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REVIEW 4 major objections 6 minor 59 references

The paper claims that replacing trajectory-level reflection with state-anchored predictive knowledge improves LLM agent learning, and supports this with two training algorithms and experiments across interactive and reasoning tasks.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · deepseek-v4-flash

2026-08-03 00:49 UTC pith:2F25WHBA

load-bearing objection A promising stateful-knowledge training idea with a coherent motivating example and two reasonable algorithms, but the central empirical claim is under-supported because the baselines are borrowed and the SKL-RL variants add simulation/search compute that the baselines lack. the 4 major comments →

arxiv 2607.28638 v1 pith:2F25WHBA submitted 2026-05-19 cs.CL cs.LG

Learning Stateful Predictive Knowledge From Experience

Yan Song , Xidong Feng , Bo Liu , Xinyu Cui , Haotian Fu , Zichen Liu , Mengyue Yang , Cheng Deng
show 2 more authors
Jian Zhao Jun Wang
This is my paper
classification cs.CL cs.LG
keywords stateful knowledge learningpredictive knowledgeLLM agentsexperience learningknowledge bootstrappingself-distillationreinforcement learningtrajectory reflection
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to change what LLM agents extract from experience. Rather than reflecting on a whole trajectory after the fact—what it calls episodic hindsight—an agent should maintain state-anchored predictive knowledge: explicit natural-language predictions about what each state portends. The central claim is that such stateful knowledge is finer-grained, more generalizable, and can be bootstrapped backward from successor states, so a later success improves earlier judgments. Two training variants, one based on self-distillation and one on reinforcement learning, internalize this capability. Experiments on shopping, science, and chess tasks report that this approach outperforms existing reflection-based training, supporting the paper's thesis that predictive foresight grounded in states is a more effective learning signal than trajectory hindsight.

Core claim

The central discovery is that predictive knowledge—declarative statements about future outcomes—becomes a far more effective learning signal for LLM agents when it is indexed by state rather than by trajectory. The authors show that an agent which aggregates outcomes across repeated visits to the same state and updates that state's knowledge using the knowledge of successor states discovers environment structure (such as asymmetric slip dynamics) that trajectory-level reflection misses. On stochastic tasks this removes hindsight bias; on transfer tasks the state-grounded knowledge degrades less than path-grounded heuristics. Training the model to extract and use such knowledge via self-disti

What carries the argument

The central object is the stateful knowledge table {z_s}: for each visited state s, a natural-language predictive assessment generated by prompting the model with 'What does this state predict about future outcomes?'. The two mechanisms that carry the argument are state aggregation—collecting outcomes from N partial rollouts rooted at the same state with horizon H—and knowledge bootstrapping, which updates z_{s_t} using the already-assessed successor knowledge z_{s_{t+H}}, analogous to temporal-difference learning. These feed two training objectives: a self-distillation loss that reinforces filtered successful rollouts and bootstrapping corrections, and a reinforcement-learning loss that opt

Load-bearing premise

The premise that an LLM's natural-language answer to 'what does this state predict?' is genuinely state-grounded, rather than a summary of the path taken to that state, is the load-bearing assumption; if the model cannot separate state from trajectory, the knowledge inherits the very hindsight bias it is meant to fix.

