Large language models (LLMs) have demonstrated promise in reasoning tasks and general decision-making in static environments. In long-term planning tasks, however, errors tend to accumulate, often leading to unsafe or inefficient behavior, limiting their use in general-purpose settings. We propose a modular actor-critic architecture in which an LLM actor is guided by LTLCrit, a trajectory-level LLM critic that communicates via linear temporal logic (LTL). Our setup combines the reasoning strengths of language models with the guarantees of formal logic. The actor selects high-level actions from natural language observations, while the critic analyzes full trajectories and proposes new LTL constraints that shield the actor from future unsafe or inefficient behavior. The architecture supports both fixed, hand-specified safety constraints and adaptive, learned soft constraints that promote long-term efficiency. Our architecture is model-agnostic: any LLM-based planner can serve as the actor, and LTLCrit serves as a logic-generating wrapper. We formalize planning as graph traversal under symbolic constraints, allowing LTLCrit to analyze failed or suboptimal trajectories and generate new temporal logic rules that improve future behavior. We evaluate our system on the Minecraft diamond-mining benchmark, achieving 100% completion rates and improving efficiency compared to baseline LLM planners. Our results suggest that enabling LLMs to supervise each other through logic is a powerful and flexible paradigm for safe, generalizable decision making.
This paper presents a formalised architecture for synthetic agents designed to retain immutable memory, verifiable reasoning, and constrained epistemic growth. Traditional AI systems rely on mutable, opaque statistical models prone to epistemic drift and historical revisionism. In contrast, we introduce the concept of the Merkle Automaton, a cryptographically anchored, deterministic computational framework that integrates formal automata theory with blockchain-based commitments. Each agent transition, memory fragment, and reasoning step is committed within a Merkle structure rooted on-chain, rendering it non-repudiable and auditably permanent. To ensure selective access and confidentiality, we derive symmetric encryption keys from ECDH exchanges contextualised by hierarchical privilege lattices. This enforces cryptographic access control over append-only DAG-structured knowledge graphs. Reasoning is constrained by formal logic systems and verified through deterministic traversal of policy-encoded structures. Updates are non-destructive and historied, preserving epistemic lineage without catastrophic forgetting. Zero-knowledge proofs facilitate verifiable, privacy-preserving inclusion attestations. Collectively, this architecture reframes memory not as a cache but as a ledger - one whose contents are enforced by protocol, bound by cryptography, and constrained by formal logic. The result is not an intelligent agent that mimics thought, but an epistemic entity whose outputs are provably derived, temporally anchored, and impervious to post hoc revision. This design lays foundational groundwork for legal, economic, and high-assurance computational systems that require provable memory, unforgeable provenance, and structural truth.
Neuro-symbolic approaches combining large language models (LLMs) with solvers excels in logical reasoning problems need long reasoning chains. In this paradigm, LLMs serve as translators, converting natural language reasoning problems into formal logic formulas. Then reliable symbolic solvers return correct solutions. Despite their success, we find that LLMs, as translators, struggle to handle lexical diversification, a common linguistic phenomenon, indicating that LLMs as logic translators are unreliable in real-world scenarios. Moreover, existing logical reasoning benchmarks lack lexical diversity, failing to challenge LLMs' ability to translate such text and thus obscuring this issue. In this work, we propose SCALe, a benchmark designed to address this significant gap through **logic-invariant lexical diversification**. By using LLMs to transform original benchmark datasets into lexically diversified but logically equivalent versions, we evaluate LLMs' ability to consistently map diverse expressions to uniform logical symbols on these new datasets. Experiments using SCALe further confirm that current LLMs exhibit deficiencies in this capability. Building directly on the deficiencies identified through our benchmark, we propose a new method, MenTaL, to address this limitation. This method guides LLMs to first construct a table unifying diverse expressions before performing translation. Applying MenTaL through in-context learning and supervised fine-tuning (SFT) significantly improves the performance of LLM translators on lexically diversified text. Our code is now available at https://github.com/wufeiwuwoshihua/LexicalDiver.
