Constraints-Guided Diffusion Reasoner for Neuro-Symbolic Learning

Chao Qu; Weidi Xu; Xuan Zhang; Yanting Miao; Yuan Qi; Zhijian Zhou

REVIEW 3 major objections 6 minor 12 references

Constraints-Guided Diffusion Reasoner for Neuro-Symbolic Learning

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that fine-tuning a masked diffusion reasoner on a binary constraint-satisfaction reward makes it solve symbolic puzzles—Sudoku, mazes, shortest paths, and preference rankings—far more accurately than supervised denoising al

desk verdict Worth taking seriously — RL fine-tuning consistently improves a masked diffusion reasoner on symbolic puzzles, but the "constraints-guided" reward is mostly exact-match and the group-advantage update has an unstated zero-variance gap. read the letter →

arxiv 2508.16524 v1 pith:XXMUI3XX submitted 2025-08-22 cs.AI

Xuan Zhang , Zhijian Zhou , Weidi Xu , Yanting Miao , Chao Qu , Yuan Qi This is my paper

classification cs.AI

keywords neuro-symboliclearningdiffusionmodelsreinforcementconstraintsatisfactionpolicyoptimizationSudokumazenavigationlogicalconsistency

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper is trying to establish that a diffusion model can be turned into a competent symbolic reasoner by training it in two stages: first, ordinary supervised denoising on solved puzzles; second, reinforcement learning that rewards only outputs satisfying the puzzle's hard constraints. The authors claim this second stage is decisive: it lifts complete-board Sudoku accuracy from 78.19 to 97.79 percent on the big kaggle set and from 8.03 to 18.25 percent on minimal-17 Sudoku, and takes maze solving from 43.30 percent to perfect accuracy at 20x20. If true, the work matters because it shows hard logical consistency can be learned as a reward signal rather than glued onto a network through solvers or constraint layers, and the resulting model still runs as a single-pass neural generator at inference. The punchline is a training recipe, not a new architecture: masked denoising plus a sparse rule-based reward and group-normalized policy updates.

What carries the argument

The central mechanism is the denoising process treated as a Markov decision process and optimized by constraints-guided policy optimization: a masked DDPM supplies the sampling distribution p_theta(x_{t-1}|x_t); the final discretized sample x_0 is scored by a binary rule-based consistency reward; advantages are normalized within each puzzle's sampled group using the group mean and standard deviation; and a clipped importance-ratio objective limits each policy update. The same machinery includes dynamic sampling that drops solved puzzles and allocates more trajectories to harder ones.

What would settle it

Force a zero-variance reward group by running the released code with a puzzle the current policy always solves, or never solves, and check whether the update crashes, skips, or uses a fallback; alternatively, replace group normalization with a fixed constant baseline and test whether the 100 percent Maze and 97.79 percent Sudoku numbers survive.

Watch

Extended reading notes

Core claim

DDReasoner represents each puzzle as a continuous one-hot distribution, keeps observed cells fixed with a mask, and denoises the rest with a U-Net-style network. Stage one trains the network to predict added noise on complete solutions. Stage two treats each denoising trajectory as a Markov decision process: states are (x_t, t), actions are the denoising steps, and only the final discretized solution receives a reward of 1 if it satisfies the task's hard constraints and 0 otherwise. A clipped policy-gradient objective with group-relative advantage, computed from a group of sampled trajectories per puzzle, updates the policy, while unsolved puzzles are re-sampled in later epochs with more tra

Load-bearing premise

The RL update assumes that within every sampled group of puzzle attempts the rewards are not all the same; if a batch is entirely correct or entirely wrong, the advantage is undefined, and the paper does not state what the implementation does in that case.

