We introduce ClosureBench, a constructive benchmark for compositional graph-relational reasoning with programmatically verified ground truth. Unlike fixed-test-set benchmarks vulnerable to data contamination, ClosureBench generates instances on demand: each task's reference answer is computed by executing a program in the Ein tensor-logic language, ensuring machine-verified correctness. The benchmark spans 26 task categories at three compositional levels (L1-L3), with difficulty controlled along three independent axes: graph size, edge density, and query depth. We evaluate models from 1.5B open weights to frontier systems (o3, GPT-4.1, Gemini 2.5, Claude Sonnet 4) and report three findings. First, because the benchmark can always supply fresh instances, it measures memorisation directly: a model fine-tuned on a fixed test set shows a 19.3 percentage-point gap between its accuracy on seen and on fresh instances, which a static test set cannot reveal. We scope this to supervised fine-tuning on answer pairs, not pretraining contamination. Second, accuracy falls as graph size and query depth increase, and the two interact: models misread the graph from its natural-language description and then reason correctly over the wrong graph, so even the strongest frontier model degrades from atomic to compositional queries. This bottleneck is a property of the reasoning rather than the input format: it persists when the graph is given as a JSON edge list or an adjacency matrix instead of prose. Third, a 4B model fine-tuned to emit executable programs rather than answers stays nearly flat across compositional levels and approaches frontier accuracy (94.3% on held-out instances) at a fraction of the token cost. This holds for two program targets, Ein and Python+NetworkX, so it is a property of verified program synthesis rather than of one language.
Sajad Movahedi, Vera Milovanović, Shlomo Libo Feigin +5cs.AI
Looped architectures provide an inductive bias toward learning step-by-step procedures for tasks that require compositional reasoning. The number of effective layers reached by looping determines the quality of the solution these models find. Like deep architectures, looped architectures are prone to a signal propagation problem induced by depth as the halting decision is postponed. In this paper, we address this signal propagation issue using pre-norm layers and residual scaling. Building on these architectural modifications, we propose FPRM, a Transformer-based Fixed-Point Reasoning Model that uses fixed-point convergence as an end-to-end halting mechanism in a looped architecture. We show that fixed-point halting allows FPRM to adapt its compute to task difficulty. FPRM is effective on common reasoning benchmarks, namely Sudoku, Maze, state-tracking, and ARC-AGI.
Question decomposition, i.e. breaking a complex query into simpler sub-queries whose answers are composed to produce a final answer, is a widely used strategy for improving LLM reasoning, yet it currently lacks a rigorous mathematical foundation. In this paper, we propose operads, mathematical structures that model many-in, one-out operations and compositions thereof, as a natural framework for describing question decomposition. We define the questions operad $Q$, in which operations correspond to question templates and composition corresponds to substitution of sub-answers, and show how QA models can be interpreted as algebras over $Q$. Beyond reframing existing practice, this operadic perspective points toward new methods, in particular a notion of operadic consistency, which measures whether a QA model's answers agree across the partial collapses of a question decomposition tree. Empirical evaluation of operadic consistency is reported in our companion paper (Bottman, Liu, and Richardson, 2026), which finds it strongly correlated with accuracy across twelve LLMs and four multi-hop QA datasets and outperforming standard temperature-based self-consistency baselines. We argue that operads are the natural mathematical home for question decomposition, and that invariants such as operadic consistency open new directions for analyzing and improving the reliability of multi-step reasoning.