Algorithmic Reasoning

There is an increasing number of algorithmic computations in theoretical physics. These, while conceptually simple, can nevertheless be time-consuming and contain subtleties that should not be overlooked. Given the recent improvement of Large Language Models (LLM), it is natural to investigate whether LLMs equipped with a computer algebra system (CAS) runtime and sufficiently informative context can reliably carry out these algorithmic tasks. In this work, we interface Claude with Maple, and apply this framework to cosmological perturbations in modified theories of gravity. We demonstrate the current capabilities of this approach, the typical failures, and how the same can be improved. We find that a frontier LLM supplied with worked examples is able to solve most test problems.

Cyber threats are rapidly increasing, expanding their impact from large-scale enterprises to government services and individual users, making robust security systems increasingly essential. However, a significant shortage of skilled cybersecurity professionals exacerbates this challenge. While recent research has explored automating tasks such as penetration testing using LLM-based agents, existing frameworks often perform poorly due to limited capability in strategy formulation, domain-specific reasoning, and accurate action and tool selection. To overcome these limitations, we propose Pen-Strategist framework, consisting of a novel domain-specific reasoning model that derives pentesting strategies via logical reasoning and a classifier that converts the strategies into actionable steps. First, we construct a reasoning dataset containing logical explanations for both strategy derivation and step selection in pentesting scenarios. We then fine-tune a Qwen-3-14B model for strategy generation using reinforcement learning. Evaluation on the test split of the dataset demonstrates a 87% improvement in strategy derivation performance compared to the baseline. Furthermore, we integrate the fine-tuned Pen-Strategist model into existing automated pentesting frameworks, such as PentestGPT, and evaluate its performance on vulnerable machines, achieving a 47.5% improvement in subtask completion while surpassing the baseline GPT-5. Further experiments on the CTFKnow benchmark show an 18% performance gain over the base model. For step prediction, we train a semantic-based CNN classifier, which outperforms commercial LLMs by 28% and enhances execution stability. Finally, we conduct a user study to qualitatively assess the generated strategies, and Pen-Strategist demonstrates superior performance compared to the Claude-4.6-Sonnet.

This article introduces S-AI-Recursive, a bio-inspired Sparse Artificial Intelligence architecture in which reasoning is operationalized as a hormonal closed-loop iteration rather than a single feed-forward pass. Building upon the S-AI foundational framework [1], the hormonal-probabilistic unification doctrine [2], and the formal mathematical methodology established in S-AI-IoT [3], the present work formalizes the Recursive Reasoning Cycle (RRC) as a dynamical system governed by two novel hormones: Clarifine, a convergence signal, and Confusionin, an uncertainty detector, whose antagonistic regulation drives iterative state refinement toward a stable cognitive equilibrium. The complete mathematical framework is developed, including recursive state dynamics, Lyapunov stability proof, entropic contraction theorem, hormonal stopping criterion with finite-time termination guarantee, Euler-Maruyama discretization with projection, primal-dual agent selection under iteration budget, and recursive engram memory with warm-start acceleration. Experimental validation on the SAI-UT+ testbench demonstrates that S-AI-Recursive achieves competitive reasoning performance on abstract and symbolic benchmarks with fewer than ten million parameters, confirming the central principle of temporal parsimony: iterative cognitive depth substitutes for architectural width.

Large language models are highly capable of answering difficult questions by retrieving, recombining, and attending to information in long contexts. For agentic tasks, an additional capability is required: the preservation of an exact state while repeatedly applying rules. We find that this reliability is absent across language models. To demonstrate, we query 126 leading model variants with the task of counting a long string of repeated characters, and we find they all cannot accurately count above a model-dependent, syntax-sensitive counting capacity threshold. Failures are abrupt and persist even with increasing model size, inference time computation, and external tool. Mechanistic probing indicates that models use a finite number of internal states to mimic counting as a rule and fail once these states are exhausted. Furthermore, such states are the basis for performing complex tasks beyond counting. These results indicate that fundamentally new model architectures are required for autonomous agents to achieve truly reliable rule following capabilities.

