A Survey on Large Language Models for Mathematical Reasoning

Despite the impressive performance across numerous tasks, Large Language Models (LLMs) often fail in solving simple decision-making tasks due to the misalignment of the knowledge in LLMs with environments. On the contrary, Reinforcement Learning (RL) agents learn policies from scratch, which makes them always align with environments but difficult to incorporate prior knowledge for efficient explorations. To narrow the gap, we propose TWOSOME, a novel general online framework that deploys LLMs as decision-making agents to efficiently interact and align with embodied environments via RL without requiring any prepared datasets or prior knowledge of the environments. First, we query the joint probabilities of each valid action with LLMs to form behavior policies. Then, to enhance the stability and robustness of the policies, we propose two normalization methods and summarize four prompt design principles. Finally, we design a novel parameter-efficient training architecture where the actor and critic share one frozen LLM equipped with LOw-Rank Adapters (LoRA) updated by PPO. We conduct extensive experiments to evaluate TWOSOME. (i) TWOSOME exhibits significantly better sample efficiency and performance compared to the conventional RL method, PPO, and prompt tuning method, SayCan, in both classical decision-making environment, Overcooked, and simulated household environment, VirtualHome. (ii) Benefiting from LLMs’ open-vocabulary feature, TWOSOME shows superior generalization ability to unseen tasks. (iii) Under our framework, there is no significant loss of the LLMs’ original ability during online PPO finetuning.a

The mathematical capabilities of AI systems are complex and multifaceted. Most existing research has predominantly focused on the correctness of AI-generated solutions to mathematical problems. In this work, we argue that beyond producing correct answers, AI systems should also be capable of, or assist humans in, developing novel solutions to mathematical challenges. This study explores the creative potential of Large Language Models (LLMs) in mathematical reasoning, an aspect that has received limited attention in prior research. We introduce a novel framework and benchmark, CreativeMath, which encompasses problems ranging from middle school curricula to Olympic-level competitions, designed to assess LLMs' ability to propose innovative solutions after some known solutions have been provided. Our experiments demonstrate that, while LLMs perform well on standard mathematical tasks, their capacity for creative problem-solving varies considerably. Notably, the Gemini-1.5-Pro model outperformed other LLMs in generating novel solutions. This research opens a new frontier in evaluating AI creativity, shedding light on both the strengths and limitations of LLMs in fostering mathematical innovation, and setting the stage for future developments in AI-assisted mathematical discovery.

Mathematical reasoning ability objectively reflects a language model's understanding of implicit knowledge in contexts, with logic being a prerequisite for exploring, articulating and establishing effective reasoning. Large language models (LLMs) have shown great potential in complex reasoning tasks represented by mathematical reasoning. However, existing mathematical datasets either focus on commonsense reasoning, assessing the model's knowledge application ability, or arithmetic problems with fixed calculation rules, evaluating the model's rapid learning capability. There is a lack of datasets that require solving problems solely through logical reasoning. As a result, the performance of LLMs in accurately understanding the implicit logical relationships in problems and deriving conclusions based solely on given conditions is hindered. To address this challenge, we construct a dataset specifically for multiple step reasoning tasks: Reasoning-Math (RMath). This dataset focuses on evaluating logical reasoning abilities with mathematical reasoning problems, covering typical problem types, including direct reasoning problems, hypothetical reasoning problems, and nested reasoning problems. Additionally, we design a standardized annotation scheme that transforms natural language descriptions of conditions into formal propositions. Other annotation contents include problem categories, proposition truth values, and proposition relationship types. This not only reduces biases caused by semantic misunderstandings during problem-solving, but also facilitates the incorporation of theoretically grounded logical reasoning methods to enhance reasoning abilities. Furthermore, we propose a normalization problem-solving framework based on propositional logic for RMath and design the problem-solving process for prompt tuning to guide LLMs to absorb mathematical logical theories and improving reasoning abilities. Finally, we evaluate RMath on several popular LLMs and present the corresponding results.

