Deductive Reasoning in Language Models

Continuous latent reasoning enables language models to perform intermediate computation without generating explicit reasoning traces. Advances in stochastic latent reasoning allow multiple paths to be sampled in continuous space, but stochasticity alone does not ensure effective exploration. A key challenge is guiding stochastic transitions between intermediate latent states toward useful reasoning paths. We introduce SubLAT, which couples a relative answer potential with Subtrajectory Balance to learn stochastic latent exploration. The relative potential evaluates the answer support of sampled states against replacement states, with a correctness signal at the terminal state. Subtrajectory Balance relates potential differences to forward and backward transition densities across subtrajectories of different lengths. The model is jointly trained with answer supervision, CoT supervision, and representation distillation. Experiments on arithmetic and deductive reasoning show increasing solution coverage with larger sampling budgets across in-domain and out-of-domain benchmarks, as well as improvements on several structural OOD benchmarks. Trajectory analyses reveal progressive divergence among sampled latent paths and higher mean relative potentials for correct trajectories across arithmetic benchmarks.

Current frontier LLMs can theoretically process long contexts with 1M tokens or more. But to what extent can they go beyond simple retrieval and perform deeper reasoning over such long contexts? We empirically investigate long-horizon reasoning capabilities of LLMs, focusing on deductive logic expressed in Prolog. We construct ProloNg, a synthetic testbed to probe Prolog Long Reasoning, which systematically varies the complexity (reasoning depth) of problems, where the hardest case has a reasoning depth of 22 and 62k context length. We study 8 reasoning models across 5 families of frontier LLMs, and find that performance degrades substantially as reasoning depth grows, with the majority of models approaching chance beyond depth 10.

Large language models (LLMs) are increasingly deployed for natural-language logical reasoning, where the final answer is easy to check but the proof behind it is not. In natural-language logical reasoning, an intermediate conclusion should follow from its premises, and the resulting derivation should support the final answer. Existing methods lack machine-checkable verification of intermediate conclusions and answer-supporting proof dependencies, so they may assign credit to invalid or answer-irrelevant steps. We propose Proof-R1, an RL framework from formal verification that trains LLMs to construct verifiable proofs for natural-language logical reasoning. Proof-R1 admits a generated conclusion into the verified proof state only when the corresponding reasoning action satisfies the proof obligations through UNSAT-based machine-checkable formal verification. Proof-R1 also recovers the answer-supporting dependency closure to trace the proof structure of the final answer and align outcome credit with the proof dependencies. Experiments demonstrate that Proof-R1 improves answer accuracy across three logical reasoning benchmarks and four backbone models and outperforms training-free agents and training-based methods in terms of reasoning-process verifiability.

Despite recent advances in large language models (LLMs), performing logically consistent deductive reasoning over extended interactions remains challenging. Tasks that require integrating evidence across multiple reasoning steps, maintaining consistency with prior inferences, and updating beliefs under new constraints can surface limitations in current models while providing a useful testbed for evaluating reasoning enhancements. In this paper, we implement a text-based, multi-agent version of the classic board game Clue as an environment to evaluate multi-step, agentic deductive reasoning. In this setting, agents must infer hidden information from a sequence of observations, maintain consistency across turns, and reason over an evolving set of logical constraints. We instantiate six LLM-based agents (GPT-4o-mini and Gemini-2.5-Flash) as players that engage in turn-based gameplay; using three agents per model family, we establish baseline performance across repeated games. We then introduce a tool-augmented approach in which a structured possibility matrix converts implicit game state from generated reasoning logs into an explicit representation of remaining possibilities. The possibility matrix encodes extended-turn memory and deductive constraints, offloading these tasks from the agent. We compare this approach against the baseline to evaluate how tool augmentation supports reasoning quality and task success for autonomous agents in a strategic reasoning environment.

Language models (LMs) struggle with logical tasks like reasoning on syllogisms. It has been shown that Knowledge Representation (KR) plays a crucial role in expressing input information to help models solve tasks. This observation motivates our study of the impact of different formal KR notations on syllogistic reasoning by extending the FOLIO and P-FOLIO datasets. Our experiments on Small Language Models (SLMs) in Supervised Fine-Tuning (SFT) and Zero-Shot (ZS) settings show that the choice of input notation can yield performances competitive with natural language while enabling faster inference. We also propose a syllogistic categorization method (SEF) and use it to enrich ZS prompts with logical definitions, which boost reasoning in small models. We open-source our framework, Common Logic Grammar Construction (CLGC), as the first Python library for automatically generating syllogisms in KR notations and defining their SEF categories.

Large Language Models (LLMs) have made significant progress in reasoning, particularly in deductive reasoning, which is crucial for high-stakes decision-making. As models improve, evaluation benchmarks should evolve to keep pace. However, existing benchmarks lack fine-grained control over logical complexity and struggle to balance semantic diversity with logical consistency. To address these issues, we propose QMFOL, an automated framework for generating monadic first-order logic reasoning tasks with quantifiable and controllable complexity. It constructs formal logical structures using conjunction and disjunction patterns, enabling precise control over reasoning depth, width, label types, and distractors. These structures are then translated into natural language via LLMs, with logical consistency ensured through round-trip verification using an external prover. Based on our framework, we build QMFOLBench, a benchmark comprising 2880 instances with 960 configurations across diverse logical and semantic dimensions. Evaluations on six large reasoning models (LRMs) and two LLMs show that performance degrades and computational overhead increases with rising logical complexity. Models perform better on True-labeled tasks than on False or Unknown ones, and exhibit sensitivity to semantic variation. Overall, QMFOL offers a scalable and reliable approach for constructing deductive reasoning benchmarks with controllable complexity, enabling more precise evaluation of reasoning capabilities in modern language models.