Editorial extensions

If this is right

A neural reasoner trained this way can satisfy exact constraints on unseen instances without invoking a SAT solver, Prolog, or a search algorithm at inference time.
Because a single wrong cell is enough to zero the reward, the RL stage strongly penalizes near-miss solutions that supervised per-position loss treats as nearly correct.
Because the reward is rule-based rather than tied to one dataset's labels, the same training procedure transfers across Sudoku datasets and maze sizes, as reported in the cross-dataset experiments.
On mazes, the fine-tuned 5.3M-parameter model reaches the same perfect accuracy as a specialized transformer baseline while using fewer parameters, less data, and less training time.
If the paper's priority claim is correct, diffusion models become a viable policy class for neuro-symbolic tasks previously dominated by constraint-loss or solver-augmented architectures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct test of whether the binary all-or-nothing consistency reward is the active ingredient would be to keep the RL loop but replace it with a smoothed degree-of-violation reward; the paper does not run this ablation, and a large gap would argue that binary feedback drives the gains.
The dynamic curriculum that drops solved puzzles and upsamples hard ones is entangled with the policy-gradient update; isolating the two would show how much of the reported improvement comes from curriculum selection rather than the clipped objective.
The same masked-diffusion-plus-rule-reward recipe should transfer to other combinatorial problems with cheap verifiers, such as scheduling or routing, where multiple valid solutions exist and supervised losses misweight global consistency.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is worth reading more for the empirical recipe than for the neuro-symbolic framing: train a masked DDPM with plain supervised learning, then fine-tune with PPO/GRPO-style policy optimization using a sparse binary reward and dynamic sampling that concentrates on unsolved puzzles. Across Sudoku, Maze, simple path, preference learning, and min-cost path, RL consistently improves over its own SL checkpoint. The cross-dataset Sudoku numbers are the most interesting result — big kaggle to multiple sol going from 3.30 to 23.60 is a real generalization gain, and the paper honestly shows that scaling up SL alone does not close that gap.

Credit where it is due: the method is stated in equations, the components are standard and properly cited, the benchmarks are external, and the limitations section openly admits that RL may be ineffective when the SL base is too weak. That last point tells me the authors have a realistic sense of scope.

Now the soft spots, in rough order of severity. First, the reward is mostly not constraint-based. Appendix B.2 step 3 says rewards come from direct comparison with a precomputed ground-truth solution for every task except multiple-sol Sudoku and min-cost path. So "constraints-guided" overstates what is happening; only those two tasks use rule-based checking. Relatedly, for unique-answer tasks the reported consistency metric is by definition the training reward check, so RL improving consistency is partly by construction. That is a framing issue rather than a fatal flaw.

Second, the stress-test about Eq. 11 lands. The advantage normalizes binary rewards by group standard deviation, and any all-fail or all-success group has std=0. The dynamic sampling scheme deliberately retains hard unsolved puzzles, so all-fail groups are not rare in early epochs. The paper never specifies the guard — epsilon, filtering, or fallback baseline. This is load-bearing for exact replication of the Table 2 and Table 3 gains, because the policy update is undefined in exactly the cases the curriculum targets. It is likely just an implementation detail, but it must be stated.

Third, the main tables have no error bars or seed variance, apart from one Maze cross-size table where the authors do report them. Given the stochasticity of RL and diffusion sampling, that is a real omission. Fourth, closely related diffusion-reasoning baselines (d1, van Krieken et al., DINGO) are cited but not compared empirically; one direct comparison would make the contribution much easier to position.

This is a solid application paper with a clear, consistent empirical trend and an honest limitations section. It deserves a serious referee, not a desk reject. I would send it to review and ask for three things: specify the zero-variance handling, reframe the reward claim, and add variance estimates.

Referee Report

3 major / 6 minor

Summary. The paper proposes DDReasoner, a masked DDPM-based reasoner for symbolic constraint-satisfaction tasks (Sudoku, Maze, simple path prediction, preference learning, minimum-cost path finding). Training is two-stage: first a basic supervised-learning (SL) checkpoint, then reinforcement learning (RL) that treats the denoising process as an MDP and optimizes a binary reward meant to encode logical consistency. The authors report large improvements of DDReasoner-RL over DDReasoner-SL, including perfect Maze accuracy up to 20x20 and large Sudoku gains, and claim this is the first RL-based training of diffusion models for symbolic logical reasoning. The paper includes a reproducibility URL, Appendix B with implementation details, and cross-dataset/generalization experiments.

