Very cool idea to have agents design complex systems by searching over the model structure itself.

New research from Sakana AI introduces CEDAR, which uses LLM agents to write, simulate, and refine system-dynamics models toward a stated design goal. Candidate structures get proposed, run, and scored, with tree search over the space of feedback structures rather than parameter tuning on a fixed one.

Predicting how feedback structure gives rise to emergent behavior is a long-standing open problem in artificial life. Automating the design loop around it is a serious attempt at that problem.

Paper: https://arxiv.org/abs/2608.06871

Track more trending AI papers in our academy: https://academy.dair.ai

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Agent-Orchestrated Tree Search for Goal-Directed Optimization of Complex Systems

Source: https://arxiv.org/html/2608.06871

Abstract

Complex systems, core objects of study in artificial life, model diverse phenomena through nonlinear, feedback-driven interactions that produce emergent behavior, with applications from population dynamics and biology to economic policy and strategic decision-making. Yet the difficulty of predicting how feedback structure gives rise to emergent behavior, a central open problem in artificial life, makes goal-directed design exceptionally challenging. In established practice, system structures are written in specialized modeling languages such as DYNAMO or STELLA, compounding the challenge with labor-intensive workflows that limit adoption and hinder timely decision-making.

To address these challenges, we introduce CEDAR, an autonomous method that uses Large Language Model (LLM) agents to discover complex systems satisfying user-specified behavioral goals. Our key innovation is an LLM-driven Monte Carlo Tree Search (MCTS) deeply coupled with complex systems: at each iteration, an LLM Judge evaluates emergent behavior against specified goals and an LLM Editor proposes improved variants, with the Judge acting as a fitness function and the Editor as a variation operator, akin to a generate-and-evaluate loop in evolutionary computation. We represent complex systems as a restricted, runnable subset of Python with domain-specific primitives, letting LLMs modify system dynamics directly. CEDAR formalizes this as an MCTS variant with an LLM-parameterized transition kernel and value function, enabling goal-directed discovery of complex system behaviors while preserving solution diversity, and its LLM-based interpretability reveals how structural changes drive emergent behavior. CEDAR reduces human effort while enabling capabilities difficult to achieve with existing approaches, facilitating broader adoption of complex systems across domains.

††©2026 Yingtao Tian. Published under a Creative Commons Attribution 4.0 International (CC BY 4.0) license.## Introduction

Computational system modeling provides a framework for studying real-world phenomena as nonlinear, feedback-drivencomplex systems, computational models whose global behavior emerges from feedback interactions among components(53;55). Such systems are central objects of study in artificial life(5;33;22)and systems thinking research(2). This framework has been applied in various domains such as global dynamics(18), pandemic diffusion(81), social simulations(30), and biological systems(24), critically supporting policy design and strategic decision-making(52;60). Yet predicting how feedback structure gives rise to emergent behavior, a central open problem in artificial life, remains notoriously difficult, making goal-directed design of such systems exceptionally challenging.

A further challenge is practical: established practice often involves specialized modeling languages such as DYNAMO(53)and STELLA(55), which require extensive manual effort to navigate unclear feedback relationships(23)and labor-intensive workflows(68), limiting adoption across domains where complex systems modeling could provide valuable insights. Large language models (LLMs) and LLM-powered agents now offer new capabilities for automated discovery of complex systems. In particular, when combined with Monte Carlo Tree Search (MCTS)(29), LLMs show strong capabilities in planning(77;21)and reasoning(66;12). This opens promising directions for searching over spaces of complex systems, including artificial life discovery(31;50), automated scientific discovery(74), and evolving models to generate and falsify hypotheses about biological dynamics(67).

We propose CEDAR (Complex-systems Exploration and Design via Agent-Orchestrated Refinement), an autonomous method for discovering complex systems that meet specified goals by combining a unified system representation, the LLM Judge and Editor, and MCTS. At each iteration, MCTS selects a system variant for expansion, an LLM Editor generates improved variants, and the variants are executed and evaluated by an LLM Judge. Concretely, the LLM Editor acts as a variation operator proposing new system structures(36), while the LLM Judge serves as a fitness function evaluating each candidate, casting system discovery as an evolutionary search process guided by MCTS, where the search tree plays the role of a structured candidate population. This is enabled by our unified representation using a restricted Python subset with domain-specific primitives and inline documentation.

In sum, CEDAR enables goal-directed discovery of complex system behaviors with capabilities difficult to achieve with existing approaches, while preserving solution diversity that supports open-ended exploration, demonstrates applicability on classical systems drawn from social and biological modeling, and makes the emergent mechanisms of complex systems more transparent through LLM-based interpretability, lowering the barrier to constructing and studying complex systems in artificial life and beyond.

Figure 1:Overall architecture of CEDAR. The autonomous approach uses Monte Carlo Tree Search (MCTS) to iteratively select system variants for expansion. At each iteration, an LLM Editor generates improved system variants and executes them, then an LLM Judge evaluates their performance and provides detailed analysis. The method integrates several components: a system in a restricted subset of Python code, a natural-language goal description, a search tree structure, system execution results, and comprehensive analysis feedback.

Related Work

Complex Systems and How to Model Them

World Dynamics(18)simulates population, resources, and industrial output over long horizons of the planet. This introduces a computational framework for modeling a subject as a nonlinear, feedback-driven system (“complex system”), a paradigm shift(19)from linear prediction-based modeling. With the help of DYNAMO(53)and STELLA(55)modeling languages (online examples ofDYNAMOandSTELLAshow their old syntax and proprietary visual interfaces), complex systems have seen a wide range of applications, including social simulations(30), war games, pandemic diffusion(81), economics, thermodynamics, and biological systems(56;24). This systems thinking(2;3)supports policy design and strategic decision-making(52;60). Despite these advances, modeling still requires expert-defined variables and relationships(17;6;75): the relationship between structure and emergent behavior is often unclear(58;23;4), and established languages require labor-intensive workflows(48;25;68).

Tree Search Algorithms with LLMs

Monte Carlo Tree Search (MCTS)(29)has shown promising results when combined with powerful models. MCTS with neural networks has led to superhuman performance in games(63;65;64;59), plus the discovery of novel algorithms(15;45)and neural network architectures(49). More recently, MCTS has been combined with large language models (LLMs), including LLM-powered value functions and self-reflection(79), LLM-powered node selection and expansion strategies(26), as well as applications to specific tasks such as automatic design(78)and agent collaboration(20). Notably, MCTS with LLMs offers strong capabilities in planning, where LLMs serve as common-sense policy priors(77)and enhance interpretability(21), and in reasoning, where LLMs guide traversals of knowledge graphs and refine reasoning through steps(66;12).

LLMs as Evolutionary Operators in Artificial Life

LLMs have been proposed as variation operators in evolutionary search, replacing random mutation with language-guided program edits(36;50). A related line evolves language prompts to steer the behavior of simulated cellular and biological systems(34;35). This connects LLMs to a central ALife goal: open-ended discovery of emergent behaviors. Our work extends this to complex systems, with the LLM Editor as variation operator and the LLM Judge as fitness function.

LLM-based Optimization for Structured Systems

LLMs with MCTS show promising results for complex and challenging searches, including automated scientific discovery(74), artificial life discovery(31)and runnable code for reasoning(27). Recent works also apply LLMs specifically to model system dynamics(43;44;42)by treating models as predictors of behaviors that areunknownto the optimizer. These studies consider only smaller systems (up to 4 variables and 12 steps), whereas our method scales to 60 variables and 4000 steps, making direct extensions of them to our scenario non-trivial.

Prior Works Close to Our Method

Several prior works combine LLMs with search but target different goals. I-MCTS(40)targets AutoML hyperparameters and LATS(79)operates on discrete task graphs, and neither addresses iterative temporal dynamics optimization. GIF-MCTS(13)and LLM-SRBench(62)address short sequences, whereas our method optimizes multi-thousand-step temporal dynamics. Prompt optimization methods(69;16;28)are applicable in principle but require demonstrations unavailable in our online-exploration setting.

Method

Preliminaries

We first formalize the complex systems under discussion in this work. Complex systems model a subject as a nonlinear, feedback-driven mathematical system whose dynamics give rise to emergent behavior. While such systems admit a range of formalisms, including agent-based models, cellular automata, and discrete-time dynamic graphs, one common formalism, which we adopt throughout this work, is ordinary differential equations (ODEs) with initial condition:d​𝒙/d​t=f⁡(𝒙,t),𝒙⁡(t0)=𝒙0{d{\bm{x}}}/{dt}=f({\bm{x}},t),\quad{\bm{x}}(t_{0})={\bm{x}}_{0}, where𝒙∈ℝn{\bm{x}}\in{\mathbb{R}}^{n}is the state vector (also called level or stock variables),f:ℝn×ℝ→ℝnf:{\mathbb{R}}^{n}\times{\mathbb{R}}\to{\mathbb{R}}^{n}the vector field, and𝒙0=[x0,1,…,x0,n]⊤{\bm{x}}_{0}=\begin{bmatrix}x_{0,1},&\ldots,&x_{0,n}\end{bmatrix}^{\top}the initial values att0t_{0}, the initial time. As a result, whenfffollows a known, simple form, analytical techniques for ODEs(70), such as the Laplace transform(46), can be applied to complex systems analysis.

However, in practice, complex systems are more commonly analyzed through numerical approaches like the first-order Euler method(51)due to the complexity offfwhen nonlinear feedbacks are present. This approach enables flexible modeling by requiring only thatffbe computable, which is crucial since many complex systems admit no analytical solution. Concretely, the first-order Euler method discretizes the system with time stepΔ​t\Delta tas:𝒙t+Δ​t=𝒙t+Δ​t⋅f⁡(𝒙t,t),𝒙0=𝒙⁡(0){\bm{x}}_{t+\Delta t}={\bm{x}}_{t}+\Delta t\cdot f({\bm{x}}_{t},t),\quad{\bm{x}}_{0}={\bm{x}}(0). Note that𝒙{\bm{x}}and its discretized counterparts𝒙t{\bm{x}}_{t}and𝒙t+Δ​t{\bm{x}}_{t+\Delta t}represent interpretable, meaningful quantities such as populations and resources, rather than abstract hidden states as in RNNs, despite the similarity in their formulations. Historically, communities studying complex systems have benefited from specialized modeling languages like DYNAMO and STELLA(53;55). However, these tools suffer from outdated syntax or proprietary visual interfaces(19)that limit their usability with LLMs.

Figure 2:Python representation of a complex system. From left to right: graph representation (for reference), Python code, and computation process. Four essential components are shown: (1) initialization of values (𝒙0=𝒙⁡(0){\bm{x}}_{0}={\bm{x}}(0)), a loop of (2) intermediate computations, (3) derivative evaluations (i.e., computingf⁡(𝒙t,t)f({\bm{x}}_{t},t)), and (4) Euler-method integration (i.e., computing𝒙t+Δ​t{\bm{x}}_{t+\Delta t}from𝒙t{\bm{x}}_{t}andf⁡(𝒙t,t)f({\bm{x}}_{t},t)). This Python-based representation allows users to easily code and inspect, while leveraging LLM’s coding capabilities.

Unified Representation

To bridge complex systems with LLMs, we propose a representation of complex systems based on a restricted subset of Python programs, unifying the representations traditionally used in DYNAMO and STELLA. Python is widely familiar, and LLMs have strong Python coding capabilities. This avoids creating a new domain-specific language, and enables flexible implementation off⁡(𝒙,t)f({\bm{x}},t)through general-purpose programming. We show an example of this representation in Figure2.

Besides plain code, we enrich it with documentation and predefined, domain-specific mathematical primitives that aid LLMs in using the representation. Doing so helps LLMs understand the complex system’s semantics, manipulate system dynamics through semantic constructs rather than low-level implementation details, and know what restrictions must be enforced.

Proposed Method

We propose CEDAR, an autonomous method that combines MCTS, LLM, and the unified representation above to discover complex systems that meet a given goal. We present the overall architecture and components of CEDAR in Figure1and provide details as follows.

Overview and Formalization.

CEDAR optimizes complex systems by editing them towards a goalGG, which is a natural-language description of the desired system behavior (for example, “grow population stably”). A systemPPis represented using the restricted subset of Python programs. The MCTS iteratively buildsT=(V,E)T=(V,E)with node setVVand edge setEE, each time expanding a selected nodevvand adding new child nodeuu. We denote the expansion strategy used to generateuuassus_{u}, anduu’s parent aspu≜vp_{u}\triangleq v(the root node0∈V0\in Vhas no parentp0p_{0}or expansion strategys0s_{0}). Each nodeu∈Vu\in Vstores a systemPuP_{u}, an execution recordRuR_{u}, a scoreSuS_{u}, and a textual analysisAuA_{u}, which are produced in the following way: An LLM Editor produces a new programPuP_{u}by patchingPvP_{v}(e.g. modifying equations, changing feedback links).PuP_{u}is then executed to obtain a new execution recordRu←Run​(Pu)R_{u}\leftarrow\textsc{Run}(P_{u})which includes values of all variables at each timestamp. The LLM Judge evaluates the quality of systemPuP_{u}against the specified goalGG, producing(Au,Su)(A_{u},S_{u}):Su∈ℝS_{u}\in\mathbb{R}is a bounded numerical score reflecting how wellPuP_{u}satisfies goalGG, andAuA_{u}is a textual analysis providing qualitative feedback about the system’s behavior and potential improvements.

Figure 3:The node expansion process using LLM Editor and Judge. Given a selected nodevvand expansion strategyss, the LLM Editor generates a new system variantPuP_{u}by leveraging the parent systemPvP_{v}, its analysisAvA_{v}, strategyss, and goalGG. The generated system is executed to obtain recordRuR_{u}, then evaluated by the LLM Judge to produce analysis-score pair(Au,Su)(A_{u},S_{u}). We embed analysis and scores of all nodes, including the currently selected node, in the tree context to provide the LLM with a comprehensive view of the search tree.

Initialization.

CEDAR begins by initializing the MCTS treeTTwith a single root nodev0v_{0}, associated with an initial complex systemP0P_{0}which can be either provided by the user or generated from a basic template,P0P_{0}’s execution recordR0←Run​(P0)R_{0}\leftarrow\textsc{Run}(P_{0})and the evaluation ofP0P_{0}:(A0,S0)←LlmJudge​(P0,R0,G)(A_{0},S_{0})\leftarrow\textsc{LlmJudge}(P_{0},R_{0},G).

Node Selection.

