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In the context of least-to-most prompting, the decomposition of an initial problem, $$p_0$$, into a set of sub-problems, $$\left\{p_1, ..., p_n\right\}$$, is represented by the formula $$\left\{p_1, ..., p_n\right\} = G(p_0)$$. The function $$G$$ in this equation symbolizes the decomposition method used to break down the original problem $$p_0$$ into its constituent parts.

Formula for Problem Decomposition in Least-to-Most Prompting

This formula provides a recursive definition for generating the $$i$$-th sub-problem, $$p_i$$. It is expressed as: $$p_i = G_i(p_0, \{p_i, a_i\})$$ Here, the sub-problem $$p_i$$ is defined by a function $$G_i$$ that takes the original problem $$p_0$$ as context, along with the sub-problem $$p_i$$ itself and its answer $$a_i$$. This structure implies that a valid sub-problem must satisfy this self-referential condition.

Recursive Formula for Sub-Problem Generation

When generating sub-problems step-by-step during a decomposition process, the formulation of each new sub-problem depends on the history of the task. The $$i$$-th sub-problem, $$p_i$$, is created by a generation function $$G_i(\cdot)$$ that considers the original problem $$p_0$$ along with the preceding sub-problems $$p_{<i}$$ and their respective solutions $$a_{<i}$$. This relationship is represented mathematically by the equation: $$p_i = G_i(p_0, \{p_{<i}, a_{<i}\})$$.

Formula for Sequential Sub-Problem Generation

A user gives a language model the following instruction: 'Create a comprehensive business plan for a new coffee shop.' The model responds by first generating a list of essential sections to write: 1. Executive Summary, 2. Company Description, 3. Market Analysis, 4. Organization and Management, 5. Products and Services, 6. Marketing and Sales Strategy, 7. Financial Projections. This process of breaking down the main task can be represented by the formula {p_1, ..., p_n} = G(p_0). Analyze this scenario and identify what p_0, G, and {p_1, ..., p_n} represent.

Decomposition of a Planning Task

In the context of breaking down a complex problem for a language model, explain the role of each component in the formula {p_1, ..., p_n} = G(p_0).

Explaining the Problem Decomposition Formula

A user wants a language model to write a comprehensive travel guide for Paris. To manage this complex task, they first prompt the model to create a list of key sections to cover, such as 'Famous Landmarks,' 'Local Cuisine,' and 'Transportation Tips.' This process of breaking down the main task into a set of smaller, structured parts can be represented by the formula {p_1, ..., p_n} = G(p_0). In this scenario, what does the function G represent?

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