1. Bibliographic Information

• Title: Where to Search: Measure the Prior-Structured Search Space of LLM Agents
• Authors: Zhuo-Yang Song (School of Physics, Peking University, Beijing 100871, China)
• Journal/Conference: This paper is a preprint available on arXiv. It has not yet undergone formal peer review for a conference or journal. arXiv is a common platform for researchers to share early-stage work.
• Publication Year: 2025 (as listed on the arXiv submission, which is likely a placeholder or future projection; the v2 version was submitted in October 2025 according to the source link format).
• Abstract: The paper addresses the challenge of understanding and measuring the effectiveness of iterative search processes driven by Large Language Models (LLMs), such as the "generate-filter-refine" paradigm. The core problem is that an agent's success depends heavily on how its search space is structured by domain-specific knowledge (priors). The author proposes a formal mathematical theory to describe and measure this search space. Key elements of the theory include: representing an agent as a fuzzy relation, defining a "safety envelope" that constrains the agent's actions, and introducing a "coverage generating function" to measure the difficulty of reaching a target state. This framework provides a geometric view of the search process and yields testable hypotheses, which are validated through a simple "majority-vote" experiment on a grid. The theory aims to provide a systematic language and tools for analyzing and comparing LLM agents.
• Original Source Link: https://arxiv.org/abs/2510.14846v2 (This link points to the preprint on arXiv.)

2. Executive Summary

• Background & Motivation (Why):

  • Core Problem: LLM-based agents are increasingly used for complex tasks like reasoning, programming, and scientific discovery using an iterative "generate-filter-refine" loop. However, there is no formal, quantitative way to understand where these agents are searching. Their effectiveness is not just about raw intelligence but about how well their search is constrained and guided by prior knowledge of the problem domain.
  • Importance & Gaps: Without a formal theory, designing and analyzing these agents relies on heuristics (e.g., prompt engineering, ad-hoc scoring). This makes it difficult to ensure safety (preventing agents from taking dangerous actions), measure search efficiency, compare different agents systematically, or understand their long-horizon behavior. The paper identifies this lack of a "unified language and quantitative tools" as a key theoretical gap hindering the development of controllable and measurable LLM agents.
  • Fresh Angle: The paper introduces a novel approach by borrowing concepts from fuzzy set theory and combinatorics to build a formal measurement system from the ground up. It moves beyond simple success/fail metrics to provide a continuous measure of "reachability difficulty" and connects it to the geometric structure of the search space.

• Main Contributions / Findings (What):

  • Formal Theory for LLM Search: The paper proposes a compact theory to model LLM-driven iterative search. The main contributions are:
    1. Fuzzy Agent Representation: An agent's behavior (transition from input $f$ to output $g$) is modeled as a fuzzy relation operator $\mu_f(g)$, which captures the likelihood or feasibility of a transition.
    2. Safety Envelope: A crisp idealized agent $\mathcal{T}_0$ is defined to represent a strict "safety envelope"—the set of all permissible transitions. Real agents must operate within this boundary.
    3. Coverage Generating Function: A mathematical tool, $P_{f,g}(p)$, is introduced to aggregate the weights of all possible paths from a start state $f$ to a target $g$. A continuation parameter $p$ is used to systematically account for paths of different lengths.
    4. Reachability Measures: From the generating function, the paper derives key metrics: the critical parameter $p_c$ and the coverage index $R_c = 1 - p_c$, which quantify the difficulty of reaching a target. It also connects these to geometric graph properties like shortest distance ($d_0$) and the number of shortest paths ($N_{d_0}$).
  • Key Findings:
    • The theory provides a testable hypothesis for LLM agents: in many tasks, their search is approximately unidirectional (they rarely loop or backtrack). This implies a specific mathematical relationship: the number of shortest paths grows slower than exponentially with the path length, formally stated as $\log N_{d_0}(f,g) \ll d_0(f,g)$.
    • A "majority-vote" experiment on a 2D grid confirms this hypothesis. The search graph generated by LLMs was indeed unidirectional, and the measured data for $(d_0, N_{d_0})$ fit the predicted trend, providing initial validation for the theory.

