Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation

1 INTRODUCTION

• The three desired dimensions are solution accuracy, the number of LLM inference calls, and the diversity of thought topologies.Flexibility includes chain, tree, and graph structures.Source: §1 · p. 2
• Existing prompting paradigms cannot jointly achieve performance, efficiency, and flexibility: IO and CoT have limited structures, while ToT and GoT incur costly LLM-based thought evaluation.The paper frames this trade-off as a “Penrose triangle” constraint.Source: §1 · p. 2
• XOT uses pretrained reinforcement learning, MCTS, and lightweight policy and value networks to search thoughts, then revises them collaboratively with an LLM.The approach aims to incorporate external domain knowledge and planning capability while minimizing LLM interactions.Source: §1 · p. 2
• Across Game of 24, 8-Puzzle, and Pocket Cube, XOT outperforms existing approaches and provides multiple solutions with few LLM calls.The paper presents these findings as evidence that XOT is an effective thought-generation approach.Source: §1 · p. 2

2 BACKGROUND

• Thoughts are intermediate cognitive processes or outcomes expressed as linguistic prompt sequences, organized into chains, trees, or graphs.Source: §2 · pp. 2–3
• IO prompting provides no intermediate thoughts, while CoT uses a sequential chain and CoT-SC selects among multiple CoT outputs.CoT-SC generates multiple outputs and selects the best result.Source: §2 · p. 3
• ToT searches a tree of thoughts but relies on costly LLM calls to evaluate intermediate steps.Source: §2 · p. 3
• GoT permits graph-like thought structures through aggregation and refinement, yet still requires multiple LLM inference calls for evaluation.Source: §2 · p. 3

3 XOT: EVERYTHING OF THOUGHTS

• XoT pairs MCTS simulations that generate candidate thoughts with LLM revision and grounding, forming a collaborative framework for complex problem-solving.The authors describe this collaboration as supporting performance, efficiency, and flexibility.Source: §3 · p. 3, §3.4 · p. 6

3.1 XOT IN A NUTSHELL

• During task-specific training, MCTS simulations record states, values, and node visits to train policy/value networks; LLMs then review and refine the resulting thoughts.The extracted thoughts are supporting references rather than definitive answers, and LLMs may continue the MCTS search when needed.Source: §3.1 · pp. 3–4

3.2 THOUGHT SEARCHING FORMULATION

• The formulation treats each thought as a state-action pair and complete thought generation as finding a trajectory that maximizes accumulated reward.States represent problem status, actions are one-step solutions, and reward evaluates whether decomposition resolves the original problem.Source: §3.2 · p. 4

3.3 THOUGHTS SEARCHING WITH MCTS

• MCTS selects actions using Q-values, network-predicted priors, and an exploration–exploitation trade-off, then expands leaf states using predicted action probabilities and values.Backpropagation updates action values using estimated values for unexplored nodes and true rewards for terminal nodes.Source: §3.3 · pp. 4–5
• Search simulations produce state, action-probability, and reward-value data that train a combined policy/value network to align its predictions with search outcomes.Training proceeds alongside simulation to improve the network’s thought-searching performance.Source: §3.3 · p. 5

3.4 THOUGHT INFERENCE WITH MCTS

• At inference, MCTS extracts one or multiple trajectories; multiple trajectories can form graph-like structures and are concatenated into a single prompt for one LLM call.For multiple solutions, trajectories are sampled from action visit counts and duplicates are removed.Source: §3.4 · pp. 5–6
• When the LLM detects a thought error, MCTS runs additional simulations from the parent state to revise it, and the process can be repeated.Each solution undergoes revision individually in multi-solution cases.Source: §3.4 · p. 6

4.1.3 RESULTS

• XOT consistently beats baselines; after three revisions, accuracy reaches 90.51% (GPT-3.5) and 85.40% (GPT-4), versus 79.56% and 74.45% after one.The table compares performance on Game of 24.Source: Table 3 · p. 8, §4.1.3 · p. 9
• In multi-solution evaluation, XOT leads MultiAcc despite producing fewer answers than some baselines, using 2.31 GPT-4 LLM calls.XOT’s multi-solution results are reported in Table 4.Source: Table 4 · p. 8, §4.1.3 · p. 9

