Zhaoyu Li

Zhaoyu Li, Jinpei Guo, Xujie Si · utoronto

Graph neural networks (GNNs) have recently emerged as a promising approach for solving the Boolean Satisfiability Problem (SAT), offering potential alternatives to traditional backtracking or local search SAT solvers. However, despite the growing volume of literature in this field, there remains a notable absence of a unified dataset and a fair benchmark to evaluate and compare existing approaches. To address this crucial gap, we present G4SATBench, the first benchmark study that establishes a comprehensive evaluation framework for GNN-based SAT solvers. In G4SATBench, we meticulously curate a large and diverse set of SAT datasets comprising 7 problems with 3 difficulty levels and benchmark a broad range of GNN models across various prediction tasks, training objectives, and inference algorithms. To explore the learning abilities and comprehend the strengths and limitations of GNN-based SAT solvers, we also compare their solving processes with the heuristics in search-based SAT solvers. Our empirical results provide valuable insights into the performance of GNN-based SAT solvers and further suggest that existing GNN models can effectively learn a solving strategy akin to greedy local search but struggle to learn backtracking search in the latent space. Our codebase is available at https://github.com/zhaoyu-li/G4SATBench.

Zhaoyu Li, Xujie Si · utoronto

We present the Neural Satisfiability Network (NSNet), a general neural framework that models satisfiability problems as probabilistic inference and meanwhile exhibits proper explainability. Inspired by the Belief Propagation (BP), NSNet uses a novel graph neural network (GNN) to parameterize BP in the latent space, where its hidden representations maintain the same probabilistic interpretation as BP. NSNet can be flexibly configured to solve both SAT and #SAT problems by applying different learning objectives. For SAT, instead of directly predicting a satisfying assignment, NSNet performs marginal inference among all satisfying solutions, which we empirically find is more feasible for neural networks to learn. With the estimated marginals, a satisfying assignment can be efficiently generated by rounding and executing a stochastic local search. For #SAT, NSNet performs approximate model counting by learning the Bethe approximation of the partition function. Our evaluations show that NSNet achieves competitive results in terms of inference accuracy and time efficiency on multiple SAT and #SAT datasets.

Zhaoyu Li, Jinpei Guo, Yuhe Jiang et al. · utoronto

Bridging logical reasoning and deep learning is crucial for advanced AI systems. In this work, we present a new framework that addresses this goal by generating interpretable and verifiable logical rules through differentiable learning, without relying on pre-specified logical structures. Our approach builds upon SATNet, a differentiable MaxSAT solver that learns the underlying rules from input-output examples. Despite its efficacy, the learned weights in SATNet are not straightforwardly interpretable, failing to produce human-readable rules. To address this, we propose a novel specification method called "maximum equality", which enables the interchangeability between the learned weights of SATNet and a set of propositional logical rules in weighted MaxSAT form. With the decoded weighted MaxSAT formula, we further introduce several effective verification techniques to validate it against the ground truth rules. Experiments on stream transformations and Sudoku problems show that our decoded rules are highly reliable: using exact solvers on them could achieve 100% accuracy, whereas the original SATNet fails to give correct solutions in many cases. Furthermore, we formally verify that our decoded logical rules are functionally equivalent to the ground truth ones.

Bin Liu, Xiaohong Liu, Qin Luo et al.

Pairwise learning underpins implicit collaborative filtering, yet its effectiveness is often hindered by sparse supervision, noisy interactions, and popularity-driven exposure bias. In this paper, we propose Variational Bayesian Personalized Ranking (VarBPR), a tractable variational framework for implicit-feedback pairwise learning that offers principled exposure controllability and theoretical interpretability. VarBPR reformulates pairwise learning as variational inference over discrete latent indexing variables, explicitly modeling noise and indexing uncertainty, and divides training into two stages: variational inference and variational learning. In the variational inference stage, we develop a variational formulation that integrates preference alignment, denoising, and popularity debiasing under a unified ELBO/regularization objective, deriving closed-form posteriors with clear control semantics: the prior encodes a target exposure pattern, while temperature/regularization strength controls posterior-prior adherence. As a result, exposure controllability becomes an endogenous and interpretable outcome of variational inference. In the variational learning stage, we propose a posterior-compression objective that reduces the ideal ELBO's computational complexity from polynomial to linear, with the approximation justified by an explicit Jensen-gap upper bound. Theoretically, we provide interpretable generalization guarantees by identifying a structural error component and revealing the opportunity cost of prioritizing certain exposure patterns (e.g., long-tail), offering a concrete analytical lens for designing controllable recommender systems. Empirically, We validate VarBPR across popular backbones; it demonstrates consistent gains in ranking accuracy, enables controlled long-tail exposure, and preserves the linear-time complexity of BPR.