3. Prerequisite Knowledge & Related Work

• Foundational Concepts:

  • Large Language Models (LLMs): These are massive neural networks trained on vast amounts of text data, capable of generating human-like text, answering questions, and performing reasoning tasks. In this paper, they are the "engine" of the agent.
  • Generate-Filter-Refine Paradigm: This is an iterative problem-solving loop.
    1. Generate: An LLM proposes one or more potential solutions or next steps.
    2. Filter: An evaluator (which could be another LLM, a piece of code, or a human) assesses the quality or validity of the proposals.
    3. Refine: Based on the evaluation, the current state is updated, and the process repeats. This is a form of guided search.
  • LLM Agent: An autonomous system that uses an LLM as its core reasoning component to perceive its environment, make decisions, and take actions to achieve a goal.
  • Domain Prior: Pre-existing knowledge or constraints specific to a task. For example, in a chess game, the rules of how pieces can move are a domain prior. In this paper, priors structure the "hypothesis space" to make search tractable.
  • Fuzzy Set Theory: A branch of mathematics that generalizes classical set theory. In classical sets, an element either belongs to a set or it doesn't. In fuzzy sets, an element can have a degree of membership between 0 and 1. A fuzzy relation extends this to relationships between elements, where $\mu_f(g)$ represents the degree to which $f$ is related to $g$.
  • Graph Theory: The study of graphs, which are mathematical structures used to model pairwise relations between objects. A directed graph consists of nodes (or vertices) and directed edges (or arcs) connecting them. A path is a sequence of edges, and the shortest distance between two nodes is the minimum number of edges in a path connecting them.
  • Generating Function: A formal power series from combinatorics whose coefficients encode a sequence of numbers. For example, the sequence $a_0, a_1, a_2, \dots$ can be represented by the function $A(x) = \sum_{n=0}^{\infty} a_n x^n$. In this paper, the coefficients are the number of paths of a given length.

• Previous Works: The paper situates itself within several research areas:

  • Reasoning and Planning with LLMs: It cites foundational works like Chain-of-Thought [3], Tree of Thoughts [2], and Graph of Thoughts [5], which are all methods to improve LLM reasoning by structuring it as a multi-step or graph-based search.
  • Tool Use and Program-Aided LLMs: Works like Toolformer [7] and PAL [10] show how LLMs can learn to use external tools or write code to solve problems, expanding their operational capabilities.
  • AI for Science: The paper references examples where iterative search has been used for scientific discovery, such as finding faster sorting algorithms [12] or discovering mathematical theorems [13].
  • Safety and Controllability: The concept of a safety envelope is related to research on making AI systems safe and aligned with human intentions, such as Constitutional AI [25] and Nemo Guardrails [29].

• Differentiation: While previous works use iterative search, this paper aims to formally measure it. It abstracts away the specifics of the LLM or the prompt and provides a general, model-agnostic framework. Instead of just proposing a new search algorithm, it offers a set of mathematical tools (fuzzy relations, generating functions, coverage index) to analyze any such algorithm. This provides a meta-level understanding of the search process itself.

4. Methodology (Core Technology & Implementation)

The paper's core contribution is a formal theory to measure the search space of LLM agents. Here is a step-by-step breakdown.

• Principles: The central idea is to model the agent's behavior as a set of possible transitions on a graph, where each transition has a "strength" or "membership value." By summing up the strengths of all possible paths between two points, we can quantify how "reachable" one point is from another.

• Steps & Procedures:

  1. Defining the Agent and its Boundaries

  • The search space is defined by an input space $C_1$ and an output space $C_2$. For iterative tasks, the output can be fed back as input, so $C_2 \subseteq C_1$.

  • Definition 1 (Ideal agent): An agent $\tau$ is formalized as a mapping that, for any input state $f \in C_1$, defines a membership function over all possible output states $g \in C_2$. $$\mu _ { f } : C _ { 2 } \to [ 0 , 1 ] , \qquad g \mapsto \mu _ { f } ( g )$$

    • Explanation: This function $\mu_f(g)$ represents the "degree of possibility" or "feasibility" that the agent transitions from state $f$ to state $g$ in one step. A value of 1 means a fully possible transition, 0 means impossible, and a value in between represents partial feasibility. This is also described as a fuzzy relation operator $\mathcal{T}(f, g) = \mu_f(g)$.

  • Definition 2 (Crisp idealized agent and safety envelope): A special type of agent where all transitions are either fully possible (1) or impossible (0). This is called a crisp agent. Such an agent, denoted $\mathcal{T}_0$, defines a safety envelope. A regular fuzzy agent $\tau$ is "constrained by safety" if its transition possibilities are always less than or equal to those of the safety envelope. $$0 \leq \mathcal{T} ( f , g ) \leq \mathcal { T } _ { 0 } ( f , g ) , \qquad \forall f , g$$

    • Explanation: $\mathcal{T}_0$ acts as a hard boundary. It defines all theoretically allowed moves. The actual agent $\mathcal{T}$ may only be able to execute a subset of these moves, and possibly with less than full confidence.