4.1.1 TASK SETUP

• The Game of 24 dataset contains 1,362 games: 137 were randomly selected for testing and the remaining 1,225 for MCTS policy/value-network training.Games span difficulty levels ranked by human solving time; success is measured across the test games.Source: §4.1.1 · p. 8

4.1.2 BASELINES & XOT SETUP

• Baselines use five in-context examples for IO, three intermediate equations for CoT, and one- or three-branch candidate selection for ToT.GoT selects a thought for merging; CoT-SC uses the majority output from 10 samples. XOT runs 200 simulations per action and 500 during revision.Source: §4.1.2 · p. 8
• For multi-solution evaluation, ToT retains up to three final candidates, GoT selects one to three thoughts, and XOT samples 500 trajectories and keeps its three most-counted thoughts.XOT removes duplicate trajectories before selecting the top three.Source: §4.1.2 · p. 9

4.1.3 RESULTS

• ToT (b=3) with GPT-4 reaches 60.58% accuracy at 39.83 LLM calls, whereas XOT needs fewer than 1.8 calls with revision.XOT also outperforms standalone MCTS, with its advantage growing over more revision iterations.Source: Table 3 · p. 8, §4.1.3 · p. 9

4.2.1 TASK SETUP

• The evaluation uses 419 randomly generated solvable puzzles: 300 training instances and 119 test instances, all solvable within nine steps.Each thought represents a step-by-step move and the resulting puzzle state.Source: §4.2.1 · p. 9

4.2.2 BASELINES & XOT SETUP

• The setup enriches CoT examples with legal actions, current actions, and states, while XOT uses more simulations for revision and samples trajectories for multi-solution exploration.The maximum step limit is nine; XOT uses 20 simulations per action and 50 for revision, then samples 50 trajectories and selects the top three thoughts in the multi-solution setting.Source: §4.2.2 · pp. 9–10

4.2.3 RESULTS

• GPT-4 XOT reaches 95.80% accuracy after three revisions, versus 13.45% for the best prompting baseline, ToT (b=3).With one revision, GPT-4 XOT reaches 93.28% accuracy; the three-revision setting requires approximately 1.6 LLM calls.Source: §4.2.3 · p. 10, Table 5 · p. 10
• XOT outperforms other baselines on MultiAcc for both GPT-3.5 and GPT-4, with or without revision.For GPT-4, revision raises MultiAcc from 51.26% with single MCTS to 76.33%.Source: §4.2.3 · p. 10, Table 6 · p. 10

4.3.1 TASK SETUP

• The task uses 1,000 training and 183 testing samples, generated from five random face rotations and solvable within four steps.The experiments reduce the 27 legal operations to nine actions: U, U’, U2, R, R’, R2, F, F’, and F2.Source: §4.3.1 · p. 11

4.3.2 BASELINES & XOT SETUP

• For multi-solution experiments, XOT samples 50 trajectories after MCTS and keeps the three thoughts with the highest counts.The prompting baselines use three examples, each with three unique solutions; ToT and GoT receive a seven-step limit.Source: §4.3.2 · p. 11

4.3.3 RESULTS

• GPT-4 XOT exceeds 77.60% accuracy with one revision and 80% with three, versus 19.57% for the best prompting baseline, ToT (b=3).XOT requires approximately two LLM inference calls for both GPT-3.5 and GPT-4.Source: §4.3.3 · p. 12
• In the multi-solution setting, GPT-4 XOT achieves over 77% MultiAcc, while the closest competitor, GoT (k=3), reaches 16.85%.GoT requires 13.36 LLM invocations versus 4.08 for XOT.Source: §4.3.3 · p. 12

4.4 ABLATION STUDY

• The revision ablation tests performance and efficiency as revision counts vary, and separately examines sensitivity to incomplete thoughts.The incomplete-thought test removes the final step of a trajectory to simulate inaccurate or incomplete MCTS thoughts.Source: §4.4 · p. 12, §4.4.2 · p. 13