Significance. If the reported results hold, the paper demonstrates a practically effective recipe for fine-tuning diffusion reasoners on structured puzzles using a group-relative policy-gradient update. Credit is due for using standard DDPM and PPO formulations, releasing code, and evaluating on multiple external benchmarks with cross-dataset transfer experiments. The main scientific value, however, rests on the claim that a constraint-satisfaction reward, rather than additional supervised fitting, drives the gains. That claim is currently not supported for most benchmarks because the implemented reward is exact match to a precomputed ground-truth solution, not a rule-based consistency check. The central mechanism therefore needs to be re-stated or re-implemented before the contribution can be assessed.

major comments (3)

[Appendix B.2, step 3; Eq. (10)] The 'constraints-guided' claim is not supported for most benchmarks. Eq. (10) defines consistency for unique-answer tasks as equivalent(Discretize(x0), a), i.e., exact equality to a precomputed ground-truth solution. Appendix B.2, step 3 then states that rewards for all tasks are derived from direct comparison with the ground truth, except Sudoku (multiple sol) and Minimum-Cost Path Finding, which use rule checking. Thus on big-kaggle, minimal-17, satnet-data, Maze, simple path, and preference learning, the binary reward is an exact-match label reward, not a rule-based consistency check. The reported 'consistent' improvements on those tasks are then partly an artifact of optimizing the evaluation metric itself. The abstract/intro claim that the reward comes from 'logical consistency of neural outputs' should be substantially qualified, or rule-based constraints should actually be used.
[Eq. (11); Algorithm 1] The advantage estimator in Eq. (11) divides by the within-group standard deviation of binary rewards. Since r(x0) in Eq. (9) is 0/1, any group with all failures or all successes has std=0, making A_i^t undefined. The curriculum in Algorithm 1 deliberately retains unsolved puzzles and assigns larger G to low-accuracy puzzles, so all-failure groups are expected on hard puzzles, especially in early epochs after the SL checkpoint has low solve rate. The paper specifies no epsilon, filtering rule, or fallback baseline for zero-variance groups. Without this, the policy update for the exact puzzles whose gains are highlighted (minimal-17, 15x15/20x20 Maze, k=30) is not well-defined, and exact replication depends on an unstated implementation detail. Please specify the guard and its effect on the gradient.
[§Experiment / Table 7; §Appendix B.4] The comparison between DDReasoner-SL and DDReasoner-RL is not always apples-to-apples in the main tables. Table 7 shows that scaling up SL (DDReasoner-SL*) already yields large gains (e.g., big-kaggle test 82.82 vs. 78.19; Maze 10x10 91.35 vs. 70.95), and the main Tables 2-3 compare against the smaller SL checkpoint. The RL-specific improvement is better isolated by the SL* column, but the paper does not consistently present SL* alongside RL in the main result tables. This is not fatal, but the reader should be able to see the RL-only gain without cross-referencing the appendix.

minor comments (6)

[Figure 1] The figure uses q_0, q_T alongside p_T, p_0, while the text defines the process with p notation. This is confusing and should be made consistent.
[Eq. (5)] The notation c ~ ppuzzle(xT) suggests c is random, but c appears to be a fixed clue/hint vector for a given puzzle. Use c_i or 'known clue vector' to avoid ambiguity.
[Algorithm 1] The line 'Compute the policy gradient ... update θ: pθold ← pθ' is ambiguous: the old policy should be updated only after the gradient update, not as part of the same statement. Specify the order explicitly.
[Eq. (12)] The objective sums min(...) over G samples but omits a 1/G normalization. This is a minor notational issue but should be fixed for clarity.
[Appendix D, Limitations] Typo: 'Chanllenging' should be 'Challenging'. Also '1.51x more parameters-efficient' is awkward; consider '1.51x more parameter-efficient'.
[Related Work / Introduction] The paper claims to be 'the first to utilize RL techniques for training diffusion models to solve symbolic logical reasoning tasks.' Given that the reference list includes d1 (Zhao et al. 2025), which applies RL to diffusion LLMs for reasoning, the novelty claim should be more carefully scoped to avoid overstatement.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the RL objective is the reported consistency metric by explicit design, but the paper's claims are empirical and benchmarked externally.

