Dynamical Systems Theory Behind a Hierarchical Reasoning Model

Current large language models (LLMs) primarily rely on linear sequence generation and massive parameter counts, yet they severely struggle with complex algorithmic reasoning. While recent reasoning architectures, such as the Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), demonstrate that compact recursive networks can tackle these tasks, their training dynamics often lack rigorous mathematical guarantees, leading to instability and representational collapse. We propose the Contraction Mapping Model (CMM), a novel architecture that reformulates discrete recursive reasoning into continuous Neural Ordinary and Stochastic Differential Equations (NODEs/NSDEs). By explicitly enforcing the convergence of the latent phase point to a stable equilibrium state and mitigating feature collapse with a hyperspherical repulsion loss, the CMM provides a mathematically grounded and highly stable reasoning engine. On the Sudoku-Extreme benchmark, a 5M-parameter CMM achieves a state-of-the-art accuracy of 93.7 %, outperforming the 27M-parameter HRM (55.0 %) and 5M-parameter TRM (87.4 %). Remarkably, even when aggressively compressed to an ultra-tiny footprint of just 0.26M parameters, the CMM retains robust predictive power, achieving 85.4 % on Sudoku-Extreme and 82.2 % on the Maze benchmark. These results establish a new frontier for extreme parameter efficiency, proving that mathematically rigorous latent dynamics can effectively replace brute-force scaling in artificial reasoning.

Large Neighborhood Search meets Iterative Neural Constraint Heuristics

Neural networks are being increasingly used as heuristics for constraint satisfaction. These neural methods are often recurrent, learning to iteratively refine candidate assignments. In this work, we make explicit the connection between such iterative neural heuristics and Large Neighborhood Search (LNS), and adapt an existing neural constraint satisfaction method-ConsFormer-into an LNS procedure. We decompose the resulting neural LNS into two standard components: the destroy and repair operators. On the destroy side, we instantiate several classical heuristics and introduce novel prediction-guided operators that exploit the model's internal scores to select neighborhoods. On the repair side, we utilize ConsFormer as a neural repair operator and compare the original sampling-based decoder to a greedy decoder that selects the most likely assignments. Through an empirical study on Sudoku, Graph Coloring, and MaxCut, we find that adapting the neural heuristic to an LNS procedure yields substantial gains over its vanilla settings and improves its competitiveness with classical and neural baselines. We further observe consistent design patterns across tasks: stochastic destroy operators outperform greedy ones, while greedy repair is more effective than sampling-based repair for finding a single high-quality feasible assignment. These findings highlight LNS as a useful lens and design framework for structuring and improving iterative neural approaches.

AS2 -- Attention-Based Soft Answer Sets: An End-to-End Differentiable Neuro-Soft-Symbolic Reasoning Architecture

Neuro-symbolic artificial intelligence (AI) systems typically couple a neural perception module to a discrete symbolic solver through a non-differentiable boundary, preventing constraint-satisfaction feedback from reaching the perception encoder during training. We introduce AS2 (Attention-Based Soft Answer Sets), a fully differentiable neuro-symbolic architecture that replaces the discrete solver with a soft, continuous approximation of the Answer Set Programming (ASP) immediate consequence operator $T_P$. AS2 maintains per-position probability distributions over a finite symbol domain throughout the forward pass and trains end-to-end by minimizing the fixed-point residual of a probabilistic lift of $T_P$, thereby differentiating through the constraint check without invoking an external solver at either training or inference time. The architecture is entirely free of conventional positional embeddings. Instead, it encodes problem structure through constraint-group membership embeddings that directly reflect the declarative ASP specification, making the model agnostic to arbitrary position indexing. On Visual Sudoku, AS2 achieves 99.89% cell accuracy and 100% constraint satisfaction (verified by Clingo) across 1,000 test boards, using a greedy constrained decoding procedure that requires no external solver. On MNIST Addition with $N \in \{2, 4, 8\}$ addends, AS2 achieves digit accuracy above 99.7% across all scales. These results demonstrate that a soft differentiable fixpoint operator, combined with constraint-aware attention and declarative constraint specification, can match or exceed pipeline and solver-based neuro-symbolic systems while maintaining full end-to-end differentiability.