  • Definition 3 (Search trajectory): A sequence of states $(f^{(0)}, f^{(1)}, \dots, f^{(n)})$ is a valid search trajectory if every step in the sequence has a non-zero transition possibility: $\mu_{f^{(i)}}(f^{(i+1)}) > 0$.

  2. Measuring Reachability with a Generating Function

  • Definition 4 (Coverage generating function): To measure the total "connectivity" between a start state $f$ and a target state $g$, the paper introduces a generating function $P_{f,g}(p)$. This function sums up the "weights" of all possible paths from $f$ to $g$. $$P _ { f , g } ( p ) : = \sum _ { n = 0 } ^ { \infty } \sum _ { { \cal S } T : f ^ { ( 0 ) } = f , \atop f ^ { ( n ) } = g } p ^ { n } \prod _ { i = 0 } ^ { n - 1 } \mu _ { f ^ { ( i ) } } \bigl ( f ^ { ( i + 1 ) } \bigr )$$

    • Symbol Explanation:
      • $P_{f,g}(p)$: The coverage generating function from $f$ to $g$.
      • $n$: The length of a path (number of steps).
      • $\mathcal{S}T$: A specific search trajectory (path) of length $n$ from $f^{(0)}=f$ to $f^{(n)}=g$.
      • $p$: The continuation parameter ($p \in [0,1]$). This is a formal variable, not a probability. It acts as a "bookkeeping" device that assigns a weight of $p^n$ to a path of length $n$.
      • $\prod_{i=0}^{n-1} \mu_{f^{(i)}}(f^{(i+1)})$: The weight of a single path, calculated by multiplying the membership values of each step along the path.
      • The $n=0$ term is 1 if $f=g$ (a zero-length path from a node to itself) and 0 otherwise.

  • Remark 1 (Operator viewpoint): If the state space is countable, the agent's transitions can be represented by a matrix $M$ where $M_{f,g} = \mathcal{T}(f,g)$. The generating function can then be written compactly as a geometric series of matrices: $$P ( p ) = \sum _ { n \geq 0 } p ^ { n } M ^ { n } = ( I - p M ) ^ { - 1 }$$

    • Explanation: This formula is analogous to the scalar geometric series $1/(1-x) = \sum x^n$. It provides a way to compute the coverage for all pairs (f,g) at once. The series converges when $p \rho(M) < 1$, where $\rho(M)$ is the spectral radius of the matrix $M$.

  3. Defining Geometric Quantities and Reachability Metrics

  • Definition 5 & 6 (Path counting and Shortest distance): On the graph defined by the safety envelope $\mathcal{T}_0$, the generating function simplifies to a path-counting polynomial: $$P _ { f , g } ^ { \mathrm { i d e a l } } ( p ) = \sum _ { n = 0 } ^ { \infty } N _ { n } ( f , g ) p ^ { n }$$ where $N_n(f,g)$ is the number of distinct paths of length $n$ from $f$ to $g$. The shortest distance $d_0(f,g)$ is the smallest $n$ for which $N_n(f,g) \ge 1$.

  • Definition 7 (Coverage index and critical parameter): This is the paper's key measure of difficulty. The critical parameter $p_c(f,g)$ is the smallest value of $p$ for which the ideal generating function $P_{f,g}^{\text{ideal}}(p)$ reaches a value of 1. $$p _ { c } ( f , g ) : = \operatorname* { i n f } \big \{ p \in [ 0 , 1 ] : { \cal P } _ { f , g } ^ { \mathrm { i d e a l } } ( p ) \geq 1 \big \}$$ The coverage index $R_c(f,g)$ is defined as: $$R _ { c } ( f , g ) : = 1 - p _ { c } ( f , g ) \in [ 0 , 1 ]$$

    • Explanation: A path is "easier" if it is short or if there are many alternative paths. Both factors cause $P^{\text{ideal}}(p)$ to grow faster with $p$. Therefore, an "easier" path from $f$ to $g$ will reach the threshold of 1 at a smaller value of $p_c$. This means a larger coverage index $R_c$. So, $R_c$ is a measure of reachability, where higher is better (easier).

  4. Deriving Testable Hypotheses

  • Assumption 1 (Approximate thresholding): The paper hypothesizes that LLM agents often conduct approximately unidirectional searches (they don't backtrack or get stuck in loops). This means the generating function is dominated by its first few terms (shorter paths).