Large language models (LLMs) have significantly advanced natural language understanding and demonstrated strong problem-solving abilities. Despite these successes, most LLMs still struggle with solving mathematical problems due to the intricate reasoning required. To support rigorous evaluation of mathematical reasoning in LLMs, we introduce the “MathOdyssey” dataset - a curated collection of 387 expert-generated mathematical problems spanning high school, university, and Olympiad-level topics. Each problem is accompanied by a detailed solution and categorized by difficulty level, subject area, and answer type. The dataset was developed through a rigorous multi-stage process involving contributions from subject experts, peer review, and standardized formatting. We provide detailed metadata and a standardized schema to facilitate consistent use in downstream applications. To demonstrate the dataset’s utility, we evaluate several representative LLMs and report their performance across problem types. We release MathOdyssey as an open-access resource to enable reproducible and fine-grained assessment of mathematical capabilities in LLMs and to foster further research in mathematical reasoning and education.

Despite being empowered with alignment mechanisms, large language models (LLMs) are increasingly vulnerable to emerging jailbreak attacks that can compromise their alignment mechanisms. This vulnerability poses significant risks to real-world applications. Existing work faces challenges in both training efficiency and generalization capabilities (i.e., Reinforcement Learning from Human Feedback and Red-Teaming). Developing effective strategies to enable LLMs to resist continuously evolving jailbreak attempts represents a significant challenge. To address this challenge, we propose a novel defensive paradigm called GuidelineLLM, which assists LLMs in recognizing queries that may have harmful content. Before LLMs respond to a query, GuidelineLLM first identifies potential risks associated with the query, summarizes these risks into guideline suggestions, and then feeds these guidelines to the responding LLMs. Importantly, our approach eliminates the necessity for additional safety fine-tuning of the LLMs themselves; only the GuidelineLLM requires fine-tuning. This characteristic enhances the general applicability of GuidelineLLM across various LLMs. Experimental results demonstrate that GuidelineLLM can significantly reduce the attack success rate (ASR) against LLM (an average reduction of 34.17% ASR) while maintaining the usefulness of LLM in handling benign queries.

Generative AI in cybersecurity: A comprehensive review of LLM applications and vulnerabilities

Generative large language models (LLMs) are a subset of transformers-based neural network architecture models. LLMs have successfully leveraged a combination of an increased number of parameters, improvements in computational efficiency, and large pre-training datasets to perform a wide spectrum of natural language processing (NLP) tasks. Using a few examples (few-shot) or no examples (zero-shot) for prompt-tuning has enabled LLMs to achieve state-of-the-art performance in a broad range of NLP applications. This article by the American Medical Informatics Association (AMIA) NLP Working Group characterizes the opportunities, challenges, and best practices for our community to leverage and advance the integration of LLMs in downstream NLP applications effectively. This can be accomplished through a variety of approaches, including augmented prompting, instruction prompt tuning, and reinforcement learning from human feedback (RLHF). Our focus is on making LLMs accessible to the broader biomedical informatics community, including clinicians and researchers who may be unfamiliar with NLP. Additionally, NLP practitioners may gain insight from the described best practices. We focus on 3 broad categories of NLP tasks, namely natural language understanding, natural language inferencing, and natural language generation. We review the emerging trends in prompt tuning, instruction fine-tuning, and evaluation metrics used for LLMs while drawing attention to several issues that impact biomedical NLP applications, including falsehoods in generated text (confabulation/hallucinations), toxicity, and dataset contamination leading to overfitting. We also review potential approaches to address some of these current challenges in LLMs, such as chain of thought prompting, and the phenomena of emergent capabilities observed in LLMs that can be leveraged to address complex NLP challenge in biomedical applications.