Many applications of large language models (LLMs) require deductive reasoning, yet models frequently produce incorrect or redundant inference steps. We frame natural language inference as a search problem where the final answer is the valid proof itself, requiring a reasoning procedure in which intermediate inferences are correct. Specifically, we investigate whether LLMs can learn to generate correct and efficient proofs with guidance from A* search -- an algorithm that guarantees an optimally efficient path to a goal. We explore two training techniques: supervised fine-tuning on execution traces from A* and reinforcement learning with A*-informed process reward models. Empirically, we find that Llama-3.2 models in the 1B--3B range benefit substantially from A* post training, going from near-zero accuracy to outperforming DeepSeek-V3.2 -- a much larger model. Our analysis uncovers a trade-off: while simple correctness rewards maximize accuracy, A*-informed signals strike a balance between accuracy and efficiency. Furthermore, we find that on larger search spaces, models trained with imperfect heuristics exhibit superior accuracy. Our results demonstrate a promising direction towards reasoning guided by principles derived from classical search algorithms.

Curriculum learning (CL), motivated by the intuition that learning in increasing order of difficulty should ease generalization, is commonly adopted both in pre-training and post-training of large language models (LLMs). The intuition of CL is particularly compelling for compositional reasoning, where complex problems are built from elementary inference rules; however, the actual impact of CL on such tasks remains largely underexplored. We present a systematic empirical study of CL for post-training of LLMs, using synthetic arithmetic and logical benchmarks where difficulty is characterized by reasoning complexity rather than surface-level proxies. Surprisingly, across multiple model families and curriculum schedules, we find no robust advantage in difficulty-based sequencing over standard random sampling in either accuracy or response length. These findings persist across both supervised fine-tuning (SFT) and reinforcement learning (RL) methods. Our study suggests that, in the context of deductive reasoning, the specific ordering of training examples plays a negligible role in achieving compositional generalization, challenging the practical utility of curriculum-based post-training.

Deducing whodunit proves challenging for LLM agents. In this paper, we implement a text-based multi-agent version of the classic board game Clue as a rule-based testbed for evaluating multi-step deductive reasoning, with six agents drawn from GPT-4o-mini and Gemini-2.5-Flash. We further investigate whether fine-tuning on structured logic puzzles transfers to improved in-game reasoning and gameplay. Across 18 simulated games, agents achieve only four correct wins, indicating difficulty in maintaining consistent deductive reasoning over the course of a full game. Additionally, we find that fine-tuning does not reliably improve performance and, in some cases, appears to increase reasoning volume without improving reasoning precision.

Recent language models exhibit significant logical reasoning abilities, yet the mechanisms supporting deductive inference remain poorly understood. This paper studies small transformer-based language models trained from scratch on multi-step deductive tasks, focusing on the distinction between horizontal reasoning, where intermediate steps are generated autoregressively, and vertical reasoning, where inference unfolds implicitly across layers before the first output token is produced. We analyze two synthetic tasks: logical consequence over chains of symbolic implications and root-to-leaf navigation in binary trees. Mechanistic interpretability reveals that Chain-of-Thought supervision enables models to learn rule-based inference rather than statistical shortcuts. In the horizontal setting, a shallow attention-only model develops interpretable circuits for rule completion, rule chaining, and final decision making, largely implemented through induction-head-like mechanisms. We further introduce a truncated pseudoinverse method to decode the information carried by queries, keys, and values. For vertical reasoning, Chain-of-Thought appears to act less as explicit step-by-step guidance and more as a form of curriculum learning, helping the model acquire increasingly complex reasoning patterns. Without Chain-of-Thought, models tend to memorize or exploit dataset biases. These results provide a low-level account of how transformers can implement deductive reasoning and suggest how Chain-of-Thought may serve different functions in horizontal and vertical reasoning.

Modern language models (LMs) can learn to perform new tasks in different ways: in instruction following, the target task is described explicitly in natural language; in few-shot prompting, the task is specified implicitly with a small number of examples; in instruction inference, LMs are presented with in-context examples and are then prompted to generate a natural language task description before making predictions. Each of these procedures may be thought of as invoking a different form of reasoning: instruction following involves deductive reasoning, few-shot prompting involves inductive reasoning, and instruction inference involves abductive reasoning. How do these different capabilities relate? Across four LMs (from the gpt and llama families) and two learning problems (involving arithmetic functions and machine translation) we find a strong dissociation between the different types of reasoning: LMs can sometimes learn effectively from few-shot prompts even when they are unable to explain their own prediction rules; conversely, they sometimes infer useful task descriptions while completely failing to learn from human-generated descriptions of the same task. Our results highlight the non-systematic nature of reasoning even in some of today's largest LMs, and underscore the fact that very different learning mechanisms may be invoked by seemingly similar prompting procedures.