nature of its operation implies that it requires a systematic approach to evaluate and compare these scores across documents. However, the specifics of how it processes and retrieves these documents can sometimes be a bit complex, as it involves a nuanced interplay of multiple factors and may not just be about merely scanning or indexing. The implications of its design extend to how it handles various data structures and the efficiency with which it can return a result set that is both relevant and optimized for the user's request. In summary, while there are clear principles guiding Fagin's algorithm, the exact mechanics and outcomes can vary depending on the context and the data involved., source_sentence: Devise an algorithm that, without consensus, implements a weaker specification of NBAC by replacing the termination property with very weak termination.

Very weak termination: If no process crashes, then all processes decide. Is a failure detector needed to implement this algorithm?
sentences: To design an algorithm that implements a weaker specification of Non-Blocking Atomic Commit (NBAC) by replacing the termination property with very weak termination, we must first clarify the concepts involved.

### Background Concepts

1. **Non-Blocking Atomic Commit (NBAC)**: A distributed commit protocol allows processes to reach a consensus without blocking in the presence of failures. The standard termination property ensures that any process that does not crash eventually decides on a value.

2. **Very Weak Termination**: In our context, this means that if no process crashes, then every process must eventually decide on a value, but we do not impose this requirement on cases with process crashes.

3. **Failure Detector**: A failure detector is a mechanism that helps the system to ascertain the health of its processes (whether they've failed or are still running). It simplifies the design of consensus protocols by helping to distinguish between live and failed processes.

### The Proposed Algorithm

Let’s outline an algorithm that achieves very weak termination:

**1. Roles and Initial Setup**:
- Each process \( P_i \) has an input value \( v_i \) and is responsible for making a decision on a common value \( d_i \).

**2. Communication Phase**:
- Each process sends its value \( v_i \) to all other processes periodically. This may begin immediately or after a predefined timeout.

**3. Value Collection Phase**:
- Each process collects the responses (i.e., the values) from other processes.
- After receiving responses from a certain number of processes (majority or all active ones), it creates a set of collected values.

**4. Decision Phase**:
- After collecting values, each process decides based on the aggregated inputs according to a predefined rule (e.g., if a majority agree on a common value, choose that; if all values are the same, choose that; otherwise, select a default value).

**5. Weak Termination Condition**:
- If no process crashes, all processes eventually decide based on their communications. If a process does not receive messages from others due to a crash or network issues, it will still rely on its own data and previously received inputs to make a decision.

### Does This Require a Failure Detector?

**Requirement of Failure Detector**:
In scenarios where we want to ensure the system can differentiate between processes that have crashed and those that are alive, a failure detector can be beneficial. It provides the necessary assurances to the processes about the liveness of others, which can help avoid unnecessary waiting.

However, **the presence of a failure detector is not strictly necessary** to achieve our goal of very weak termination. The algorithm could still function using timeout-based communication:
- Each process may wait for a certain duration for responses before making a decision independently.
- This inherently introduces a way to account for failures even without using an explicit failure detector.

### Conclusion

The algorithm outlined above achieves very weak termination without the need for a consensus. Processes communicate, aggregate values, and make decisions based on responses or independently if no responses are received within a set timeframe. While failure detectors can facilitate improved decision-making, they are not a strict necessity in this context—a timeout-based mechanism suffices for our design requirements., To analyze the given binary encoding and the provided options, let's break it down step by step:

1. **Understanding the Encoding**: 
   The encoding for the symbols \( a, b, c, d \) is of the form \( ??0 \), where the first two bits are unknown. This means that the last bit of each encoding is fixed as '0', while the first two bits can vary, leading to different combinations.

2. **Total Combinations**: 
   Since there are 2 bits that can vary (each can be either '0' or '1'), there are \( 2^2 = 4 \) possible combinations for the first two bits: 
   - 00
   - 01
   - 10
   - 11

   Thus, we can encode \( a, b, c, d \) uniquely using these combinations, as long as each symbol gets a different combination of the first two bits.

3. **Evaluating the Options**:

   - **Option 1: "The encoding is uniquely-decodable."**
     - Since each symbol can be assigned a unique combination of the first two bits (e.g., \( 00, 01, 10, 11 \)), and all code-words end with '0', this encoding is indeed uniquely decodable. 