Large language models (LLMs) have achieved significant progress in mathematical reasoning, especially in elementary math. However, they remain indisposed on tackling complex questions at high-school or college levels, which put forward a more advanced requirement of mastering relevant mathematical theorems. For we humans, whether selecting the appropriate theorems according to the provided question is a crucial factor affecting the quality of the ultimate solutions, yet which has been neglected by previous research in the field of LLM reasoning. In this paper, we propose a novel approach to enhance the LLM's capability of utilizing the mathematical theorems to specific problems, which we refer to as Theorem Rationale (TR). To this end, a new dataset encompassing problem-theorem-solution triples is deliberately established for transferring principles of TR. Furthermore, we develop an evolving strategy to boost hierarchical instructions oriented on the theorems to alleviate difficulty in acquiring the curated data and facilitate the digestion of theorem application from various perspectives. Evaluations on a wide range of public datasets exhibit that the model fine-tuned with our dataset achieves consistent improvements at varying mathematical levels compared to the backbone. And further ablation studies illustrate the effectiveness of our proposed evolutionary strategies on enhancing the model's capability of math problem-solving. Overall, extensive experiments reveal the potential of our proposed method which highlights the significance of aligning the problems with the concrete theorems for LLMs to alleviate hallucination and improve the models' mathematical reasoning capabilities.

The growing interest in advanced large language models (LLMs) like ChatGPT has sparked debate about how best to use them in various human activities. However, a neglected issue in the debate concerning the applications of LLMs is whether they can reason logically and follow rules in novel contexts, which are critical for our understanding and applications of LLMs. To address this knowledge gap, this study investigates five LLMs (ChatGPT-4o, Claude, Gemini, Meta AI, and Mistral) using word ladder puzzles to assess their logical reasoning and rule-adherence capabilities. Our two-phase methodology involves (1) explicit instructions about word ladder puzzles and rules regarding how to solve the puzzles and then evaluate rule understanding, followed by (2) assessing LLMs’ ability to create and solve word ladder puzzles while adhering to rules. Additionally, we test their ability to implicitly recognize and avoid HIPAA privacy rule violations as an example of a real-world scenario. Our findings reveal that LLMs show a persistent lack of logical reasoning and systematically fail to follow puzzle rules. Furthermore, all LLMs except Claude prioritized task completion (text writing) over ethical considerations in the HIPAA test. Our findings expose critical flaws in LLMs’ reasoning and rule-following capabilities, raising concerns about their reliability in critical tasks requiring strict rule-following and logical reasoning. Therefore, we urge caution when integrating LLMs into critical fields and highlight the need for further research into their capabilities and limitations to ensure responsible AI development.

We survey a current, heated debate in the artificial intelligence (AI) research community on whether large pretrained language models can be said to understand language—and the physical and social situations language encodes—in any humanlike sense. We describe arguments that have been made for and against such understanding and key questions for the broader sciences of intelligence that have arisen in light of these arguments. We contend that an extended science of intelligence can be developed that will provide insight into distinct modes of understanding, their strengths and limitations, and the challenge of integrating diverse forms of cognition.

Reinforcement Learning with Human Feedback (RLHF) is a methodology designed to align Large Language Models (LLMs) with human preferences, playing an important role in LLMs alignment. Despite its advantages, RLHF relies on human annotators to rank the text, which can introduce potential security vulnerabilities if any adversarial annotator (i.e., attackers) manipulates the ranking score by upranking any malicious text to steer the LLM adversarially. To assess the red-teaming of RLHF against human preference data poisoning, we propose RankPoison, a poisoning attack method on candidates' selection of preference rank flipping to reach certain malicious behaviors (e.g., generating longer sequences, which can increase the computational cost). With poisoned dataset generated by RankPoison, we can perform poisoning attacks on LLMs to generate longer tokens without hurting the original safety alignment performance. Moreover, applying RankPoison, we also successfully implement a backdoor attack where LLMs can generate longer answers under questions with the trigger word. Our findings highlight critical security challenges in RLHF, underscoring the necessity for more robust alignment methods for LLMs.