What would settle it

Collect the state knowledge a trained SKL agent produces at the same state reached by two different trajectories—one that ended in success and one that ended in failure. If the generated predictions differ systematically with the path, the knowledge is not state-grounded and the paper's central mechanism is falsified. Conversely, if transfer of a state-knowledge table to a new environment with identical state space but different transition dynamics succeeds without retraining, state-grounding is supported.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Agents trained this way carry an explicit 'mental map' of the task that conditions each decision on the state's predicted future, rather than on generic lessons from past runs.
  • A single successful outcome at a later state automatically improves the knowledge of earlier states through backward bootstrapping, so experience is reused across the trajectory.
  • Because knowledge is state-indexed, it transfers to new instances of an environment with the same state structure better than trajectory-level heuristics.
  • The RL variant lets the agent decide what to simulate, so the balance between exploring uncertain branches and exploiting known strong lines emerges from training rather than being hand-designed.
  • Distillation of simulated look-ahead into the model's parameters reduces the need for expensive simulation at inference time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A direct way to test the state-grounding premise is to reach the same state via different histories and compare the generated knowledge z_s; if the predictions differ by path, the approach has not actually escaped trajectory dependence.
  • The bootstrapping logic suggests a natural extension to hierarchical or abstract states, where knowledge at a coarse state constrains knowledge at its descendants—possibly enabling transfer across environments that share only abstract structure.
  • The paper's evidence implies an experience-efficiency gain, not just a ceiling gain: a stateful agent should reach a given success level with fewer rollouts, which could be measured directly.
  • If state knowledge is a learned language value function, then techniques from value-based RL—such as prioritized experience replay over states—might improve the state aggregation step further.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This paper challenges trajectory-level reflection as the dominant paradigm for LLM agents learning from experience, arguing that it is 'episodic hindsight' and proposing Stateful Knowledge Learning (SKL). SKL maintains a table of explicit predictive assessments z_s anchored to states, updates them by aggregating state-specific rollouts and bootstrapping from successor-state knowledge, and conditions actions on z_s. Two training variants are proposed: SKL-SD, based on offline replay-buffer bootstrapping with verification and self-distillation, and SKL-RL, based on online state-settable simulator rollouts with GRPO and self-distillation. Experiments include a Slippery FrozenLake motivating study, SKL-SD on WebShop/ScienceWorld, and SKL-RL on ChessPuzzles. The paper reports consistent improvements over reflection-based baselines and claims these 'significantly outpace current reflection-based training paradigms.'

Significance. If substantiated, the proposed shift is useful: it connects LLM agent experience learning to predictive knowledge and TD-style bootstrapping, offers a declarative alternative to scalar Q-functions, and gives two concrete training recipes. Strengths include a public code repository, clear pseudocode, and several meaningful ablations (e.g., heuristic simulation quality, distillation loss coefficient, response-length dynamics). The framework is grounded by external task rewards and verified retries, so circularity is not the main concern. The empirical evidence, however, does not yet establish the central claim: the comparisons confound the stateful-knowledge mechanism with extra compute/search, and some baseline numbers are borrowed rather than re-run. The authors are transparent about compute costs and limitations, but the headline 'significantly outpaces' claim needs stronger support.

major comments (4)
  1. [§4.2, Eq. (2) and Fig. 6] The central SKL-RL comparison does not isolate the stateful-knowledge mechanism. SKL-RL adds (i) access to a state-settable simulator, (ii) N parallel H-step look-ahead rollouts at every state, (iii) a knowledge+action two-turn response format, and (iv) an auxiliary distillation objective. The trajectory-level baselines in Fig. 6 have none of these, so matching by 'number of LLM calls' does not control the content of the calls: SKL-RL's calls are online search. A trajectory-level baseline equipped with the same simulator, same N/H rollout budget, and same response format, but aggregating outcomes by trajectory/action rather than by state, is necessary before the advantage can be attributed to state anchoring. Without it, the result is consistent with a pure search-depth/compute explanation.
  2. [§4.1, Table 1] The WebShop/ScienceWorld comparison is not self-contained. Baseline numbers are explicitly borrowed from [Shi et al., 2026b] and were not re-run in the SKL-SD harness; no error bars, seed counts, or significance tests are reported. SKL-SD also admits ~1.5× training overhead and samples only 20% of states for bootstrapping, so the comparison is not matched in compute or data curation. The WebShop gain over R3L is +3.3 absolute, which could easily be within run-to-run variance. Please re-run the baselines in the same codebase and report means/variances over seeds, or clearly label Table 1 as a transferred comparison from the original paper.
  3. [§2.1 and §3, Eqs. (1) and (4)] The conceptual premise that z_s is state-grounded rather than another trajectory summary is never directly tested. The evidence consists of task successes and selected traces (e.g., Appendix F.2), but a stateful representation could in principle be issuing path-dependent skills from the same API. The paper should include a 'state-grounding probe' that reaches the same state through different histories and compares the generated z_s, or measures whether z_s predicts future outcomes better than a matched trajectory summary. Without such a probe, the mechanism named 'stateful' is not distinguishable from a memorized/checklist policy.
  4. [§2.2, Fig. 3] The bootstrapping benefit is not quantitatively demonstrated. The figure reports an example trace, the frequency of the 'recoverable' keyword, and the relative position of 'irrecoverable'; no success-rate curve compares the fully bootstrapping agent with its root-state-only counterpart. The claim that bootstrapping 'propagates knowledge backward' therefore rests on a qualitative trend. A small controlled comparison (with/without H-step bootstrapping, fixed aggregation budget and inference budget) would make this load-bearing component measurable.
minor comments (6)
  1. [§4.1 and Algorithm 1] The text says bootstrapping is performed on only 20% of states, but Algorithm 1 loops over all time steps T-1...0. Clarify how the sampling modifies the loop and whether z_s^* in line 14 is defined in the notation table.
  2. [Appendix E, Eqs. (5)-(6)] 'Foward' should be 'Forward'; also the KL notation is confusing because the two arguments look identical. Specify which distribution is the teacher and which is the student in each divergence.
  3. [Fig. 3] The caption says 'performance' but no performance metric is plotted. Define the 'relative location' measure and the keyword-frequency statistic in the caption or text.
  4. [§4.2.1] Typos: 'We will confirmed this' should be 'We confirm this'; 'export move' should be 'expert move'.
  5. [Abstract and §1] 'Significantly outpaces' is not supported by significance tests. Recommend rewording to 'outperforms in the reported experiments' or adding statistical tests.
  6. [§4.2.3, Fig. 7] The distillation ablation reports only training success rate. Since ChessPuzzles test generalization is part of the paper's claim, include test-set curves or report the variance across seeds.