Recursive reasoning systems alternate between acquiring new evidence and refining an accumulated understanding. Two design choices are typically left implicit: how to represent the evolving reasoning state, and when to stop iterating. This paper addresses both. We represent the reasoning state as an epistemic state graph encoding extracted claims, evidential relations, open questions, and confidence weights. We define the order-gap as the distance between the states reached by expand-then-consolidate versus consolidate-then-expand; a small order-gap suggests that the two orderings agree and further iteration is unlikely to help. Our main result gives a necessary and sufficient condition for the linearised order-gap to be non-degenerate near the fixed point, showing when the criterion is informative rather than algebraically vacuous. This is a local condition, not a global convergence guarantee. We apply the framework to recursive reasoning systems and sketch its application to agent loops, tree-of-thought reasoning, theorem proving, and continual learning.

We demonstrate that a small transformer can learn to execute programs in MicroPy, a simplified yet computationally universal programming language. Given procedure definitions together with an expression to evaluate, the transformer predicts small-step execution using PENCIL scaffolding for space-efficient execution within a bounded context window. After training on randomly generated, meaningless MicroPy programs, the learned transformer generalizes to various human-written programs including bit copying and flipping, binary addition and multiplication, and SAT verification and solving. We note that the trained model can achieve out-of-distribution generalization; i.e., evaluate novel programs from distribution on programs. Since MicroPy can express any computation, our results provide empirical evidence that a standard transformer can be trained to act as a universal computer.

We study learned memory tokens as a computational scratchpad for a single-block Universal Transformer with Adaptive Computation Time (ACT) on Sudoku-Extreme, a combinatorial reasoning benchmark. Memory tokens are empirically necessary: no configuration without them reaches non-trivial performance. The optimal count has a sharp lower threshold (T=0 always fails, T=8 reliably succeeds) followed by a stable plateau (T=8-32, 57.4% +/- 0.7% exact-match) and a dilution boundary at T=64. Under halt-side pressure (lambda warmup), mean halt drops monotonically with memory size across the plateau (from 11.6 at T=8 to 8.3 at T=64), showing that memory tokens and ponder depth substitute as resources at fixed accuracy. We also identify a router initialization trap that causes the majority of training runs to fail: both default zero-bias and Graves' recommended positive bias settle into a shallow halt equilibrium the model cannot escape. Inverting the bias to -3 ("deep start") eliminates the failure mode, and ablation shows the trap is inherent to ACT initialization rather than an artifact of our architecture. With reliable training, ACT yields an order of magnitude lower seed variance than fixed-depth processing (+/-0.7 vs +/-9.3 pp); lambda warmup recovers 34% of compute at matched accuracy; and attention heads specialize into memory readers, constraint propagators, and integrators across recursive depth. Code: https://github.com/che-shr-cat/utm-jax.

Self-consistency boosts inference-time performance by sampling multiple reasoning traces in parallel and voting. However, in constrained domains like math and code, this strategy is compute-inefficient because it samples with replacement, repeatedly revisiting the same high-probability prefixes and duplicate completions. We propose Distinct Leaf Enumeration (DLE), a deterministic decoding method that treats truncated sampling as traversal of a pruned decoding tree and systematically enumerates distinct leaves instead of sampling with replacement. This strategy improves inference efficiency in two ways. Algorithmically, it increases coverage of the truncated search space under a fixed budget by exploring previously unvisited high-probability branches. Systemically, it reuses shared prefixes and reduces redundant token generation. Empirically, DLE explores higher-quality reasoning traces than stochastic self-consistency, yielding better performance on math, coding, and general reasoning tasks.