full rationale

The derivation chain is self-contained and does not reduce to its inputs. Stage 1 uses the standard masked DDPM objective (Eq. 6). Stage 2 formulates denoising as an MDP following Black et al. (2024) and uses a GRPO-style group advantage (Eq. 11) with a standard clipped PPO objective (Eq. 12); these are external, established methods, not self-citations. The reward in Eqs. 9-10 is rule-based or exact-match against ground truth, and the reported evaluation metrics (exact, consistent) are the same quantities. That overlap is not a hidden circular step: the paper explicitly states that it uses constraint-consistency as the reward to improve constraint-consistency, and the empirical content is whether this reward can be optimized and whether it generalizes to held-out puzzles and cross-dataset transfer (Tables 2, 3, 10, 11). No parameter is fitted to the test set, no load-bearing result is justified only by an author's prior work, and no uniqueness theorem is imported. The potential zero-variance issue in Eq. 11 is an implementation/reproducibility ambiguity, not a circularity. The paper's own stated limitations (e.g., RL may be ineffective if SL cannot learn underlying relationships) further show that the authors do not treat RL as guaranteed by construction. Therefore no significant circularity is present.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim does not derive a constant or law; the expense is paid in training hyperparameters and modeling assumptions. The most consequential items are the reward definition (exact match vs rule check), the curriculum thresholds, and the unstated zero-variance handling in Eq 11. No new physical or conceptual entities are introduced.

free parameters (6)

diffusion timesteps T = 20
Chosen uniformly for all tasks (Appendix B.2); not derived from data or theory.
initial group size G_initial = 64
Set for RL sampling; 64/128/256 tiers with gamma=4 (Appendix B.2).
sample scaling factor gamma = 4
Controls extra sampling for hard puzzles; no principled selection is given.
binarization threshold = 0.5
Used to discretize Maze and path outputs; changing the threshold would shift reported accuracy.
SL data/epoch budget per task = task-specific
Table 8/9: different batch sizes, sample counts, and learning rates; the SL-to-RL transition is a manual tuning choice discussed in Appendix D.
PPO clip epsilon = not reported
Clipping range 1 +/- epsilon is used in Eq 12, but the value is never given and is needed to reproduce.

assumptions (5)

domain assumption The denoising process can be modeled as an MDP with p_theta(x_{t-1}|x_t) as the policy and a terminal reward r(x0).
Inherited from Black et al. 2024; treats stochastic sampling as a policy without formal justification in this discrete setting.
domain assumption For unique-answer tasks, logical consistency is equivalent to exact equality with the precomputed ground-truth solution a (Eq 10).
This makes the reward supervised rather than a pure rule check, and it depends on solver correctness and on uniqueness guarantees for each dataset.
standard math The standard DDPM objective and the PPO clipped surrogate are valid optimization targets for this setting.
Equations 1-4 are taken from Ho et al. 2020 and Schulman et al. 2017; the paper does not re-derive them.
domain assumption Generated Maze datasets via DFS are acyclic and have unique solutions (Appendix B.1).
If a maze has multiple valid paths, the exact-match reward would penalize valid alternatives and bias training.
domain assumption A U-Net with constant spatial resolution can represent position-dependent reasoning rules for puzzles of varying size.
Architecture choice in Appendix B.2; no proof or analysis is given that this capacity suffices.

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Cite this review

Pith. "Pith review of Constraints-Guided Diffusion Reasoner for Neuro-Symbolic Learning." pith.science (2026). https://pith.science/paper/XXMUI3XX