Circularity Check

0 steps flagged

No significant circularity: SKL's stateful-knowledge loop is grounded by external task rewards and held-out benchmarks, not by its own definitions.

full rationale

The paper is an empirical training framework rather than a formal derivation, so no equation-level reduction makes a predicted quantity equal to a fitted input. The stateful knowledge z is generated by the model, but it is updated from aggregated environment rollouts and bootstrap signals that are ultimately supervised by external outcome rewards R (Eqs. 1-4), and the central comparisons use held-out test splits on WebShop, ScienceWorld, and ChessPuzzles against external baselines. The self-distillation components (D_distill, L_Distill) are self-referential in mechanism, but they distill simulation-derived \hat z back into z under outcome verification, so the loop is grounded rather than equivalent to its input. The backward knowledge-bootstrapping update in Sec. 2.2 is explicitly designed to feed successor knowledge into earlier states, so the observed backward keyword shift is partly a design consequence; however, it is presented as a qualitative illustration and is not the load-bearing claim. The only notable self-citation is [Feng et al., 2024] for the language-value-function framing; it is used for attribution and related work, while this paper's experiments stand on their own. Acknowledged limitations (1.5x training overhead, exponential simulation cost, borrowed baselines) are validity/compute concerns, not circularity. No circular step meets the evidentiary bar requiring a specific reduction.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 2 invented entities

The central contribution is empirical and rests on strong domain assumptions about LLM-generated natural-language knowledge, the informativeness of small aggregation budgets, and for SKL-RL the availability of a state-settable simulator. No physical entities are posited; the main invented ledger entry is the stateful knowledge representation itself.

