LLM-First Search Framework

• LLM‐First Search is a paradigm that uses large language models as fuzzy agents to iteratively explore hypothesis spaces with built-in safety constraints.
• The framework formalizes search using quantitative measures like the coverage generating function, critical parameters, and geometric interpretations to guide effective exploration.
• Its applications span diverse domains including reasoning, program synthesis, neural architecture search, image geolocation, and semantic segmentation.

LLM‐First Search (LFS) is an iterative, LLM–driven paradigm that treats problem solving as a structured search process. In LFS, the LLM is not only responsible for generating content but is also imbued with the capacity to guide the exploration of a hypothesis or solution space. This approach formalizes the search process by integrating domain priors into an operationally structured hypothesis space. As a result, LFS has been applied in domains as diverse as reasoning, program discovery, hardware‐aware neural architecture search, image geolocation, and even semantic segmentation.

1. Formal Foundations and the Fuzzy Agent Model

The core abstraction of LFS is to view an LLM‐based agent as a fuzzy relation operator on input and output states. Let $C_1,\, C_2$ denote the possible agent inputs and outputs, respectively. An ideal (possibly stochastic) agent $T$ is defined by $T(f, g) = p_f(g)$ where $p_f: C_2 \to [0,1]$ represents a membership function indicating the allowed or likely transition from an input $f$ to an output $g$. In the crisp (binary) case, $p_f(g) \in \{0,1\}$. The agent operates within a fixed safety envelope $T_0$; that is, all transitions satisfy

$0 \leq T(f, g) \leq T_0(f, g),\quad \forall f,\,g.$

This constraint enforces that only transitions pre-approved by domain priors are executed, thus embedding safety and prior knowledge into the search process.

2. Coverage Generating Function and Reachability Metrics

To capture the dynamics of multi‐step search, LFS introduces the concept of a coverage generating function. For iterative search where outputs are continuously recycled as inputs, the coverage generating function from an initial state $f$ to a target state $g$ is given by

$$P_{f,g}(p) = \sum_{n=0}^{\infty} \left(\sum_{\text{all paths } S_T: f(0)=f,\;f(n)=g} \prod_{i=0}^{n-1} T(f(i), f(i+1))\right) p^n,$$

with a continuation parameter $p \in [0,1]$ that serves as a weight for longer trajectories.

The search space is further quantified by defining the critical parameter $p_c(f, g)$ as the infimum of $p$ for which

$P_{f,g}(p) \geq 1,$

and the associated coverage index

$R_c(f, g) = 1 - p_c(f, g).$

A larger $R_c$ indicates that reaching $g$ from $f$ is easier. Moreover, the safety envelope $T_0$ naturally induces a directed graph with nodes corresponding to states and edges corresponding to allowed transitions; key geometric quantities include the shortest path distance $d_0(f, g)$ and the number of such shortest paths $N_{d_0}(f, g)$. For small $p$, the following lemma holds: $$\lim_{p \to 0^+} \frac{P_{f,g}(p)}{p^{d_0(f,g)}} = N_{d_0}(f, g).$$ This result underscores how search difficulty can be approximated by the length of the shortest reasoning path when path diversity is low.

3. Geometric Interpretation and Transitivity in Search Graphs

The induced search graph provides a geometric view of LFS. Here, the safety envelope governs the construction of a directed graph in which the shortest path distance $d_0(f, g)$ and the path count $N_{d_0}(f, g)$ play central roles. In such graphs, a key transitivity property holds: for any intermediate state $h$,

$P_{f,g}(p) \geq P_{f,h}(p) \cdot P_{h,g}(p),$

which implies

$$p_c(f, g) \leq \max\big(p_c(f, h),\, p_c(h, g)\big) \quad \text{and} \quad R_c(f, g) \geq \min\big(R_c(f, h),\, R_c(h, g)\big).$$

This means that introducing waypoints cannot increase the overall difficulty of search beyond that determined by the most challenging segment. In practice, when closed walks and path diversity are limited, the search difficulty is dominated by the shortest path length, as indicated by the empirical trend $\log N_{d_0} \ll d_0$.