Zhaoyu Li, Jialiang Sun, Logan Murphy et al. · utoronto

Theorem proving is a fundamental aspect of mathematics, spanning from informal reasoning in natural language to rigorous derivations in formal systems. In recent years, the advancement of deep learning, especially the emergence of large language models, has sparked a notable surge of research exploring these techniques to enhance the process of theorem proving. This paper presents a comprehensive survey of deep learning for theorem proving by offering (i) a thorough review of existing approaches across various tasks such as autoformalization, premise selection, proofstep generation, and proof search; (ii) an extensive summary of curated datasets and strategies for synthetic data generation; (iii) a detailed analysis of evaluation metrics and the performance of state-of-the-art methods; and (iv) a critical discussion on the persistent challenges and the promising avenues for future exploration. Our survey aims to serve as a foundational reference for deep learning approaches in theorem proving, inspiring and catalyzing further research endeavors in this rapidly growing field. A curated list of papers is available at https://github.com/zhaoyu-li/DL4TP.

Bowen Li, Zhaoyu Li, Qiwei Du et al.

Recent years have witnessed the rapid development of Neuro-Symbolic (NeSy) AI systems, which integrate symbolic reasoning into deep neural networks. However, most of the existing benchmarks for NeSy AI fail to provide long-horizon reasoning tasks with complex multi-agent interactions. Furthermore, they are usually constrained by fixed and simplistic logical rules over limited entities, making them far from real-world complexities. To address these crucial gaps, we introduce LogiCity, the first simulator based on customizable first-order logic (FOL) for an urban-like environment with multiple dynamic agents. LogiCity models diverse urban elements using semantic and spatial concepts, such as IsAmbulance(X) and IsClose(X, Y). These concepts are used to define FOL rules that govern the behavior of various agents. Since the concepts and rules are abstractions, they can be universally applied to cities with any agent compositions, facilitating the instantiation of diverse scenarios. Besides, a key feature of LogiCity is its support for user-configurable abstractions, enabling customizable simulation complexities for logical reasoning. To explore various aspects of NeSy AI, LogiCity introduces two tasks, one features long-horizon sequential decision-making, and the other focuses on one-step visual reasoning, varying in difficulty and agent behaviors. Our extensive evaluation reveals the advantage of NeSy frameworks in abstract reasoning. Moreover, we highlight the significant challenges of handling more complex abstractions in long-horizon multi-agent scenarios or under high-dimensional, imbalanced data. With its flexible design, various features, and newly raised challenges, we believe LogiCity represents a pivotal step forward in advancing the next generation of NeSy AI. All the code and data are open-sourced at our website: https://jaraxxus-me.github.io/LogiCity/

Zenan Li, Zhaoyu Li, Kaiyu Yang et al.

Mathematical reasoning demands two critical, complementary skills: constructing rigorous proofs for true statements and discovering counterexamples that disprove false ones. However, current AI efforts in mathematics focus almost exclusively on proof construction, often neglecting the equally important task of finding counterexamples. In this paper, we address this gap by fine-tuning large language models (LLMs) to reason about and generate counterexamples. We formalize this task as formal counterexample generation, which requires LLMs not only to propose candidate counterexamples but also to produce formal proofs that can be automatically verified in the Lean 4 theorem prover. To enable effective learning, we introduce a symbolic mutation strategy that synthesizes diverse training data by systematically extracting theorems and discarding selected hypotheses, thereby producing diverse counterexample instances. Together with curated datasets, this strategy enables a multi-reward expert iteration framework that substantially enhances both the effectiveness and efficiency of training LLMs for counterexample generation and theorem proving. Experiments on three newly collected benchmarks validate the advantages of our approach, showing that the mutation strategy and training framework yield significant performance gains.

Zenan Li, Yifan Wu, Zhaoyu Li et al. · microsoft-research, utoronto

Autoformalization, the task of automatically translating natural language descriptions into a formal language, poses a significant challenge across various domains, especially in mathematics. Recent advancements in large language models (LLMs) have unveiled their promising capabilities to formalize even competition-level math problems. However, we observe a considerable discrepancy between pass@1 and pass@k accuracies in LLM-generated formalizations. To address this gap, we introduce a novel framework that scores and selects the best result from k autoformalization candidates based on two complementary self-consistency methods: symbolic equivalence and semantic consistency. Elaborately, symbolic equivalence identifies the logical homogeneity among autoformalization candidates using automated theorem provers, and semantic consistency evaluates the preservation of the original meaning by informalizing the candidates and computing the similarity between the embeddings of the original and informalized texts. Our extensive experiments on the MATH and miniF2F datasets demonstrate that our approach significantly enhances autoformalization accuracy, achieving up to 0.22-1.35x relative improvements across various LLMs and baseline methods.