Large language models often appear strong on symbolic and algorithmic tasks, yet this apparent strength can hide brittle behaviour when problems become longer, harder, or slightly out of distribution. A major limitation of current reasoning benchmarks is that many primarily test whether a model can produce a valid answer, while paying less attention to whether the solution is minimal, robust, and stable under controlled difficulty scaling. We introduce RecurrReason, a difficulty-controlled benchmark of four recurrent logic puzzles (Tower of Hanoi, River Crossing, Block World, and Checkers Jumping) with BFS-optimal trajectories and a single interpretable difficulty parameter $N \in \{1,\dots,10\}$, totalling 10{,}817 unique puzzles and 285{,}933 moves. We benchmark two Transformer families, an encoder-decoder model (T5-style) and a decoder-only model (GPT-2-style), under consistent data splits and evaluation criteria, training on $N{=}1$ to $7$ and evaluating on both held-out in-distribution instances and harder out-of-distribution instances at $N{=}8$ to $10$. Fine-tuned pre-trained T5 achieves 97.27% validation and 81.00% OOD accuracy on Block World; all models score 0.00% on River Crossing under all conditions. Failure mode analysis reveals that architecture is a stronger determinant of success than scale. Pre-training transfers only to puzzles with locally structured transition functions. Our code and dataset will be open-sourced upon acceptance.

Large Language Models (LLMs) have demonstrated strong potential for many mathematical problems. However, their performance on graph algorithmic tasks is still unsatisfying, since graphs are naturally more complex in topology and often require systematic multi-step reasoning, especially on larger graphs. Motivated by this gap, we propose GraphDC, a Divide-and-Conquer multi-agent framework for scalable graph algorithm reasoning. Specifically, inspired by Divide-and-Conquer design, GraphDC decomposes an input graph into smaller subgraphs, assigns each subgraph to a specialized agent for local reasoning, and uses a master agent to integrate the local outputs with inter-subgraph information to produce the final solution. This hierarchical design reduces the reasoning burden on individual agents, alleviates computational bottlenecks, and improves robustness on large graph instances. Extensive experiments show that GraphDC consistently outperforms existing methods on graph algorithm reasoning across diverse tasks and scales, especially on larger instances where direct end-to-end reasoning is less reliable.

Recent advances in agentic frameworks have enabled AI agents to perform complex reasoning and decision-making. However, evidence comparing their reasoning performance, efficiency, and practical suitability remains limited. To address this gap, we empirically evaluate 22 widely used agentic frameworks across three reasoning benchmarks: BBH, GSM8K, and ARC. The frameworks were selected from 1,200 GitHub repositories collected between January 2023 and July 2025 and organized into a taxonomy based on architectural design. We evaluated them under a unified setting, measuring reasoning accuracy, execution time, computational cost, and cross-benchmark consistency. Our results show that 19 of the 22 frameworks completed all three benchmarks. Among these, 12 showed stable performance, with mean accuracy of 74.6-75.9%, execution time of 4-6 seconds per task, and cost of 0.14-0.18 cents per task. Poorer results were mainly caused by orchestration problems rather than reasoning limits. For example, Camel failed to complete BBH after 11 days because of uncontrolled context growth, while Upsonic consumed USD 1,434 in one day because repeated extraction failures triggered costly retries. AutoGen and Mastra also exhausted API quotas through iterative interactions that increased prompt length without improving results. We also found a sharp drop in mathematical reasoning. Mean accuracy on GSM8K was 44.35%, compared with 89.80% on BBH and 89.56% on ARC. Overall, this study provides the first large-scale empirical comparison of agentic frameworks for reasoning-intensive software engineering tasks and shows that framework selection should prioritize orchestration quality, especially memory control, failure handling, and cost management.

