Large Language Model for Science: A Study on P vs. NP

In this work, we use large language models (LLMs) to augment and accelerate research on the P versus NP problem, one of the most important open problems in theoretical computer science and mathematics. Specifically, we propose Socratic reasoning, a general framework that promotes in-depth thinking with LLMs for complex problem-solving. Socratic reasoning encourages LLMs to recursively discover, solve, and integrate problems while facilitating self-evaluation and refinement. Our pilot study on the P vs. NP problem shows that GPT-4 successfully produces a proof schema and engages in rigorous reasoning throughout 97 dialogue turns, concluding "P $\neq$ NP", which is in alignment with (Xu and Zhou, 2023). The investigation uncovers novel insights within the extensive solution space of LLMs, shedding light on LLM for Science.

Super Solutions of the Model RB

The concept of super solution is a special type of generalized solutions with certain degree of robustness and stability. In this paper we consider the $(1,1)$-super solutions of the model RB. Using the first moment method, we establish a "threshold" such that as the constraint density crosses this value, the expected number of $(1,1)$-super solutions goes from $0$ to infinity.

Exact Phase Transitions of Model RB with Slower-Growing Domains

The second moment method has always been an effective tool to lower bound the satisfiability threshold of many random constraint satisfaction problems. However, the calculation is usually hard to carry out and as a result, only some loose results can be obtained. In this paper, based on a delicate analysis which fully exploit the power of the second moment method, we prove that random RB instances can exhibit exact phase transition under more relaxed conditions, especially slower-growing domain size. These results are the best by using the second moment method, and new tools should be introduced for any better results.