Enhancing Mathematical Reasoning Through Autonomously Learning Knowledge

Enabling machines to solve mathematical problems is a vital endeavor in developing intelligence that emulates human-like thinking and reasoning. However, most existing approaches focus on reconstructing human comprehension of problems, which are still far from enough since they neglect the fundamental human ability to learn knowledge from experiences. In this article, we focus on empowering models with the cognitive capacity to autonomously learn knowledge from mathematical problem-solving. We first propose a Cognitive Solver (CogSolver) that contains an intelligent BRAIN-ARM framework as the cognitive structure and operates the knowledge learning process in Store-Apply-Update steps inspired by two cognitive science theories. The BRAIN system stores three basic types of mathematical knowledge, and the ARM system applies them organically in answer reasoning process. After solving problems, the BRAIN updates its stored knowledge based on the ARM's feedback, with knowledge filters to eliminate redundancies and foster a more rational knowledge base. Our CogSolver carries out the above three steps iteratively, emulating a more human-like behavior. Furthermore, in order to overcome knowledge forgetting during the learning process, we extend CogSolver to CogSolver+ by incorporating an essential knowledge Recall mechanism, which is inspired by another prominent cognitive theory. We first discuss and fuse three crucial factors in simulating human memory replay. Then, we propose a influenced-based method with a theoretical guarantee of efficiency to consolidate the updated knowledge. Experiments on three math word problem benchmarks demonstrate the improvements of our CogSolver and CogSolver+ in answer reasoning and clearly illustrate how they acquire knowledge, leading to superior interpretability.