Youyuan Zhang, Jialiang Sun, Hangrui Bi et al.

We introduce DreamProver, an agentic framework that leverages a "wake-sleep" program induction paradigm to discover reusable lemmas for formal theorem proving. Existing approaches either rely on fixed lemma libraries, which limit adaptability, or synthesize highly specific intermediate lemmas tailored to individual theorems, thereby lacking generality. DreamProver addresses this gap through an iterative two-stage process. In the wake stage, DreamProver attempts to prove theorems from a training set using the current lemma library while proposing new candidate lemmas. In the "sleep" stage, it abstracts, refines, and consolidates these candidates to compress and optimize the library. Through this alternating cycle, DreamProver progressively evolves a compact set of high-level, transferable lemmas that can be effectively used to prove unseen theorems in related domains. Experimental results demonstrate that DreamProver substantially improves proof success rates across a diverse set of mathematical benchmarks, while also producing more concise proofs and reducing computational cost.

Zenan Li, Zhaoyu Li, Wen Tang et al. · utoronto

Large language models (LLMs) can prove mathematical theorems formally by generating proof steps (\textit{a.k.a.} tactics) within a proof system. However, the space of possible tactics is vast and complex, while the available training data for formal proofs is limited, posing a significant challenge to LLM-based tactic generation. To address this, we introduce a neuro-symbolic tactic generator that synergizes the mathematical intuition learned by LLMs with domain-specific insights encoded by symbolic methods. The key aspect of this integration is identifying which parts of mathematical reasoning are best suited to LLMs and which to symbolic methods. While the high-level idea of neuro-symbolic integration is broadly applicable to various mathematical problems, in this paper, we focus specifically on Olympiad inequalities (Figure~1). We analyze how humans solve these problems and distill the techniques into two types of tactics: (1) scaling, handled by symbolic methods, and (2) rewriting, handled by LLMs. In addition, we combine symbolic tools with LLMs to prune and rank the proof goals for efficient proof search. We evaluate our framework on 161 challenging inequalities from multiple mathematics competitions, achieving state-of-the-art performance and significantly outperforming existing LLM and symbolic approaches without requiring additional training data.

Zenan Li, Yunpeng Huang, Zhaoyu Li et al. · utoronto

Neuro-symbolic systems combine the abilities of neural perception and logical reasoning. However, end-to-end learning of neuro-symbolic systems is still an unsolved challenge. This paper proposes a natural framework that fuses neural network training, symbol grounding, and logical constraint synthesis into a coherent and efficient end-to-end learning process. The capability of this framework comes from the improved interactions between the neural and the symbolic parts of the system in both the training and inference stages. Technically, to bridge the gap between the continuous neural network and the discrete logical constraint, we introduce a difference-of-convex programming technique to relax the logical constraints while maintaining their precision. We also employ cardinality constraints as the language for logical constraint learning and incorporate a trust region method to avoid the degeneracy of logical constraint in learning. Both theoretical analyses and empirical evaluations substantiate the effectiveness of the proposed framework.

Shuai Guo, Jielei Chu, Lin Ma et al.

Generative Flow Networks (GFlowNets or GFNs) are probabilistic models predicated on Markov flows, and they employ specific amortization algorithms to learn stochastic policies that generate compositional substances including biomolecules, chemical materials, etc. With a strong ability to generate high-performance biochemical molecules, GFNs accelerate the discovery of scientific substances, effectively overcoming the time-consuming, labor-intensive, and costly shortcomings of conventional material discovery methods. However, previous studies rarely focus on accumulating exploratory experience by adjusting generative structures, which leads to disorientation in complex sampling spaces. Efforts to address this issue, such as LS-GFN, are limited to local greedy searches and lack broader global adjustments. This paper introduces a novel variant of GFNs, the Dynamic Backtracking GFN (DB-GFN), which improves the adaptability of decision-making steps through a reward-based dynamic backtracking mechanism. DB-GFN allows backtracking during the network construction process according to the current state's reward value, thereby correcting disadvantageous decisions and exploring alternative pathways during the exploration process. When applied to generative tasks involving biochemical molecules and genetic material sequences, DB-GFN outperforms GFN models such as LS-GFN and GTB, as well as traditional reinforcement learning methods, in sample quality, sample exploration quantity, and training convergence speed. Additionally, owing to its orthogonal nature, DB-GFN shows great potential in future improvements of GFNs, and it can be integrated with other strategies to achieve higher search performance.

