提出可衡量大模型搜索空间的数学理论，帮AI更好规划探索路径。
Where to Search: Measure the Prior-Structured Search Space of LLM Agents
- 用模糊关系建模智能体行为，定义安全约束范围
- 通过延续参数量化多步推理的可达性难度
- 为大模型搜索提供可计算、可验证的评估框架
基于大语言模型的生成-过滤-精炼迭代范式在人工智能与科学交叉领域已取得进展，但搜索效率高度依赖于如何将领域先验信息编码为结构化的假设空间。本文提出一种紧凑的公理化理论，用于描述和度量由领域先验引导的LLM辅助迭代搜索过程。将智能体建模为输入输出间的模糊关系算子，以捕捉可行状态转移，并受固定安全包络约束；为描述多步推理/搜索，通过单一延续参数加权所有可达路径并求和，得到覆盖生成函数，进而诱导出可达性难度度量，并提供由安全包络诱导图上的几何解释。进一步给出最简可检验推论，并通过两个实例验证。该理论为度量智能体及其搜索空间提供了可操作的语言与工具，系统性地构建了由大模型实现的迭代搜索的形式化描述。
原文摘要 · Abstract (English)
The generate-filter-refine (iterative paradigm) based on large language models (LLMs) has achieved progress in reasoning, programming, and program discovery in AI+Science. However, the effectiveness of search depends on where to search, namely, how to encode the domain prior into an operationally structured hypothesis space. To this end, this paper proposes a compact formal theory that describes and measures LLM-assisted iterative search guided by domain priors. We represent an agent as a fuzzy relation operator on inputs and outputs to capture feasible transitions; the agent is thereby constrained by a fixed safety envelope. To describe multi-step reasoning/search, we weight all reachable paths by a single continuation parameter and sum them to obtain a coverage generating function; this induces a measure of reachability difficulty; and it provides a geometric interpretation of search on the graph induced by the safety envelope. We further provide the simplest testable inferences and validate them via two instantiation. This theory offers a workable language and operational tools to measure agents and their search spaces, proposing a systematic formal description of iterative search constructed by LLMs.
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