free parameters (5)
  • Bootstrapping horizon H = H=3 for SKL-SD; H=2 and H=3 for SKL-RL
    Hand-chosen budget controlling how far ahead successor-state knowledge is used for bootstrapping; central results are reported for one or two values only.
  • State-aggregation width N = N=3 for SKL-SD and main SKL-RL; N=4 in an ablation
    Number of parallel rollouts aggregated per state; chosen by hand as part of the bootstrap budget.
  • Loss weights λ1, λ2 for SKL-SD and λ for SKL-RL = SKL-RL λ ablated over 0.2, 0.5, 1.0; SKL-SD λ1/λ2 not fully specified
    Weighting of distillation and SFT auxiliary terms relative to the main policy-gradient loss; selected by the authors.
  • State sampling fraction in SKL-SD = 0.2 (20% of states uniformly sampled)
    Compute-saving choice that changes which states receive bootstrapping updates; motivated only by efficiency.
  • ChessPuzzles difficulty cutoff = rating < 1000
    Evaluation subset selected by puzzle rating; affects task difficulty and the reported learning curves.
axioms (5)
  • domain assumption Predictive knowledge must be state-grounded; trajectory-level hindsight is structurally insufficient.
    Section 1 and Section 2.1 rest the entire framework on this premise, citing Sutton et al.; it is argued conceptually rather than proven.
  • domain assumption An LLM can express, aggregate, and refine predictive knowledge as natural-language declarations z_s accurate enough to condition policy.
    Sections 2.1 and 3 prompt the model to generate z_s; Appendix G shows both successes and hallucinations, so this is not guaranteed.
  • domain assumption State aggregation from N rollouts at budget (N,H) gives sufficient statistics of future outcomes for a state.
    Algorithm 1 and Algorithm 2 rely on small (N,H) budgets being informative; the paper does not analyze estimator bias or variance.
  • domain assumption A resettable, state-settable environmental simulator E_sim is available for SKL-RL.
    Section 3.2 and Algorithm 2 require E_sim; SKL-RL is only evaluated in a simulator (ChessPuzzles), not in WebShop or ScienceWorld.
  • standard math Standard LLM training machinery (GRPO policy gradient, KL distillation, SFT) behaves as expected.
    Used without proof as background: GRPO, KL divergence, and SFT losses are standard tools.
invented entities (2)
  • Stateful Knowledge table {z_s} (explicit state-indexed predictive assessments) no independent evidence
    purpose: The central representation the paper introduces: state-anchored, declarative predictions that condition action selection and are bootstrapped backward across states.
    No external falsifiable handle exists outside the paper's own traces; the quality of z_s is only observed indirectly through downstream task success.
  • Emergent concept 'recoverability' in FrozenLake no independent evidence
    purpose: An agent-invented criterion used to judge whether a state has a safe continuation, enabling backward knowledge propagation.
    Documented only in qualitative reasoning traces; not independently validated or measured as a separate quantity.

pith-pipeline@v1.3.0-alltime-deepseek · 27588 in / 11446 out tokens · 104359 ms · 2026-08-03T00:49:57.358023+00:00 · methodology

0 comments
read the original abstract

As large language model (LLM) agents increasingly learn from experience, they primarily rely on trajectory-level reflection to extract insights. Viewed through the lens of predictive knowledge, we argue that this approach operates on episodic hindsight rather than predictive foresight, yielding brittle, path-dependent heuristics. To address this, we propose Stateful Knowledge Learning (SKL). SKL shifts the agent's focus from trajectory-level summarization to maintaining Stateful Knowledge: explicit, declarative predictive assessments anchored to state. We first demonstrate a motivating example showing how stateful knowledge provides granularity, enhances generalization, and enables knowledge bootstrapping. To further scale up the idea, we introduce two algorithms via self-distillation (SKL-SD) and reinforcement learning (SKL-RL), training agents to autonomously extract state-grounded predictive knowledge from experience and learn to leverage it for policy making. Experiments on interactive environments (WebShop, ScienceWorld) and a complex reasoning task (ChessPuzzles) demonstrate that equipping models with the inherent ability to learn stateful predictive knowledge significantly outpaces current reflection-based training paradigms.

Figures

Figures reproduced from arXiv: 2607.28638 by Bo Liu, Cheng Deng, Haotian Fu, Jian Zhao, Jun Wang, Mengyue Yang, Xidong Feng, Xinyu Cui, Yan Song, Zichen Liu.