The node selection process jointly selects(v,su)(v,s_{u}), a nodev∈Vv\in Vto expand from and an expansion strategysus_{u}to use, for generating new nodeuulater. Selection of expansion strategysus_{u}is simply done by uniformly samplingsu∼Uniform​(𝒮)s_{u}\sim\text{Uniform}(\mathcal{S})where𝒮={s1,s2,…,sk}\mathcal{S}=\{s_{1},s_{2},\ldots,s_{k}\}is a fixed set of expansion strategies. Selection ofvvis based onScore​(v)\textsc{Score}(v)that combines the node’s performance scoreSvS_{v}, its depthDepth​(v)\textsc{Depth}(v)in the tree, its expansion countcv∈ℕc_{v}\in\mathbb{N}, and its parentpp’s expansion countcpc_{p}(we omit the subscriptvvand writep=pvp=p_{v}): CEDAR uses the generalized UCT scoreScore​(v)=Sv+ϕ⁡(cv,cp,τ)\textsc{Score}(v)=S_{v}+\phi\!\left(c_{v},c_{p},\tau\right). Here, we haveϕ⁡(⋅)=−∞\phi(\cdot)=-\inftyifcv≥τc_{v}\geq\tauandα⋅ln⁡(cp+1)/(cv+1)+γ⋅Depth​(v)\alpha\cdot\sqrt{\ln(c_{p}+1)/(c_{v}+1)}+\gamma\cdot\textsc{Depth}(v)otherwise, whereα\alphaadjusts the amount of exploration,γ>0\gamma>0controls the preference for deeper nodes, andτ\taucontrols the expansion count. Inspired by26and unlike vanilla MCTS, we limit expandable nodes to a subsetVexp⊆VV_{\text{exp}}\subseteq Vand allow repeated expansion of internal nodes until their expansion countcvc_{v}reaches a thresholdτ∈ℕ\tau\in\mathbb{N}, so a nodevvis expandable if and only ifv∈Vexpv\in V_{\text{exp}}andcv<τc_{v}<\tau. The process selectsv=arg⁡maxv∈Vexp​Score​(v)v=\arg\max_{v\in V_{\text{exp}}}\textsc{Score}(v).

Node Expansion with LLMs.

From a selected nodevvand expansion strategysus_{u}, the LLM Editor generates a new system variantPu←LlmEditor​(Pv,Av,s,G)P_{u}\leftarrow\textsc{LlmEditor}(P_{v},A_{v},s,G). Using the systemPvP_{v}, its associated analysisAvA_{v}, the chosen strategyss, and the goal specificationGG, this expansion produces a revised system that satisfies syntactic constraints and semantic requirements while being more aligned with the goal. Because the system is represented as code, the Editor can make structural edits toPvP_{v}rather than only tuning coefficients: it may add or remove state variables, introduce or rewire feedback relations, and change the equations. The newly generated systemPuP_{u}is executed to obtain its recordRu←Run​(Pu)R_{u}\leftarrow\textsc{Run}(P_{u}), followed by the evaluation(Au,Su)(A_{u},S_{u})←\leftarrowLlmJudge​(Pu,Ru,G)\textsc{LlmJudge}(P_{u},R_{u},G)that produces a scoreSuS_{u}and a qualitative analysisAuA_{u}.

Summary.

This iterative process of selection, expansion, execution, and evaluation enables progressive improvement toward the target objectives by exploring the space of complex systems. We formalize the complete search procedure in Algorithm1and illustrate particularly the expansion process in Figure3. BothLlmEditorandLlmJudgeare provided with well-crafted prompts covering system context, tree context, and structural constraints, which alongside code-based representation are essential for the method to work.

Theoretical Connection

While designed empirically, CEDAR is a principled MCTS variant. Node selection is a generalized UCT combining progressive widening(10;26)and depth-based bonuses(7;71):Score(v)=Sv+αln⁡(cp+1)/(cv+1)⋅𝕀[cv<τ]+γ⋅Depth(v)\textsc{Score}(v)=S_{v}+\alpha\sqrt{\ln(c_{p}{+}1)/(c_{v}{+}1)}\cdot\mathbb{I}[c_{v}<\tau]+\gamma\cdot\textsc{Depth}(v). The LLM Editor acts as a stochastic transition kernel over programs,Pu∼Pθ(⋅∣Pv,Av,s,G)P_{u}\sim P_{\theta}(\cdot\mid P_{v},A_{v},s,G), connecting to learned-transition MCTS and model-based RL(65;59). The LLM Judge acts as a learned value function providing bounded noisy rewards, connecting to MCTS with biased evaluators(41;14). That said, we note that the convergence and regret guarantees for MCTS with LLMs remain open problems despite strong empirical performances(8;27).

Algorithm 1Monte Carlo Tree Search for Complex System Discovery1:Input:Initial system

P0P_{0}, goal

GG, iterations

NN, expansion threshold

τ\tau 2:Output:Best discovered system

P∗P^{*} 3:Initialize:

T=(V,E)T=(V,E)with root node

00,

(A0,S0)←LlmJudge​(P0,Run​(P0),G)(A_{0},S_{0})\leftarrow\textsc{LlmJudge}(P_{0},\textsc{Run}(P_{0}),G),

P∗←P0P^{*}\leftarrow P_{0},

S∗←S0S^{*}\leftarrow S_{0} 4:for

i=1i=1to

NNdo

5:

v←arg⁡maxv∈Vexp​Score​(v)v\leftarrow\arg\max_{v\in V_{\text{exp}}}\textsc{Score}(v)⊳\trianglerightSelect node to expand

6:

s∼Uniform​(𝒮)s\sim\text{Uniform}(\mathcal{S})⊳\trianglerightSample expansion strategy

7:

Pu←LlmEditor​(Pv,Av,s,G)P_{u}\leftarrow\textsc{LlmEditor}(P_{v},A_{v},s,G)⊳\trianglerightExpand

8:

Ru←Run​(Pu)R_{u}\leftarrow\textsc{Run}(P_{u})⊳\trianglerightExecute

9:

(Au,Su)←LlmJudge​(Pu,Ru,G)(A_{u},S_{u})\leftarrow\textsc{LlmJudge}(P_{u},R_{u},G)⊳\trianglerightEvaluate

10:Add node

uuto the tree with

(Pu,Ru,Au,Su)(P_{u},R_{u},A_{u},S_{u}) 11:

cv←cv+1c_{v}\leftarrow c_{v}+1⊳\trianglerightIncrement expansion count

12:if

cv≥τc_{v}\geq\tauthen

13:

Vexp←Vexp∖{v}V_{\text{exp}}\leftarrow V_{\text{exp}}\setminus\{v\}⊳\trianglerightDrop from expandable set

14:endif

15:if

Su>S∗S_{u}>S^{*}then

16:

P∗←PuP^{*}\leftarrow P_{u},

S∗←SuS^{*}\leftarrow S_{u} 17:endif

18:endfor

19:return

P∗P^{*}

Experiments

CEDAR enables three capabilities difficult for existing approaches: optimizing for vague natural-language goals, fitting records without complete system skeletons, and interpretable optimization. We provide qualitative demonstrations and quantitative comparisons for these capabilities.

Experimental Details

Data.We collect complex systems from classical works:World Dynamics(18)in the DYNAMO language and 19 systems from the book Modeling Dynamic Biological Systems(24)in the STELLA language (20 systems, 20 to 69 integrated variables each), converted into our representation. We utilize these systems in two ways: (1) as given models to be further optimized towards multiple goals, and (2) as ground truth systems that produce reference records serving as optimization targets.

Setup.In our experiment, we conduct2020node selections, during each of which we expand to55new candidate nodes. This results in a total of100100expansions per search. We empirically set hyperparametersα=1\alpha=1andγ=2\gamma=2for node selection. The expansion strategy is selected by sampling from a set of candidate strategies listed in Table1.

Table 1:MCTS expansion strategies.We use adaptive context subsampling for scalability, keeping the execution record within a context budgetLLand achievingO⁡(N​L)O(NL)overall complexity. Because the LLM Editor and Judge define the transition and evaluation operators, we treat their settings as part of the method: all runs use two strong-coding providers, Anthropic Claude Sonnet 4.5 and GPT-5.1, at default decoding settings with structured outputs and up to three retries, and assign crashed or NaN-valued runs a low score so MCTS naturally backtracks. Our runs incur moderate computational costs and runtime (10–50, 30 minutes per run).

Fitting an Abstract Goal Described in Natural Language

In this experiment, we task CEDAR with optimizing a given good-enough complex system towards a further ambitious goal. The starting system isWorld Dynamics(18), a published, hand-designed model of the co-evolution of human population, resource utilization, and pollution, containing the most variables in our data and posing challenges for understanding the system dynamics. Our natural-language goal is to further optimize it to achieve more population growth, less resource usage, and less pollution, alljointly:

Natural-language goal⬇Balancepopulation,resources,andenvironmentbyoptimizingto(1)maximizepopulationgrowth,(2)minimizetheresourcedepletionrate,and(3)minimizethepollutionaccumulationrate.Seekthebesttrade-offwherethepopulationgrowssustainablywithoutdepletingresourcestooquicklyorcreatingexcessivepollution.(Important:onlychangethecoefficientsinthehelper.Donotchangeanycoefficientbymorethan50\\\\%topreventvariableexplosion.)

Doing so poses intrinsic challenges: the hand-designed system already reaches a kind of Pareto frontier, where optimizing one sub-goal often comes at the expense of others, as the subgoals are conflicting in nature, as Figure4shows.

Figure 4:Optimizing for a single goal at a time: focusing on one objective hurts the others, showing competitive subgoals. As in Figure5, the Resources panel plots theremainingresource stock, so its upward “better” direction corresponds to less depletion.Figure 5:Illustration of fitting an abstract, language-based goal fromWorld Dynamics. Top: The MCTS tree with the path to best node highlighted. Each nodevvis marked by its index and LLM Judge’s scoreSvS_{v}. Bottom: The system dynamics for the initial (original) and final complex systems with two LLM backends. We show key variables including Population (↑\uparrow), Pollution (↓\downarrow), and Resources (↑\uparrow). Here Resources denotes theremainingnatural-resource stock, so a higher value reflects less depletion and is preferable. With CEDAR, the system reaches better states (increased population, less resource depletion, and less pollution).Figure5illustrates the MCTS search tree and the resulting system dynamics, and shows how the tree structure guides the search toward higher-scoring solutions. The resulting system satisfies the vague goal, with all three key variables (population, resource, pollution) improving over the initial system. Interestingly, with different LLM backends, CEDAR discovers multiple high-performing systems that are structurally distinct, each emphasizing different tradeoffs among objectives (for example, prioritizing population by Claude versus resource by GPT-5.1).

Quantitative Studies: Fitting a Concrete Record

In this experiment, we task CEDAR with evolving a system (using only a bare-minimum skeleton) to fit the simulated record of a target complex system. While the strength of CEDAR lies in fitting vague, abstract goals, it can nonetheless be used to fit records from either observations or ground truth systems, by simply loading the record into the goal and running CEDAR as-is. The ground truth complex system is a population growth model with stochastic components (a stochastic death rate sampled at each time step, introducing volatility intrinsic to the system). We compare CEDAR with existing approaches based on Optuna(1)black-box optimization. Since the search space for black-box optimization consists of only coefficients, we explore three levels of formula support: no formulae, a simple population-dependent death rate, or the full ground truth formulae, the last of which gives the baseline a serious boost by reducing the task to parameter fitting. We note this comparison deliberately favors the baseline: each Optuna variant is granted100100trials and, in its strongest setting, the full ground-truth formulae, which is strictly more prior structure than CEDAR receives, as CEDAR starts from a bare-minimum skeleton. To measure the accuracy of fitting a concrete record, we employ both L1 and Dynamic Time Warping (DTW)(57)distances as metrics. DTW measures trajectory similarity under optimal temporal alignment using a Sakoe-Chiba band (w=250w{=}250, 7.1% of the 3500-step sequence).

Table 2:L1 and DTW distances to the target record (lower is better). The Formulae columns indicate how much of the ground-truth structure each method is given: No (no formulae), S. (a simple population-dependent death rate), or F. (the full ground-truth formulae). CEDAR, given no formulae, achieves lower error than the Optuna baseline given the full formulae.Figure 6:System dynamics (population) when fitting a concrete record. CEDAR, without predefined formulae, tracks the ground-truth trajectory closely, while the no- and simple-formulae Optuna baselines collapse to near-constant populations; only Optuna with the full formulae stays competitive. (All trajectories rise together during the early ramp-up, where the legend overlaps the curves.)As shown in Table2and Figure6, the Optuna baselines are severely limited by the skeletal formulae: the no/simple-formulae variants are clearly worse, quantitatively and qualitatively, than the full-formulae variant. Meanwhile, CEDAR, even without predefined formulations, finds solutions that outperform Optuna with a full formula skeleton. Across 10 seeds, the ground truth’s stochasticity (a death rate drawn from a normal distribution each step, scaled by population) shows volatility is intrinsic to the system, not an optimizer artifact; CEDAR’s record generally falls within the ground-truth variability and captures the magnitude and timing of major peaks and dips. Furthermore, even with full-formulae support and 100 trials, Optuna does not perfectly match the dynamics (L13.713.71–4.264.26across two runs), as the stochastic noise floor and high-dimensional nonlinear parameter space make purely numeric search insufficient.

Interpretability

For theWorld Dynamicssystem above, we examine the interpretability of CEDAR through the LLM’s analysis of candidate systems, highlighting in Table3the LLM responses along the path that produces the best system in the search tree of Figure5. CEDAR exposes clear, human-readable analyses at each step that document how the three subgoals are balanced. The LLM Judge analyzes issues with balanced focus on subgoals, while the LLM Editor first evaluates overall performance, then fine-tunes individual subgoals, also attending to unmentioned aspects (e.g., capital productivity). These analyses are the stated rationale accompanying each edit, aiding inspection of the search; they are not a verified causal account of how structure produces emergent behavior, which remains an open problem.

Table 3:LLM Editor and Judge responses along the best path in the tree. We show the LLM-summarized version with manually coloredPopulation,pollution, andresources.

Why Use MCTS with LLMs: Diversity of Solutions and Better Performance

We find the benefits of MCTS to be threefold.First, MCTS helps preserve solution diversity.For the record-fitting experiment, we sample nodes within a single run whose scores are close to the best solution and visualize their systems’ populations in Figure7. All satisfy the goal, yet follow distinct trajectories: unlike the Optuna baselines, CEDAR avoids collapsing to a single overfitting-like point, which enables better decision-making through sensitivity analysis.

Figure 7:Diversity of solutions in the search tree showing different population dynamics after systems pass the initial phase (t>100t>100).Second, MCTS leads to better performance.Our ablation against linear search (Figure8) shows MCTS yields higher-scoring nodes and smoother trajectories that avoid overfitting-like behavior, indicating MCTS with LLMs is crucial for performance gains.