Youyuan Zhang, Zehua Liu, Zenan Li et al. · utoronto

In this paper, we consider the conditional generation problem by guiding off-the-shelf unconditional diffusion models with differentiable loss functions in a plug-and-play fashion. While previous research has primarily focused on balancing the unconditional diffusion model and the guided loss through a tuned weight hyperparameter, we propose a novel framework that distinctly decouples these two components. Specifically, we introduce two variables ${x}$ and ${z}$, to represent the generated samples governed by the unconditional generation model and the guidance function, respectively. This decoupling reformulates conditional generation into two manageable subproblems, unified by the constraint ${x} = {z}$. Leveraging this setup, we develop a new algorithm based on the Alternating Direction Method of Multipliers (ADMM) to adaptively balance these components. Additionally, we establish the equivalence between the diffusion reverse step and the proximal operator of ADMM and provide a detailed convergence analysis of our algorithm under certain mild assumptions. Our experiments demonstrate that our proposed method ADMMDiff consistently generates high-quality samples while ensuring strong adherence to the conditioning criteria. It outperforms existing methods across a range of conditional generation tasks, including image generation with various guidance and controllable motion synthesis.

Honghua Dong, Qidong Su, Yubo Gao et al.

Large Language Models (LLMs) have become increasingly capable of handling diverse tasks with the aid of well-crafted prompts and integration of external tools, but as task complexity rises, the workflow involving LLMs can be complicated and thus challenging to implement and maintain. To address this challenge, we propose APPL, A Prompt Programming Language that acts as a bridge between computer programs and LLMs, allowing seamless embedding of prompts into Python functions, and vice versa. APPL provides an intuitive and Python-native syntax, an efficient parallelized runtime with asynchronous semantics, and a tracing module supporting effective failure diagnosis and replaying without extra costs. We demonstrate that APPL programs are intuitive, concise, and efficient through three representative scenarios: Chain-of-Thought with self-consistency (CoT-SC), ReAct tool use agent, and multi-agent chat. Experiments on three parallelizable workflows further show that APPL can effectively parallelize independent LLM calls, with a significant speedup ratio that almost matches the estimation.

Fanfan Wang, Zixiang Ding, Rui Xia et al.

Emotion cause analysis has received considerable attention in recent years. Previous studies primarily focused on emotion cause extraction from texts in news articles or microblogs. It is also interesting to discover emotions and their causes in conversations. As conversation in its natural form is multimodal, a large number of studies have been carried out on multimodal emotion recognition in conversations, but there is still a lack of work on multimodal emotion cause analysis. In this work, we introduce a new task named Multimodal Emotion-Cause Pair Extraction in Conversations, aiming to jointly extract emotions and their associated causes from conversations reflected in multiple modalities (text, audio and video). We accordingly construct a multimodal conversational emotion cause dataset, Emotion-Cause-in-Friends, which contains 9,272 multimodal emotion-cause pairs annotated on 13,509 utterances in the sitcom Friends. We finally benchmark the task by establishing a baseline system that incorporates multimodal features for emotion-cause pair extraction. Preliminary experimental results demonstrate the potential of multimodal information fusion for discovering both emotions and causes in conversations.

Zhaoyu Li, Binghong Chen, Xujie Si

Interactive theorem proving is a challenging and tedious process, which requires non-trivial expertise and detailed low-level instructions (or tactics) from human experts. Tactic prediction is a natural way to automate this process. Existing methods show promising results on tactic prediction by learning a deep neural network (DNN) based model from proofs written by human experts. In this paper, we propose NeuroTactic, a novel extension with a special focus on improving the representation learning for theorem proving. NeuroTactic leverages graph neural networks (GNNs) to represent the theorems and premises, and applies graph contrastive learning for pre-training. We demonstrate that the representation learning of theorems is essential to predict tactics. Compared with other methods, NeuroTactic achieves state-of-the-art performance on the CoqGym dataset.

Bruce D'Ambrosio, Tony Fountain, Zhaoyu Li

We report on an experimental investigation into opportunities for parallelism in beliefnet inference. Specifically, we report on a study performed of the available parallelism, on hypercube style machines, of a set of randomly generated belief nets, using factoring (SPI) style inference algorithms. Our results indicate that substantial speedup is available, but that it is available only through parallelization of individual conformal product operations, and depends critically on finding an appropriate factoring. We find negligible opportunity for parallelism at the topological, or clustering tree, level.

Zhaoyu Li, Bruce D'Ambrosio

Given a belief network with evidence, the task of finding the I most probable explanations (MPE) in the belief network is that of identifying and ordering the I most probable instantiations of the non-evidence nodes of the belief network. Although many approaches have been proposed for solving this problem, most work only for restricted topologies (i.e., singly connected belief networks). In this paper, we will present a new approach for finding I MPEs in an arbitrary belief network. First, we will present an algorithm for finding the MPE in a belief network. Then, we will present a linear time algorithm for finding the next MPE after finding the first MPE. And finally, we will discuss the problem of finding the MPE for a subset of variables of a belief network, and show that the problem can be efficiently solved by this approach.