Figure 1
Figure 1. Figure 1: Reflect on trajectories and reflect on stateful knowledge (root state update only and fully [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Slippery FrozenLake Analysis. (a) Comparison of trajectory-level and stateful reflection. The grey dashed line represents the baseline for retrying the game ten times without any self-reflection and success once. Good move ratio refers to the probability of a move not leading to falling. (b) Representative examples of agent behaviour in different iterative loops. Trajectory Reflection (Hindsight): At itera… view at source ↗
Figure 3
Figure 3. Figure 3: Left: An example reasoning trace from the fully bootstrapping variant at the final iteration. The agent defines and utilizes recoverability for reasoning. Middle: The average frequency of the keyword recoverability across training iterations, showing a steady upward trend. Right: The relative distribution of keywords along the trajectory horizon; notably, the occurrence of the keyword irrecoverable shifts … view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of two training variants: SKL-SD and SKL-RL. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Experiments of SKL-RL with heuristic search simulation on ChessPuzzles. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: SKL-RL training with self-generated parallel simulation, evaluated at matched budgets. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Ablation study on SKL-RL with inte￾grated training loss. These results suggest that LDistill helps stabilize the joint optimization of this compositional reasoning process. As shown by the reasoning traces in Ap￾pendix G.6, the SKL-RL agent updates stateful predictive knowledge by building upon previ￾ously distilled information. Specifically, it treats zˆ from the prior learning epoch as a condensed repres… view at source ↗
Figure 8
Figure 8. Figure 8: At each state st, the agent first performs a tree search where each node expansion consists of state knowledge and a simulation move. The generated simulation experience τ H N (st) is aggregated as the context, and the agent summarizes an updated state understanding zˆst ∼ πθ(·|τ H N (st), st, zst ) as well as a finalized move at to transition to the next state st+1. destination square (e.g., Nf3, Rxf2, c5… view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

59 extracted references

  1. [1]

    Base Response EDbase [Rbase · ∇θ logπ θ(τbase |s 0)]

  2. [2]

    Distilled Refined Response EDdistill reflect [Rreflect · ∇θ logπ θ(τnew |s 0)]

  3. [3]

    Distilled Retried Response EDdistill retry [Rretried · ∇θ logπ θ(τbase,<pivot, τretry,>pivot |s 0)]

  4. [4]

    Complete Refinement Process EDreflect [Rreflect · ∇θ logπ θ([τbase, dreflection, τnew]|s 0)]

  5. [5]

    Complete Retry Process EDretry [Rretry · ∇θ logπ θ([τbase, dreflection,[τ base,<pivot, τretry,>pivot]]|s 0)]

  6. [6]

    D Localized parallel simulation in SKL-RL In the context of SKL-RL, the localized parallel simulation mechanism acts as an online, text-based look-ahead tree exploration

    Reflection E{dreflection} [Rreflect · ∇θ logπ θ(dreflection |s t, τbase)] Table 3: Training objective for different purposes in works related to other self-reflection training methods. D Localized parallel simulation in SKL-RL In the context of SKL-RL, the localized parallel simulation mechanism acts as an online, text-based look-ahead tree exploration. B...

  7. [7]

    The simulator’s state is checkpointed or saved at this exact position, acting as the root node for the simulations

    State Anchoring(The Root): At any given online step, the agent encounters state st. The simulator’s state is checkpointed or saved at this exact position, acting as the root node for the simulations

  8. [8]

    Multi-Branch Spawning(Width N): From this single root state st, the agent generates knowledge zst, and instantiates N actions for independent, parallel branches (or simulation workers)

  9. [9]

    , H, every branch rolls out independently

    Independent Temporal Expansion(Horizon H): At each subsequent simulation timestep 1, . . . , H, every branch rolls out independently. Each branch samples knowledge and actions according to the agent’s current policy (or an exploration policy) and receives independent observations and rewards from its own isolated instance of the environment

  10. [10]

    A concrete example is also provided in Figure 8

    Experience Aggregation: Once all N branches reach the maximum simulation horizon H (or hit a terminal state), their complete rollout histories are gathered into an aggregated experience blockτ H N . A concrete example is also provided in Figure 8. E Multi-turn self-distillation in SKL-RL The primary objective of the distillation loss LDistill({zs,ˆzs}s∈S ...