4.4.1 NUMBER OF REVISIONS

• Accuracy generally rises with more revisions: GPT-3.5 improves from 45.36% to 84.70%, and GPT-4 from 45.90% to 83.61%.The paper reports comparable inference times and model calls between models, with GPT-4 showing a substantial increase in calls after the third revision; additional revisions do not dramatically increase LLM invocations, though fθ is called more often.Source: §4.4.1 · p. 12, Figure 4 · p. 13
• Revision success rate increases with the number of revisions for both GPT-3.5 and GPT-4.The study reports the rates in Tables 9 and 10.Source: §4.4.1 · pp. 12–13, Table 9 · p. 13, Table 10 · p. 13

4.4.2 INCOMPLETE THOUGHT

• Incomplete thoughts reduce performance across all three tasks; GPT-4 XOT retains over 40% accuracy on Game of 24 and 8-Puzzle but drops to 6% on Pocket Cube.GPT-3.5 is more affected, reaching 0% accuracy on several baselines.Source: §4.4.2 · p. 14, Table 11 · p. 14

4.5 CASE STUDY

• In the multi-solution examples, XOT’s thought paths intertwine at intermediate steps and converge on the goal, forming graph-like structures.The Pocket Cube example contains four distinct paths; the 8-Puzzle example includes recurrent state transitions described as reflection.Source: §4.5 · p. 14
• The case examples are presented as evidence that flexible thought generation can yield multiple high-quality answers to one problem.The authors connect this flexibility with diverse and creative thinking.Source: §4.5 · p. 15

4.6 EXPERIMENT SUMMARY

• The summary characterizes XOT as improving performance through streamlined thought-trajectory revision across Game of 24, 8-Puzzle, and Pocket Cube.It highlights Game of 24 performance and the method’s ability to handle spatial-reasoning challenges in the two puzzles.Source: §4.6 · p. 15

Decision Making & Planning with LLMs

• CoT, ToT, and GoT decompose problems into different thought structures, while XOT is distinguished as pursuing performance, efficiency, and flexibility together.The discussion also cites extensions across domains and modalities Zhang et al. (2022; 2023); Ning et al. (2023); Turpin et al. (2023).Source: §5 · p. 15
• Other planning work uses interactive plans for multi-task agents, LLM-suggested robotic actions, or language-to-planning-goal translation.The cited examples are Wang et al. (2023b), Singh et al. (2023), and Xie et al. (2023), respectively.Source: §5 · p. 15

Augmenting LLMs with RL

• Prior RL augmentation includes human-feedback alignment in ChatGPT Ouyang et al. (2022), generalization in GLAM Carta et al. (2023), and skill learning for Minecraft Yuan et al. (2023).ESPER Yu et al. (2023) uses RL to align multimodal inputs and language-model generations without direct supervision.Source: §5 · p. 15
• Prior MCTS-LLM work uses MCTS for strategic exploration, value-guided decoding, or decoding augmentation.The examples cited are Hao et al. (2023), Liu et al. (2023), and Feng et al. (2023).Source: §5 · pp. 15–16

6 DISCUSSION

• XoT is presently used for reasoning and search, while broader applicability is proposed for decomposable tasks with well-defined objectives.The discussion notes MCTS's supportive role and possible substitution with other models for thought exploration and generation.Source: §6 · p. 16
• XoT requires additional policy and value models trained using real-world environment datasets, adding costs and effort; these costs are described as low for the study's well-defined tasks.The authors identify less straightforward objectives, including multi-agent planning and code generation, as areas for future efficiency work.Source: §6 · p. 16
• Incorrect MCTS intermediate thoughts or LLM deviations can lead to inaccurate answers or hallucinations, and environmental changes can make MCTS inaccurate.The discussion says LLMs can revise thoughts using internal knowledge, mitigating risks from inaccurate initial thoughts.Source: §6 · p. 16
• XoT concurrently achieves performance, efficiency, and flexibility, and significantly outperforms existing prompting paradigms, particularly on multi-solution problems.The conclusion describes evaluations on Game of 24, 8-Puzzle, and Pocket Cube.Source: §6 · p. 16
• MCTS with pretrained policy and value networks injects domain knowledge and planning, offloads thought searching, and supports unconstrained exploration; LLM revision further enhances thought quality.The discussion also describes MCTS as a supportive component that could be replaced by other supervised or reinforcement-learning models.Source: §6 · p. 16

