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A 3D Characterization Framework for Intelligent Sequential Decision Making

Sadig Gojayev, Carolina Fortuna

arXiv:2610.11696v1cs.AI

TL;DR

Comparisons of sequential decision-making methods often lack a unified link among formulation, human guidance, skill, and computational cost. The paper applies a three-dimensional characterization framework to representative graph-based, reinforcement-learning, and LLM-based Tower of Hanoi methods. It finds that weakly constrained LLM action spaces shift complexity to inference-time verification and bring substantially higher runtime and memory costs than graph-based and reinforcement-learning approaches.

  • Problem

    Comparisons across sequential-decision paradigms often lack an explicit connection among method formulation, human-imposed guidance, decision-making skill, and computational cost.

    Source: §I · p. 2, §II · p. 2

  • Method

    The paper characterizes methods by their MDP formulation, human prior across design stages, and skill–cost trade-offs, comparing representative graph-based, reinforcement-learning, and LLM-based Tower of Hanoi solvers.

    Source: §I · p. 2

  • Results

    Weakly constrained LLM action spaces shift complexity to inference-time verification, with substantially higher runtime and memory costs than graph-based and reinforcement-learning approaches under the same task scaling.

    Source: §I · p. 2

  • Takeaways & Limitations

    The comparison highlights inference-time verification as a major locus of complexity for the LLM-based methods studied.

    Source: §I · p. 2

Abstract

from arXiv · show

Puzzles are widely used to evaluate the reasoning capabilities of artificial intelligence (AI) systems for sequential decision making, yet approaches originating from different paradigms are rarely compared under unified conditions. To address this gap, we introduce a three-dimensional characterization framework that enables the analysts of AI methods by 1) projecting them to the Markov decision process (MDP) sequential decision making formalism, 2) degree of autonomy through human prior ranking of their designs and, 3) skill and computational cost. Using this framework, we analyze how representative graph-based, reinforcement learning, and large language model (LLM)-based approaches differ in their design choices and performance characteristics, instantiated respectively by Neurosolver, forward-backward reinforcement learning (FBRL), and automated thought-of-search (AutoToS), including a double-agent extension of thought-of-search (DA-ToS). The analysis relies on the Tower of Hanoi puzzle that provides a controlled benchmark with well-defined rules and scalable complexity, enabling consistent comparison across increasing problem sizes. The 3D characterization reveals that LLM-based methods, due to their weakly constrained action-space design, shift complexity from architecture to inference-time verification, leading to substantially higher memory and runtime costs than Neurosolver and FBRL.

I. INTRODUCTION

The paper addresses the lack of unified comparisons of sequential decision-making methods that connect their formulations, human guidance, skill, and computational cost. It introduces a three-dimensional framework and applies it to representative graph-based, reinforcement-learning, and LLM-based Tower of Hanoi solvers.

I. INTRODUCTION

  • The Tower of Hanoi provides a sequential-decision puzzle in which each move changes the state, subject to the rule that only a smaller disc may sit on a larger disc.The puzzle starts with all discs on the first peg and aims to move them to the last peg.Source: §I · p. 1
  • The framework jointly characterizes methods by their projection to the MDP formalism, human prior across design stages, and skill–cost trade-offs.The human-prior rubric assigns High, Medium, or Low levels across representation/mechanism, knowledge/learning, and guidance/decision/validation.Source: §I · p. 2
  • The comparison instantiates graph-based solving with Neurosolver, reinforcement learning with FBRL, and LLM planning with AutoToS, including DA-ToS's modular code generation and repair.The methods are compared using the Tower of Hanoi puzzle.Source: §I · p. 2
  • Weakly constrained LLM action spaces shift complexity to inference-time verification, yielding substantially higher runtime and memory costs than graph-based and reinforcement-learning approaches under the same task scaling.The compared methods include Neurosolver, FBRL, AutoToS, and DA-ToS.Source: §I · p. 2

II. RELATED WORK

Prior comparisons largely emphasize outcome metrics, while leaving links among method formulation, human guidance, decision-making skill, and computational cost underdeveloped. The proposed framework compares methods through MDP components, human-prior levels, and skill–cost trade-offs.

II. RELATED WORK

  • Existing comparisons largely emphasize outcome measures and lack unified analysis connecting representation, decision mechanisms, human guidance, and computational cost across paradigms.Prior theoretical frameworks formulate methods through MDP optimization but do not connect that formulation to decision-making skill and computational cost.Source: §II · p. 2

III. PROPOSED 3D CHARACTERIZATION FRAMEWORK

A. Projecting through Markov Decision Process

  • An MDP represents sequential decision making through states, actions, transitions, rewards, and goal states.The transition function gives the probability of reaching a next state after an action in the current state.Source: A. Projecting through Markov Decision… · pp. 2–3

B. Human Prior in AI for Sequential Decision Making

The framework assesses human know-how embedded during architecture design, training, and inference, alongside decision-making skill and computational cost. It treats performance as a sequence-level concern and focuses cost measurement on time and memory.