@misc{pith2026250816524,
  author       = {Pith},
  title        = {Pith review of: Constraints-Guided Diffusion Reasoner for Neuro-Symbolic Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XXMUI3XX}},
  note         = {Machine review of arXiv:2508.16524}
}

read the original abstract

Enabling neural networks to learn complex logical constraints and fulfill symbolic reasoning is a critical challenge. Bridging this gap often requires guiding the neural network's output distribution to move closer to the symbolic constraints. While diffusion models have shown remarkable generative capability across various domains, we employ the powerful architecture to perform neuro-symbolic learning and solve logical puzzles. Our diffusion-based pipeline adopts a two-stage training strategy: the first stage focuses on cultivating basic reasoning abilities, while the second emphasizes systematic learning of logical constraints. To impose hard constraints on neural outputs in the second stage, we formulate the diffusion reasoner as a Markov decision process and innovatively fine-tune it with an improved proximal policy optimization algorithm. We utilize a rule-based reward signal derived from the logical consistency of neural outputs and adopt a flexible strategy to optimize the diffusion reasoner's policy. We evaluate our methodology on some classical symbolic reasoning benchmarks, including Sudoku, Maze, pathfinding and preference learning. Experimental results demonstrate that our approach achieves outstanding accuracy and logical consistency among neural networks.

Figures

Figures reproduced from arXiv: 2508.16524 by the authors.

**Figure 2.** Overall workflow of DDReasoner’s RL training. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Examples of a valid Sudoku solution and two in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: (a) An example of one-hot representation of Sudoku puzzles; (b) An example of one-hot representation of [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Examples of reward curve during RL training on Sudoku and Maze. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: The efficiency comparison results on 5 × 5 Maze between DDReasoner and Baseline. set DDReasoner-SL DDReasoner-RL M: 20, N: 10 51.08 ± 0.36 98.19 ± 0.36 M: 20, N: 5 39.43 ± 0.73 99.03 ± 0.03 M: 5, N: 10 4.60 33.55 M: 10, N: 20 12.60 52.45 [PITH_FULL_IMAGE:figures/full_…

**Figure 7.** Figure 7: Examples of the effect of RL training on specific Sudoku cases. The blue numbers denote the hints already provided [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: Examples of the effect of RL training on specific Maze cases. [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

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Reference graph

Works this paper leans on

12 extracted references · 10 canonical work pages

[1]

Problems that rank lower by this metric are then sampled more intensively during the current epoch

Select data : From the second epoch onwards, the re- maining challenging data is then processed in mini- batches, in which puzzles are ranked in an ascending or- der according to their accc,mask in the previous epoch. Problems that rank lower by this metric are then sampled more intensively during the current epoch

work page
[2]

Sample trajectories: The pre-trained modelpθSL is used to generate G solution trajectories, capturing the hidden layers of intermediate steps xt across T steps as well as the final predicted solution x0

work page
[3]

arXiv:2311.12908

Diffusion Model Alignment Using Direct Preference Optimization. arXiv:2311.12908. Wang, P.; Donti, P. L.; Wilder, B.; and Kolter, J. Z. 2019. SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver. CoRR, abs/1905.12149. Wang, W.; and Pan, S. J. 2019. Integrating Deep Learn- ing with Logic Fusion for Information E...

arXiv 2019
[4]

The fine-tuned model pθold is then updated

Fine-tune with RL methods : The rewards collected in each trajectory i are used to calculate the Advantage ˆAi t, while the importance sampling ratio I i t is calculated us- ing the hidden layers xi t at each step t. The fine-tuned model pθold is then updated. At the end of each epoch, the updated model pθ is used to perform new sampling, facilitating an ...

work page
[6]

Calculate rewards: The reward for each generated so- lution x0 is primarily ascertained by its adherence to the problem’s logical hard constraints

work page
[8]

During the initial epoch, each puz- zle is sampled 64 times

In step 1, we set the hyperparameter Ginitial to 64 and the scaling factor γ to 4. During the initial epoch, each puz- zle is sampled 64 times. For subsequent epochs, the num- ber of samples per puzzle is dynamically adjusted within the range of [64, 256]. To prevent GPU memory overload and ensure training efficiency, we establish three discrete sampling ...

work page
[9]

In step 3, to enhance computational efficiency and train- ing speed, rewards for all tasks are derived from di- rect comparison with the ground truth, except for Su- doku (multiple sol) and Minimum-Cost Path Finding. Given that puzzles in tasks Sudoku (multiple sol) and Minimum-Cost Path Finding can admit multiple valid so- lutions satisfying all constrai...

work page
[10]