Standard Transformers have a fixed computational depth, limiting their ability to generalize to tasks that require variable-depth reasoning. The usual remedy, Chain-of-Thought (CoT), spends tokens to reason, inflating the key--value cache and making latency grow with the step count, so memory becomes the limiting cost when reasoning is served over large query batches. We study a depth-recurrent Transformer that decouples computational depth from parameter count by iterating a shared-weight block, so that each added reasoning step costs flat memory and linear latency, with no token generation. Three ingredients keep the recurrence stable for 20+ thinking steps: a silent thinking objective that supervises only the final output, LayerScale initialization, and an identity-biased gate that opens a gradient highway across steps. We characterize it on three compositional domains with decreasing structural bias: graph reachability (adjacency masking), nested boolean logic (relative positioning), and unstructured relational text (no positional cue). We find a \emph{computational frontier}: accuracy climbs once the thinking-step count meets the task's complexity, reaching near-perfect performance on the two structured tasks and a lower plateau on unstructured text. How it climbs depends on the structural bias---abruptly from chance on the graph task, gradually on the other two. Depth recurrence extrapolates beyond the training range: it succeeds on the graph task where fixed-depth models barely extrapolate, and on the two sequence tasks comes within two points of fixed-depth Transformers that use $4$--$6.4\times$ more parameters. On the graph task, whose adjacency mask makes propagation depth verifiable, intermediate per-step supervision---a standard recipe for deep iterative models---consistently \emph{harms} this extrapolation. We release the code for reproducibility.

Modern language models reason within bounded context, an inherent constraint that poses a fundamental barrier to long-horizon reasoning. We identify recursion as a core principle for overcoming this barrier, and propose recursive models as a minimal realization, where the model can recursively invoke itself to solve subtasks in isolated contexts. We prove that any computable problem admits a recursive decomposition of reasoning in which each subtask requires only exponentially smaller active context than standard autoregressive models; this strictly surpasses any context management approach confined to a single sequence, such as summarization. We further generalize our framework to modern agentic systems with arbitrary context processing and control flows, and prove that recursive models can achieve optimal power within this broader class. Experimentally, we test two settings: fine-tuning a pretrained base model for recursive SAT solving, and training a small model from scratch on Go traces generated by exact game-tree search. Both show improved long-horizon accuracy with small active contexts.

Humans can apply ideas learned in one context to substantially different situations. We call this process of searching for and constructing novel recombinations of learned relationships to solve new problems \textit{out-of-distribution (OOD) reasoning}. The capacity for large language models (LLMs) to support OOD reasoning underpins proposals for generally intelligent systems. However, this property is challenging to measure because most problems admit many decompositions, some involving shallow subproblems and others involving subproblems that may have been memorized from the training data. This makes it difficult to determine how much compositional reasoning a model must actually perform. Uncertainty about training distributions, how to measure a datapoint's distance from a training distribution, and the exponential number of ways to decompose most tasks make it intractable to robustly measure any modern LLM's OOD-reasoning capacity on standard natural-language tasks. In this work, we introduce a benchmark in which a model progressively solves a Boolean circuit over $GF(2)$ from data. This benchmark is both minimal, isolating OOD reasoning from these confounding factors, and general, as any computable function can be represented as a sufficiently large $GF(2)$ polynomial. We find that standalone models' next-step accuracy collapses as depth grows. In contrast, tool use through the synthesis and execution of code prevents this collapse in both small and frontier LLMs. These results indicate that synthesizing tools play a crucial role in supporting OOD reasoning.

Learning algorithmic computation often requires explicit relational intermediate states, yet many graph processors maintain their primary states on individual entities. We introduce \fnet and \textbf{Pivotal Attention} (PA), which maintain ordered pair states and update a target relation $(i,k)$ by attending over candidates formed from $(i,j)$ and $(j,k)$ for every pivot $j$. Motivated by the pair-and-pivot structure of Floyd--Warshall, PA learns relation composition and pivot weighting in parallel rather than executing its ordered min-plus recurrence. The \kfnet{k} framework extends this operation to ordered $k$-tuples, with Self-Attention and PA as its $k=1$ and $k=2$ cases at the attention-operation level. Under atomic tuple initialization and invariant readout, we show that \kfnet{k} is no more graph-discriminative than k-FWL; on BREC, each evaluated variant matches the success set of its corresponding WL reference. \fnet further achieves 96.64% mean accuracy under the reported CLRS-30 protocol and a 99.8% optimality rate with 10 samples on held-out non-metric TSP instances.