SummaryLanguage has long been an essential tool for human reasoning. The rise of large language models (LLMs) has led to research on their application in complex reasoning tasks. Researchers are exploring the concept of “thought,” which represents intermediate reasoning steps, allowing LLMs to emulate humanlike reasoning processes. Recent work has applied reinforcement learning (RL) to train LLMs by searching for high-quality reasoning trajectories through trial-and-error exploration. In parallel, studies also demonstrate that allowing LLMs to “think” with longer chains of intermediate tokens at test time can also substantially improve reasoning accuracy. The combination of training and test-time advancements outlines a path toward large reasoning models. This survey reviews recent progress in LLM reasoning. It covers foundational concepts behind LLMs and the key technical components that contribute to the development of large reasoning models, and it highlights popular open-source projects for building these models. The survey concludes by discussing ongoing challenges and future research directions in this field.

Direct Preference Optimization (DPO) using an implicit reward model has proven to be an effective alternative to reinforcement learning from human feedback (RLHF) for fine-tuning preference aligned large language models (LLMs). However, the overall preference annotations of responses do not fully capture the finegrained quality of model outputs in complex multi-step reasoning tasks, such as mathematical reasoning. To address this limitation, we introduce a novel algorithm called Step-level Value Preference Optimization (SVPO). Our approach employs Monte Carlo Tree Search (MCTS) to automatically annotate step-level preferences for multi-step reasoning. Furthermore, from the perspective of learning-to-rank, we train an explicit value model to replicate the behavior of the implicit reward model, complementing standard preference optimization. This value model enables the LLM to generate higher reward responses with minimal cost during inference. Experimental results demonstrate that our method achieves state-of-the-art performance on both in-domain and out-ofdomain mathematical reasoning benchmarks. Our code is available at https://github. com/MARIO-Math-Reasoning/Super_MARIO.

Offensive cybersecurity is changing because of AI-enabled automation, especially in penetration testing platforms like Kali Linux. There is little academic research evaluating the comparative performance, operational risks, and governance requirements of various tools, such as multi-agent testers like BreachSeek, Large Language Model (LLM) systems like PentestGPT and RapidPen, and Reinforcement Learning (RL) agents like DeepExploit. In order to close this gap, this study presents unique tests carried out on a simulated 20-node business environment and suggests a unified analytical methodology for assessing AI-assisted penetration testing. We assess RL-based, LLM-based, and hybrid multi-agent systems in terms of accuracy, exploitation speed, false-positive rates, failure modes, and adversarial brittleness using standardized attack chains. Results indicate that LLM-driven orchestrators perform better than RL agents in reconnaissance and decision-making (avg. 4.3 min to shell vs. 7.8 min), although RL agents suffer from sparse-reward inefficiencies and provide more consistent payload selection. In line with the EU AI Act, ISO/IEC 42001, and the NIST AI Risk Management Framework (AI RMF), the study incorporates a structured ethical and governance analysis that highlights vulnerabilities including dual-use misuse pathways, over-automation, and hallucinated exploit paths. This work promotes a research-based framework for safe, responsible adoption in cybersecurity operations and offers a fair, fact-based assessment of AI-powered penetration testing.

Large language models (LLMs) excel in natural language generation but often confidently produce incorrect responses, especially in tasks like mathematical reasoning. Chain-of-thought prompting, self-verification, and multi-agent debate are among the strategies proposed to improve the reasoning and factual accuracy of LLMs. Building on Du et al.’s multi-agent debate framework, we find that multi-agent debate helps at any model scale, and that diversity of thought elicits stronger reasoning in debating LLMs. Across various model sizes, performance on mathematical reasoning tasks benefits most when diverse trained models are used. Remarkably, after 4 rounds of debate, a diverse set of medium-capacity models (Gemini-Pro, Mixtral 7B×8, and PaLM 2-M) outperforms GPT-4 on the GSM-8K benchmark, scoring 91% accuracy. By comparison, when 3 instances of Gemini-Pro are used, performance only reaches 82%. Finally, this diverse set of medium-capacity models sets a new state-of-the-art performance on the ASDiv benchmark (94%). These results underscore the idea that the future of AI is agentic, with diverse cooperating agents yielding emergent capabilities beyond even the most powerful individual models.