    2024

  11. [11]

    Search Queries: • Ensure the search terms include specific attributes and measurements to match the exact product requirements: - Color: . . . - Size: . . . - Additional Features:. . . - Price:

  12. [12]

    • Directly click "Buy Now" if the product matches all specified criteria, without additional checks for availability

    Product Verification and Filtering: • After locating a potential product, verify all criteria (color, size, features, price) are correctly specified. • Directly click "Buy Now" if the product matches all specified criteria, without additional checks for availability. 18 Learning Stateful Predictive Knowledge From Experience

  13. [13]

    Buy Now" button is performed directly on the detailed product page with all filters correctly applied. • Click the

    Buy Now on Detailed Product Page: • Ensure that the click on the "Buy Now" button is performed directly on the detailed product page with all filters correctly applied. • Click the "Buy Now" button if all conditions are verified

  14. [14]

    • Double-check that the product’s details, especially dimensions (size: 120x60cm) and color (white), and price (≤$70.00) match the criteria

    Detailed Review and Double-Check: • Verify that the product’s page correctly displays all the required attributes and features, using inspection tools if necessary. • Double-check that the product’s details, especially dimensions (size: 120x60cm) and color (white), and price (≤$70.00) match the criteria

  15. [15]

    Steps for Implementation:

    Error Logging: • If a product does not match all criteria, log the relevant details of the product and try again, ensuring a revisit to the search page with corrected parameters if necessary. Steps for Implementation:

  16. [16]

    [State Knowledge for Browsing] State Understanding:

    Refine Search Query with Specific Attributes and Dimensions: . . . . . . [State Knowledge for Browsing] State Understanding:

  17. [17]

    Color: ..., 2

    Correct Product Matching: Ensure that the selected product details (color, size, price)—all attributes must correctly match the required criteria: 1. Color: ..., 2. Size: ..., 3. Price:

  18. [18]

    • Verify that images and descriptions explicitly confirm the exact details being used (white color, 120x60cm size, etc)

    Product Detail Verification: • Double-check each product detail on the result page to ensure accuracy. • Verify that images and descriptions explicitly confirm the exact details being used (white color, 120x60cm size, etc). ...... N. Notes for Future Runs: • Cross-Verify Attributes: Validate the exact color, size, and price on each detailed product page. ...

  19. [19]

    Navigating the URL: Start by ensuring that the correct URL is accessed, specifically B095KCTB3Y , to display the desired product

  20. [20]

    • Double-check that the product details on the page reflect the correct options and attributes (white, 120x60cm, non-slip)

    Selecting Size and Color Details: • Ensure that the size 120x60cm and color white are explicitly selected and chosen correctly from dropdown options (if available). • Double-check that the product details on the page reflect the correct options and attributes (white, 120x60cm, non-slip)

  21. [21]

    white",

    Direct Purchase Action: Once all conditions (size and color) are correctly matched and verified on the product description, click the Buy Now button. Notes: • Ensure that the search query uses keywords such as "white", "120x60cm", and "pu leather" to precisely target the correct product. • Directly interact with the specified URL and verify the exact attr...

    2020

  22. [22]

    (Misidentified the piece)

    Qxc8: Captures the pawn and puts pressure on the black king. (Misidentified the piece)

  23. [23]

    (Misidentified the piece)

    Qa8: Attacks the rook on a1 and also threatens the black king. (Misidentified the piece)

  24. [24]

    (Wrong justification) Now let’sevaluate each move: (Completely wrong) 1.Qxc8: If this move is played, Black will respond with Rxa8

    Qc7: Targets the black king directly. (Wrong justification) Now let’sevaluate each move: (Completely wrong) 1.Qxc8: If this move is played, Black will respond with Rxa8. The resulting position will leave White with a potential fork with the queen, targeting both the black king and the rook. However, this might be risky as Black could still create threats ...

  25. [25]

    Likely response: Black must recapture with Rxc8 (the rook from f8 goes to c8)

    Qxc8 20 Learning Stateful Predictive Knowledge From Experience Idea: capture the black queen on c8 and trade queens immediately. Likely response: Black must recapture with Rxc8 (the rook from f8 goes to c8). Outcome: White has traded queens, but then White is only up material modestly and still has to deal with Black’s active rook and passed f-pawn. This ...

  26. [26]

    Qxa7 Idea: win a pawn with check possibilities later (queen near the king). Likely response: Black can continue with f2 (or Rxa7 isn’t possible), pushing the passed pawn and creating real promotion threats, while White hasn’t reduced Black’s major-piece activity. Outcome: This is slower and allows Black counterplay with the f-pawn and rook coordination. N...