  • Assumption 2 (Basic lower bound and testable inference): The generating function is at least as large as its first non-zero term, which corresponds to the shortest paths. $$P _ { f , g } ^ { \mathrm { i d e a l } } ( p ) \geq N _ { d _ { 0 } } ( f , g ) p ^ { d _ { 0 } ( f , g ) }$$ From this, a bound on the critical parameter $p_c$ is derived. By setting the right side to 1, we get $p_c(f,g) \le (N_{d_0}(f,g))^{-1/d_0(f,g)}$. This leads to a key testable inequality. For difficult paths where $R_c$ is small and $d_0$ is large, this simplifies to: $$d _ { 0 } ( f , g ) \cdot R _ { c } ( f , g ) \ \stackrel { > } { \sim } \ \log N _ { d _ { 0 } } ( f , g )$$ And because $R_c \ll 1$ in these difficult cases, it implies a fundamental property of the search space: $$\log N _ { d _ { 0 } } ( f , g ) \ll d _ { 0 } ( f , g )$$

    • Explanation: This is the central testable prediction. It states that the logarithm of the number of shortest paths grows much more slowly than the length of the shortest path itself. In other words, path diversity is limited, and the search complexity is dominated by the sheer distance to the target, not by a combinatorial explosion of choices at each step.

5. Experimental Setup

The paper provides a "majority-vote instantiation" to validate the theory.

• Task Space (Dataset):

  • The experiment is conducted on a two-dimensional grid, $G_N = \{0, \dots, N-1\}^2$.

  • Three grid sizes are used: $N=3, 5, 8$.

  • The corresponding target points $t$ for each grid size are specified.

  • Origin/Characteristics: This is a synthetic "toy" problem, chosen for its simplicity and because its geometric properties (like distance) are easy to compute and visualize. It serves as a minimal, controllable environment to test the formal concepts.

    The paper includes a table in the Appendix with the specific grid sizes and target points, which is transcribed below.

  Table 1: Three grid sizes and corresponding target points $t$.

  number	N	t
  1	3	(1,2)
  2	5	(3, 4)
  3	8	(6, 7)

• Evaluation Metrics: The experiment does not use traditional machine learning metrics like accuracy or F1-score. Instead, it measures the geometric properties of the search graph predicted by the theory.

  1. Shortest Distance ($d_0(f,t)$):
     • Conceptual Definition: The minimum number of steps (unit moves up, down, left, or right) required to get from a starting node $f$ to the target node $t$ on the graph of allowed moves. It measures the most efficient path's length.
     • Computation: Computed using a Breadth-First Search (BFS) algorithm starting from $f$ on the directed graph $\mathcal{G}_t$.
  2. Number of Shortest Paths ($N_{d_0}(f,t)$):
     • Conceptual Definition: The total count of distinct paths that have the minimum possible length, $d_0$. This metric quantifies the "path diversity" or redundancy available to the agent for finding the most efficient solution.
     • Computation: Computed simultaneously with the BFS by keeping track of how many ways each node can be reached at its shortest distance from the source.

• Baselines:

  • There are no competing models or methods used as baselines. The "baseline" is the theoretical prediction derived in Assumption 2: $\log N_{d_0} \ll d_0$. The experiment aims to see if the empirical data from the LLM-generated graph conforms to this theoretical upper-trend.

• Implementation Details (The "Majority-Vote Instantiation"):

  1. Ideal Agent Construction: For a given grid with start $f$ and target $t$, an LLM is prompted to choose the next move (up, down, left, right). This process is repeated 5 times ($m=5$). If a single move receives a strict majority (e.g., is chosen 3+ times), that move is considered the agent's deterministic choice from $f$. If there's no majority, the agent makes no move from $f$. This is done for a set of 8 different LLMs (gpt-5-mini, gpt-5, qwen3, etc.).
  2. Aggregate Agent: The final ideal agent $\mu_f^{(t)}(g)$ is created by averaging the decisions of all 8 LLMs. For a transition from $f$ to $g$, its membership value is the fraction of LLMs that chose $g$ as the next step from $f$.
  3. Crisp Agent (Safety Envelope): The crisp idealized agent $\mu_f^{0,(t)}(g)$ is obtained by taking the support of the ideal agent. A directed edge from $f$ to $g$ exists if at least one LLM voted for that move ($\mu_f^{(t)}(g) > 0$). This forms the directed graph $\mathcal{G}_t$.
  4. Prompt: A specific prompt template was used, instructing the LLM to act as an ant on the grid trying to reach food, and to return the next position in JSON format.

6. Results & Analysis

The analysis focuses on the structure of the generated graph $\mathcal{G}_t$ and the relationship between the measured geometric quantities $d_0$ and $N_{d_0}$.