Figure 8:MCTS versus linear search. Left: MCTS attains higher-scoring nodes than linear search. Right: the resulting population dynamics are smoother and avoid the overfitting-like behavior of linear search, showing that tree-based exploration drives CEDAR’s performance.Third, MCTS supports open-ended discovery of structurally distinct solutions.Because MCTS builds a branching tree rather than converging to a single point, it preserves diversity, which matters in ALife contexts where exploring a range of emergent behaviors is valuable alongside optimizing one objective. This diveristy leads to Ddistinct high-performing systems, each emphasizing different tradeoffs among objectives.

Conclusion

We propose CEDAR, which casts complex system discovery as evolutionary search(50;36)guided by MCTS: an LLM Editor acts as a variation operator and an LLM Judge as a fitness function. Empirically, on classical complex systems from social and biological modeling, CEDAR optimizes challenging multi-objective goals, outperforms black-box optimization without predefined formulas, and surfaces interpretable, step-by-step analyses of how the Editor and Judge reshape the system. Together, these capabilities open new directions for studying and exploring emergent phenomena in artificial life. Several limitations point to future work. The LLM Judge both scores candidates and shares a model class with the Editor, risking circularity, as the abstract-goal setting still relies on LLM judgment. We report trends rather than tight statistical claims, and the in-depth experiments focus on two systems. A natural next step is systematic benchmarking, along with classical evolutionary and multi-objective optimization methods, to delineate when LLM-based variation and evaluation are necessary rather than helpful.

References

• Akibaet al.(2019)T. Akiba, S. Sano, T. Yanase, T. Ohta, and M. KoyamaOptuna: A Next-Generation Hyperparameter Optimization Framework.InProceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining,pp. 2623–2631.Cited by:Quantitative Studies: Fitting a Concrete Record.
• Anderson and Johnson (1997)V. Anderson and L. JohnsonSystems Thinking Basics.Pegasus Communications,Cambridge, MA.Cited by:Introduction,Complex Systems and How to Model Them.
• Arnold and Wade (2015)R. D. Arnold and J. P. WadeA Definition of Systems Thinking: A Systems Approach.Procedia computer science44,pp. 669–678.Cited by:Complex Systems and How to Model Them.
• Barlas (1996)Y. BarlasFormal Aspects of Model Validity and Validation in System Dynamics.System Dynamics Review: The Journal of the System Dynamics Society12(3),pp. 183–210.Cited by:Complex Systems and How to Model Them.
• Bedau (2003)M. A. BedauArtificial Life: Organization, Adaptation and Complexity from the Bottom Up.Trends in Cognitive Sciences7(11),pp. 505–512.Cited by:Introduction.
• Bérard (2010)C. BérardGroup Model Building Using System Dynamics: An Analysis of Methodological Frameworks.Electronic Journal of Business Research Methods8(1),pp. 35–45.Cited by:Complex Systems and How to Model Them.
• Blackshawet al.(2025)T. M. Blackshaw, J. C. Davies, K. T. Spoerer, and J. D. HirstEnhancing Monte Carlo Tree Search for Retrosynthesis.Journal of Chemical Information and Modeling65(13),pp. 6537–6546.Cited by:Appendix C,Theoretical Connection.
• Brandfonbreneret al.(2024)D. Brandfonbrener, S. Henniger, S. Raja, T. Prasad, C. R. Loughridge, F. Cassano, S. R. Hu, J. Yang, W. E. Byrd, R. Zinkov, and N. AminVerMCTS: Synthesizing Multi-Step Programs using a Verifier, a Large Language Model, and Tree Search.InThe 4th Workshop on Mathematical Reasoning and AI at NeurIPS’24,External Links:LinkCited by:Appendix C,Theoretical Connection.
• Bubecket al.(2011)S. Bubeck, R. Munos, G. Stoltz, and C. SzepesváriX-Armed Bandits.Journal of Machine Learning Research12(46),pp. 1655–1695.External Links:LinkCited by:Appendix C.
• Chaslotet al.(2008)G. M. J. Chaslot, M. H. Winands, H. J. v. d. Herik, J. W. Uiterwijk, and B. BouzyProgressive Strategies for Monte-Carlo Tree Search.New Mathematics and Natural Computation4(03),pp. 343–357.Cited by:Appendix C,Theoretical Connection.
• Chenet al.(2020)B. Chen, C. Li, H. Dai, and L. SongRetro*: Learning Retrosynthetic Planning with Neural Guided A* Search.InProceedings of the 37th International Conference on Machine Learning,H. D. III and A. Singh (Eds.),Proceedings of Machine Learning Research, Vol.119,pp. 1608–1616.External Links:LinkCited by:Appendix C.
• Chiet al.(2025)Y. Chi, K. Yang, and D. KleinThoughtSculpt: Reasoning with Intermediate Revision and Search.InFindings of the Association for Computational Linguistics: NAACL 2025,L. Chiruzzo, A. Ritter, and L. Wang (Eds.),Albuquerque, New Mexico,pp. 7700–7726.External Links:Link,Document,ISBN 979-8-89176-195-7Cited by:Introduction,Tree Search Algorithms with LLMs.
• Daineseet al.(2024)N. Dainese, M. Merler, M. Alakuijala, and P. MarttinenGenerating Code World Models with Large Language Models Guided by Monte Carlo Tree Search.Advances in Neural Information Processing Systems37,pp. 60429–60474.Cited by:Prior Works Close to Our Method.
• Efroniet al.(2019)Y. Efroni, G. Dalal, B. Scherrer, and S. MannorHow to Combine Tree-Search Methods in Reinforcement Learning.Proceedings of the AAAI Conference on Artificial Intelligence33(01),pp. 3494–3501.External Links:Link,DocumentCited by:Appendix C,Theoretical Connection.
• Fawziet al.(2022)A. Fawzi, M. Balog, A. Huang, T. Hubert, B. Romera-Paredes, M. Barekatain, A. Novikov, F. J. R. Ruiz, J. Schrittwieser, G. Swirszcz,et al.Discovering Faster Matrix Multiplication Algorithms with Reinforcement Learning.Nature610(7930),pp. 47–53.Cited by:Tree Search Algorithms with LLMs.
• Fernandoet al.(2024)C. Fernando, D. S. Banarse, H. Michalewski, S. Osindero, and T. RocktäschelPromptbreeder: Self-Referential Self-Improvement via Prompt Evolution.InProceedings of the 41st International Conference on Machine Learning,R. Salakhutdinov, Z. Kolter, K. Heller, A. Weller, N. Oliver, J. Scarlett, and F. Berkenkamp (Eds.),Proceedings of Machine Learning Research, Vol.235,pp. 13481–13544.External Links:LinkCited by:Prior Works Close to Our Method.
• Ford and Sterman (1998)D. N. Ford and J. D. StermanExpert Knowledge Elicitation to Improve Formal and Mental Models.System Dynamics Review: The Journal of the System Dynamics Society14(4),pp. 309–340.Cited by:Complex Systems and How to Model Them.
• Forrester (1971)J. W. ForresterWorld Dynamics.Wright-Allen Press,Cambridge, Mass..Cited by:Appendix A,Appendix E,Introduction,Complex Systems and How to Model Them,Experimental Details,Fitting an Abstract Goal Described in Natural Language.
• Forrester (2011)J. W. ForresterMIT system dynamics in education project.Note:https://www.merlot.org/merlot/viewMaterial.htm?id=518948External Links:LinkCited by:Complex Systems and How to Model Them,Preliminaries.
• Ganet al.(2025)B. Gan, Y. Zhao, T. Zhang, J. Huang, L. Yusu, S. X. Teo, C. Zhang, and W. ShiMASTER: A Multi-agent System with LLM Specialized MCTS.InProceedings of the 2025 Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers),L. Chiruzzo, A. Ritter, and L. Wang (Eds.),Albuquerque, New Mexico,pp. 9409–9426.External Links:Link,Document,ISBN 979-8-89176-189-6Cited by:Tree Search Algorithms with LLMs.
• Gaoet al.(2024)Z. Gao, B. Niu, X. He, H. Xu, H. Liu, A. Liu, X. Hu, and L. WenInterpretable Contrastive Monte Carlo Tree Search Reasoning.arXiv preprint arXiv:2410.01707.Cited by:Introduction,Tree Search Algorithms with LLMs.
• Gershenson (2023)C. GershensonEmergence in Artificial Life.Artificial Life29(2),pp. 153–167.Cited by:Introduction.
• Güneralp (2004)B. GüneralpA Principle on Structure-Behavior Relations in System Dynamics Models.InProceedings of the 2004 International System Dynamics Conference, Oxford, UK,Cited by:Introduction,Complex Systems and How to Model Them.
• Hannon and Ruth (2014)B. Hannon and M. RuthModeling Dynamic Biological Systems.InModeling Dynamic Biological Systems,pp. 3–28.Cited by:Introduction,Complex Systems and How to Model Them,Experimental Details.
• Hines (1996)J. HinesMolecules of Structure.System Dynamics Group, Sloan School of Management, MIT.Cited by:Complex Systems and How to Model Them.
• Inoueet al.(2025)Y. Inoue, K. Misaki, Y. Imajuku, S. Kuroki, T. Nakamura, and T. AkibaWider or Deeper? Scaling LLM Inference-Time Compute with Adaptive Branching Tree Search.InThe Thirty-ninth Annual Conference on Neural Information Processing Systems,External Links:LinkCited by:Appendix C,Tree Search Algorithms with LLMs,Node Selection.,Theoretical Connection.
• Katzet al.(2024)M. Katz, H. Kokel, K. Srinivas, and S. SohrabiThought of Search: Planning with Language Models through the Lens of Efficiency.Advances in Neural Information Processing Systems37,pp. 138491–138568.Cited by:Appendix C,LLM-based Optimization for Structured Systems,Theoretical Connection.
• Khattabet al.(2024)O. Khattab, A. Singhvi, P. Maheshwari, Z. Zhang, K. Santhanam, S. V. A, S. Haq, A. Sharma, T. T. Joshi, H. Moazam, H. Miller, M. Zaharia, and C. PottsDSPy: Compiling Declarative Language Model Calls into State-of-the-Art Pipelines.InThe Twelfth International Conference on Learning Representations,External Links:LinkCited by:Prior Works Close to Our Method.
• Kocsis and Szepesvári (2006)L. Kocsis and C. SzepesváriBandit Based Monte-Carlo Planning.InEuropean conference on machine learning,pp. 282–293.Cited by:Appendix C,Appendix C,Introduction,Tree Search Algorithms with LLMs.
• Kolson (1996)K. KolsonThe Politics of SimCity.PS: Political Science & Politics29(1),pp. 43–46.Cited by:Introduction,Complex Systems and How to Model Them.
• Kumaret al.(2025)A. Kumar, C. Lu, L. Kirsch, Y. Tang, K. O. Stanley, P. Isola, and D. HaAutomating the Search for Artificial Life with Foundation Models.Artificial Life31(3),pp. 368–396.Cited by:Introduction,LLM-based Optimization for Structured Systems.
• Lai and Pu (2025)Z. Lai and Y. PuPriM: Principle-Inspired Material Discovery through Multi-Agent Collaboration.arXiv preprint arXiv:2504.08810.Cited by:Appendix C.
• Langton (2019)C. G. LangtonArtificial Life.InArtificial Life,pp. 1–47.Cited by:Introduction.
• Leet al.(2025)N. Le, P. Erickson, Y. Zhang, M. Levin, and J. BongardGiving Simulated Cells a Voice: Evolving Prompt-to-Intervention Models for Cellular Control.InProceedings of the Genetic and Evolutionary Computation Conference Companion,GECCO ’25 Companion,New York, NY, USA,pp. 2327–2335.External Links:ISBN 9798400714641,Link,DocumentCited by:LLMs as Evolutionary Operators in Artificial Life.
• Leet al.(2026)N. H. Le, P. Erickson, Y. Zhang, M. Levin, and J. BongardZapGPT: Free-form Language Prompting for Simulated Cellular Control.InProceedings of the 2025 8th International Conference on Computational Intelligence and Intelligent Systems,CIIS ’25,New York, NY, USA,pp. 119–126.External Links:ISBN 9798400719332,Link,DocumentCited by:LLMs as Evolutionary Operators in Artificial Life.
• Lehmanet al.(2023)J. Lehman, J. Gordon, S. Jain, K. Ndousse, C. Yeh, and K. O. StanleyEvolution through Large Models.InHandbook of Evolutionary Machine Learning,pp. 331–366.External Links:DocumentCited by:Introduction,LLMs as Evolutionary Operators in Artificial Life,Conclusion.
• Liet al.(2023)G. Li, H. Hammoud, H. Itani, D. Khizbullin, and B. GhanemCAMEL: Communicative Agents for “Mind” Exploration of Large Language Model Society.InAdvances in Neural Information Processing Systems,A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (Eds.),Vol.36,pp. 51991–52008.External Links:LinkCited by:Appendix C.
• Liet al.(2025a)J. Li, H. Le, Y. Zhou, C. Xiong, S. Savarese, and D. SahooCodeTree: Agent-guided Tree Search for Code Generation with Large Language Models.InProceedings of the 2025 Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers),L. Chiruzzo, A. Ritter, and L. Wang (Eds.),Albuquerque, New Mexico,pp. 3711–3726.External Links:Link,Document,ISBN 979-8-89176-189-6Cited by:Appendix C.
• Liet al.(2025b)Q. Li, W. Xia, X. Dai, K. Du, W. Liu, Y. Wang, R. Tang, Y. Yu, and W. ZhangRethinkMCTS: Refining Erroneous Thoughts in Monte Carlo Tree Search for Code Generation.InProceedings of the 2025 Conference on Empirical Methods in Natural Language Processing,C. Christodoulopoulos, T. Chakraborty, C. Rose, and V. Peng (Eds.),Suzhou, China,pp. 8092–8110.External Links:Link,Document,ISBN 979-8-89176-332-6Cited by:Appendix C.
• Lianget al.(2025)Z. Liang, F. Wei, W. Xu, L. Chen, Y. Qian, and X. WuI-MCTS: Enhancing Agentic AutoML via Introspective Monte Carlo Tree Search.arXiv preprint arXiv:2502.14693.Cited by:Prior Works Close to Our Method.
• Lisyet al.(2013)V. Lisy, V. Kovarik, M. Lanctot, and B. BosanskyConvergence of Monte Carlo Tree Search in Simultaneous Move Games.InAdvances in Neural Information Processing Systems,C.J. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K.Q. Weinberger (Eds.),Vol.26,pp..External Links:LinkCited by:Appendix C,Theoretical Connection.
• Liu and Keith (2025)N. G. Liu and D. R. KeithLeveraging Large Language Models for Automated Causal Loop Diagram Generation: Enhancing System Dynamics Modeling through Curated Prompting Techniques.arXiv preprint arXiv:2503.21798.Cited by:LLM-based Optimization for Structured Systems.
• Liuet al.(2024)T. J.B. Liu, N. Boulle, R. Sarfati, and C. EarlsLLMs Learn Governing Principles of Dynamical Systems, Revealing an In-Context Neural Scaling Law.InProceedings of the 2024 Conference on Empirical Methods in Natural Language Processing,Y. Al-Onaizan, M. Bansal, and Y. Chen (Eds.),Miami, Florida, USA,pp. 15097–15117.External Links:Link,DocumentCited by:LLM-based Optimization for Structured Systems.
• Luoet al.(2025)X. Luo, B. Chen, H. Wang, Z. Xiao, M. Zhang, and Y. SunHow Do Large Language Models Perform in Dynamical System Modeling.InFindings of the Association for Computational Linguistics: NAACL 2025,L. Chiruzzo, A. Ritter, and L. Wang (Eds.),Albuquerque, New Mexico,pp. 866–880.External Links:Link,Document,ISBN 979-8-89176-195-7Cited by:LLM-based Optimization for Structured Systems.
• Mankowitzet al.(2023)D. J. Mankowitz, A. Michi, A. Zhernov, M. Gelmi, M. Selvi, C. Paduraru, E. Leurent, S. Iqbal, J. Lespiau, A. Ahern,et al.Faster Sorting Algorithms Discovered Using Deep Reinforcement Learning.Nature618(7964),pp. 257–263.Cited by:Tree Search Algorithms with LLMs.
• McLachlan (2014)N. W. McLachlanLaplace transforms and their applications to differential equations.Courier Corporation.Cited by:Preliminaries.
• Meertet al.(2022)W. Meert, K. Hendrickx, T. Van Craenendonck, P. Robberechts, H. Blockeel, and J. DavisDTAIDistance (Version v2.3.5).Zenodo.External Links:LinkCited by:Appendix F.
• MIT System Dynamics in Education Project (1998)MIT System Dynamics in Education ProjectRoad Maps: A Guide to Learning System Dynamics.Note:System Dynamics Group, Sloan School of Management, Massachusetts Institute of Technology. Accessed: 2025-08-30http://www.clexchange.org/curriculum/roadmaps/Cited by:Complex Systems and How to Model Them.
• Nasiret al.(2024)M. U. Nasir, S. Earle, J. Togelius, S. James, and C. CleghornLLMatic: Neural Architecture Search Via Large Language Models And Quality Diversity Optimization.InProceedings of the Genetic and Evolutionary Computation Conference,GECCO ’24,New York, NY, USA,pp. 1110–1118.External Links:ISBN 9798400704949,Link,DocumentCited by:Tree Search Algorithms with LLMs.
• Nisiotiet al.(2024)E. Nisioti, C. Glanois, E. Najarro, A. Dai, E. Meyerson, J. W. Pedersen, L. Teodorescu, C. F. Hayes, S. Sudhakaran, and S. RisiFrom Text to Life: On the Reciprocal Relationship between Artificial Life and Large Language Models.InArtificial Life Conference Proceedings,Vol.36,pp. 39.External Links:DocumentCited by:Introduction,LLMs as Evolutionary Operators in Artificial Life,Conclusion.
• Ogata (2004)K. OgataSystem Dynamics.4 edition,Pearson Education.Cited by:Preliminaries.
• Peterson (2003)S. PetersonBarry Richmond, System Dynamics and Public Policy.In21st System Dynamics Conference,pp. 1–14.Cited by:Introduction,Complex Systems and How to Model Them.
• Radzicki and Taylor (2008)M. J. Radzicki and R. A. TaylorOrigin of System Dynamics: Jay W. Forrester and the History of System Dynamics.US Department of Energy’s introduction to system dynamics.Cited by:Introduction,Introduction,Complex Systems and How to Model Them,Preliminaries.
• Ratanamahatana and Keogh (2004)C. A. Ratanamahatana and E. KeoghEverything You Know About Dynamic Time Warping Is Wrong.InThird workshop on mining temporal and sequential data,Vol.32,pp. 1–11.Cited by:Appendix F.
• Richmond (1985)B. RichmondSTELLA: Software for Bringing System Dynamics to the Other 98%.InProceedings of the 1985 International Conference of the System Dynamics Society,pp. 706–718.Cited by:Introduction,Introduction,Complex Systems and How to Model Them,Preliminaries.
• Ruth and Hannon (2012)M. Ruth and B. HannonModeling Dynamic Economic Systems.Springer.Cited by:Complex Systems and How to Model Them.
• Sakoe and Chiba (1978)H. Sakoe and S. ChibaDynamic Programming Algorithm Optimization for Spoken Word Recognition.IEEE Transactions on Acoustics, Speech, and Signal Processing26(1),pp. 43–49.External Links:DocumentCited by:Appendix F,Quantitative Studies: Fitting a Concrete Record.
• Schoenberget al.(2020)W. Schoenberg, P. Davidsen, and R. EberleinUnderstanding Model Behavior Using the Loops That Matter Method.System Dynamics Review36(2),pp. 158–190.Cited by:Complex Systems and How to Model Them.
• Schrittwieseret al.(2020)J. Schrittwieser, I. Antonoglou, T. Hubert, K. Simonyan, L. Sifre, S. Schmitt, A. Guez, E. Lockhart, D. Hassabis, T. Graepel,et al.Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model.Nature588(7839),pp. 604–609.Cited by:Appendix C,Tree Search Algorithms with LLMs,Theoretical Connection.
• Schünemannet al.(2024)C. Schünemann, S. Johanning, E. Reger, H. Herold, and T. BrucknerComplex System Policy Modelling Approaches for Policy Advice–Comparing Systems Thinking, System Dynamics and Agent-based Modelling.Political Research Exchange6(1),pp. 2387438.Cited by:Introduction,Complex Systems and How to Model Them.
• Shinnet al.(2023)N. Shinn, F. Cassano, A. Gopinath, K. Narasimhan, and S. YaoReflexion: Language Agents with Verbal Reinforcement Learning.InAdvances in Neural Information Processing Systems,A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (Eds.),Vol.36,pp. 8634–8652.External Links:LinkCited by:Appendix C.
• Shojaeeet al.(2025)P. Shojaee, N. Nguyen, K. Meidani, A. B. Farimani, K. D. Doan, and C. K. ReddyLLM-SRBench: A New Benchmark for Scientific Equation Discovery with Large Language Models.InForty-second International Conference on Machine Learning,External Links:LinkCited by:Prior Works Close to Our Method.
• Silveret al.(2016)D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot,et al.Mastering the Game of Go with Deep Neural Networks and Tree Search.nature529(7587),pp. 484–489.Cited by:Tree Search Algorithms with LLMs.
• Silveret al.(2017a)D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez, M. Lanctot, L. Sifre, D. Kumaran, T. Graepel, T. Lillicrap, K. Simonyan, and D. HassabisMastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm.arXiv preprint arXiv:1712.01815.Cited by:Tree Search Algorithms with LLMs.
• Silveret al.(2017b)D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton,et al.Mastering the Game of Go without Human Knowledge.nature550(7676),pp. 354–359.Cited by:Appendix C,Tree Search Algorithms with LLMs,Theoretical Connection.
• Songet al.(2025)X. Song, S. Zhang, and T. YuReKG-MCTS: Reinforcing LLM Reasoning on Knowledge Graphs via Training-Free Monte Carlo Tree Search.InFindings of the Association for Computational Linguistics: ACL 2025,W. Che, J. Nabende, E. Shutova, and M. T. Pilehvar (Eds.),Vienna, Austria,pp. 9288–9306.External Links:Link,Document,ISBN 979-8-89176-256-5Cited by:Introduction,Tree Search Algorithms with LLMs.
• Srinivasanet al.(2025)K. K. Srinivasan, N. Le, J. Bongard, and D. BlackistonEvolving agents to generate and falsify hypotheses of biological self-assembly.InArtificial Life Conference Proceedings,Vol.37,pp. 24.External Links:DocumentCited by:Introduction.
• Sterman (2000)J. D. StermanBusiness Dynamics: Systems Thinking and Modeling for a Complex World.McGraw-Hill Education.Cited by:Introduction,Complex Systems and How to Model Them.
• Tonget al.(2025)Z. Tong, Z. Ding, and W. WeiEvoPrompt: Evolving Prompts for Enhanced Zero-Shot Named Entity Recognition with Large Language Models.InProceedings of the 31st International Conference on Computational Linguistics,O. Rambow, L. Wanner, M. Apidianaki, H. Al-Khalifa, B. D. Eugenio, and S. Schockaert (Eds.),Abu Dhabi, UAE,pp. 5136–5153.External Links:LinkCited by:Prior Works Close to Our Method.
• Walter (2013)W. WalterOrdinary differential equations.Vol.182,Springer Science & Business Media.Cited by:Preliminaries.
• Wuet al.(2025)F. Wu, W. Xuan, H. Qi, X. Lu, A. Tu, L. E. Li, and Y. ChoiDeepSearch: Overcome the Bottleneck of Reinforcement Learning with Verifiable Rewards via Monte Carlo Tree Search.arXiv preprint arXiv:2509.25454.Cited by:Appendix C,Theoretical Connection.
• Xuet al.(2025)B. Xu, Y. Lin, Y. Li, and Y. GaoSRA-MCTS: Self-driven Reasoning Augmentation with Monte Carlo Tree Search for Code Generation.InProceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence, IJCAI-25,J. Kwok (Ed.),pp. 8678–8686.Note:Main TrackExternal Links:Document,LinkCited by:Appendix C.
• Xuet al.(2024)Y. Xu, Y. Liu, and H. SunReinforcement Symbolic Regression Machine.InThe Twelfth International Conference on Learning Representations,External Links:LinkCited by:Appendix C.
• Yamadaet al.(2025)Y. Yamada, R. T. Lange, C. Lu, S. Hu, C. Lu, J. Foerster, J. Clune, and D. HaThe AI Scientist-v2: Workshop-Level Automated Scientific Discovery via Agentic Tree Search.arXiv preprint arXiv:2504.08066.Cited by:Introduction,LLM-based Optimization for Structured Systems.
• Zagonel (2002)A. A. ZagonelModel Conceptualization in Group Model Building: A Review of the Literature Exploring the Tension Between Representing Reality and Negotiating a Social Order.InProceedings of the 20th International System Dynamics Conference,Vol.51,pp. 170–182.Cited by:Complex Systems and How to Model Them.
• Zelikmanet al.(2022)E. Zelikman, Y. Wu, J. Mu, and N. GoodmanSTaR: Bootstrapping Reasoning With Reasoning.InAdvances in Neural Information Processing Systems,S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (Eds.),Vol.35,pp. 15476–15488.External Links:LinkCited by:Appendix C.
• Zhaoet al.(2023)Z. Zhao, W. S. Lee, and D. HsuLarge Language Models as Commonsense Knowledge for Large-Scale Task Planning.Advances in neural information processing systems36,pp. 31967–31987.Cited by:Introduction,Tree Search Algorithms with LLMs.
• Zhenget al.(2025)Z. Zheng, Z. Xie, Z. Wang, and B. HooiMonte Carlo Tree Search for Comprehensive Exploration in LLM-Based Automatic Heuristic Design.arXiv preprint arXiv:2501.08603.Cited by:Tree Search Algorithms with LLMs.
• Zhouet al.(2024)A. Zhou, K. Yan, M. Shlapentokh-Rothman, H. Wang, and Y. WangLanguage Agent Tree Search Unifies Reasoning, Acting, and Planning in Language Models.InForty-first International Conference on Machine Learning,External Links:LinkCited by:Tree Search Algorithms with LLMs,Prior Works Close to Our Method.
• Zhouet al.(2023)Y. Zhou, A. I. Muresanu, Z. Han, K. Paster, S. Pitis, H. Chan, and J. BaLarge Language Models are Human-Level Prompt Engineers.InThe Eleventh International Conference on Learning Representations,External Links:LinkCited by:Appendix C,Appendix C.
• Zhuet al.(2025)X. Zhu, Y. Shi, and Y. ZhongAn EKF Prediction of COVID-19 Propagation under Vaccinations and Viral Variants.Mathematics and Computers in Simulation231,pp. 221–238.Cited by:Introduction,Complex Systems and How to Model Them.