Algorithmic reasoning -- the ability to perform step-by-step logical inference -- is a synthetic benchmark for evaluating multi-step reasoning abilities, designed for graph neural networks and also for transformer models. Prior work has evaluated reasoning for executing a single algorithmic task, whereas a more desirable objective is to perform multiple algorithmic reasoning tasks simultaneously. We start by noting that this is inherently difficult due to differences arising from the execution traces of the algorithms (such as depth- vs. breadth-first search), which cause interference when they are trained together. In this paper, we introduce {branching neural networks}, a new architecture for multitask algorithmic reasoning. The main idea is to search for a recursive tree-structured partition of $n$ algorithmic tasks into a $k$-ary tree (divided into $L$ layers). Naive search requires $O(k^{nL})$ complexity; we develop an algorithm that reduces this to $O(nL)$ by solving a convex relaxation at each layer to approximate an optimal partition. Our approach clusters these tasks using gradient-based affinity and can be used on top of any base model. We validate our approach on algorithmic reasoning benchmarks and their extensions with text descriptions. We show that gradient-based affinity scores help estimate true performance with less than 5% error, measured across eight different architectures with up to 34 billion parameters. On the CLRS benchmark, our approach outperforms existing graph neural networks by 3.7% and baselines by 1.2%, while reducing runtime by 48% and memory usage by 26%. The learned branching structure shows a hierarchical clustering of related algorithms. On three text-based graph reasoning benchmarks, our approach improves over baseline methods by 3.2%. Finally, we validate our approach for overlapping community detection.

We investigate the performance of large language models (LLMs) on repetitive deterministic prediction tasks and study how the sequence accuracy rate (SAR) scales with output length. Each such task involves the repetition of the same operation $N$ times. Examples of such tasks include letter replacement in letter strings following a given rule, integer addition, and multiplication of string operators in many-body quantum mechanics. If the LLM performs the task by a simple repetition algorithm, the success rate would follow an exponential decay with sequence length. In contrast, our experiments on leading LLMs reveal a crossover that is sharper than exponential: $-\log\mathrm{SAR}$ grows super-linearly with $N$, and accuracy collapses around a characteristic length $N_*$, the accuracy cliff that separates reliable from unreliable generation. The hypothesis of independent per-step errors is rejected for every model and task we studied. The crossover is well described by a double-exponential accumulation law, $\mathrm{SAR}=\exp(-β_0 Nα^{N-1})$, whose crossover scale $N_*$ does not depend on the functional form chosen to fit it. To interpret this behaviour we introduce a minimal effective model in which step-correctness variables interact through dense random couplings and compete with an external field set by the prompt. Solved by direct enumeration, the model reproduces the super-linear error accumulation and the accuracy cliff qualitatively, and it assigns to each model--task pair two interpretable parameters, an intrinsic error rate and an error-accumulation factor.

We investigate how embedding dimension affects the emergence of an internal "world model" in a transformer trained with reinforcement learning to perform bubble-sort-style adjacent swaps. Models achieve high accuracy even with very small embedding dimensions, but larger dimensions yield more faithful, consistent, and robust internal representations. In particular, higher embedding dimensions strengthen the formation of structured internal representation and lead to better interpretability. After hundreds of experiments, we observe two consistent mechanisms: (1) the last row of the attention weight matrix monotonically encodes the global ordering of tokens; and (2) the selected transposition aligns with the largest adjacent difference of these encoded values. Our results provide quantitative evidence that transformers build structured internal world models and that model size improves representation quality in addition to end performance. We release our metrics and analyses, which can be used to probe similar algorithmic tasks.

Tool-integrated reasoning (TIR) has become a key approach for improving large reasoning models (LRMs) on complex problems. Prior work has mainly studied when to invoke tools, while overlooking how tools are applied. We identify two common patterns: a calculator pattern that uses code for direct computation, and an algorithmic pattern that encodes problems as programs. Misaligned choices often cause failures even when reasoning is sound. We propose a two-stage framework that first builds code competence from both patterns and then aligns pattern selection with teacher preferences. Across challenging math datasets, our pattern-aware method substantially improves both code usage and accuracy, for instance raising Code@1 on MATH500 from 64.0% to 70.5% and on AIME24 from 26.7% to 50.0%. These gains highlight the effectiveness of a pattern-aware approach for tool-integrated reasoning.