Self-Aware Markov Models for Discrete Reasoning

Standard masked discrete diffusion models face limitations in reasoning tasks due to their inability to correct their own mistakes on the masking path. Since they rely on a fixed number of denoising steps, they are unable to adjust their computation to the complexity of a given problem. To address these limitations, we introduce a method based on learning a Markov transition kernel that is trained on its own outputs. This design enables tokens to be remasked, allowing the model to correct its previous mistakes. Furthermore, we do not need a fixed time schedule but use a trained stopping criterion. This allows for adaptation of the number of function evaluations to the difficulty of the reasoning problem. Our adaptation adds two lightweight prediction heads, enabling reuse and fine-tuning of existing pretrained models. On the Sudoku-Extreme dataset we clearly outperform other flow based methods with a validity of 95%. For the Countdown-4 we only need in average of 10 steps to solve almost 96% of them correctly, while many problems can be solved already in 2 steps.

EndoCoT: Scaling Endogenous Chain-of-Thought Reasoning in Diffusion Models

Recently, Multimodal Large Language Models (MLLMs) have been widely integrated into diffusion frameworks primarily as text encoders to tackle complex tasks such as spatial reasoning. However, this paradigm suffers from two critical limitations: (i) MLLMs text encoder exhibits insufficient reasoning depth. Single-step encoding fails to activate the Chain-of-Thought process, which is essential for MLLMs to provide accurate guidance for complex tasks. (ii) The guidance remains invariant during the decoding process. Invariant guidance during decoding prevents DiT from progressively decomposing complex instructions into actionable denoising steps, even with correct MLLM encodings. To this end, we propose Endogenous Chain-of-Thought (EndoCoT), a novel framework that first activates MLLMs' reasoning potential by iteratively refining latent thought states through an iterative thought guidance module, and then bridges these states to the DiT's denoising process. Second, a terminal thought grounding module is applied to ensure the reasoning trajectory remains grounded in textual supervision by aligning the final state with ground-truth answers. With these two components, the MLLM text encoder delivers meticulously reasoned guidance, enabling the DiT to execute it progressively and ultimately solve complex tasks in a step-by-step manner. Extensive evaluations across diverse benchmarks (e.g., Maze, TSP, VSP, and Sudoku) achieve an average accuracy of 92.1%, outperforming the strongest baseline by 8.3 percentage points.

Diffusion LLMs can think EoS-by-EoS

Diffusion LLMs have been proposed as an alternative to autoregressive LLMs, excelling especially at complex reasoning tasks with interdependent sub-goals. Curiously, this is particularly true if the generation length, i.e., the number of tokens the model has to output, is set to a much higher value than is required for providing the correct answer to the task, and the model pads its answer with end-of-sequence (EoS) tokens. We hypothesize that diffusion models think EoS-by-EoS, that is, they use the representations of EoS tokens as a hidden scratchpad, which allows them to solve harder reasoning problems. We experiment with the diffusion models LLaDA1.5, LLaDA2.0-mini, and Dream-v0 on the tasks Addition, Entity Tracking, and Sudoku. In a controlled prompting experiment, we confirm that adding EoS tokens improves the LLMs' reasoning capabilities. To further verify whether they serve as space for hidden computations, we patch the hidden states of the EoS tokens with those of a counterfactual generation, which frequently changes the generated output to the counterfactual. The success of the causal intervention underscores that the EoS tokens, which one may expect to be devoid of meaning, carry information on the problem to solve. The behavioral experiments and the causal interventions indicate that diffusion LLMs can indeed think EoS-by-EoS.