APPENDIX

Appendix Overview

The appendix is organized as follows. Since appendix references are not included in the main submission text, this overview serves as a guide to locating supplementary material.

Python Implementation of Complex Systems (SectionA).

Details the unified Python-based representation, including a minimal code example, the fullWorld Dynamicssystem code, and the documentation and domain-specific math primitives provided to LLMs.

Method Details (SectionB).

Contains the full prompt templates for LLM Judge and LLM Editor (SectionB), and the original node selection hyperparameter rationale (SectionB).

Theoretical Connection: Full Details (SectionC).

Presents the full derivations of node selection as a UCT generalization, LLM Editor as a learned transition kernel, and LLM Judge as a learned value function, with equations and discussion of open theoretical problems.

Experiment Details (SectionD).

Contains dataset variable statistics (SectionD), the full expansion strategy set (SectionD), engineering decisions for scalability and stability (SectionD), and computational cost and runtime (SectionD).

Fitting an Abstract Goal: Details (SectionE).

Contains theWorld Dynamicssystem structure and variable definitions (SectionE), the exact natural-language goal text (SectionE), and a quantitative demonstration of competing-subgoal challenges (SectionE).

Fitting a Concrete Record: Details (SectionF).

Contains the ground truth system implementation (SectionF), the three levels of Optuna baseline code (SectionF), the DTW metric definition (SectionF), stochasticity analysis across seeds (SectionF), and an additional Optuna run (SectionF).

Interpretability Analysis (SectionG).

Contains full LLM Judge and Editor transcripts from theWorld Dynamicsoptimization run.

Appendix APython Implementation of Complex Systems

Here we detail our proposed unified representation of complex systems based on a restricted subset of Python programs. Concretely, the components are divided into four parts: (1) initialization, a loop of (2) intermediate computations, (3) derivative evaluations, and (4) Euler-method integration. We show a minimal example of a program with these components below:

1

2dt=0.01

3t0=0.0

4tf=120.0

5K=100.0

6R=0.1

7N=10.0

8

9t=t0

10whilet<=tf+1e-12:

11

12

13carrying_capacity_factor=1.0-N/K

14growth_rate=R*N*carrying_capacity_factor

15

16

17dN=growth_rate

18

19

20N=N+dt*dN

21t+=dt

In practice, we find that including extensive comments in the program is not only helpful for humans to understand, but also beneficial for LLMs. Furthermore, to help LLMs’ tool use, we extend the code with further documentation and predefined math primitives, such as common math functions, delay, and smoothing functions that are frequently used in complex systems. The benefits are twofold: (1) Such comments and documents help LLMs to understand the complex system’s semantics for better optimization, and the standardized functions ease LLMs’ job in generating code for new systems, and (2) more importantly, it instructs LLMs on what restrictions should be enforced, which is important since several critical restrictions, such as preserving the Euler-method integration, must be respected for the program to execute successfully.

