Measuring in-context algorithmic reasoning in language models against an exact Bayes-optimal standard
Authors: Hector Zenil, Luan Ozelim
Organizations: Oxford Immune Algorithmics, Oxford University Innovation & London Institute for Healthcare Engineering, U.K · Department of Biomedical Computing, School of Biomedical Engineering and Imaging Sciences & King’s Institute for AI, King’s College London, U.K
Abstract
Whether large language models perform genuine algorithmic reasoning or mere pattern completion is hard to test, because most benchmarks lack a ground truth for correct inductive inference. We introduce F-ICL, an in-context-learning benchmark that supplies one exactly. Using the Turing-complete machine F, complement-symmetrised into sF to remove output-polarity bias, we exhaustively enumerate all 1.5 billion programs of length L≤13 and compute the Bayes-optimal posterior in closed form under a bounded universal (Levin--Solomonoff) prior; models are scored by how closely their served distributions approach it at matched evidence. Each task is paired with its bitwise complement, on which the optimum scores identically, so an original-twin gap isolates the model's inductive bias. Across 105 serving configurations spanning 37 open models (0.8B--675B) and frontier systems from four laboratories, models answer up to 92% of queries correctly, yet 45 of 46 models yield distributions farther from the optimum than a keystroke reference, and their behaviour is bracketed by low-order prefix statistics fitted only on visible evidence. That reference is itself an algorithmic mixture, induced by a print-only machine with no loops, so the panel's implied measure sits closer to a loop-free mixture than to the loop-bearing optimum, independently of the reference machine. Updating is also non-monotone, which no prior explains: a Bayes-rational solved set can only grow in this realisable, noiseless setting, yet added examples produce 6,545 solved-to-unsolved transitions against 13,702 gains. The gap is not predicted by accuracy (Spearman ρ=−0.19, p=0.21), does not close with scale or across frontier generations in the serving modes that expose distributions, and is widened by instruction and reasoning post-training. F-ICL is released as an open, reproducible benchmark and toolkit.
For tool-augmented language models, comparing natural-language reasoning with code-execution pipelines is difficult because the comparison changes both the intermediate representation and the execution mechanism. We separate these factors with an intermediate intervention: the model expresses its reasoning as executable code, and the language model simulates that code in context to produce an answer. On a 40-task verifiable algorithmic benchmark, deterministic code execution outperforms natural-language reasoning by +31.6pp. We observe that the intermediate intervention is not meaningfully different from natural-language reasoning (+0.15pp). These results suggest that, in our evaluated setting, changing the intermediate representation alone does not explain the tool-use advantage, providing evidence for the performance gains requiring reliable external execution. We formalize this intuition with a simple statistical decision-theoretic model that characterizes when execution dominates end-to-end risk in our disentangled trace-generation/execution regime. We validate our theory using a reconstruction intervention that leverages a proxy language model to infer natural-language reasoning traces from code representations, recovering performance comparable to the original natural-language reasoning pipeline. All experiments are at https://github.com/TerryTong-Git/ToolProj.
High pass rates on established programming benchmarks such as HumanEval and LiveCodeBench do not always show whether a model can reason about algorithms. Many fixed benchmarks eventually become part of the public training ecosystem through released problem statements, editorials, and generated solutions, allowing later models to improve partly by exposure rather than by stronger algorithmic ability. We introduce ALGOBENCH, a framework that automatically builds novel algorithmic problems from known competitive-programming problems through structured constraint-shifting transformations. Each accepted ALGOBENCH variant is traceable to a source problem, but must make the original reference algorithm fail. Beyond pass@k, we introduce complexity-aware metrics -- including OPTT, OPTS, TRAPRATE, GAPT, and CONSENS -- to test whether a solution is not only functionally correct but also asymptotically suitable for the generated problem. Experiments across multiple LLMs and prompting strategies show that performance drops sharply on ALGOBENCH variants, retrieval can increase reuse of the old algorithm, and many correct-looking solutions fail to meet the required complexity. Error analysis shows that failures are mainly algorithmic rather than implementation-level, suggesting that ALGOBENCH evaluates adaptation beyond functional correctness.
Should we trust Large Language Models (LLMs) with high accuracy? LLMs achieve high accuracy on reasoning benchmarks, but correctness alone does not reveal the quality of the reasoning used to produce it. This highlights a fundamental limitation of outcome-based evaluation: models may arrive at correct answers through flawed reasoning, and models with substantially different reasoning capabilities can nevertheless exhibit similar benchmark accuracy, for example due to memorization or over-optimization. In this paper, we ask: given existing benchmarks, can we move beyond outcome-based evaluation to assess the quality of reasoning itself? We seek metrics that (1) differentiate models with similar accuracy and (2) are robust to variations in input prompts and generation configurations. To this end, we propose a reasoning score that evaluates reasoning traces along dimensions such as faithfulness, coherence, utility, and factuality. A remaining question is how to aggregate this score across multiple sampled traces. Naively averaging them is undesirable, particularly in long-horizon settings, where the number of possible trajectories grows rapidly, and low-confidence correct traces are more likely to be coincidental. To address this, we introduce the Filtered Reasoning Score (FRS), which computes reasoning quality using only the top-K% most confident traces. Evaluating with FRS, models that are indistinguishable under standard accuracy exhibit significant differences in reasoning quality. Moreover, models with higher FRS on one benchmark tend to perform better on other reasoning benchmarks, in both accuracy and reasoning quality. Together, these findings suggest that FRS complements accuracy by capturing a model's transferable reasoning capabilities. We open source our evaluation codebase: https://github.com/Manas2006/benchmark_reproducibility.