Recursive Inference Machines for Neural Reasoning

Neural reasoners such as Tiny Recursive Models (TRMs) solve complex problems by combining neural backbones with specialized inference schemes. Such inference schemes have been a central component of stochastic reasoning systems, where inference rules are applied to a stochastic model to derive answers to complex queries. In this work, we bridge these two paradigms by introducing Recursive Inference Machines (RIMs), a neural reasoning framework that explicitly incorporates recursive inference mechanisms inspired by classical inference engines. We show that TRMs can be expressed as an instance of RIMs, allowing us to extend them through a reweighting component, yielding better performance on challenging reasoning benchmarks, including ARC-AGI-1, ARC-AGI-2, and Sudoku Extreme. Furthermore, we show that RIMs can be used to improve reasoning on other tasks, such as the classification of tabular data, outperforming TabPFNs.

Symbol-Equivariant Recurrent Reasoning Models

Reasoning problems such as Sudoku and ARC-AGI remain challenging for neural networks. The structured problem solving architecture family of Recurrent Reasoning Models (RRMs), including Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), offer a compact alternative to large language models, but currently handle symbol symmetries only implicitly via costly data augmentation. We introduce Symbol-Equivariant Recurrent Reasoning Models (SE-RRMs), which enforce permutation equivariance at the architectural level through symbol-equivariant layers, guaranteeing identical solutions under symbol or color permutations. SE-RRMs outperform prior RRMs on 9x9 Sudoku and generalize from just training on 9x9 to smaller 4x4 and larger 16x16 and 25x25 instances, to which existing RRMs cannot extrapolate. On ARC-AGI-1 and ARC-AGI-2, SE-RRMs achieve competitive performance with substantially less data augmentation and only 2 million parameters, demonstrating that explicitly encoding symmetry improves the robustness and scalability of neural reasoning. Code is available at https://github.com/ml-jku/SE-RRM.

Can Continuous-Time Diffusion Models Generate and Solve Globally Constrained Discrete Problems? A Study on Sudoku

Can standard continuous-time generative models represent distributions whose support is an extremely sparse, globally constrained discrete set? We study this question using completed Sudoku grids as a controlled testbed, treating them as a subset of a continuous relaxation space. We train flow-matching and score-based models along a Gaussian probability path and compare deterministic (ODE) sampling, stochastic (SDE) sampling, and DDPM-style discretizations derived from the same continuous-time training. Unconditionally, stochastic sampling substantially outperforms deterministic flows; score-based samplers are the most reliable among continuous-time methods, and DDPM-style ancestral sampling achieves the highest validity overall. We further show that the same models can be repurposed for guided generation: by repeatedly sampling completions under clamped clues and stopping when constraints are satisfied, the model acts as a probabilistic Sudoku solver. Although far less sample-efficient than classical solvers and discrete-geometry-aware diffusion methods, these experiments demonstrate that classic diffusion/flow formulations can assign non-zero probability mass to globally constrained combinatorial structures and can be used for constraint satisfaction via stochastic search.

Speed is Confidence

Biological neural systems must be fast but are energy-constrained. Evolution's solution: act on the first signal. Winner-take-all circuits and time-to-first-spike coding implicitly treat when a neuron fires as an expression of confidence. We apply this principle to ensembles of Tiny Recursive Models (TRM). By basing the ensemble prediction solely on the first to halt rather than averaging predictions, we achieve 97.2% puzzle accuracy on Sudoku-Extreme while using 10x less compute than test-time augmentation (the baseline achieves 86.1% single-pass, 97.3% with TTA). Inference speed is an implicit indication of confidence. But can this capability be manifested as a training-only cost? Evidently yes: by maintaining K = 4 parallel latent states during training but backpropping only through the lowest-loss "winner," a single model achieves 96.9% +/- 0.6% puzzle accuracy with a single forward pass-matching TTA performance without any test-time augmentation. As in nature, this work was also resource constrained: all experimentation used a single RTX 5090. This necessitated efficiency and compelled our invention of a modified SwiGLU which made Muon viable. With Muon and K = 1 training, we exceed TRM baseline performance in 7k steps (40 min). Higher accuracy requires 36k steps: 1.5 hours for K = 1, 6 hours for K = 4.