Large Language Models have demonstrated remarkable capabilities in natural language processing tasks, including mathematical problem-solving that requires multi-step logical reasoning. However, challenges persist in automating the identification of key mathematical concepts, understanding their interrelations, and formalizing proofs within a rigorous framework. We present a novel framework that leverages knowledge graphs to augment LLMs to construct and formalize mathematical proofs. Our results demonstrate significant performance improvements across multiple datasets, with using knowledge graphs, achieving up to a 34% success rate on the MUSTARDSAUCE dataset on o1-mini and consistently outperforming baseline approaches by 2-11% across different models. We show how this approach bridges the gap between natural language understanding and formal logic proof systems and achieve elevated results for foundation models over baseline.
This paper introduces a formal framework for modeling observer-dependent collapse dynamics and temporal identity drift within artificial and mathematical systems, grounded entirely in the symbolic foundations of Alpay Algebra. Building upon the fixed-point emergence structures developed in Alpay Algebra I and II, this third installment formalizes the observer-coupled {\phi}-collapse process through transfinite categorical flows and curvature-driven identity operators. We define a novel temporal drift mechanism as a recursive deformation of identity signatures under entangled observer influence, constructing categorical invariants that evolve across fold iterations. The proposed system surpasses conventional identity modeling in explainable AI (XAI) by encoding internal transformation history into a symbolic fixed-point structure, offering provable traceability and temporal coherence. Applications range from AI self-awareness architectures to formal logic systems where identity is not static but dynamically induced by observation. The theoretical results also offer a mathematically rigorous basis for future AI systems with stable self-referential behavior, positioning Alpay Algebra as a next-generation symbolic framework bridging category theory, identity logic, and observer dynamics.
Process Reward Models (PRMs), which provide step-by-step feedback on the reasoning generated by Large Language Models (LLMs), are receiving increasing attention. However, two key research gaps remain: collecting accurate step-level error labels for training typically requires costly human annotation, and existing PRMs are limited to math reasoning problems. In response to these gaps, this paper aims to address the challenges of automatic dataset creation and the generalization of PRMs to diverse reasoning tasks. To achieve this goal, we propose FoVer, an approach for training PRMs on step-level error labels automatically annotated by formal verification tools, such as Z3 for formal logic and Isabelle for theorem proof, which provide automatic and accurate verification for symbolic tasks. Using this approach, we synthesize a training dataset with error labels on LLM responses for formal logic and theorem proof tasks without human annotation. Although this data synthesis is feasible only for tasks compatible with formal verification, we observe that LLM-based PRMs trained on our dataset exhibit cross-task generalization, improving verification across diverse reasoning tasks. Specifically, PRMs trained with FoVer significantly outperform baseline PRMs based on the original LLMs and achieve competitive or superior results compared to state-of-the-art PRMs trained on labels annotated by humans or stronger models, as measured by step-level verification on ProcessBench and Best-of-K performance across 12 reasoning benchmarks, including MATH, AIME, ANLI, MMLU, and BBH. The datasets, models, and code are provided at https://github.com/psunlpgroup/FoVer.
Cybersecurity demands rigorous and scalable techniques to ensure system correctness, robustness, and resilience against evolving threats. Automated reasoning, encompassing formal logic, theorem proving, model checking, and symbolic analysis, provides a foundational framework for verifying security properties across diverse domains such as access control, protocol design, vulnerability detection, and adversarial modeling. This survey presents a comprehensive overview of the role of automated reasoning in cybersecurity, analyzing how logical systems, including temporal, deontic, and epistemic logics are employed to formalize and verify security guarantees. We examine SOTA tools and frameworks, explore integrations with AI for neural-symbolic reasoning, and highlight critical research gaps, particularly in scalability, compositionality, and multi-layered security modeling. The paper concludes with a set of well-grounded future research directions, aiming to foster the development of secure systems through formal, automated, and explainable reasoning techniques.