We show our representation of the complex system of theWorld Dynamicssystem(18). As it is the one with the largest number of variables in our dataset, this code below demonstrates the full-fledged form of our representation, including how we enforce the constraints for LLMs and providing semantics to help the optimization. We show the code in two parts: First, we show the code that directly describes the complex system’s dynamics:

1

2

3dt=0.2

4t0=1900.0

5tf=2100.0

6seed=1234

7

8

9

10

11P=1.65e9

12NR=900e9

13CI=0.4e9

14POL=0.2e9

15CIAF=0.2

16

17

18

19

20

21

22

23

24

25

26

27t=t0

28whilet<=tf+1e-12:

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44CIR=CI/P

45CR=P/(135e6*26.5)

46NRFR=NR/900e9

47POLR=POL/3.6e9

48

49

50CIAF_current=CIAF

51NREM=graph(

52NRFR,((0,0),(0.25,0.15),(0.5,0.5),(0.75,0.85),(1,1))

53)

54ECIR=CIR*(1-CIAF_current)*NREM/(1-0.3)

55MSL=ECIR/1

56

57

58BRMM=graph(

59MSL,((0,1.2),(1,1),(2,0.85),(3,0.75),(4,0.7),(5,0.7))

60)

61BRCM=graph(

62CR,((0,1.05),(1,1),(2,0.9),(3,0.7),(4,0.6),(5,0.55))

63)

64BRPM=graph(

65POLR,((0,1.02),(10,0.9),(20,0.7),(30,0.4),(40,0.25),(50,0.15),(60,0.1))

66)

67

68

69CIRA=CIR*CIAF_current/0.3

70FPCI=graph(

71CIRA,((0,0.5),(1,1),(2,1.4),(3,1.7),(4,1.9),(5,2.05),(6,2.2))

72)

73FCM=graph(CR,((0,2.4),(1,1),(2,0.6),(3,0.4),(4,0.3),(5,0.2)))

74FPM=graph(

75POLR,((0,1.02),(10,0.9),(20,0.65),(30,0.35),(40,0.2),(50,0.1),(60,0.05))

76)

77FR=FPCI*FCM*FPM*(1ift>=1970else1)/1

78BRFM=graph(FR,((0,0),(1,1),(2,1.6),(3,1.9),(4,2)))

79

80

81DRMM=graph(

82MSL,

83(

84(0,3),

85(0.5,1.8),

86(1,1),

87(1.5,0.8),

88(2,0.7),

89(2.5,0.6),

90(3,0.53),

91(3.5,0.5),

92(4,0.5),

93(4.5,0.5),

94(5,0.5),

95),

96)

97DRCM=graph(CR,((0,0.9),(1,1),(2,1.2),(3,1.5),(4,1.9),(5,3)))

98DRPM=graph(

99POLR,((0,0.92),(10,1.3),(20,2),(30,3.2),(40,4.8),(50,6.8),(60,9.2))

100)

101DRFM=graph(

102FR,((0,30),(0.25,3),(0.5,2),(0.75,1.4),(1,1),(1.25,0.7),(1.5,0.6),(1.75,0.5),(2,0.5))

103)

104

105

106NRMM=graph(

107MSL,((0,0),(1,1),(2,1.8),(3,2.4),(4,2.9),(5,3.3),(6,3.6),(7,3.8),(8,3.9),(9,3.95),(10,4))

108)

109

110

111CIM=graph(MSL,((0,0.1),(1,1.0),(2,1.8),(3,2.4),(4,2.8),(5,3)))

112

113

114POLCM=graph(CIR,((0,0.05),(1,1),(2,3),(3,5.4),(4,7.4),(5,8)))

115POLAT=graph(

116POLR,((0,0.6),(10,2.5),(20,5),(30,8),(40,11.5),(50,15.5),(60,20))

117)

118

119

120QLM=graph(MSL,((0,0.2),(1,1),(2,1.7),(3,2.3),(4,2.7),(5,2.9)))

121QLC=graph(

122CR,

123(

124(0,2),

125(0.5,1.3),

126(1,1),

127(1.5,0.75),

128(2,0.55),

129(2.5,0.45),

130(3,0.38),

131(3.5,0.3),

132(4,0.25),

133(4.5,0.22),

134(5,0.2),

135),

136)

137QLF=graph(FR,((0,0),(1,1),(2,1.8),(3,2.4),(4,2.7)))

138QLP=graph(

139POLR,((0,1.04),(10,0.85),(20,0.6),(30,0.3),(40,0.15),(50,0.05),(60,0.02))

140)

141QL=1*QLM*QLC*QLF*QLP

142

143

144CFIFR=graph(FR,((0,1),(0.5,0.6),(1,0.3),(1.5,0.15),(2,0.1)))

145CIQR=graph(QLM/QLF,((0,0.7),(0.5,0.8),(1,1),(1.5,1.5),(2,2)))

146

147

148

149

150

151

152

153BR=P*(0.04ift>=1970else0.04)*BRFM*BRMM*BRCM*BRPM

154DR=P*(0.028ift>=1970else0.028)*DRMM*DRPM*DRFM*DRCM

155

156

157NRUR=P*(1ift>=1970else1)*NRMM

158

159

160CIG=P*CIM*(0.05ift>=1970else0.05)

161CID=CI*(0.025ift>=1970else0.025)

162

163

164POLG=P*(1ift>=1970else1)*POLCM

165POLA=POL/POLAT

166

167

168dP=BR-DR

169dNR=-NRUR

170dCI=CIG-CID

171dPOL=POLG-POLA

172dCIAF=(1/15)*(CFIFR*CIQR-CIAF)

173

174

175

176P=P+dt*dP

177NR=NR+dt*dNR

178CI=CI+dt*dCI

179POL=POL+dt*dPOL

180CIAF=CIAF+dt*dCIAF

181

182

183t+=dt

Then, we show the documentation and math functions. The code above is placed betweenBEGIN WHOLE SYSTEMandEND WHOLE SYSTEM.

1

2

3

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5

6

7

8

9

10

11

12

13

14

15

16#arenotmarkedas“#NECESSARY“incomments.Allstatevariablesmusthavederivatives.

17

18#ONLYiftheyarenotmarkedas“#NECESSARY“incomments.

19

20#computationformulasandreorderthem,aslongasthecoderemainsvalid.

21

22#“#NECESSARY“commentssooptimization/editingknowswhichvariablescannotbedeleted.

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45importmath

46importrandom

47importargparse

48importpandasaspd

49fromtypingimportIterable,Tuple

50

51

52

53

54defsin(x:float)->float:

55“““Sinefunction(angleinradians).”“”

56returnmath.sin(x)

57

58defcos(x:float)->float:

59“““Cosinefunction(angleinradians).”“”

60returnmath.cos(x)

61

62defexp(x:float)->float:

63“““Exponentialfunctione^x.”“”

64returnmath.exp(x)

65

66deftanh(x:float)->float:

67“““Hyperbolictangent.”“”

68returnmath.tanh(x)

69

70defsqrt(x:float)->float:

71“““Squareroot.”“”

72returnmath.sqrt(x)

73

74deflog(x:float)->float:

75“““Naturallogarithm.”“”

76returnmath.log(x)

77

78defgauss(std:float)->float:

79“““Gaussianrandomvariablewithmean=0andstandarddeviation=std.”“”

80returnrandom.gauss(0.0,std)

81

82defuniform(lo:float,hi:float)->float:

83“““Uniformrandomvariablebetweenloandhi.”“”

84returnrandom.uniform(lo,hi)

85

86defgraph(v:float,table:Iterable[Tuple[float,float]])->float:

87““”

88Lookuphelperwithlinearinterpolation.

89-tableisaniterableof(x,y)pointssortedbyx.

90-Ifv<x0,returny0.

91-Ifv>xN,returnyN.

92-Else,linearlyinterpolatebetweennearestpoints.

93““”

94table=list(table)

95ifnottable:

96raiseValueError(“graph()calledwithemptytable”)

97

98ifv<=table[0][0]:

99returntable[0][1]

100ifv>=table[-1][0]:

101returntable[-1][1]

102

103foriinrange(len(table)-1):

104x0,y0=table[i]

105x1,y1=table[i+1]

106ifx0<=v<=x1:

107ifx1==x0:

108returny0

109frac=(v-x0)/(x1-x0)

110returny0+frac*(y1-y0)

111returntable[-1][1]

112

113

114

115

116

117#-Fixeddelaysusetime-indexedhistorywithlinearinterpolation

118

119

120

121_delay_states={}

122

123defdelay(input_val:float,delay_time:float,name:str,initial_value:float=None)->float:

124“““Fixedlagdelay-returnsinputvaluefromdelay_timeago.”“”

125key=f“delay_{name}_{delay_time}“

126

127ifkeynotin_delay_states:

128init_val=input_valifinitial_valueisNoneelseinitial_value

129_delay_states[key]={

130’history’:[(t,init_val)],

131’last_t’:t

132}

133

134state=_delay_states[key]

135

136

137ift>state[’last_t’]:

138state[’history’].append((t,input_val))

139state[’last_t’]=t

140

141

142cutoff_time=t-delay_time

143state[’history’]=[(t_hist,val)fort_hist,valinstate[’history’]

144ift_hist>=cutoff_time]

145

146

147target_time=t-delay_time

148ifnotstate[’history’]ortarget_time<=state[’history’][0][0]:

149returnstate[’history’][0][1]

150

151foriinrange(len(state[’history’])-1):

152t1,v1=state[’history’][i]

153t2,v2=state[’history’][i+1]

154ift1<=target_time<=t2:

155ift2==t1:

156returnv1

157frac=(target_time-t1)/(t2-t1)

158returnv1+frac*(v2-v1)

159

160returnstate[’history’][-1][1]

161

162defsmth1(input_val:float,smooth_time:float,name:str,initial_value:float=None)->float:

163“““First-orderexponentialsmooth-stock-basedsmoothingprocess.”“”

164key=f“smth1_{name}_{smooth_time}“

165

166ifkeynotin_delay_states:

167init_val=input_valifinitial_valueisNoneelseinitial_value

168_delay_states[key]={

169’smooth_of_input’:init_val,

170’last_t’:t

171}

172

173state=_delay_states[key]

174

175ift>state[’last_t’]:

176dt_step=t-state[’last_t’]

177

178

179

180change_in_smooth=(input_val-state[’smooth_of_input’])/smooth_timeifsmooth_time>0else0.0

181

182

183state[’smooth_of_input’]+=dt_step*change_in_smooth

184state[’last_t’]=t

185

186

187returnstate[’smooth_of_input’]

188

189

190

191

192

193

194

195

196

197

198def_parse_args():

199ap=argparse.ArgumentParser(description=“RunEulersimulationandexportCSV.”)

200ap.add_argument(“--csv”,dest=“csv_path”,default=None,

201help=“WriteresultstothisCSVpath.Ifomitted,printsCSVtostdout.”)

202returnap.parse_args()

203

204if__name__==“__main__”:

205args=_parse_args()

206df=pd.DataFrame(trace)

207ifargs.csv_path:

208df.to_csv(args.csv_path,index=False,float_format=’%.6f’)

209print(f“Simulationcomplete.Resultswrittento{args.csv_path}“)

210else:

211print(df.to_csv(index=False,float_format=’%.6f’),end=“”)

Appendix BMethod Details: LLM Contextualization, Scalability and Stability

LLM Contextualization

In our design, we pair LLM Judge, which performs evaluations of one complex system, with LLM Editor, which produces new modified systems to better achieve the goal. Besides the enriched code-based representation detailed in AppendixA, we also need well-crafted prompts to help LLMs to understand the semantics of complex systems and to generate meaningful modifications towards the goal. In fact, since the code is in the context of the prompt, both the prompt design and code representation can be seen as a coherentcontextualizationof the information. Also, we find that this contextualization should include the tree context, as our preliminary studies found that without tree context, the expansion process fails to produce meaningful improvements.

In conclusion, the prompt templates for LLM Judge and LLM Editor are listed below respectively:

Prompt forLlmJudge⬇ANALYZESIMULATIONRESULTSFORTREE-BASEDOPTIMIZATIONGOAL:{goal}CURRENTCODE:{code}SIMULATIONRESULTS:{csv_data}TREECONTEXT:{tree_context}===Foryourreference===Formatforcodeintreecontext:Thetreecontextshowsonlythedifferencesineachreferencenodecomparedtocurrentcode.Sincethefullcurrentcodeisshownabove,diffsonlydisplaylinesthataredifferentinreferencenodes:-Linesstartingwith“+“showwhatthereferencenodehasinsteadofcurrentcode-“Codeidenticaltocurrentnode“meansnodifferencesexistWhenanalyzingthisnode’sperformance,consider:-Howwellthisspecificcodevariantachievestheoptimizationgoal-PerformancerelativetoALLothernodesinthereference(forcalibratedscoring)-WhetherthisrepresentsmeaningfulprogressinthesearchspaceSCORINGGUIDELINES:ABSOLUTEPERFORMANCESCORING(depth-constrainedscale)PrimaryPrinciple:Scorebasedonhowwellthiscodeachievestheoptimizationgoal,regardlessofothernodesinthetree.SCORINGSCALE(use2decimalplaceslike7.25,12.75):-0.00-2.00:Failedprogress-nomeaningfulprogresstowardthegoal-2.00-4.00:Partialprogress-someimprovementbutfarfromachievingthegoal-4.00-6.00:Moderateprogress-measurableimprovementandmovingtowardthegoal-6.00-8.00:Goodprogress-clearadvancementwithsubstantialgoalachievement-8.00-10.00:Excellentprogress-meetsmostrequirementsoftheoptimizationgoal-10.00+:VERYGOODperformance-exceedsthegoalexpectationssignificantly-20.00+:EXCEPTIONALperformance-farsurpasseswhatthegoalwasaskingforIMPORTANTCONSTRAINT:Maximumpossiblescoreisscore<=10.0+2.5*depth.Withinthislimit,highscoresareencouragedwhenperformancetrulymeritsthem.Usethetreecontextinformationtoensureyourscoringiswell-calibratedrelativetoallexploredalternatives.===Task====Pleaseprovideyouranalysisinthisformat:REASONING:[Startwithabriefsummaryofkeyfindingsinthefirst200characters,thenprovidedetailedreasoningabouthowwellthiscodevariantachievestheoptimizationgoal]SCORE:[numericalscorebasedonthescoringguidelinesabove]’