B. Human Prior in AI for Sequential Decision Making

  • Human prior measures how much human know-how is embedded during architecture design, training, and inference.These stages are further divided into representation and mechanism; knowledge input and learning; and guidance, decision, and validation.Source: B. Human Prior in AI for Sequential… · p. 3
  • The architecture-design stage concerns model structure before learning or execution, while the training stage concerns skill acquisition and human shaping of that process.The inference stage concerns model behavior after training, including guidance, decision production, and output checks.Source: B. Human Prior in AI for Sequential… · p. 3

C. Performance and Computational Cost

  • The skill–cost dimension assesses decision-making performance against the computational resources required to achieve it.The framework uses time and memory as its primary cost indicators and also provides theoretical memory estimates.Source: C. Performance and Computational Cost · p. 3
  • Performance includes single-step metrics as well as measures characterizing the entire decision sequence; computational cost can include memory, operations, time, and energy.The paper focuses its empirical cost measurements on time and memory.Source: C. Performance and Computational Cost · p. 3

IV. METHODS

The methods section selects LLM-based agents, a graph-based method, and an RL approach for analysis. AutoToS uses LLM-generated functions that are iteratively checked and refined with search.

IV. METHODS

  • The study analyzes LLM-based agents, a graph-based learning method, and a reinforcement-learning approach.Source: §IV · p. 3

1) Thought-of-Search (ToS):

  • AutoToS uses an LLM to generate a successor function and a goal predicate for search-based planning.Source: 1) Thought-of-Search (ToS) · pp. 3–4
  • Table III is identified as presenting levels of human prior at the inference stage.Source: Table III · p. 4
  • Breadth-First Search checks generated functions for correctness and optimality, and detected errors prompt regeneration until soundness and completeness are achieved.Source: 1) Thought-of-Search (ToS) · p. 4

2) DA-ToS:

DA-ToS separates code generation from error repair in a test-driven feedback loop. The section also describes LLM memory estimation, experimental choices, and Neurosolver’s graph-based planning and memory scaling.

2) DA-ToS:

  • DA-ToS separates code generation from repair: a Repair agent interprets compilation errors and test failures and feeds targeted guidance to the Coding agent until tests pass.Source: 2) DA-ToS · p. 4

3) Estimated Memory:

  • LLM inference memory is estimated as model weights plus a KV cache, whose size grows linearly with conversation token count while weight memory remains constant.Source: 3) Estimated Memory · p. 4

4) Experimental Setting for LLM-based Agents:

B. Neurosolver

  • Neurosolver represents puzzle states as graph nodes, propagates activity backward from the goal, then reconstructs a valid path forward from the start.Source: B. Neurosolver · p. 5

1) Estimated Memory:

  • Neurosolver’s estimated memory grows exponentially with disk count, with an additional linear factor from the number of possible moves.Source: 1) Estimated Memory · p. 5

2) Experimental Setting for Neurosolver:

The Neurosolver setup averages ten independent runs after 100 random-exploration iterations per configuration. The section also describes FBRL’s forward/backward learning and decomposes its memory requirements.

2) Experimental Setting for Neurosolver:

C. Forward-Backward Reinforcement Learning

  • FBRL combines real forward trajectories and imagined backward trajectories within a DDQN framework, using prioritized replay to sample transitions with higher temporal-difference errors more often.Source: C. Forward-Backward Reinforcement… · p. 5

1) Estimated Memory:

  • FBRL memory comprises the Q-network, backward model, and replay buffer; the network terms scale roughly linearly with disk count, while the buffer term is independent of it and may dominate for large capacity.Source: 1) Estimated Memory · p. 5

2) Experimental Setting for FBRL:

FBRL uses string-based state representations and a 10,000-transition replay buffer in the reported setup. Its memory estimate uses specified network and buffer parameters, while Table IV compares selected models’ MDP representations.

2) Experimental Setting for FBRL:

  • FBRL encodes each disk’s peg location as a binary tuple and uses a replay buffer with capacity 10,000 transitions.Source: 2) Experimental Setting for FBRL · p. 5
  • The FBRL memory estimate uses n = 4, 12-dimensional inputs and actions, hidden widths 64 and 100, two-layer networks, and 10,000 buffer entries stored as 32-bit floats.Source: 2) Experimental Setting for FBRL · p. 5
  • Table IV compares selected Tower of Hanoi models by their state, action, transition, reward, and goal representations.Source: Table IV · p. 6

V. ANALYSIS FROM THE MDP PERSPECTIVE

The MDP projection makes the methods’ state, action, transition, reward, and goal designs comparable. ToS methods use token-sequence states and probabilistic language-model transitions, while Neurosolver and FBRL encode puzzle states and legal moves directly.