With the rapid advancement of Artificial Intelligence (AI), Large Language Models (LLMs) have significantly impacted a wide array of domains, including healthcare, engineering, science, education, and mathematical reasoning. Among these, mathematical reasoning remains a particularly challenging capability, often requiring multi-step logic and abstract generalization. While prior work has explored LLM performance on reasoning tasks, comprehensive evaluations that span both depth and breadth across model families remain limited. In this study, we present a systematic evaluation of mathematical reasoning abilities across eight leading LLMs, including two recent DeepSeek models, using three independent benchmark datasets. Our analyses reveal several key findings: (1) DeepSeek-R1 performs competitively with o1 across most domains and achieves the highest accuracy on the MMLU Formal Logic benchmark; (2) distilled variants, such as DeepSeek-1.5B, exhibit substantial performance degradation; and (3) Gemini 2.0 Flash achieves the lowest response latency. Beyond quantitative metrics, we explore how architectural choices, training paradigms, and optimization strategies contribute to variation in reasoning performance. These findings provide new insights into the capabilities and limitations of current LLMs in mathematical domains, and offer guidance for the development of future models better aligned with rigorous reasoning demands.
Sustainable agricultural production aligns with several sustainability goals established by the United Nations (UN). However, there is a lack of studies that comprehensively examine sustainable agricultural practices across various products and production methods. Such research could provide valuable insights into the diverse factors influencing the sustainability of specific crops and produce while also identifying practices and conditions that are universally applicable to all forms of agricultural production. While this research might help us better understand sustainability, the community would still need a consistent set of vocabularies. These consistent vocabularies, which represent the underlying datasets, can then be stored in a global food systems datahub. The standardized vocabularies might help encode important information for further statistical analyses and AI/ML approaches in the datasets, resulting in the research targeting sustainable agricultural production. A structured method of representing information in sustainability, especially for wheat production, is currently unavailable. In an attempt to address this gap, we are building a set of ontologies and Knowledge Graphs (KGs) that encode knowledge associated with sustainable wheat production using formal logic. The data for this set of knowledge graphs are collected from public data sources, experimental results collected at our experiments at Kansas State University, and a Sustainability Workshop that we organized earlier in the year, which helped us collect input from different stakeholders throughout the value chain of wheat. The modeling of the ontology (i.e., the schema) for the Knowledge Graph has been in progress with the help of our domain experts, following a modular structure using KNARM methodology. In this paper, we will present our preliminary results and schemas of our Knowledge Graph and ontologies.
Linguistic evaluations of how well LMs generalize to produce or understand novel text often implicitly take for granted that natural languages are generated by symbolic rules. Grammaticality is thought to be determined by whether sentences obey such rules. Interpretation is believed to be compositionally generated by syntactic rules operating on meaningful words. Semantic parsing is intended to map sentences into formal logic. Failures of LMs to obey strict rules have been taken to reveal that LMs do not produce or understand language like humans. Here we suggest that LMs' failures to obey symbolic rules may be a feature rather than a bug, because natural languages are not based on rules. New utterances are produced and understood by a combination of flexible, interrelated, and context-dependent constructions. We encourage researchers to reimagine appropriate benchmarks and analyses that acknowledge the rich, flexible generalizations that comprise natural languages.
This comprehensive survey examines the integration of knowledge-based approaches into autonomous driving systems, with a focus on trajectory prediction and planning. We systematically review methodologies for incorporating domain knowledge, traffic rules, and commonsense reasoning into these systems, spanning purely symbolic representations to hybrid neuro-symbolic architectures. In particular, we analyze recent advancements in formal logic and differential logic programming, reinforcement learning frameworks, and emerging techniques that leverage large foundation models and diffusion models for knowledge representation. Organized under a unified literature survey section, our discussion synthesizes the state-of-the-art into a high-level overview, supported by a detailed comparative table that maps key works to their respective methodological categories. This survey not only highlights current trends -- including the growing emphasis on interpretable AI, formal verification in safety-critical systems, and the increased use of generative models in prediction and planning -- but also outlines the challenges and opportunities for developing robust, knowledge-enhanced autonomous driving systems.
In response to the call for agent-based solutions that leverage the ever-increasing capabilities of the deep models' ecosystem, we introduce Hive -- a comprehensive solution for selecting appropriate models and subsequently planning a set of atomic actions to satisfy the end-users' instructions. Hive operates over sets of models and, upon receiving natural language instructions (i.e. user queries), schedules and executes explainable plans of atomic actions. These actions can involve one or more of the available models to achieve the overall task, while respecting end-users specific constraints. Notably, Hive handles tasks that involve multi-modal inputs and outputs, enabling it to handle complex, real-world queries. Our system is capable of planning complex chains of actions while guaranteeing explainability, using an LLM-based formal logic backbone empowered by PDDL operations. We introduce the MuSE benchmark in order to offer a comprehensive evaluation of the multi-modal capabilities of agent systems. Our findings show that our framework redefines the state-of-the-art for task selection, outperforming other competing systems that plan operations across multiple models while offering transparency guarantees while fully adhering to user constraints.