Scaling to greater complexity and broader domains : Future work can focus on extending DDReasoner-RL to tackle higher-dimensional problems, e.g., visual Su- doku and visual generation related to physical consis- tency. While scaling up to more complex spaces may necessitate the addition of an auxiliary neural network modules and logical constraint functio...

work page

Show all 12 references

[11]

Enhancing interpretability and explainability : While DDReasoner-RL demonstrates strong performance, fur- ther research into understanding its decision-making pro- cess during the denoising steps could provide valuable insights into how neural networks internalize and apply lo...
[12]

Refining training dynamics and efficiency: Further ex- ploration into optimizing the interplay between the SL and RL stages, including the precise timing of the tran- sition and the design of even more sophisticated reward or curriculum learning strategies for the RL phase, co...
[2023]

In The Eleventh International Conference on Learning Representations (ICLR)

Learning where and when to reason in neuro-symbolic inference. In The Eleventh International Conference on Learning Representations (ICLR). Dhariwal, P.; and Nichol, A. Q. 2021. Diffusion Models Beat GANs on Image Synthesis. In Beygelzimer, A.; Dauphin, Y .; Liang, P.; and Vau...

2021 arXiv
[2025]

arXiv:2505.13138

Neurosymbolic Diffusion Models. arXiv:2505.13138. Wallace, B.; Dang, M.; Rafailov, R.; Zhou, L.; Lou, A.; Pu- rushwalkam, S.; Ermon, S.; Xiong, C.; Joty, S.; and Naik, N

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Problems that rank lower by this metric are then sampled more intensively during the current epoch

Select data : From the second epoch onwards, the re- maining challenging data is then processed in mini- batches, in which puzzles are ranked in an ascending or- der according to their accc,mask in the previous epoch. Problems that rank lower by this metric are then sampled more intensively during the current epoch

work page

[2] [2]

Sample trajectories: The pre-trained modelpθSL is used to generate G solution trajectories, capturing the hidden layers of intermediate steps xt across T steps as well as the final predicted solution x0

work page

[3] [3]

arXiv:2311.12908

Diffusion Model Alignment Using Direct Preference Optimization. arXiv:2311.12908. Wang, P.; Donti, P. L.; Wilder, B.; and Kolter, J. Z. 2019. SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver. CoRR, abs/1905.12149. Wang, W.; and Pan, S. J. 2019. Integrating Deep Learn- ing with Logic Fusion for Information E...

arXiv 2019

[4] [4]

The fine-tuned model pθold is then updated

Fine-tune with RL methods : The rewards collected in each trajectory i are used to calculate the Advantage ˆAi t, while the importance sampling ratio I i t is calculated us- ing the hidden layers xi t at each step t. The fine-tuned model pθold is then updated. At the end of each epoch, the updated model pθ is used to perform new sampling, facilitating an ...

work page

[5] [6]

Calculate rewards: The reward for each generated so- lution x0 is primarily ascertained by its adherence to the problem’s logical hard constraints

work page

[6] [8]

During the initial epoch, each puz- zle is sampled 64 times

In step 1, we set the hyperparameter Ginitial to 64 and the scaling factor γ to 4. During the initial epoch, each puz- zle is sampled 64 times. For subsequent epochs, the num- ber of samples per puzzle is dynamically adjusted within the range of [64, 256]. To prevent GPU memory overload and ensure training efficiency, we establish three discrete sampling ...

work page

[7] [9]

In step 3, to enhance computational efficiency and train- ing speed, rewards for all tasks are derived from di- rect comparison with the ground truth, except for Su- doku (multiple sol) and Minimum-Cost Path Finding. Given that puzzles in tasks Sudoku (multiple sol) and Minimum-Cost Path Finding can admit multiple valid so- lutions satisfying all constrai...

work page

[8] [10]

Scaling to greater complexity and broader domains : Future work can focus on extending DDReasoner-RL to tackle higher-dimensional problems, e.g., visual Su- doku and visual generation related to physical consis- tency. While scaling up to more complex spaces may necessitate the addition of an auxiliary neural network modules and logical constraint functio...

work page