Neural algorithmic reasoning (NAR) is a paradigm that trains neural networks to execute classic algorithms by supervised learning. Despite its successes, important limitations remain: inability to construct valid solutions without post-processing and to reason about multiple correct ones, poor performance on combinatorial NP-hard problems, and inapplicability to problems for which strong algorithms are not yet known. To address these limitations, we reframe the problem of learning algorithm trajectories as a Markov decision process, which imposes structure on the solution construction procedure and unlocks the powerful tools of imitation and reinforcement learning (RL). We propose the GNARL framework, encompassing the methodology to translate problem formulations from NAR to RL and a learning architecture suitable for a wide range of graph-based problems. We achieve high rates of reaching correct states on several CLRS-30 problems and performance matching or exceeding much narrower NAR approaches for NP-hard problems. Remarkably, GNARL remains applicable when no expert algorithm is available, though reward-driven learning can exhibit greater variability than direct supervision.

Best-of-n sampling improves the accuracy of large language models (LLMs) and large reasoning models (LRMs) by generating multiple candidate solutions and selecting the one with the highest reward. The key challenge for reasoning tasks is designing a scoring function that can identify correct reasoning chains without access to ground-truth answers. We propose Probabilistic Confidence Selection And Ranking (PiCSAR): a simple, training-free method that scores each candidate generation using the joint log-likelihood of the reasoning and final answer. The joint log-likelihood of the reasoning and final answer naturally decomposes into reasoning confidence and answer confidence. PiCSAR achieves substantial gains across diverse benchmarks (+10.18 on MATH500, +9.81 on AIME2025), outperforming baselines with at least 2x fewer samples in 16 out of 20 comparisons. Our analysis reveals that correct reasoning chains exhibit significantly higher reasoning and answer confidence, justifying the effectiveness of PiCSAR.

We introduce PuzzleJAX, a GPU-accelerated puzzle game engine and description language designed to support rapid benchmarking of tree search, reinforcement learning, and LLM reasoning abilities. Unlike existing GPU-accelerated learning environments that provide hard-coded implementations of fixed sets of games, PuzzleJAX allows dynamic compilation of any game expressible in its domain-specific language (DSL). This DSL follows PuzzleScript, which is a popular and accessible online game engine for designing puzzle games. In this paper, we validate in PuzzleJAX several hundred of the thousands of games designed in PuzzleScript by both professional designers and casual creators since its release in 2013, thereby demonstrating PuzzleJAX's coverage of an expansive, expressive, and human-relevant space of tasks. By analyzing the performance of search, learning, and language models on these games, we show that PuzzleJAX can naturally express tasks that are both simple and intuitive to understand, yet often deeply challenging to master, requiring a combination of control, planning, and high-level insight.

Large language models (LLMs) demonstrate remarkable reasoning capabilities in tasks such as algorithmic coding and mathematical problem-solving. Recent methods have improved reasoning through expanded corpus and multistage training combining reinforcement learning and supervised fine-tuning. Although some methods suggest that small but targeted dataset can incentivize reasoning via only distillation, a reasoning scaling laws is still taking shape, increasing computational costs. To address this, we propose a data-efficient distillation framework (DED) that optimizes the Pareto frontier of reasoning distillation. Inspired by the on-policy learning and diverse roll-out strategies of reinforcement learning, the key idea of our approach is threefold: (1) We identify that benchmark scores alone do not determine an effective teacher model. Through comprehensive comparisons of leading reasoning LLMs, we develop a method to select an optimal teacher model. (2) While scaling distillation can enhance reasoning, it often degrades out-of-domain performance. A carefully curated, smaller corpus achieves a balanced trade-off between in-domain and out-of-domain capabilities. (3) Diverse reasoning trajectories encourage the student model to develop robust reasoning skills. We validate our method through evaluations on mathematical reasoning (AIME 2024/2025, MATH-500) and code generation (LiveCodeBench), achieving state-of-the-art results with only 0.8k carefully curated examples, bypassing the need for extensive scaling. Our systematic analysis demonstrates that DED outperforms existing methods by considering factors beyond superficial hardness, token length, or teacher model capability. This work offers a practical and efficient pathway to advanced reasoning while preserving general capabilities.

