Can Large Language Models Reinvent Foundational Algorithms?

LLMs have shown strong potential to advance scientific discovery. Whether they possess the capacity for foundational innovation, however, remains an open question. In this work, we focus on a prerequisite for foundational innovation: textitcan LLMs reinvent foundational algorithms in computer science? We use LLM unlearning methods to suppress direct recall of the target algorithm and let the model reason with the remaining knowledge to recover it. Although unlearning does not guarantee full knowledge removal, LLMs fail to recover nearly half of the target algorithms. Notably, even suppressing the mention of the algorithm's name during decoding without unlearning makes the models' recovery rate drop dramatically (19--39%), suggesting their overreliance on memorized knowledge. We observe that recoverable algorithms tend to be simple in structure or core ideas, whereas the others are less straightforward. We also introduce a generative verifier that sustains models' reasoning strength, helping to avoid the ``thought collapse'' phenomenon. Taken together, by treating the unlearned model's recovery rate as an approximate upper bound, our empirical results suggest that current LLM systems still have limited ability to make foundational algorithm innovation. Our code is available at https://github.com/Algo-Reinvention/algo-reinvention.