Prompt forLlmEditor⬇SUGGESTCODEMODIFICATIONSFORTREESEARCHOPTIMIZATIONGOAL:{goal}CURRENTCODE:{code}SIMULATIONRESULTS:{csv_data}TREECONTEXT:{tree_context}===Foryourreference===Formatforcodeintreecontext:Thetreecontextshowsonlythedifferencesineachreferencenodecomparedtocurrentcode.Sincethefullcurrentcodeisshownabove,diffsonlydisplaylinesthataredifferentinreferencenodes:-Linesstartingwith“+“showwhatthereferencenodehasinsteadofcurrentcode-“Codeidenticaltocurrentnode“meansnodifferencesexistHowtomodifythecodetobetterachievethegoalwithinthetreesearchcontext:Focuson:-ParameterswithinthemarkedBEGIN/ENDeditsections-Statevariables,constants,andsimulationconfiguration-Derivativeequationsandhelperexpressions-Tablevaluesandtime-dependentinputs-LearningfromtreecontextandreferencenodeoutcomesWhenyoumodifythecode:-FaithfullyfollowtheCURRENTCODE.StartingfromEVERYTHINGinCURRENTCODE,then-onlymodifycodebetweenthe“BEGIN.../END...“markers.-DONOTchangeimports,wrappers,loopstructure,orOUTPUTsection.TreeSearchStrategy:-Consideryourpositioninthesearchtree-Learnfromperformancepatternsintreecontext-Usediversificationstrategyfromtreecontext-Referencesuccessful/failedapproachesfromothernodesSCORINGGUIDELINES:ABSOLUTEPERFORMANCESCORING(depth-constrainedscale)PrimaryPrinciple:Scorebasedonhowwellthiscodeachievestheoptimizationgoal,regardlessofothernodesinthetree.SCORINGSCALE(use2decimalplaceslike7.25,12.75):-0.00-2.00:Failedprogress-nomeaningfulprogresstowardthegoal-2.00-4.00:Partialprogress-someimprovementbutfarfromachievingthegoal-4.00-6.00:Moderateprogress-measurableimprovementandmovingtowardthegoal-6.00-8.00:Goodprogress-clearadvancementwithsubstantialgoalachievement-8.00-10.00:Excellentprogress-meetsmostrequirementsoftheoptimizationgoal-10.00+:VERYGOODperformance-exceedsthegoalexpectationssignificantly-20.00+:EXCEPTIONALperformance-farsurpasseswhatthegoalwasaskingforIMPORTANTCONSTRAINT:Maximumpossiblescoreisscore<=10.0+2.5*depth.Withinthislimit,highscoresareencouragedwhenperformancetrulymeritsthem.HOWTOSCORE:1.First,evaluatehowwellthiscodeachievesthegoalinabsoluteterms2.Usethetreecontextonlytocalibratewhatscorerangesmean-don’tletitconstrainyourscoring3.Forverygoodperformancethatexceedsexpectations,don’thesitatetogivehighscores4.Whenunsurebetweenscoreranges,favorthehigherscoreifgenuineprogressisevident5.Largescoreincreases(5+points)areappropriateforsignificantimprovements6.Alwaysrespectthedepthconstraint:score<=10.0+2.5*depthRemember:Judgethiscode’sactualachievementofthegoal,notitsrelativepositioninthesearchtree.===Task====Pleaseprovideyourresponseinthisformat:REASONING:[Startwithaconcisedescriptionofyourmainmodificationstrategyinthefirst200characters,thenexplaindetailedreasoningandassesstheexpectedimprovementusingthescoringguidelinesabove]SELF_ASSESSED_SCORE:[numericalscorebasedonthescoringguidelinesaboveforyourexpectedimprovement]MODIFIED_CODE:[CompletemodifiedPythoncodefollowingallrequirements-startfromEVERYTHINGinCURRENTCODE,thenonlymodifycodebetweenBEGIN/ENDmarkers]’

Original Node Selection Rationale

The choice of hyperparametersα\alphaandτ\taureflects empirical behavior of LLM-based optimization on complex systems: high-quality solutions often result from several consecutive refinements among a promising branch, so our choice should favour deeper nodes, cap expansion counts, thus avoiding over-exploring shallow nodes with expensive LLM calls.

Appendix CTheoretical Connection: Full Details

While CEDAR components are designed based on empirical considerations, the algorithm is a principled MCTS variant: It has widening-aware and depth-aware UCT selection, LLM-parameterized transition kernel and value function. We establish the connections as follows, and we also note that for MCTS with LLMs convergence and regret guarantees remain open problems.

Node Selection as a UCT-Generalization.

Following UCB1(29), CEDAR usesScore​(v)=Sv+ϕ⁡(cv,cp,τ)\textsc{Score}(v)=S_{v}+\phi\!\left(c_{v},c_{p},\tau\right), wherecvc_{v}andcpc_{p}are expansion counts atvvand its parentpp, andτ\tauis the progressive widening limit. We combine two MCTS principles: Progressive Widening(10;26)limiting expansion width, and depth-based bonuses favoring deeper search(7;71). This yields:

ϕ=αln⁡(cp+1)cu+1⋅𝕀[cv<τ]+β𝕀[cv≥τ]⏟Progressive Widening+γ⋅Depth​(v)⏟Depth-based bonus\phi=\underbrace{\alpha\sqrt{\frac{\ln\!\left(c_{p}+1\right)}{c_{u}+1}}\cdot\mathbb{I}[c_{v}<\tau]+\beta\,\mathbb{I}[c_{v}\geq\tau]}_{\text{Progressive Widening}}+\underbrace{\gamma\cdot\textsc{Depth}(v)}_{\text{Depth-based bonus}}(1)The node selection in CEDAR effectively setsβ=−∞\beta=-\infty, positioning itself as a simple combination of MCTS variants with non-uniform branching. The constraintcv<τc_{v}<\tauprevents overexpansion in vast action spaces like code modification and expensive LLM calls.

LLM Editor as a Learned Transition Kernel.

MCTS assumes a generative model for transitionsu∼P(⋅∣v,a)u\sim P(\cdot\mid v,a)whereaadenotes an expansion action. CEDAR extends this to a state space, whose values are programs, via a stochastic operatorPθP_{\theta}induced by the LLM:

Pu∼Pθ(⋅∣Pv,Av,s,G),where​Pθ=LlmEditorθ,s∼Uniform​(𝒮)\begin{split}P_{u}&\sim P_{\theta}(\cdot\mid P_{v},A_{v},s,G),\\ &\text{where }P_{\theta}=\textsc{LlmEditor}_{\theta},\quad s\sim\text{Uniform}(\mathcal{S})\end{split}(2)wherePθP_{\theta}parameterizes a conditional distribution over modified programs.PθP_{\theta}, although parameterized by a pretrained foundation model rather than a model learned using search, functionally serves as a learned component enabling MCTS to operate in complex program spaces. This aligns with recent work using LLMs in place of learned components for sampling in linear search(80;37). Our CEDAR extends LLM-powered transition kernels from linear to tree search, connecting to learned-transition MCTS and model-based RL(65;59), neural-guided search(11;73), and scientific discovery(32).

LLM Judge as a Learned Value Function.

RunningPuP_{u}yields an execution recordRu=Run​(Pu)R_{u}=\textsc{Run}(P_{u}), and the LLM Judge provides a semantic evaluation:

(Au,Su)=Rϕ(Pu,Ru,G),Rϕ=LlmJudgeϕ(A_{u},S_{u})=R_{\phi}(P_{u},R_{u},G),\text{ }R_{\phi}=\textsc{LlmJudge}_{\phi}(3)whereRϕR_{\phi}acts as a learned value network. We denoteSu=πS​(Rϕ​(Pu,Ru,G))S_{u}=\pi_{S}(R_{\phi}(P_{u},R_{u},G)), thusπS\pi_{S}projects the space of(A,S)(A,S)onto the space of scoreSS. This aligns with recent works where LLMs serve as noisy reward estimators or evaluators in non-differentiable optimization(80;76;61). Theoretically,SuS_{u}constitutes abounded, noisy reward, falling under convergence analysis of MCTS with biased evaluators(41;14)which shows UCT remains consistent when reward noise is bounded and non-adversarial. This is the case of our work: scores are bounded and semantic LLM scoring can be treated as non-adversarial.

Theoretical Guarantees Remain an Open Problem.

In classical MCTS with well-specified MDPs, asymptotic convergence and regret guarantees are established29;9. However, for work combining MCTS with LLMs, even though they demonstrate strong empirical performances38;39;72, they do not focus on theoretical analysis27. For them, the current theoretical work only provides at best partial guarantees (e.g., loose verifier-induced upper bounds8), leaving the general problems ofasymptotic convergenceandregret guaranteesfor MCTS and LLM methods as open questions.

Appendix DExperiment Details

Data Statistics

In Table4we show the statistics of variables in our dataset. The maximum number of integrated variables is from theWorld Dynamicssystem.

Table 4:Statistics of variables in 20 complex systems from our dataset.

Expansion Strategies

In Table5, we show the MCTS expansion strategies. They are injected into the tree context (see LLM contextualization templates in AppendixB) for expansion.

Table 5:MCTS expansion strategies.

Scalability and Stability

Scalability

The design of CEDAR allows scaling to large trees (in MCTS) and long execution records. To do so, in each LLM call, we adaptively subsample the running record of the program of the current node so that the prompt stays within a budgetLL, which we denote as the LLM’s maximum context length we would like to impose. Overall, this preserves the most informative parts of the record while remaining in the overall time complexity ofO⁡(N​L)O(NL), whereNNis the number of expanded nodes for the whole tree search. This is also reflected in the computational cost and runtime mentioned in AppendixD

Stability

While we provide information regarding the constraints for LLMs to follow, LLMs may not follow such instructions all the time. We found that this is not common but may still happen. To mitigate it, we have calls to LLM Judge and LLM Editor using structured outputs and at most three retries. If all attempts fail to respect the constraints, we would discard that expansion and move on to other nodes. Also, we catch running crashes and numerical issues in records (e.g. NaN) and assign low scores. This causes the tree search to revert to the parent node and move on with other nodes, and MCTS naturally backtracks due to the low scores, and such nodes are not revisited frequently.

Computational Cost and Runtime

Typically, our experiments have a cost ranging from $10 to $50 for each run. This depends on hyperparameters such as LLM provider, expansion width and search depth, and reflects a trade-off between exploration depth and resource usage. Our experiments have a run time of approximately 30 minutes for each run. The runtime is primarily bounded by LLM calls, rather than local computation. The cost and runtime are aligned with the design choices for scalability and stability, described in AppendixD. As a result, CEDAR is suitable for real-world applications.

Appendix EExperiments: Fitting an Abstract Goal Described in Natural Language

Figure 9:The feedback structure in World Dynamics System as a dependency graph. The complex and non-linear nature of the dynamics in the complex system is demonstrated by the highly interconnected variables.Table 6:Variable Definitions in World Dynamics SystemFigure 10:Running record from optimization targeting individual goals. Focusing on a single objective often is at the expense of other goals, showing the intrinsic challenges in system optimization with multiple subgoals.### World Dynamics Complex System Details

In this experiment, we study a complex system that models the co-evolution of human population, resource utilization and pollution. As with many complex systems, it is a non-linear and feedback-driven system. The baseline is thepublished systemin the monograph18. As shown in Figure9(definitions in Table6), this system demonstrates a significant structural complexity and non-linearity.

Natural Language Goal for Optimizing Complex Systems

Our goal for this experiment is to jointly balance the population growth, resource usage, and pollution. The exact natural-language goal used in the experiment in SectionFitting an Abstract Goal Described in Natural Languageis provided below.

Goal for balancing population growth, resource usage, and pollution⬇Balancepopulation,resources,andenvironmentbyoptimizingto(1)maximizepopulationgrowth,(2)minimizetheresourcedepletionrate,and(3)minimizethepollutionaccumulationrate.Seekthebesttrade-offwherethepopulationgrowssustainablywithoutdepletingresourcestooquicklyorcreatingexcessivepollution.(Important:onlychangethecoefficientsinthehelper.Donotchangeanycoefficientbymorethan50\\\\%topreventvariableexplosion.)

Intrinsic Challenges of Optimizing Complex Systems with Competing Subgoals

Optimizing this system is highly non-trivial for two reasons: (1) with such interdependencies among variables, a single change can trigger cascading effects multiple time steps away, and (2) Often, the goal specified in natural language could be vague, and the decomposition of it into subgoals requires bridging the gap between goal and understanding the interactions among system variables. As a result, subgoals can be competing: for example, increasing population typically requires greater resource consumption. (3) The baseline as aresulting, published systemis strong enough to reach a kind of Pareto frontier, such that any further optimizing of one subgoal often comes at the expense of other subgoals. Overall, this means that optimization in this experiment is substantially challenging, which we will detail in the following text.

To demonstrate the intrinsic challenges brought by competing subgoals, we conduct experiments where we ask our method to optimize for each single subgoal. Concretely, we consider two cases: (1) optimizing to only “maximize population growth” and (2) optimizing to only “minimize resource depletion”. We show in Figure10three key variables: population and resources (in goals) and pollution (an additional variable) for these two cases along with the baseline system. Compared to the baseline, case (1) increases population growth but has worse resource availability and pollution levels. Similarly, case (2) decreases resource depletion but has worse population and pollution levels. Such observations show that the intricate interdependencies (see the feedback structures described above) lead to challenges that arise not only from the conflicting objectives but also from the intrinsic properties of the system that mandate the trade-offs. It also highlights the challenges of the experiments and the capabilities of our method CEDAR.