   - **Option 2: "The encoding is uniquely-decodable but not prefix-free."**
     - A prefix-free code means that no code-word is a prefix of another code-word. In this case, because all code-words end with '0', any combination of the first two bits will not create a prefix situation, as no code can be a prefix of another that differs in the last bit. Thus, while the encoding is uniquely decodable, it is also prefix-free.

   - **Option 3: "We do not possess enough information to say something about the code."**
     - This option can be discarded, as we have enough information to determine that the encoding can be made uniquely decodable with the given structure.

   - **Option 4: "It does not satisfy Kraft's Inequality."**
     - Kraft’s Inequality relates to the lengths of code-words in a uniquely decodable code. In this case, all code-words are of the same length (3 bits), and thus they can satisfy Kraft's Inequality. Since we can uniquely assign combinations, this option is not applicable.

4. **Comparing Options 1 and 2**:
   - Both options 1 and 2 claim that the encoding is uniquely decodable. However, option 2 states that it is not prefix-free, which, upon evaluation, is incorrect because all the code-words have a uniform structure that does not lead to any prefix conflicts. Therefore, option 1 is the correct and more accurate choice.

**Final Conclusion**:
The best answer is: **"The encoding is uniquely-decodable."**, To implement an algorithm that achieves the weaker specification of NBAC with very weak termination, we can utilize a consensus algorithm that allows for termination if no failures occur. 

**Algorithm Outline:**

1. **Initialization**: Each process starts with a proposed value.
2. **Vote Phase**: Each process collects proposed values from all others.
3. **Decision Phase**: 
   - If a process has received proposals from all other processes and its own value is not contradicted by others, it decides on a value.
   - If a process finds no disagreement among proposals, it can proceed to make a decision.
4. **Termination**:
   - If no process crashes, all processes will eventually decide based on the collected proposals.

**Need for a Failure Detector**:
In this algorithm, a failure detector is not necessary to implement the requirement of very weak termination. Since the specification guarantees that if no processes crash, all processes will eventually decide, we do not need the additional overhead of detecting failures to reach a consensus. This allows the system to reach a decision among processes reliably without distinguishing live and failed processes in the decision-making process, as long as they continue operating correctly.

In summary, while traditional consensus algorithms often use failure detection to handle crashes, in this algorithm focused on very weak termination, we can achieve the goal without it, assuming no failures occur., source_sentence: Select which statements are true regarding SCFGs.A penalty will be applied for any incorrect answers.
sentences: The true statements regarding SCFGs are A and B. 

Reason: A is true because the probabilities of rules with the same left-hand side must sum to 1 to maintain valid probability distributions. B is also true because lexical rules in SCFGs represent the probabilities of generating terminal symbols, analogous to emission probabilities in Hidden Markov Models (HMMs) used for Part-of-Speech (PoS) tagging. The other statements are either incorrect or not universally applicable to all SCFGs., The true statements regarding Stochastic Context-Free Grammars (SCFGs) are:

A: The sum over all the probabilities of the rules of a SCFG that share the same left-hand side should sum up to 1, which is a fundamental property of probabilistic grammars. 
B: The probabilities of lexical rules of a SCFG correspond to emission probabilities of Hidden Markov Models (HMMs) for Part-of-Speech (PoS) tagging, indicating a similarity in how both types of models handle probabilities associated with observed events. 

The other statements either misrepresent SCFG properties or are incorrect., To avoid hitting the API rate limit while still allowing for testing, I propose implementing a local mock service that can simulate Google authentication responses, enabling us to conduct thorough integration tests without making actual API calls. This mock service could be switched on or off via a configuration flag, allowing us to preserve our API calls for customer use while still ensuring the robustness of our authentication integration.
pipeline_tag: sentence-similarity
library_name: sentence-transformers
metrics: cosine_accuracy
model-index: name: SentenceTransformer based on allenai/scibert_scivocab_uncased
results: task: type: triplet
name: Triplet
dataset: name: val eval
type: val-eval
metrics: type: cosine_accuracy
value: 0.6229373216629028
name: Cosine Accuracy
verified: no
Likes: 0 | All-time downloads: 37
Created: 2025-06-05 17:27:58