In human-AI interaction, effective communication relies on aligning the AI agent's model with the human user's mental model, a process known as model reconciliation. However, existing model reconciliation approaches predominantly assume deterministic models, overlooking the fact that human knowledge is often uncertain or probabilistic. To bridge this gap, we present a probabilistic model reconciliation framework that resolves inconsistencies in MPE outcome probabilities between an agent's and a user's models. Our approach is built on probabilistic logic programming (PLP) using ProbLog, where explanations are generated as cost-optimal model updates that reconcile these probabilistic differences. We develop two search algorithms - a generic baseline and an optimized version. The latter is guided by theoretical insights and further extended with greedy and weighted variants to enhance scalability and efficiency. Our approach is validated through a user study on explanation types and computational experiments showing that the optimized version consistently outperforms the generic baseline.

Autoformalization, the automatic translation of mathematical content from natural language into machine-verifiable formal languages, has seen significant progress driven by advances in large language models (LLMs). Nonetheless, a primary barrier to further improvements is the limited availability of parallel corpora that map informal mathematical text to its formal counterpart. To address this limitation, we propose ATLAS (Autoformalizing Theorems through Lifting, Augmentation, and Synthesis of Data), a novel data generation framework designed to produce large-scale, high-quality parallel corpora of theorem statements. Distinct from prior approaches, ATLAS begins with a concept repository, accelerates the improvement of the student model through expert iteration combined with knowledge distillation, and introduces two novel augmentation strategies that exploit the structural characteristics of formal languages. Running the proposed ATLAS framework for 10 iterations, we construct an undergraduate-level dataset of 117k theorem statements and develop the ATLASTranslator by fine-tuning Llama3.1-8B-Instruct with LoRA. This model establishes a new state of the art, demonstrating statistically significant improvements over both the Herald Translator and the Kimina-Autoformalizer across all benchmarks (p < 0.05, two-sided t-test). Furthermore, we demonstrate that the full-parameter fine-tuning of a stronger base model on the ATLAS dataset leads to superior performance.

The integration of logic programs with embedding models resulted in a class of neurosymbolic frameworks that jointly learn symbolic rules and representations for the symbols in the logic (constant or predicate). The key idea that enabled this integration was the differentiable relaxation of unification, the algorithm for variable instantiation during inference in logic programs. Unlike unification, its relaxed counterpart exploits the similarity between symbols in the embedding space to decide when two symbols are semantically equivalent. We show that this similarity between symbols violates the transitive law of equivalence, leading to undesirable side effects in learning and inference. To alleviate those side effects, we are the first to revamp the well-known possible world semantics of probabilistic logic programs into new semantics called equivalence semantics. In our semantics, a probabilistic logic program induces a probability distribution over all possible equivalence relations between symbols, instead of a probability distribution over all possible subsets of probabilistic facts. We propose a factorization of the equivalence distribution using latent random variables and characterize its expressivity. Additionally, we propose both exact and approximate techniques for reasoning in our semantics. Experiments on well-known benchmarks show that the equivalence semantics leads to neurosymbolic models with up to 42% higher results than state-of-the-art baselines.

We take two key steps in automating the open-ended discovery of new mathematical theories, a grand challenge in artificial intelligence. First, we introduce FERMAT, a reinforcement learning (RL) environment that models concept discovery and theorem-proving using a set of symbolic actions, opening up a range of RL problems relevant to theory discovery. Second, we explore a specific problem through FERMAT: automatically scoring the interestingness of mathematical objects. We investigate evolutionary algorithms for synthesizing nontrivial interestingness measures. In particular, we introduce an LLM-based evolutionary algorithm that features function abstraction, leading to notable improvements in discovering elementary number theory and finite fields over hard-coded baselines.

Large language models (LLMs) show remarkable promise for democratizing automated reasoning by generating formal specifications. However, a fundamental tension exists: LLMs are probabilistic, while formal verification demands deterministic guarantees.