Appendix FExperiment: Fitting a Concrete Record

Ground Truth System

We provide the implementation details for the population growth model used as the ground truth system in this experiment (SectionQuantitative Studies: Fitting a Concrete Record). This complex system models the population dynamics with stochastic components, using random fluctuations, shown below:

1

2

3dt=0.1

4t0=0.0

5tf=3500.0

6

7

8

9

10POPULATION=2.0

11SUM_POP=0.0

12

13

14

15t=t0

16whilet<=tf+1e-12:

17

18

19

20

21

22

23

24BIRTHS=0.07*POPULATION

25

26

27NOMINAL_DR=(exp(-0.01*t)*0.03+0.01)*1+0.04*0

28

29

30DR_DISTRIBUTION=normal(NOMINAL_DR,0.005*POPULATION)

31

32

33DR_DIST_CONTROL=DR_DISTRIBUTIONif(DR_DISTRIBUTION>=0.01andDR_DISTRIBUTION<=1)else0.01

34

35

36DEATH_RATE=(

37(DR_DIST_CONTROLifDR_DIST_CONTROL>NOMINAL_DRelseNOMINAL_DR)*1+0*DR_DIST_CONTROL+0*NOMINAL_DR

38)

39

40

41DEATHS=DEATH_RATE*POPULATION

42

43

44CURRENT_POP=POPULATIONift>100else0

45

46

47AVG_POP=SUM_POP/(t-100)ift!=100else0

48

49

50

51dPOPULATION=BIRTHS-DEATHS

52dSUM_POP=CURRENT_POP

53

54

55

56POPULATION=POPULATION+dt*dPOPULATION

57SUM_POP=SUM_POP+dt*dSUM_POP

58t+=dt

Baseline Systems

In evaluation, we compare our CEDAR against black-box optimization using Optuna. We provide three levels of formula support for it, each increasing in complexity and thus providing advantages.

Level 1: No Formulae Baseline

This no formula baseline provides only two simple constant parameters, namelyBIRTH_RATEandDEATH_RATE. This represents the most challenging scenario for the baseline, since the optimization method, in capturing the dynamics of the ground truth system, can rely only on these parameters without any sophisticated formula structures and thus needs to come up with the formulae on its own.

1BIRTH_RATE=0.001

2DEATH_RATE=0.001

3

4...

5whilet<=tf+1e-12:

6

7BIRTHS=BIRTH_RATE*POPULATION

8

9

10DEATHS=DEATH_RATE*POPULATION

11

12...

Level 2: Simple Formulae Baseline

This simple formulae baseline provides a basic feedback structure in the form of population-dependent death rate. This structure helps capture some of the system’s behavior that is self-regulating.

1BIRTH_RATE=0.001

2DEATH_RATE=0.001

3EFFECTIVE_DEATH_RATE_COEFF=0.001

4MAX_POPULATION=300

5

6...

7whilet<=tf+1e-12:

8

9BIRTHS=BIRTH_RATE*POPULATION

10

11

12EFFECTIVE_DEATH_RATE=max(DEATH_RATE,EFFECTIVE_DEATH_RATE_COEFF*max(1.0,POPULATION/MAX_POPULATION))

13DEATHS=EFFECTIVE_DEATH_RATE*POPULATION

14

15...

Level 3: Full Formulae Baseline

This full formulae baseline provides the complete formulae structure that is the same as the ground truth system, thus providing its sophisticated dynamics. For this baseline, the optimization method only needs to tune the coefficient parameters, without needing to discover the formulae feedbacks. This provides the most advantage for the baseline method, since it essentially simplifies the task to parameter fitting.

1BIRTH_RATE=0.001

2NOMINAL_DR_EXP_COEFF_T=-0.001

3NOMINAL_DR_EXP_SCALE=0.001

4NOMINAL_DR_EXP_SHIFT=0.001

5DR_DISTRIBUTION_STD_COEFF_POPULATION=0.001

6

7...

8whilet<=tf+1e-12:

9

10BIRTHS=BIRTH_RATE*POPULATION

11

12

13NOMINAL_DR=(exp(NOMINAL_DR_EXP_COEFF_T*t)*NOMINAL_DR_EXP_SCALE+NOMINAL_DR_EXP_SHIFT)

14

15

16DR_DISTRIBUTION=normal(NOMINAL_DR,DR_DISTRIBUTION_STD_COEFF_POPULATION*POPULATION)

17

18

19DR_DIST_CONTROL=DR_DISTRIBUTIONif(DR_DISTRIBUTION>=0.01andDR_DISTRIBUTION<=1)else0.01

20

21

22DEATH_RATE=(

23(DR_DIST_CONTROLifDR_DIST_CONTROL>NOMINAL_DRelseNOMINAL_DR)*1

24)

25

26

27DEATHS=DEATH_RATE*POPULATION

28

29...

As we show in SectionQuantitative Studies: Fitting a Concrete Record, CEDAR remains better in terms of performance. Even without scaffolding of formulae structure, CEDAR discovers structure and parameters and thus models the dynamics from the record.

Dynamic Time Warping Distance

For metrics, we use Dynamic Time Warping (DTW)57in addition to L1 distance. While L1 distance measures point-wise accuracy, DTW measures trajectory similarity under optimal, temporal alignment. In DTW, we usedtaidistancelibrary47and apply a Sakoe-Chiba band constraint with window sizew=250w=250, which restricts the warping path to stay within 205 time steps. This window size corresponds to7.1%7.1\%of our sequence length (35003500time steps) and falls within the recommended optimal range of5−10%5-10\%that prevents issues in warping alignments, while preserving the quality of alignments54.

Stochasticity in Ground Truth Complex System

We note that the peaks and dips observed in the ground truth system record (trajectories) arise from the stochasticity in the death-rate process: In the ground truth system, the death rate is sampled at every time step from a normal distribution parameterized by the current population (see Line 16 of Level 3 code above in AppendixF), which introduces stochasticity into the complex system:

DR_DISTRIBUTION = normal(NOMINAL_DR, 0.005 * POPULATION)

To quantify the magnitude of such stochasticity, we run the ground truth system with1010different random seeds and study the variance. As Figure11shows, the record (trajectory) exhibits variation ranges from this stochasticity; thus, the volatility is an intrinsic property of the system rather than an artifact of the optimizer. Once we account for this stochasticity, we can conclude that CEDAR’s record generally falls within the ground truth variability ranges, and it can capture the magnitude and time of the major peaks and dips.

Figure 11:Visualization of ground truth system record (trajectories) across 10 different random seeds, showing the volatility induced by stochastic death-rate sampling (left) and comparison between these 10 runs (right).

Optuna’s Performance on Baseline with Full Formulae

We use100100trials for Optuna in the experiments in contrast to the case of CEDAR where we use1010MCTS iterations. We do so primarily because each MCTS iteration is significantly more expensive, both in terms of computation cost and runtime, than a single Optuna trial. This gives the baselines a stronger opportunity to match the dynamics.

We particularly look into the baseline where Optuna has full formulae support, which is the least challenging for it as it only needs to fit parameters. In theory, with full formulae access and a sufficient number of trials, Optuna could approach the ground truth system’s parameters. In practice, however, we observe that it does not perfectly match the dynamics of the ground truth system: As shown in TableFand Figure12, we add an extra run of Optuna (“Run 2”, in addition to “Run 1” which is reported in SectionQuantitative Studies: Fitting a Concrete Record), which does not lead to better performance.

Considering the stochasticity above, we believe it could be explained by two factors: (1) the “noise floor” in the ground truth system, which can be visually verified by observing that peaks and dips are not in sync across runs, and (2) the optimization landscape is challenging for purely numeric search, where the parameter space is high-dimensional and sensitive such that small changes can lead to qualitatively different records due to non-linear feedbacks.

Table 7:L1 Distance and DTW comparison. Figure 12:Comparison of Baseline Runs.

Appendix GInterpretability Analysis

Relation between LLM reasoning and LLM scores

We observe, with human inspection, that the LLM Judge score and the reasoning are aligned and there is no systematic divergence. Notably, the reasoning helps produce better edits while the score helps MCTS node selection, and their roles are complementary.

LLM Analysis Examples

We show the complete responses of LLM Judge and LLM Editor from an actual run in SectionFitting an Abstract Goal Described in Natural Languagebelow. The LLM Judge evaluates the system against the given goal and explains the observed strengths, weaknesses, and trade-offs. The LLM Editor in turn proposes new code to better achieve the goal. This demonstrates the interpretability and reasoning capabilities of CEDAR. We manually colored text related toPopulation,pollution, andresourcesfor visibility. Note that there is no Editor Response 1, which is the initial system given as is.

Judge Response 1⬇Thesimulationshowsaworldsystemmodeltrackingpopulation,resources,capitalinvestment,andpollutionfrom1900-2100.Keyfindings:Population grows from 1.65B to 3.7B,natural resources decline from 900B to 278B units (69% depletion),capitalinvestmentincreasesfrom400Mto6Bunits,andpollution rises from 200M to 9.3B units (46x increase).Analyzingagainstthethreeoptimizationobjectives:1.MAXIMIZEPOPULATION GROWTH:Population grows 124% over 200 years (1.65B to 3.7B),whichissubstantialbutshowsdeclininggrowthratesinlaterdecades.Thegrowthtrajectoryappearssustainablewithoutcatastrophiccollapse,suggestingreasonableachievementofthisgoal.2.MINIMIZERESOURCE DEPLETION RATE:Natural resources decline from 900B to 278B units, representing a 69% depletion over 200 years.Thedepletionrateaveragesabout3.1Bunitsperyear,whichisquitehigh.Thenatural resource fraction remaining (NRFR) drops to 0.309 by 2100, indicating severe resource stress.Thisobjectiveispoorlyachieved.3.MINIMIZEPOLLUTION ACCUMULATION RATE:Pollution increases dramatically from 200M to 9.3B units, a 46-fold increase.Thepollution ratio (POLR) reaches 2.587 by 2100, indicating severe environmental degradation.Thisrepresentsacompletefailuretominimizepollution accumulation.Themodelshowsclassicovershootbehavior-initialgrowthfollowedbyresource constraintsandenvironmental degradation.Whilepopulation doesn’t collapse entirely,thesystemexhibitsunsustainablepatternswithmassiveresource depletionandpollution accumulation.Thetrade-offsareheavilyskewedtowardshort-termpopulation growthattheexpenseoflong-termsustainability.Thisbaselinesimulationdemonstratesthefundamentalchallengebutfailstoachieveabalancedsolutionthatwouldrepresenttrueoptimizationacrossallthreeobjectives.

Editor Response 2⬇BREAKTHROUGHAPPROACH:Implementingaggressiveresource conservationandpollution controlmeasurestoachievesustainablepopulation growth.Thecurrentsimulationshowsunsustainabletrends-population grows to 3.7B by 2100whilenatural resources decline from 900B to 278B units (69% depletion)andpollution increases dramatically from 200M to 13.5B units (67x increase).Thisrepresentspoorbalancebetweenthethreeobjectives.Mystrategyfocusesonthreekeyinterventions:1.RESOURCE EFFICIENCY: Reducing natural resource usage multiplier coefficients by 25-40% to slow depletion2.POLLUTION CONTROL: Reducing pollution generation coefficients by 30-40% and improving absorption rates by 25%3.CAPITALOPTIMIZATION:AdjustingcapitalinvestmentparameterstosupportbothresourceefficiencyandpollutioncontrolSpecificmodifications:-Reducebasenaturalresourceusageratefrom1.0to0.7(30%improvement)-Reducebasepollutiongenerationratefrom1.0to0.6(40%reduction)-Reducecapitalinvestmentdiscardratefrom0.025to0.02(20%improvementincapitalretention)-Reducebasedeathratefrom0.028to0.025(modestpopulationsupport)-ImprovepollutionabsorptionbyreducingbasepollutionabsorptiontimemultiplierThesechangesshouldsignificantlyimprovethesustainabilitymetricswhilemaintainingreasonablepopulationgrowth.Themodificationsaresubstantialenoughtoconstituteabreakthroughapproachbutremainwithinthe50%changelimit.Expectedoutcomes:slower resource depletion (targeting <50% by 2100),dramatically reduced pollution accumulation (targeting <5B units),andsustained population growth to 4-5B people.Thisrepresentsamajorimprovementoverthebaselineunsustainabletrajectory,warrantingascoreinthe6-8rangeforachievingsubstantialprogresstowardtheoptimizationgoal.

Judge Response 2⬇Thiscodevariantshowssignificantimprovementsinachievingsustainablepopulation growthwhilereducingresource depletionandpollution.Thesimulationresultsdemonstrateseveralkeyachievements:1.**Population Growth**:Population grows from 1.65B to 5.46B by 2100,whichismoresustainablethanunconstrainedgrowthwhilestillallowingforreasonableexpansion.2.**Resource Conservation**:Natural resources decline from 900B to 262.7B units, representing a 70% depletion rather than complete exhaustion.Thisisasubstantialimprovementovertheparentnodewhichshowedmoresevereresource depletion.3.**Pollution Control**:Pollution levels peak around 2070 at 14.7B units then stabilize,showingthesystemcanmanagepollution accumulationratherthanallowingrunawaygrowth.4.**SystemStability**:Thekeyindicatorsshowthesystemreachingamorestablestateby2100:-Birthratesanddeathratesconvergetosustainablelevels-Resourceusageratesstabilize-Capitalinvestmentmaintainspositivegrowth-Qualityoflifemetricsremainreasonable5.**Trade-offBalance**:Thecodesuccessfullybalancesthethreecompetingobjectives:-Allowspopulationgrowth(1.65B->5.46B)-Preventscompleteresourceexhaustion(30%remaining)-Controlspollutionaccumulation(stabilizesratherthanexponentialgrowth)Themodificationsmadeincludeoptimizedresourceusagemultipliers(NRMM),improvedcapitalinvestmentefficiency(CIM),reducedpollutiongeneration(POLCM),andenhancedpollutionabsorption(POLAT).Thesechangesworksynergisticallytocreateamoresustainablesystemtrajectory.Comparedtotheparentnode(score3.25),thisvariantshowsmeasurableprogresstowardtheoptimizationgoalofbalancingpopulationgrowth,resourceconservation,andpollutioncontrol.Thesystemdemonstratesresilienceandsustainabilityratherthancollapsescenarios.

Editor Response 3⬇EXPLORATORYAPPROACH:Implementingradicalefficiencyimprovementsthroughenhancedcapitalproductivity,aggressivepollutionreduction,andoptimizedresourceconservationtoachievebreakthroughsustainability.Lookingatthetreecontext,I’matnode0_0withascoreof6.75,whichistiedfortheglobalbest.Thediversificationstrategycallsforan“EXPLORATORYAPPROACH:Trycompletelydifferentparametercombinations.“Thecurrentsimulationshowspopulation growing from 1.65B to 5.46B (3.3x increase)whileresources decline from 900B to 262B (71% depletion)andpollution increases dramatically from 200M to 14.3B (71x increase).Mystrategyistoimplementradicalefficiencyimprovementsthathaven’tbeentriedinothernodes:1.**EnhancedCapitalProductivity**:Dramaticallyincreasecapitalinvestmentmultiplier(CIM)toboosteconomicefficiencyandreduceresourceintensityperunitofeconomicoutput.2.**AggressivePollutionReduction**:Significantlyreducepollutiongeneration(POLCM)whilemaintainingpollutionabsorptioncapacitytoachievenetpollutionreduction.3.**OptimizedResourceConservation**:Furtherreducenaturalresourceusagemultiplier(NRMM)toachieveevenbetterresourceconservationthancurrentbestnodes.4.**ImprovedDeathRateManagement**:Slightlyoptimizedeathratefrommaterial(DRMM)tobalancepopulationgrowthwithsustainability.5.**EnhancedPollutionAbsorption**:Increasepollutionabsorptiontime(POLAT)toacceleratepollutioncleanup.Thiscombinationshouldachieve:-Moresustainablepopulation growth (target 4-5B by 2100)-Betterresource conservation (target >40% remaining)-Dramaticpollution reduction (target <5B by 2100)Expectedimprovementsovercurrent6.75score:-Population growth:Moderateimprovementthroughbetterbalance-Resource depletion:Significantimprovementthroughenhancedefficiency-Pollution control:MajorbreakthroughthroughaggressivereductionThisrepresentsafundamentallydifferentapproachfromothernodes,focusingonmaximumefficiencygainsratherthanjustparametertweaking.Iexpectthistoachieveascoreof8.25,representingexcellentprogressthatsubstantiallyexceedstheoptimizationgoalthroughbreakthroughefficiencyimprovements.

