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The expansion operation within a search algorithm is formally expressed as the function $$C' = \text{Expand}(C, f)$$. Generally, $$C'$$ is the new set of states derived from the current set, $$C$$, using an expansion rule, $$f$$. In the context of prompt optimization, this formula is applied where $$C$$ is an existing set of prompts, $$f$$ is a model (typically a Large Language Model), and $$C'$$ is the newly generated set of prompts. The expansion is performed by instructing the LLM to create new, relevant prompts based on the initial set.

Expansion Function in Search Algorithms

In the context of a search algorithm, the process of discovering new states can be represented by the function $C' = \text{Expand}(C, f)$. Briefly explain the role of each of the three main components in this expression: $C$, $f$, and $C'$.\n\n(Note: Use of LaTeX for mathematical notation is supported, e.g., $C'$)

Deconstructing the Expansion Function

A team is optimizing a prompt for a customer service chatbot. Their initial prompt set consists of a single prompt: `{'You are a helpful assistant.'}`. To improve it, they use a large language model, giving it the instruction: 'Rewrite the following prompt to be more empathetic and specific to a retail context. Generate 5 variations.' The model then produces a new set of five more detailed prompts. Based on the formal expression $C' = \text{Expand}(C, f)$, identify what each component ($C$, $f$, and $C'$) represents in this specific scenario.

Applying the Expansion Function in Prompt Engineering

A search algorithm is exploring a graph with the following direct connections: Node A is connected to Node B and Node C; Node B is connected to Node A and Node D; Node C is connected to Node A; Node D is connected to Node B. The algorithm's current set of states is `C = {Node A}`. If the expansion function is defined as `C' = Expand(C, f)`, where the rule `f` is 'identify all unique nodes that are directly connected to any node in the current set `C`', what is the resulting new set of states `C'

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