Reasoning, the process of devising and executing complex goal-oriented action sequences, remains a critical challenge in AI. Current large language models (LLMs) primarily employ Chain-of-Thought (CoT) techniques, which suffer from brittle task decomposition, extensive data requirements, and high latency. Inspired by the hierarchical and multi-timescale processing in the human brain, we propose the Hierarchical Reasoning Model (HRM), a novel recurrent architecture that attains significant computational depth while maintaining both training stability and efficiency. HRM executes sequential reasoning tasks in a single forward pass without explicit supervision of the intermediate process, through two interdependent recurrent modules: a high-level module responsible for slow, abstract planning, and a low-level module handling rapid, detailed computations. With only 27 million parameters, HRM achieves exceptional performance on complex reasoning tasks using only 1000 training samples. The model operates without pre-training or CoT data, yet achieves nearly perfect performance on challenging tasks including complex Sudoku puzzles and optimal path finding in large mazes. Furthermore, HRM outperforms much larger models with significantly longer context windows on the Abstraction and Reasoning Corpus (ARC), a key benchmark for measuring artificial general intelligence capabilities. These results underscore HRM's potential as a transformative advancement toward universal computation and general-purpose reasoning systems.

We introduce BigO(Bench), a novel coding benchmark designed to evaluate the capabilities of generative language models in understanding and generating code with specified time and space complexities. This benchmark addresses the gap in current evaluations that often overlook the ability of models to comprehend and produce code constrained by computational complexity. BigO(Bench) includes tooling to infer the algorithmic complexity of any Python function from profiling measurements, including human- or LLM-generated solutions. BigO(Bench) also includes of set of 3,105 coding problems and 1,190,250 solutions from Code Contests annotated with inferred (synthetic) time and space complexity labels from the complexity framework, as well as corresponding runtime and memory footprint values for a large set of input sizes. We present results from evaluating multiple state-of-the-art language models on this benchmark, highlighting their strengths and weaknesses in handling complexity requirements. In particular, token-space reasoning models are unrivaled in code generation but not in complexity understanding, hinting that they may not generalize well to tasks for which no reward was given at training time.

Modern machine learning models excel at detecting individual actions, sounds, or scene attributes from short, localized observations. However, many real-world tasks, such as in smart cities and healthcare, require reasoning over high-level complex events (CEs): spatiotemporal, rule-governed patterns of short-term atomic events (AEs). Complex event detection (CED) is challenging due to long temporal dependencies, generalization beyond the training horizon, sparse CE-level supervision without temporally aligned fine-grained AE labels, and cognitively demanding annotation, as CE labels often depend on ordering, duration, negation, and completion-time semantics. These challenges are further amplified in an online setting that requires causal, streaming inference with limited computation. We identify the primary bottleneck in online CED as learning robust CE rules, and propose a Neural Algorithmic Reasoning framework that decouples rule learning from low-level sensor semantics by (i) generating large-scale synthetic AE-level concept traces to pretrain a Mamba-based CE-rule reasoner, and (ii) introducing an adapter that learns to map raw sensor inputs into the reasoner's latent space using limited, labeled sensor data. We introduce a controlled simulator-generated online multilabel CED testbed built from real-world multimodal sensor clips and rule-generated CE labels, with stress-test settings that vary sensor noise, distribution shift, and the window size used to segment streaming sensor sequences. Experiments on this controlled benchmark show that NAROCE is competitive with the strongest baselines and often outperforms them under these stress tests and longer-horizon generalization, while using 5x fewer labeled sensor sequences and 10-20x fewer FLOPs than all non-Mamba baselines. Code and dataset available at https://github.com/nesl/naroce_dailyoce.