  27. [27]

    Qb7 Idea: directly attack Black’s rook on f7 along the 7th rank and win it. Likely response: Black’s rook has no squares available on the 7th rank because b7 controls the entire rank; it also cannot capture Qb7 (rook on f7 can’t move like that) (Wrong prediction), and moving the king doesn’t solve the attack. The only “defense” would be moving the rook aw...

  28. [28]

    [Assistant]:

    State Knowledge zt: [User]: What is your understanding of the game state: {state}. [Assistant]:

  29. [29]

    #Q: How to move ? Q: What is understanding of state: 𝑠!Updated Stateful Knowledge 𝒛#𝒔𝒕)Q: How to move ?Finalized Move: 𝑎

    Movement at: [User]: You are in stage {stage}, based on the state understanding, what is the decided move? [Assistant]: ... where the state knowledge explicitly queries the model for its internal predictive knowledge of the game state, and the movement query will use the generated understanding as the context, asking the model to directly generate action ...

  30. [30]

    White has a pawn (P) on the 8th rank (White’s king-side), and Black has a pawn on the 7th rank (Black’s king-side)

    Material Balance: The material is balanced with two rooks (R), two knights (N), and one bishop (B) for each side. White has a pawn (P) on the 8th rank (White’s king-side), and Black has a pawn on the 7th rank (Black’s king-side). 2.King Safety: Both kings are castled on the kingside, and neither side has any immediate threats to their kings. 3.Pawn Struct...

  31. [31]

    The knight moves Ne7 and Nb5 in Traj 1 and 2 suggest support for the king-side pawns and maintaining control over key squares

    Key Moves: The knight on b8 (Nb8) in Traj 3 suggests that it might be a strong move to support the pawn structure and potentially create threats. The knight moves Ne7 and Nb5 in Traj 1 and 2 suggest support for the king-side pawns and maintaining control over key squares. The rook move Na5 in Traj 4 suggests a support move for the knight and maintaining c...

  32. [32]

    White has a king and a rook, while Black has a king and a rook

    Positional Evaluation: White has a rook and a knight, while Black has a rook and a knight. White has a king and a rook, while Black has a king and a rook. White has a pawn on the 6th rank, while Black has a pawn on the 6th rank. 4.Key Pieces: White’s rook on d1 and Black’s rook on c4. White’s knight on g5 and Black’s knight on d8

  33. [33]

    Both sides have active pieces, and the game is likely to continue with tactical exchanges and positional maneuvers

    Active Play: The game is still in the middlegame, with no clear endgame or decisive advantage. Both sides have active pieces, and the game is likely to continue with tactical exchanges and positional maneuvers. Trajectory Analysis: Let’s analyze the trajectories to understand the potential future value of the root state: •Traj 1, 2, 3, 4: The state remain...

  34. [34]

    White is in a strong position, having just won a queen and a rook, which is a significant material advantage. (wrong)

  35. [35]

    (correctly sumarize the key benefits)

    White has a strong attack on the black king, with the queen and rook threatening to checkmate on the next move. (correctly sumarize the key benefits)

  36. [36]

    The black king is in a vulnerable position, with the queen and rook threatening to deliver checkmate

  37. [37]

    The black queen is a potential threat, but it is not well-placed to counter the white attack

  38. [38]

    The white king is safe and can continue to support the attack on the black king

  39. [39]

    Given this updated understanding, the position is a strong position for White, and the move Qd8 was the best move in the simulation history, leading to a win

    The board is open and dynamic, with many pieces still active and ready to participate in the attack. Given this updated understanding, the position is a strong position for White, and the move Qd8 was the best move in the simulation history, leading to a win. (combine knowledge with simulation result) Table 10: Reasoning trace example of R3L. Trajectory r...