Large language models (LLMs) are capable of solving a wide range of tasks, yet they have struggled with reasoning. To address this, we propose $\textbf{Additional Logic Training (ALT)}$, which aims to enhance LLMs' reasoning capabilities by program-generated logical reasoning samples. We first establish principles for designing high-quality samples by integrating symbolic logic theory and previous empirical insights. Then, based on these principles, we construct a synthetic corpus named $\textbf{Formal Logic Deduction Diverse}$ ($\textbf{FLD}$$_{\times 2}$), comprising numerous samples of multi-step deduction with unknown facts, diverse reasoning rules, diverse linguistic expressions, and challenging distractors. Finally, we empirically show that ALT on FLD$_{\times2}$ substantially enhances the reasoning capabilities of state-of-the-art LLMs, including LLaMA-3.1-70B. Improvements include gains of up to 30 points on logical reasoning benchmarks, up to 10 points on math and coding benchmarks, and 5 points on the benchmark suite BBH.
Federated Learning (FL) offers a promising solution to the privacy concerns associated with centralized Machine Learning (ML) by enabling decentralized, collaborative learning. However, FL is vulnerable to various security threats, including poisoning attacks, where adversarial clients manipulate the training data or model updates to degrade overall model performance. Recognizing this threat, researchers have focused on developing defense mechanisms to counteract poisoning attacks in FL systems. However, existing robust FL methods predominantly focus on computer vision tasks, leaving a gap in addressing the unique challenges of FL with time series data. In this paper, we present FLORAL, a defense mechanism designed to mitigate poisoning attacks in federated learning for time-series tasks, even in scenarios with heterogeneous client data and a large number of adversarial participants. Unlike traditional model-centric defenses, FLORAL leverages logical reasoning to evaluate client trustworthiness by aligning their predictions with global time-series patterns, rather than relying solely on the similarity of client updates. Our approach extracts logical reasoning properties from clients, then hierarchically infers global properties, and uses these to verify client updates. Through formal logic verification, we assess the robustness of each client contribution, identifying deviations indicative of adversarial behavior. Experimental results on two datasets demonstrate the superior performance of our approach compared to existing baseline methods, highlighting its potential to enhance the robustness of FL to time series applications. Notably, FLORAL reduced the prediction error by 93.27% in the best-case scenario compared to the second-best baseline. Our code is available at https://anonymous.4open.science/r/FLORAL-Robust-FTS.
Formal logic has long been applied to natural language reasoning, but this approach can sometimes lead to conclusions that, while logically entailed, are factually inconsistent with the premises or are not typically inferred by humans. This study introduces the concept of "rulebreakers", which refers to instances where logical entailment diverges from factually acceptable inference. We present RULEBREAKERS, a novel dataset for evaluating Large Language Models' (LLMs) ability to distinguish between rulebreakers and non-rulebreakers. Focusing on modus tollens and disjunctive syllogism, we assess six state-of-the-art LLMs using RULEBREAKERS, measuring their performance in terms of token-level exact accuracy and model confidence. Our findings reveal that while most models perform poorly to moderately in recognizing rulebreakers, they demonstrate a latent ability to distinguish rulebreakers when assessed by their confidence levels. Further analysis suggests that the failure to recognize rulebreakers is potentially associated with the models' world knowledge and their attention distribution patterns. This research highlights the limitation of LLMs' reasoning capabilities, and contributes to the ongoing discussion on reasoning in LLMs.