Premise selection is a critical bottleneck in interactive theorem proving, particularly with large libraries. Existing methods, primarily relying on semantic embeddings, often fail to effectively leverage the rich structural information inherent in mathematical expressions. This paper proposes a novel framework for premise selection based on the structure of expression trees. The framework enhances premise selection ability by explicitly utilizing the structural information of Lean expressions and by means of the simplified tree representation obtained via common subexpression elimination. Our method employs a multi-stage filtering pipeline, incorporating structure-aware similarity measures including the Weisfeiler-Lehman kernel, tree edit distance, Constnode Jaccard similarity, and collapse-match similarity. An adaptive fusion strategy combines these metrics for refined ranking. To handle large-scale data efficiently, we incorporate cluster-based search space optimization and structural compatibility constraints. Comprehensive evaluation on a large theorem library extracted from Mathlib4 demonstrates that our method significantly outperforms existing premise retrieval tools across various metrics. Experimental analysis, including ablation studies and parameter sensitivity analysis, validates the contribution of individual components and highlights the efficacy of our structure-aware approach and multi-metric fusion.

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 $\texttt{past}$ 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.

Automated Theorem Proving (ATP) in formal languages remains a formidable challenge in AI, demanding rigorous logical deduction and navigating vast search spaces. While large language models (LLMs) have shown promising performance, existing stepwise provers often suffer from biased search guidance, leading to inefficiencies and suboptimal proof strategies. This paper introduces the Multi-Perspective Search Prover (MPS-Prover), a novel stepwise ATP system designed to overcome these limitations. MPS-Prover incorporates two key innovations: a highly effective post-training data curation strategy that prunes approximately 40\% of redundant training data without sacrificing performance, and a multi-perspective tree search mechanism. This search integrates a learned critic model with strategically designed heuristic rules to diversify tactic selection, prevent getting trapped in unproductive states, and enhance search robustness. Extensive evaluations demonstrate that MPS-Prover achieves state-of-the-art performance on multiple challenging benchmarks, including miniF2F and ProofNet, outperforming prior 7B parameter models. Furthermore, our analyses reveal that MPS-Prover generates significantly shorter and more diverse proofs compared to existing stepwise and whole-proof methods, highlighting its efficiency and efficacy. Our work advances the capabilities of LLM-based formal reasoning and offers a robust framework and a comprehensive analysis for developing more powerful theorem provers.

Modern language models (LMs) exhibit strong deductive reasoning capabilities, yet standard evaluations emphasize correctness while overlooking a key aspect of reasoning: . In real-world reasoning scenarios, much of the available information is irrelevant, and effective deductive inference requires identifying and ignoring such distractions. We propose a framework for assessing LM reasoning efficiency through the lens of logic programming, introducing a simple method to align proofs written in natural language---as generated by an LM---with shortest proofs found by executing the logic program. Efficiency is quantified by measuring how well a model avoids unnecessary inference. Empirically, we construct a dataset of math word problems injected with various number of irrelevant axioms that vary in semantic overlap with the goal theorem. We find that current LMs show marked accuracy declines under such conditions---even with minimal, domain-consistent distractions---and the proofs they generate frequently exhibit detours through irrelevant inferences.

Given a Standard Operating Procedure (SOP) describing a target system, SpecMAS parses the specification, identifies relevant operational modes, variables, transitions, and properties, and generates a formal model in NuSMV code syntax, an industry-standard symbolic model checker. A dedicated reasoning agent extracts both explicit and implicit properties from the SOP, and verification is performed via temporal logic model checking. If any properties fail to verify, an autonomous debugging agent analyzes counterexamples and iteratively corrects the model until all properties are satisfied. This closed-loop system design guarantees provable correctness by construction and advances the state of the art in automated, interpretable, and deployable verification pipelines. We demonstrate the generality, correctness, and practical feasibility of SpecMAS across a set of representative case studies and propose a new benchmark dataset for the evaluation and comparison of model checking performance.