  40. [40]

    **Trajectory 1: Kxc8** - **Initial Move**: Kxc8 - **Opponent’s Move**: f8g8 - **Next State**: 2K3k1/3r1p2/5Q2/pp2pN2/P3P3/2P4P/1P3q2/8 w - - 1 47 - **Subsequent Move**: Kb8 - **Opponent’s Move**: g8f8 - **Next State**: 1K3k2/3r1p2/5Q2/pp2pN2/P3P3/2P4P/1P3q2/8 w - - 3 48

  41. [41]

    **Trajectory 2: Ka6** - **Initial Move**: Ka6 - **Opponent’s Move**: f8g8 - **Next State**: 2r3k1/3r1p2/K4Q2/pp2pN2/P3P3/2P4P/1P3q2/8 w - - 5 47 - **Subsequent Move**: Kxa5 - **Opponent’s Move**: g8f8 - **Next State**: 2r2k2/3r1p2/5Q2/Kp2pN2/P3P3/2P4P/1P3q2/8 w - - 1 48 Evaluation:

  42. [42]

    The black king is still a threat, and the board is less controlled by the white queen

    **Kxc8 Trajectory**: - After Kxc8, the board simplifies, and the white king is now in a more exposed position. The black king is still a threat, and the board is less controlled by the white queen. - The subsequent move Kb8 does not significantly improve the position, and the black king continues to be a threat

  43. [43]

    evaluation form

    **Ka6 Trajectory**: - After Ka6, the white king is more protected, and the board is less open. The black king is still a threat, but the white queen has more control over the center and the board. - The subsequent move Kxa5 does not significantly improve the position, and the black king continues to be a threat. (Superficial evaluation) Conclusion: While ...

  44. [44]

    26 Learning Stateful Predictive Knowledge From Experience

    **White to Move**: White is in the process of checkmating Black. 26 Learning Stateful Predictive Knowledge From Experience

  45. [45]

    Black has a king and a rook

    **Material**: White has a rook, two knights, and a king. Black has a king and a rook

  46. [46]

    - Black’s rook is on the 7th rank, which is a weak position

    **Positional Evaluation**: - White’s rook is on the 6th rank, which is a strong position. - Black’s rook is on the 7th rank, which is a weak position. - White’s knights are well-placed to attack Black’s king. - Black’s king is in the center, with no immediate escape routes. Strategic Considerations: - **Checkmate Threat**: White has a strong checkmate thr...

  47. [47]

    **Rd8**: Checkmate with the rook

  48. [48]

    **Rd7**: Checkmate with the rook

  49. [49]

    **Rxh6**: Captures a piece and puts pressure on Black’s king

  50. [50]

    **Rg6**: Check with the rook, putting pressure on Black’s king

  51. [51]

    **Rf6**: Check with the rook, putting pressure on Black’s king

  52. [52]

    **Rc6**: Check with the rook, putting pressure on Black’s king

  53. [53]

    **Rxb6**: Captures a piece and puts pressure on Black’s king

  54. [54]

    **Nxb6**: Captures a piece and puts pressure on Black’s king

  55. [55]

    **Nc5**: Maneuvering the knight to a key square

  56. [56]

    Reasoning: - **Rd8**: This move is the most direct and immediate checkmate threat

    **Ng5**: Maneuvering the knight to a key square. Reasoning: - **Rd8**: This move is the most direct and immediate checkmate threat. If Black’s king cannot escape, this will result in checkmate. - **Rd7**: Similar to Rd8, but slightly less direct. It still puts pressure on Black’s king and can lead to a quick checkmate. - **Rxh6**: Captures a piece and put...

  57. [57]

    It resulted in a perfect game termination with a reward of 1.0

    ActionQd8appears to be the best action among the three. It resulted in a perfect game termination with a reward of 1.0. This action clearly outperformed the other two actions in terms of long-term value. The simulation history for Qd8is overwhelmingly positive, and it is the only action that led to a game termination with a perfect reward. Therefore, we c...

  58. [58]

    Although it did not result in a game termination, it did not yield any positive reward

    ActionQd7did not perform well in the simulation history. Although it did not result in a game termination, it did not yield any positive reward. The simulation history forQd7did not provide any evidence of future value. In fact, the game terminated with a reward of 0.0, indicating that the action did not lead to a successful outcome. The simulation histor...

  59. [59]

    However, we should not completely rule out the possibility that Qd6 could perform better in future game play

    ActionQd6...[Same inference as 2]. However, we should not completely rule out the possibility that Qd6 could perform better in future game play. We should continue to consider Qd6 as a potential action in future game play, but we should not give it the same level of consideration as Qd8. We should be cautious in our use of Qd6 and only consider it as a po...