Language models lack the notion of interchangeable tokens: symbols that are semantically equivalent yet distinct, such as bound variables in formal logic. This limitation prevents generalization to larger vocabularies and hinders the model's ability to recognize alpha-equivalence, where renaming bound variables preserves meaning. We formalize this machine learning problem and introduce alpha-covariance, a metric for evaluating robustness to such transformations. To tackle this task, we propose a dual-part token embedding strategy: a shared component ensures semantic consistency, while a randomized component maintains token distinguishability. Compared to a baseline that relies on alpha-renaming for data augmentation, our approach demonstrates improved generalization to unseen tokens in linear temporal logic solving, propositional logic assignment prediction, and copying with an extendable vocabulary, while introducing a favorable inductive bias for alpha-equivalence. Our findings establish a foundation for designing language models that can learn interchangeable token representations, a crucial step toward more flexible and systematic reasoning in formal domains. Our code and project page are available at https://necrashter.github.io/interchangeable-token-embeddings
Autonomous agents often face the challenge of interpreting uncertain natural language instructions for planning tasks. Representing these instructions as Linear Temporal Logic (LTL) enables planners to synthesize actionable plans. We introduce CoT-TL, a data-efficient in-context learning framework for translating natural language specifications into LTL representations. CoT-TL addresses the limitations of large language models, which typically rely on extensive fine-tuning data, by extending chain-of-thought reasoning and semantic roles to align with the requirements of formal logic creation. This approach enhances the transparency and rationale behind LTL generation, fostering user trust. CoT-TL achieves state-of-the-art accuracy across three diverse datasets in low-data scenarios, outperforming existing methods without fine-tuning or intermediate translations. To improve reliability and minimize hallucinations, we incorporate model checking to validate the syntax of the generated LTL output. We further demonstrate CoT-TL's effectiveness through ablation studies and evaluations on unseen LTL structures and formulas in a new dataset. Finally, we validate CoT-TL's practicality by integrating it into a QuadCopter for multi-step drone planning based on natural language instructions.
Logical reasoning is a crucial task for Large Language Models (LLMs), enabling them to tackle complex problems. Among reasoning tasks, multi-step reasoning poses a particular challenge. Grounded in the theory of formal logic, we have developed an automated method, Multi-step Deduction (MuseD), for deductive reasoning data. MuseD has allowed us to create training and testing datasets for multi-step reasoning. Our generation method enables control over the complexity of the generated instructions, facilitating training and evaluation of models across different difficulty levels. Through RLHF training, our training data has demonstrated significant improvements in logical capabilities for both in-domain of out-of-domain reasoning tasks. Additionally, we have conducted tests to assess the multi-step reasoning abilities of various models.
Large Language Models (LLMs) have revolutionized natural language processing, yet they struggle with inconsistent reasoning, particularly in novel domains and complex logical sequences. This research introduces Proof of Thought, a framework that enhances the reliability and transparency of LLM outputs. Our approach bridges LLM-generated ideas with formal logic verification, employing a custom interpreter to convert LLM outputs into First Order Logic constructs for theorem prover scrutiny. Central to our method is an intermediary JSON-based Domain-Specific Language, which by design balances precise logical structures with intuitive human concepts. This hybrid representation enables both rigorous validation and accessible human comprehension of LLM reasoning processes. Key contributions include a robust type system with sort management for enhanced logical integrity, explicit representation of rules for clear distinction between factual and inferential knowledge, and a flexible architecture that allows for easy extension to various domain-specific applications. We demonstrate Proof of Thought's effectiveness through benchmarking on StrategyQA and a novel multimodal reasoning task, showing improved performance in open-ended scenarios. By providing verifiable and interpretable results, our technique addresses critical needs for AI system accountability and sets a foundation for human-in-the-loop oversight in high-stakes domains.
Game theory is a powerful framework for reasoning about strategic interactions, with applications in domains ranging from day-to-day life to international politics. However, applying formal reasoning tools in such contexts is challenging, as these scenarios are often expressed in natural language. To address this, we introduce a framework for the autoformalization of game-theoretic scenarios, which translates natural language descriptions into formal logic representations suitable for formal solvers. Our approach utilizes one-shot prompting and a solver that provides feedback on syntactic correctness to allow LLMs to refine the code. We evaluate the framework using GPT-4o and a dataset of natural language problem descriptions, achieving 98% syntactic correctness and 88% semantic correctness. These results show the potential of LLMs to bridge the gap between real-life strategic interactions and formal reasoning.