Standard autoregressive decoding in large language models (LLMs) is inherently short-sighted, often failing to find globally optimal reasoning paths due to its token-by-token generation process. While inference-time strategies like foresight sampling attempt to mitigate this by simulating future steps, they typically rely on ad-hoc heuristics for valuing paths and pruning the search space. This paper introduces Martingale Foresight Sampling (MFS), a principled framework that reformulates LLM decoding as a problem of identifying an optimal stochastic process. By modeling the quality of a reasoning path as a stochastic process, we leverage Martingale theory to design a theoretically-grounded algorithm. Our approach replaces heuristic mechanisms with principles from probability theory: step valuation is derived from the Doob Decomposition Theorem to measure a path’s predictable advantage, path selection uses Optional Stopping Theory for principled pruning of suboptimal candidates, and an adaptive stopping rule based on the Martingale Convergence Theorem terminates exploration once a path’s quality has provably converged. Experiments on six reasoning benchmarks demonstrate that MFS surpasses state-of-the-art methods in accuracy while significantly improving computational efficiency. Code will be released at https://github.com/miraclehetech/EACL2026-Martingale-Foresight-Sampling.

Text-driven indoor scene generation and editing require an intermediate representation that language models can both produce and revise. Existing LLM-based systems often rely on scene graphs or global constraint lists, which are compact but underspecify local geometry and make instruction-based edits difficult to localize. We frame the task as structured program generation and local program repair, and propose the Hierarchical Descriptive Scene Language (HDSL), an XML/CSS-style domain-specific language in which rooms, regions, objects, and support surfaces form a tree with parent-relative local coordinates. Our pipeline generates HDSL subtrees recursively with bounded verification, grounds non-virtual nodes through multimodal asset retrieval, and applies force-directed layout optimization to repair boundary and collision errors. For editing, Hierarchical Retrieval-Augmented Generation retrieves the relevant subtree, rewrites only that local context, and merges the result back through a deterministic three-way merge. In our reproduced benchmark, HDSL improves average object coverage, text-scene alignment, and generation time over full text-to-scene baselines while remaining competitive with layout-only reproductions on geometry metrics. On thirty paired editing instructions, HRAG reduces token use by 67.6% and runtime by 68.1%, and cuts non-target removals by 87.5% over full-scene rewriting; attribute ablations confirm that local coordinates and virtual/leaf flags are necessary for valid, locality-preserving edits.