We propose Modal Logical Neural Networks (MLNNs), a neurosymbolic framework that integrates deep learning with the formal semantics of modal logic, enabling reasoning about necessity and possibility. Drawing on Kripke semantics, we introduce specialized neurons for the modal operators $\Box$ and $\Diamond$ that operate over a set of possible worlds, enabling the framework to act as a differentiable ``logical guardrail.'' The architecture is highly flexible: the accessibility relation between worlds can either be fixed by the user to enforce known rules or, as an inductive feature, be parameterized by a neural network. This allows the model to optionally learn the relational structure of a logical system from data while simultaneously performing deductive reasoning within that structure. This versatile construction is designed for flexibility. The entire framework is differentiable from end to end, with learning driven by minimizing a logical contradiction loss. This not only makes the system resilient to inconsistent knowledge but also enables it to learn nonlinear relationships that can help define the logic of a problem space. We illustrate MLNNs on four case studies: grammatical guardrailing, axiomatic detection of the unknown, multi-agent epistemic trust, and detecting constructive deception in natural language negotiation. These experiments demonstrate how enforcing or learning accessibility can increase logical consistency and interpretability without changing the underlying task architecture.

Logical reasoning is a core challenge in natural language understanding and a fundamental capability of artificial intelligence, underpinning scientific discovery, mathematical theorem proving, and complex decision-making. Despite the remarkable progress of large language models (LLMs), most current approaches still rely on forward reasoning paradigms, generating step-by-step rationales from premises to conclusions. However, such methods often suffer from redundant inference paths, hallucinated steps, and semantic drift, resulting in inefficient and unreliable reasoning. In this paper, we propose a novel framework, Hypothesis-driven Backward Logical Reasoning (HBLR). The core idea is to integrate confidence-aware symbolic translation with hypothesis-driven backward reasoning. In the translation phase, only high-confidence spans are converted into logical form, such as First-Order Logic (FOL), while uncertain content remains in natural language. A translation reflection module further ensures semantic fidelity by evaluating symbolic outputs and reverting lossy ones back to text when necessary. In the reasoning phase, HBLR simulates human deductive thinking by assuming the conclusion is true and recursively verifying its premises. A reasoning reflection module further identifies and corrects flawed inference steps, enhancing logical coherence. Extensive experiments on five reasoning benchmarks demonstrate that HBLR consistently outperforms strong baselines in both accuracy and efficiency.

Transformer-based language models (LMs) have achieved widespread empirical success, but their theoretical expressive power remains only partially understood. In this work, we analyze a restricted idealization of fixed-precision transformers with strict future masking, soft attention, and no positional encodings. We establish that this class of models is exactly as expressive as a specific fragment of linear temporal logic that contains only a single temporal operator: the past operator. We further connect this fragment to established classes in formal language theory, automata theory, and algebra, yielding a unified framework for understanding transformer expressivity under this idealization. Finally, we present empirical results that align closely with our theory: transformers trained on languages within their characterized expressive capacity generalize reliably across sequence lengths, while they consistently fail to generalize on languages beyond it.

When the distributions of the training and test data do not coincide, the problem of understanding generalization becomes considerably more complex, prompting a variety of questions. Prior work has shown that, for some fixed learning methods, there are scenarios where training on a distribution different from the test distribution improves generalization. However, these results do not account for the possibility of choosing, for each training distribution, the optimal learning algorithm, leaving open whether the observed benefits stem from the mismatch itself or from suboptimality of the learner. In this work, we address this question in full generality. That is, we study whether it is always optimal for the training distribution to be identical to the test distribution when the learner is allowed to be optimally adapted to the training distribution. Surprisingly, assuming the existence of one-way functions, we find that the answer is no. That is, matching distributions is not always the best scenario. Nonetheless, we also show that when certain regularities are imposed on the target functions, the standard conclusion is recovered in the case of the uniform distribution.

Recent advances in natural language processing (NLP), particularly large language models (LLMs), have motivated the automatic translation of natural language statements into formal logic without human intervention. This enables automated reasoning and facilitates debugging, finding loop invariants, and adhering to specifications in software systems. However, hallucinations-incorrect outputs generated by LLMs are challenging, particularly for logical translation tasks requiring precision. This work introduces a novel framework that inputs English sentences, converts them into logical expressions, and then translates them into Conjunctive Normal Form (CNF) for satisfiability solving. It employs classical NLP techniques with self-defined grammar, symbolic computation libraries, and a fine-tuned language model to reduce hallucinations. In the early experiments, we observed that the fine-tuned model, trained on different grammar settings, could intentionally correct the same types of hallucinations made by the original model. Thus, it provides reliable CNF generation.