4. Operational Tools and Testable Predictions

LFS provides a quantitative framework for evaluating and guiding LLM-based search. The coverage generating function and the derived coverage index $R_c$ yield testable predictions. For example, the epoch needed to hit a target state $g$ from a source $f$ is modeled by

$$\text{epoch}_{hit}(f \rightarrow g) \sim \frac{1}{R_c(f,g)} \sim d_0(f,g).$$

Moreover, an assumption based on the lowest-term approximation

$P_{f,g}(p) \geq N_{d_0}(f,g) p^{d_0(f,g)}$

leads to the inequality

$d_0(f, g) \cdot R_c(f, g) \geq \log N_{d_0}(f, g).$

These formal tools enable designers to quantify the trade-off between constraint satisfaction (via the safety envelope) and the ease of exploration (reachability), thereby guiding agent training and policy design.

A summary of the key definitions is provided in the table below:

Concept	Formalization	Intuition
Fuzzy Agent	$T(f, g) = p_f(g)$	Weighted transition operations
Safety Envelope	$0 \leq T(f, g) \leq T_0(f, g)$	Constrained, pre-approved transitions
Coverage Generating Function	$$P_{f,g}(p) = \sum_{n=0}^\infty \left(\sum_{\text{paths}} \prod T\right)p^n$$	Sum over all paths weighted by length
Critical Parameter	$p_c(f,g) = \inf\{p : P_{f,g}(p) \geq 1\}$	Threshold where cumulative transition reaches 1
Coverage Index	$R_c(f, g) = 1 - p_c(f, g)$	"Ease" of reaching target state
Shortest Distance	$d_0(f,g)$	Fewest steps required to reach $g$
Path Diversity	$N_{d_0}(f, g)$	Number of distinct shortest paths

5. Applications and Interrelations with Other Search Paradigms

LLM-First Search is not merely a theoretical framework; it has been applied across various domains. In reasoning and program discovery, LFS supports iterative generate–filter–refine cycles where the LLM autonomously directs search. In hardware-aware neural architecture search, methods such as PEL-NAS incorporate LFS concepts by partitioning the search space into niches and co-evolving architecture prompts, drastically reducing compute from days to minutes. Other applications include image geolocation systems that leverage object-to-object semantic reasoning and few-shot segmentation frameworks (e.g., DSV-LFS) where LFS initiates high-level semantic reasoning before localizing responses via dense visual matching.

Furthermore, several recent works (e.g., AgentSwift, R-Search, and LiteSearch) integrate LFS with reinforcement learning strategies and tree search techniques. These diverse instantiations of LFS illustrate its versatility across tasks—from combinatorial code generation to multi-hop question answering—by emphasizing dynamic, self-guided exploration that adapts to task difficulty and model uncertainty.

6. Practical Implications and System Design

For practitioners and system designers, LFS offers a unified language for measuring and balancing two competing objectives: safety (constraint satisfaction) and reachability (ease of exploration). Metrics such as the coverage index $R_c$ provide concrete targets for shaping training rewards and adjusting safety envelopes. In agent design, intermediate waypoints are encouraged to mitigate search bottlenecks via transitivity, and empirical evaluations validate the use of shortest path length as a proxy for search difficulty.

LLM-First Search systems benefit from:

• Dynamic policy control that removes the need for fixed exploration hyperparameters.
• Direct integration of LLM-generated heuristics to inform expansion and pruning decisions.
• Scalability across problem domains by leveraging intrinsic model knowledge.

These principles serve as guiding design criteria in developing robust LLM agents that must navigate large, complex hypothesis spaces efficiently.

7. Conclusion

LLM‐First Search represents an evolution in leveraging LLMs for complex reasoning tasks. By formalizing agents as fuzzy relation operators constrained by safety envelopes and characterizing search via the coverage generating function and associated metrics, LFS provides both a language and a toolkit for system optimization. Its applications—from program synthesis and neural architecture search to autonomous object search in unstructured environments—demonstrate that embedding domain priors into the search space can significantly improve both efficiency and overall performance. This framework bridges heuristic agent engineering and rigorous mathematical analysis, marking an important step toward autonomous, self-guided reasoning systems in next‐generation AI research.