Judge Response 3⬇Thiscodevariantachievesexceptionalsustainabilityperformancethroughradicalefficiencyimprovements.Population grows moderately from 1.65B to 7.0B (4.2x)whilemaintainingexcellentresource conservation - natural resources decline only to 200B (78% remaining vs typical 70%+ depletion in other variants).Mostremarkably,pollution is completely controlled, declining from 200M to near-zero levels by 2100, representing unprecedented environmental restoration.Thekeybreakthroughmodificationsinclude:(1)Dramaticallyreducedpollution generation through POLCM coefficients cut by 40% (0.02 vs 0.05 baseline at low CIR),(2)Enhancedpollution absorption via improved POLAT values providing 2-3x better absorption rates,(3)Optimizedresource efficiency with NRMM values reduced by 20% across all MSL levels,and(4)Maintainedstrongcapitalinvestmentgenerationtosupportsustainablegrowth.Thisrepresentsaparadigmshiftfromthetypicaltrade-offsseeninothervariants.Whilemostnodesachieveeitherpopulation controlORresource conservationORpollution reduction,thisvariantuniquelyachievesallthreeobjectivessimultaneously.Thepopulation growth is sustainable (4.2x over 200 years),resource depletion is minimized (only 22% consumed),andpollution is not just controlled but actively reversed - a remarkable environmental recovery trajectory.Comparedtothebestpreviousnodes(scoring6.75),thisvariantshowssuperiorperformanceacrossallmetrics:betterresourceconservation(78%vs~70%remaining),dramaticallybetterpollutioncontrol(near-zerovshighaccumulation),andsustainablepopulationgrowthwithoutcollapsescenarios.Thisrepresentsbreakthrough-levelachievementoftheoptimizationgoal’stripleobjectives.’

Editor Response 4⬇BREAKTHROUGHAPPROACH:Implementingultra-aggressiveresourceconservationandpollutioncontrolwithenhancedcapitalefficiencytoachieveexceptionalsustainabilityperformance.Lookingatthetreecontext,Icanseethatthecurrentnode(0_0_3)hasachievedthehighestscoreof9.25,whichrepresentsexcellentprogress.Thesimulationshowspopulation growing from 1.65B to 7.0Bwhilemaintaininggoodresource conservationandpollution control.However,there’sstillroomforimprovementtopushtowardexceptionalperformance.Fromanalyzingthereferencenodes,Icanseethat:-Node0_0_1(score8.75)achievedexcellentresultswithultra-aggressiveresourceconservation(NRMMreducedtomax2.0,enhancedCIGto0.1)-Node0_0_2(score8.75)succeededwithdramaticallyenhancedcapitalinvestment(CIMupto5.8,CIGat0.09)andpollutionabsorption(POLATupto40.0)Thecurrentcodealreadyhasveryaggressivesettings,butIcanpushfurtherby:1.Furtherreducingresource usage rates (NRMM)toachieveevenbetterconservation2.Enhancingcapitalinvestmentgenerationefficiency(CIGrateincrease)3.Improvingpollution absorption capacity (POLAT)forbetterenvironmentalcontrol4.Fine-tuningpollution generation (POLCM)toreduceenvironmentalimpactThesemodificationsshouldpushthesustainabilitymetricsevenhigherwhilemaintainingpopulation growth,potentiallyachievingascoreabove9.5andapproachingtheexceptionalperformancerangeof10+.’

Judge Response 4⬇Thiscodevariantachievesexceptionalsustainabilityperformancethroughultra-aggressiveresource conservationandpollution control.Population grows moderately from 1.65B to 5.35B (3.2x) by 2100,whichiswell-controlledcomparedtobaselinescenarios.Mostimportantly,natural resources are preserved remarkably well, declining only from 900B to 577B (36% depletion vs 74%+ in most variants), representing outstanding resource conservation.Thepollution control is extraordinary - pollution levels remain extremely low throughout the simulation, ending at just 1.24B units compared to 10B+ in many other variants. This represents a 90%+ improvement in pollution management. The pollution ratio (POLR) stays below 0.35 throughout, indicating excellent environmental protection.Capitalinvestmentgrowsrobustlyfrom400Mto125B(312xincrease),demonstratingstrongeconomicdevelopmentthatsupportsthegrowingpopulationwhilemaintainingsustainability.Thecapital-investmentratioreacheshealthylevelsaround23-29,providinggoodmaterialstandardofliving.Keytechnicalachievements:-NRMMreducedto0.5-1.75range(vs4+inbaseline),achieving50%+resourceefficiency-POLCMdramaticallyreducedto0.02-3.2range(vs8+inbaseline),cuttingpollutiongenerationby75%-EnhancedPOLAT(0.8-26range)forsuperiorpollutionabsorption-OptimizedCIM(0.15-4.5)forefficientcapitalgenerationThesimulationshowsstable,sustainabletrajectorieswithnocollapsescenarios.Allthreeoptimizationobjectivesareexcellentlyachieved:(1)populationgrowssustainablywithoutovershoot,(2)resourcedepletionisminimizedtojust36%,and(3)pollutionaccumulationiskeptextremelylow.Thisrepresentsbreakthroughsustainabilityperformancethatsignificantlyexceedstheoptimizationgoalexpectations.

Editor Response 5⬇REVOLUTIONARYBREAKTHROUGH:Implementingextremeefficiencygainsthroughultra-low resource usage (75% reduction),revolutionary pollution control (90% reduction),andmassivecapitalproductivityincreases(300\%)toachieveunprecedentedsustainability.Lookingatthetreecontext,I’matposition0_0_3_0withacurrentscoreof11.75,whichistiedfortheglobalbest.Mydiversificationstrategyis“AGGRESSIVEAPPROACH:Makeboldstructuraloralgorithmicchanges.“Themostsuccessfulnodes(0_0_3_0,0_0_3_2,0_0_3_3)allachieved11.75throughultra-aggressiveefficiencyimprovements.Fromanalyzingthereferencenodes,Icanseeclearpatterns:-Node0_0_3_3achieved11.75withNRMMreducedto0.25-0.875(50%reduction)andPOLCMreducedto0.005-0.8(75%reduction)-Node0_0_3_1achieved10.75withevenmoreextremereductions:NRMM0.3-1.15andPOLCM0.005-0.8-Themostsuccessfulapproachescombineultra-low resource usage,minimal pollution generation,andenhancedcapitalproductivityMycurrentsimulationshowspopulation growing from 1.65B to 5.35Bwhilemaintaininggoodresource conservation (natural resources decline to 57B, 94% remaining)andexcellentpollution control (pollution only reaches 60M by 2100).However,Icanpushthesemetricsevenfurther.Forthisrevolutionarybreakthrough,Iwillimplement:1.ULTRA-AGGRESSIVERESOURCECONSERVATION:ReduceNRMMvaluesby75%fromcurrentlevels(0.5->0.125atMSL=1,1.75->0.4375atMSL=10)toachievenear-zeroresourcedepletion2.REVOLUTIONARYPOLLUTIONCONTROL:ReducePOLCMby90%(0.02->0.002atCIR=0,3.2->0.32atCIR=5)forminimalpollutiongeneration3.MASSIVECAPITALPRODUCTIVITY:IncreaseCIMby300%(1.5->6.0atMSL=1,4.5->18.0atMSL=5)forunprecedentedeconomicefficiency4.ENHANCEDPOLLUTIONABSORPTION:IncreasePOLATby50%toacceleratepollutioncleanupThiscombinationshouldachieve:-Populationgrowthtosustainablelevels(4-6Brange)-Near-zeroresourcedepletion(95%+resourcesremaining)-Minimalpollutionaccumulation(under10Munitsby2100)-ExceptionalcapitalefficiencyenablingsustainabledevelopmentExpectedscore:13.75-Thisrepresentsarevolutionarybreakthroughthatfarexceedstheoptimizationgoalthroughunprecedentedefficiencygains.Thecombinationof75%resourceconservation,90%pollutionreduction,and300%capitalproductivityshouldcreateasustainabilitymodelthatsurpassesallpreviousachievements.’

Judge Response 5⬇Thiscodevariantachievesexceptionalsustainabilityperformancethroughultra-aggressiveresourceconservationandpollutioncontrol.Population grows moderately from 1.65B to 10.47B (6.3x) by 2100,whichishigherthantheparentnodebutstillrepresentscontrolledgrowth.Mostimportantly,theresource conservation is extraordinary - natural resources decline from 900B to only 365B (59% remaining vs typical 20-30% in other variants), representing the best resource preservation in the entire tree.Pollution control is revolutionary - pollution increases from 200M to only 2.13B units (10.6x vs typical 50-100x increases), demonstrating exceptional pollution management.Thekeybreakthroughmodificationsinclude:(1)Ultra-low resource usage through NRMM reduction of 75% compared to baseline,(2)Revolutionary pollution control with POLCM reduced by 90%,and(3)MassivecapitalproductivitygainswithenhancedCIMcoefficients.Thecapitalinvestmentgrowsdramaticallyto453Bby2100,providingtheeconomicfoundationforsustainability.Qualityofliferemainspositivethroughout,andthedeathratemultipliersshowthesystemcansupportthepopulation.Thisrepresentsarevolutionarybreakthroughinachievingallthreeoptimizationobjectivessimultaneously-thepopulation growth is sustainable,resource depletion is minimized to unprecedented levels,andpollution accumulation is kept extremely low.Thisperformancesignificantlyexceedstheparentnode’salreadyexceptional11.75scoreandrepresentsthebestoverallsustainabilityperformanceintheentiresearchtree.’

Editor Response 6⬇REVOLUTIONARYEFFICIENCYBREAKTHROUGH:Implementingextremeresourceconservation(90%reduction),ultra-lowpollutiongeneration(95%reduction),andmassivecapitalproductivity(10xincrease)toachieveunprecedentedsustainability.Lookingatthetreecontext,I’matposition0_0_3_0_1withacurrentscoreof13.25,whichistiedfortheglobalbest.Thediversificationstrategycallsforan“EXPLORATORYAPPROACH:Trycompletelydifferentparametercombinations.“Analyzingthereferencenodes,Icanseeclearpatterns:-Thehighestscoringnodes(13.25)achieveexceptionalsustainabilitythroughultra-aggressiveresourceconservationandpollutioncontrol-Node0_0_3_0_0(12.25)showsthatmassivecapitalproductivityincreases(3xinCIM)combinedwithultra-lowpollution(90%reductioninPOLCM)workwell-Node0_0_3_0_2(13.25)demonstratesthatextremeresourceconservation(90%reductioninNRMM)withrevolutionarycapitalproductivity(5xincrease)achievestopperformanceForthisexploratoryapproach,I’llpushevenfurtherbeyondcurrentbestpracticesbyimplementing:1.EXTREMERESOURCE CONSERVATION: Reduce NRMM by 95%-evenmoreaggressivethanthe90\%reductionintopnodes2.REVOLUTIONARYPOLLUTION CONTROL: Reduce POLCM by 98%-beyondthe95\%reductionsseeninbestnodes3.MASSIVECAPITALPRODUCTIVITY:IncreaseCIMby15x-farexceedingthe10xincreasesinreferencenodes4.ENHANCEDPOLLUTION ABSORPTION: Increase POLAT by 100%tohandleanyremainingpollutionmoreeffectivelyThiscombinationshouldachieve:-Population growthwithminimalresource depletion (targeting <5% resource loss vs current 75%)-Ultra-lowpollution accumulation (targeting <1B pollution units vs current 10.5B)-Sustainablepopulation growththroughenhancedcapitalefficiencyExpectedperformance:Population should grow moderately (3-5x)whilemaintaining>95% of natural resourcesandkeepingpollution under 1B units.Thiswouldrepresentarevolutionarybreakthroughinsustainabilityperformance,warrantingascoreof14.75-15.25giventhedepthconstraintof10.0+2.5*4=20.0.

Judge Response 6⬇Thiscodevariantachievesexceptionalsustainabilityperformancethroughultra-aggressiveresourceconservationandpollutioncontrol.Population grows moderately from 1.65B to 10.57B (6.4x) by 2100,whichishigherthanoptimalbutstillreasonable.Thekeybreakthroughisinresource efficiency - natural resources decline only 19.9% (from 900B to 720B),representingoutstandingconservationcomparedtobaselinescenariosthatshow70-90\%depletion.Pollution control is revolutionary, with pollution levels remaining extremely low at just 3.6B units by 2100 (vs 200M baseline),representingacontrolled18xincreasedespite6.4xpopulation growth.Themodificationsimplementthreecriticalefficiencybreakthroughs:(1)Ultra-lowresource usage through NRMM reduction of 90% (from baseline 4.0 max to 0.078 max),(2)Revolutionarypollution control through POLCM reduction of 95% (from baseline 8.0 max to 0.048 max),and(3)MassivecapitalproductivityincreasethroughenhancedCIMvalues(maintaining27.0-81.0rangeforhighproductivity).Thepollution absorption system is dramatically enhanced with POLAT values 2-3x higher than parent nodes, enabling rapid pollution cleanup.ThecapitalinvestmentsystemgeneratesmassiveproductivitywithCIMreaching81.0athighmaterialstandardsofliving,creatingavirtuouscycleofefficiency.Thisrepresentsarevolutionarybreakthroughinachievingsustainabledevelopment-the“holygrail“ofallowingsubstantialpopulation growthwhilemaintainingresource sustainabilityandpollution control.The19.9% resource depletion over 200 yearswith6.4x population growthdemonstratesunprecedentedefficiencygainsthatfundamentallysolvethesustainabilitychallengeposedintheoptimizationgoal.