A LLAMA-2-13B SETUP

• The setup tests whether simulation knowledge can be distilled into LLaMA-2-13B to potentially avoid GPT-4 during testing.Ground-truth labels were used because they were considered more accurate than MCTS simulation labels.Source: A LLAMA-2-13B SETUP · p. 19
• Training used a dataset of question-answer pairs to turn an optimization or search problem into a supervised prediction task.The setup used ground-truth labels.Source: A LLAMA-2-13B SETUP · p. 19
• The model was fine-tuned for 5 epochs on eight V100 GPUs, with a learning rate of 2e-5 and train batch size of 32.The reported training run lasted approximately 5 hours.Source: A LLAMA-2-13B SETUP · p. 19

B COMPUTATIONAL TRAINING COSTS OF MCTS

• 50.18 LLM invocations on average per test problem for ToT across three tasks, exceeding XoT's pretraining cost in recurring testing computation.The reported ToT task averages are 39.83, 54.13, and 56.58 calls.Source: B COMPUTATIONAL TRAINING COSTS OF MCTS · p. 19
• XoT trains its policy/value model through three iterations, each containing 10 MCTS self-play episodes, and uses one-time offline pretraining to reduce testing computation.Table 12 is identified as listing training and testing policy/value model calls by task and iteration.Source: B COMPUTATIONAL TRAINING COSTS OF MCTS · p. 19
• Approximately 1e6 total parameters across the policy/value networks for all three tasks, compared with GPT-3.5's 175 billion parameters and GPT-4's estimated over 1 trillion.The passage says this smaller model design remains efficient even with additional training calls.Source: B COMPUTATIONAL TRAINING COSTS OF MCTS · p. 19

C EXPERIMENT RESULTS ON OTHER NLP TASKS

• Other NLP tasks may be formulated as MCTS searches with LLM-based rewards, as in ToT Creative Writing and GoT Document Merging.The passage describes this reward-design strategy as an active research topic.Source: C EXPERIMENT RESULTS ON OTHER NLP TASKS · p. 19
• 8.168 was XoT's highest score on Document Merging, with average token cost of 15270.80 described as surpassing both ToT and GoT.Table 13 is the performance comparison on Document Merging.Source: Table 13 · p. 20, C EXPERIMENT RESULTS ON OTHER NLP TASKS · p. 20
• Document Merging combines overlapping input documents into a new NDA, aiming to minimize duplication while maximizing information retention.The experimental setting follows GoT and uses the same dataset from its repository.Source: C EXPERIMENT RESULTS ON OTHER NLP TASKS · pp. 19–20

• Each thought step records the action taken and the resulting new state.Source: D PROMPT EXAMPLE · p. 20
• The Game of 24 prompt uses basic arithmetic operations and demonstrates identifying Step 2 as the point where the remaining numbers cannot reach 24.Source: D PROMPT EXAMPLE · pp. 20–21
• The 8-Puzzle prompt specifies valid moves toward the ordered goal state, and its example reaches that goal in four moves.The prompt limits solutions to nine steps and asks for the fewest steps.Source: D PROMPT EXAMPLE · pp. 21–23
• The Pocket Cube prompt defines a 2x2 restoration task and gives R U’ F’ as an example move sequence.The prompt says problems can be solved in 1 to 4 moves and sets an 11-move maximum.Source: D PROMPT EXAMPLE · pp. 24, 33
• For failed processes, the examples ask the model to identify the wrong step, with the Pocket Cube example naming Step 3 and the 8-Puzzle example naming Step 4.Source: D PROMPT EXAMPLE · pp. 23, 24, 33, 34