We introduce IndiMathBench, a human-verified benchmark designed to evaluate mathematical theorem proving, curated using an AI-powered human-assisted pipeline for formalizing natural language problems in Lean. IndiMathBench is composed of 312 formal Lean 4 theorems paired with their corresponding informal problem statements, sourced from Indian Mathematics Olympiads. Through category-based retrieval, iterative compiler feedback, and multi-model ensembles, our pipeline generates candidate formalizations that experts efficiently validate via an interactive dashboard with automated quality summaries. Evaluation across multiple frontier models demonstrates that autoformalization remains challenging, with substantial gaps between syntactic validity and semantic correctness, while theorem proving success rates remain low even with iterative refinement, demonstrating that \benchmark~presents a challenging testbed for mathematical reasoning. IndiMathBench is available at https://github.com/prmbiy/IndiMathBench.

Automated theorem proving in Euclidean geometry, particularly for International Mathematical Olympiad (IMO) level problems, remains a major challenge and an important research focus in Artificial Intelligence. In this paper, we present a highly efficient method for geometry theorem proving that runs entirely on CPUs without relying on neural network-based inference. Our initial study shows that a simple random strategy for adding auxiliary points can achieve silver-medal level human performance on IMO. Building on this, we propose HAGeo, a Heuristic-based method for adding Auxiliary constructions in Geometric deduction that solves 28 of 30 problems on the IMO-30 benchmark, achieving gold-medal level performance and surpassing AlphaGeometry, a competitive neural network-based approach, by a notable margin. To evaluate our method and existing approaches more comprehensively, we further construct HAGeo-409, a benchmark consisting of 409 geometry problems with human-assessed difficulty levels. Compared with the widely used IMO-30, our benchmark poses greater challenges and provides a more precise evaluation, setting a higher bar for geometry theorem proving.

Large Language Models (LLMs) have shown significant promise in automated theorem proving, yet progress is often constrained by the scarcity of diverse and high-quality formal language data. To address this issue, we introduce Spark-Prover-X1, a 7B parameter model trained via an three-stage framework designed to unlock the reasoning potential of more accessible and moderately-sized LLMs. The first stage infuses deep knowledge through continuous pre-training on a broad mathematical corpus, enhanced by a suite of novel data tasks. Key innovation is a "CoT-augmented state prediction" task to achieve fine-grained reasoning. The second stage employs Supervised Fine-tuning (SFT) within an expert iteration loop to specialize both the Spark-Prover-X1-7B and Spark-Formalizer-X1-7B models. Finally, a targeted round of Group Relative Policy Optimization (GRPO) is applied to sharpen the prover's capabilities on the most challenging problems. To facilitate robust evaluation, particularly on problems from real-world examinations, we also introduce ExamFormal-Bench, a new benchmark dataset of 402 formal problems. Experimental results demonstrate that Spark-Prover achieves state-of-the-art performance among similarly-sized open-source models within the "Whole-Proof Generation" paradigm. It shows exceptional performance on difficult competition benchmarks, notably solving 27 problems on PutnamBench (pass@32) and achieving 24.0\% on CombiBench (pass@32). Our work validates that this diverse training data and progressively refined training pipeline provides an effective path for enhancing the formal reasoning capabilities of lightweight LLMs. We will release both Spark-Prover-X1-7B and Spark-Formalizer-X1-7B, along with the ExamFormal-Bench dataset, in the near future.

The Weighted First-Order Model Counting Problem (WFOMC) asks to compute the weighted sum of models of a given first-order logic sentence over a given domain. Conditioning WFOMC on evidence -- fixing the truth values of a set of ground literals -- has been shown impossible in time polynomial in the domain size (unless $\mathsf{\#P \subseteq FP}$) even for fragments of logic that are otherwise tractable for WFOMC without evidence. In this work, we address the barrier by restricting the binary evidence to the case where the underlying Gaifman graph has bounded treewidth. We present a polynomial-time algorithm in the domain size for computing WFOMC for the two-variable fragments $\text{FO}^2$ and $\text{C}^2$ conditioned on such binary evidence. Furthermore, we show the applicability of our algorithm in combinatorial problems by solving the stable seating arrangement problem on bounded-treewidth graphs of bounded degree, which was an open problem. We also conducted experiments to show the scalability of our algorithm compared to the existing model counting solvers.

Automating the translation of natural language to first-order logic (FOL) is crucial for knowledge representation and formal methods, yet remains challenging. We present a systematic evaluation of fine-tuned LLMs for this task, comparing architectures (encoder-decoder vs. decoder-only) and training strategies. Using the MALLS and Willow datasets, we explore techniques like vocabulary extension, predicate conditioning, and multilingual training, introducing metrics for exact match, logical equivalence, and predicate alignment. Our fine-tuned Flan-T5-XXL achieves 70% accuracy with predicate lists, outperforming GPT-4o and even the DeepSeek-R1-0528 model with CoT reasoning ability as well as symbolic systems like ccg2lambda. Key findings show: (1) predicate availability boosts performance by 15-20%, (2) T5 models surpass larger decoder-only LLMs, and (3) models generalize to unseen logical arguments (FOLIO dataset) without specific training. While structural logic translation proves robust, predicate extraction emerges as the main bottleneck.