Collaborative Research: MFAI: Mathematical Foundations of Chain-of-Thought: Demystifying Reasoning Capabilities of Transformers

Artificial intelligence (AI) is driving rapid advances across science, technology, and society, yet the most powerful AI systems, such as large language models, remain difficult to explain, predict, and trust. These models often solve complex reasoning tasks in mathematics, coding, and planning by generating intermediate chain-of-thought steps. However, a mathematical understanding of how such reasoning capabilities emerge, when they generalize, and why they sometimes fail remains limited. This project addresses these foundational gaps by developing mathematical and algorithmic principles for understanding and improving reasoning in transformer-based AI systems. The resulting theory and methods will support the development of more reliable, interpretable, and efficient AI systems, helping to turn empirical progress in artificial intelligence into principled and broadly usable technology. The project will also train students across data science, computer science, and electrical engineering, and will produce open educational materials, research artifacts, and benchmarks for the broader community. The goal of this project is to develop a mathematical foundation of chain-of-thought in transformers, the backbone of large language models, by studying training dynamics and generalization properties of both supervised and reinforcement learning. The project will study how transformers acquire reasoning abilities from training data, how training procedures affect generalization to longer or more complex reasoning tasks, how reinforcement learning improves reasoning behavior, and how bidirectional reconstruction can help recover missing reasoning steps from partial information. The research will lead to new synergies in optimization, statistics, reinforcement learning, information theory, and systems, and will contribute open-source artifacts and benchmarks. Expected outcomes include new mathematical theory of the capabilities, limitations, and emerging properties of reasoning in transformers, as well as principled algorithms for more efficient training and deployment of next-generation AI systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.