• Core Results:

  Figure 1: Visualization of the Search Graph

  该图像是论文中图1的示意图，展示了大小为5的网格图结构$\,\mathcal { G } _ { ( 3 , 4 ) }$。红色箭头表示可达的有向边，透明度编码理想代理$\,\mu ^ { ( t ) }$的隶属度。该图为单向图，严格减少曼哈顿距离至目标点。

  • Description: This image shows the directed graph $\mathcal{G}_t$ generated by the LLM agents for a 5x5 grid with the target at (3,4). The red arrows are the allowed transitions.
  • Analysis: The graph is clearly unidirectional. All arrows point in a general direction that reduces the Manhattan distance to the target. There are no cycles or backward moves. This provides strong visual evidence for Assumption 1, which posits that LLM-induced search is often approximately unidirectional. The graph is also anisotropic, meaning the moves are not uniform; certain directions are preferred over others, as seen by the varying transparency (membership value) of the arrows.

  Figure 2: Validation of the Theoretical Inequality

  该图像是图2，展示了不同起始节点$f$与对应目标$t$的最短路径长度$d_0$与最短路径数量$N_{d_0}$的关系。图中颜色和标记区分了$N=3,5,8$，虚线表示经验上界$N_d = ext{exp}(d)$。

  • Description: This is a log-linear scatter plot showing the shortest distance $d_0(f,t)$ on the x-axis and the number of shortest paths $N_{d_0}(f,t)$ on the y-axis (log scale) for all reachable start-target pairs across the three grid sizes. The black dashed line represents the boundary case $\log N_{d_0} = d_0$.
  • Analysis: The key finding is that all data points lie below the black dashed line. This empirically confirms the testable inequality derived from the theory: $\log N_{d_0}(f,g) \ll d_0(f,g)$. This result supports the paper's central hypothesis that for these types of tasks, the search complexity is dominated by path length ($d_0$) rather than path diversity ($N_{d_0}$). The fit appears even better for larger values of $d_0$, which is consistent with the theoretical derivation for the "small-$R_c$ limit" (i.e., for more difficult-to-reach targets).

• Ablations / Parameter Sensitivity: The paper does not contain ablation or parameter sensitivity studies. The experiment serves as a singular proof-of-concept for the theory.

7. Conclusion & Reflections

• Conclusion Summary: The paper successfully introduces a compact and formal theory to describe and measure the search space of LLM agents. It formalizes agents as fuzzy relation operators, constrains them within a safety envelope, and uses a coverage generating function to derive a quantitative measure of reachability difficulty ($R_c$). The theory predicts that for unidirectional search, path complexity is dominated by distance, not diversity ($\log N_{d_0} \ll d_0$). This prediction was validated in a simple majority-vote experiment on a 2D grid, showing that the framework provides a workable and testable language for analyzing LLM agents.

• Limitations & Future Work (as stated by the author):

  • The primary limitation is that detailed experimental validation on more complex, real-world tasks is left for future work.
  • The author plans to further test the effectiveness of the proposed measures ($p_c, R_c, d_0, N_{d_0}$).
  • A key future direction is to connect these theoretical indicators to practical applications, such as designing reward functions in reinforcement learning or creating training signals for curriculum learning, where tasks are ordered by difficulty.

• Personal Insights & Critique:

  • Novelty and Significance: The paper's main strength is its novelty. It provides a much-needed theoretical foundation in a field dominated by empirical heuristics. Bridging concepts from fuzzy logic, combinatorics, and graph theory to analyze LLM agents is a powerful and elegant approach. If validated more broadly, these measures could become standard tools for agent evaluation.
  • Transferability: The theory is abstract and model-agnostic, suggesting it could be applied to any iterative search process, not just those driven by LLMs. However, the core assumption of "approximately unidirectional" search might not hold for all problem types. For tasks requiring extensive backtracking or exploration of cyclical states (e.g., solving a maze with dead ends), the theory's predictions might not apply as directly.
  • Untested Assumptions: The "majority-vote" instantiation is a clever but simplified way to construct an agent. It's unclear how well the findings would generalize to more sophisticated agents that use stochastic sampling, complex scoring functions, or memory.
  • Potential Improvements: The theory could be extended to handle stochastic agents by incorporating probabilities directly into the path weights, rather than using binary membership from majority voting. Furthermore, the practical computability of the generating function $P_{f,g}(p)$ could be a bottleneck in very large state spaces, and the paper could benefit from discussing approximation techniques.
  • Open Questions: How does the choice of LLM affect the geometry of the search space? Would a more capable model like GPT-4 create a graph with a higher coverage index ($R_c$) for the same task compared to a smaller model? The proposed framework makes such questions quantifiable and empirically testable, which is its most significant contribution.