  • The MDP comparison organizes each method by State, Action, Transition, Reward, and Goal.Source: §V · p. 6
  • DA-ToS represents state with the prompt, current code, and repair-agent feedback; each action invokes a language model to generate code or feedback.Source: §V · p. 6
  • AutoToS uses the prompt, candidate code, and verifier feedback as state, with no separate Repair agent; verifier output is appended directly to the state.Source: §V · p. 6
  • Neurosolver encodes each disk’s peg position, permits legal moves, and targets the destination-peg state in 2^n − 1 moves.Source: §V · p. 6
  • FBRL represents disk locations as a 3n-bit string, uses deterministic transitions, and rewards valid moves, penalizes illegal moves, and gives a terminal goal reward.Source: §V · p. 7
  • ToS/LLM methods use a much larger token-sequence state space and probabilistic structures, while Neurosolver and FBRL use graph-encoded states and transitions; ToS uses binary verifier rewards versus continuous rewards in the others.Source: §V · p. 7

VI. ANALYSIS FROM THE HUMAN PRIOR PERSPECTIVE

The human-prior analysis compares architecture, training, and inference-stage guidance across the four methods. ToS methods generally require more human insight than FBRL and Neurosolver, particularly for verification and inference guidance.

  • AutoToS and DA-ToS use implicit, function-based representations with Medium architectural prior, while Neurosolver and FBRL use low-level disk-position encodings with Low prior.Source: §VI · p. 7
  • All four methods have High human prior in their reasoning mechanisms, which are explicitly designed rather than autonomously discovered.Source: §VI · p. 7
  • Training prior varies by method: FBRL has High prior from human-defined rewards and constraints, while Neurosolver has Low prior and explores valid moves without human-provided training data or rewards.Source: §VI · p. 7
  • At inference, AutoToS has High guidance, DA-ToS and Neurosolver Medium guidance, and FBRL Low guidance; decision-making prior is Low for all four.Source: §VI · p. 8
  • AutoToS and DA-ToS use human-defined test cases for explicit validation, whereas validation is not applicable to Neurosolver and FBRL because correctness is implicit in goal satisfaction.Source: §VI · p. 8
  • ToS methods require more human insight overall than FBRL and Neurosolver, which do not require verification after the final decision under their constrained designs.Source: §VI · p. 8

VII. ANALYSIS FROM THE SKILL AND COMPUTATIONAL COST PERSPECTIVE

The cost analysis compares DA-ToS and AutoToS configurations on Tower of Hanoi instances with two to seven disks. DA-ToS incurs higher token and runtime costs, while AutoToS typically converges with fewer attempts and lower overhead.

VII. ANALYSIS FROM THE SKILL AND COMPUTATIONAL COST PERSPECTIVE

  • The evaluation covers three-peg instances with two to seven disks and reports mean ± Median Absolute Deviation across attempts or steps, token use, execution time, and estimated and observed memory.Source: §VII · p. 8

A. LLM-based Agents

  • For LLM-based approaches, reported results combine generation of the Successor and Goal functions; the optimal case requires one step for each function.Source: A. LLM-based Agents · p. 8
  • DA-ToS with DeepSeek+Gemma uses over 17k tokens and about 209 seconds, while DeepSeek+Mistral uses roughly 6.5k tokens and about 32 seconds.The Mistral-based setup also requires fewer steps and has lower run-to-run variance; quantized memory use is 17.73 GB → 15.13 GB for DeepSeek+Gemma and 15.12 GB → 14.00 GB for DeepSeek+Mistral.Source: A. LLM-based Agents · p. 8
  • AutoToS typically converges in one to three attempts, and Qwen and Yi use fewer than 1k tokens on average, finish in about 25–35 seconds, and produce the optimal step size.DeepSeek uses more tokens and refinement cycles; AutoToS has substantially lower overhead than DA-ToS because it uses one LLM without a separate repair agent.Source: A. LLM-based Agents · p. 8

B. Neurosolver

Across the evaluated Tower of Hanoi methods, LLM-based approaches shift complexity toward inference-time verification and have higher memory and runtime costs than graph-based and reinforcement-learning approaches. Neurosolver maintains optimal step counts, while FBRL’s replay buffer dominates its memory use as puzzle size increases.

B. Neurosolver

  • Neurosolver consistently achieves the optimal 2^n − 1 steps, with runtime rising from 12.06 seconds for two disks to 60.40 seconds for seven and memory from 0.005 MB to 0.321 MB.Memory growth reflects the exponential state space; the reported behavior aligns with the theoretical scaling law.Source: B. Neurosolver · pp. 8–9

C. Forward-backward Reinforcement Learning

  • FBRL runtime rises from about 2.4 seconds for two disks to over 31 seconds for four, while replay-buffer storage grows from about 0.32 MB to 0.92 MB and dominates network memory.Its dual exploration combines forward trajectories from the initial state with backward rollouts from the goal.Source: C. Forward-backward Reinforcement… · p. 9
  • The comparison concludes that LLM-based approaches require more memory, whereas Neurosolver scales better in time and memory than FBRL in the reported comparison.Source: C. Forward-backward Reinforcement… · p. 9, §VIII · p. 9

VIII. CONCLUSION

  • LLM-based methods enforce correctness through inference-time verification and correction loops, with substantially higher memory and runtime costs than graph-based and reinforcement-learning methods.The paper presents the framework as a basis for analyzing these trade-offs beyond Tower of Hanoi.Source: §VIII · p. 9, Table VI · p. 9
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