We conduct a novel theoretical analysis to demonstrate CODA's

Markov Condition, are not independent of each other .

Based onLemma 5.1anditsproof, weknownthatthePMP oftheauxiliary control system, (S.2), is exactly the differential PMP equations (13). Thus below, we only look at the differential PMP equationsin(S.2). In the system identification experiment, we collect a total number of five trajectories from systems (in Table 2) with dynamics known, wherein different trajectoriesξo = {xo0:T,u0:T 1}havedifferent initial conditionsx0 andhorizonsT (T ranges from10to20),with randominputsu0:T 1 drawnfromuniformdistribution. In fact, throughout the entire learning process, PDP always guarantees that the policyconstraint isperfectly respected (as the forward pass strictly follows the policy). Please seeAppendix Fig. S4for validation.

Automated decision-making under uncertainty requires balancing exploitation and exploration. Classical methods treat these separately using heuristics, while Active Inference unifies them through Expected Free Energy (EFE) minimization. However, EFE minimization is computationally expensive, limiting scalability. We build on recent theory recasting EFE minimization as variational inference, formally unifying it with Planning-as-Inference and showing the epistemic drive as a unique entropic contribution. Our main contribution is a novel message-passing scheme for this unified objective, enabling scalable Active Inference in factored-state MDPs and overcoming high-dimensional planning intractability.

We present a novel approach to neural code generation that incorporates real-time execution signals into the language model generation process. While large language models (LLMs) have demonstrated impressive code generation capabilities, they typically do not utilize execution feedback during inference, a critical signal that human programmers regularly leverage. Our method, Execution-Guided Classifier-Free Guidance (EG-CFG), dynamically incorporates execution signals as the model generates code, providing line-by-line feedback that guides the generation process toward executable solutions. EG-CFG employs a multi-stage process: first, we conduct beam search to sample candidate program completions for each line; second, we extract execution signals by executing these candidates against test cases; and finally, we incorporate these signals into the prompt during generation. By maintaining consistent signals across tokens within the same line and refreshing signals at line boundaries, our approach provides coherent guidance while preserving syntactic structure. Moreover, the method naturally supports native parallelism at the task level in which multiple agents operate in parallel, exploring diverse reasoning paths and collectively generating a broad set of candidate solutions. Our experiments across diverse coding tasks demonstrate that EG-CFG significantly improves code generation performance compared to standard approaches, achieving state-of-the-art results across various levels of complexity, from foundational problems to challenging competitive programming and data science tasks. Our code is available at: https://github.com/boazlavon/eg_cfg

We conduct a novel theoretical analysis to demonstrate CODA's