Large-scale pre-trained language models (PLMs) bring new opportunities to challenging problems, especially those that need high-level intelligence, such as the math word problem (MWPs). However, directly applying existing PLMs to MWPs can fail as the generation process lacks sufficient supervision and thus lacks fast adaptivity as humans. We notice that human reasoning has a dual reasoning framework that consists of an immediate reaction system (system 1) and a delicate reasoning system (system 2), where the entire reasoning is determined by their interaction. This inspires us to develop a cooperative reasoning-induced PLM for solving MWPs, called Cooperative Reasoning (CoRe), resulting in a human-like reasoning architecture with system 1 as the generator and system 2 as the verifier. In our approach, the generator is responsible for generating reasoning paths, and the verifiers are used to supervise the evaluation in order to obtain reliable feedback for the generator. We evaluate our CoRe framework on several mathematical reasoning datasets and achieve decent improvement over state-of-the-art methods, up to 9.6% increase over best baselines.

Recent advances in pre-trained language models (PLMs) have facilitated the development ofcommonsense reasoning tasks. However, existing methods rely on multi-hop knowledgeretrieval and thus suffer low accuracy due toembedded noise in the acquired knowledge. In addition, these methods often attain highcomputational costs and nontrivial knowledgeloss because they encode the knowledge independently of the PLM, making it less relevant to the task and thus resulting in a poorlocal optimum. In this work, we propose MultiView Knowledge Retrieval with Prompt Tuning (MVP-Tuning). MVP-Tuning leveragessimilar question-answer pairs in the training setto improve knowledge retrieval and employsa single prompt-tuned PLM to model knowledge and input text jointly. We conduct our experiments on five commonsense reasoning QAbenchmarks to show that MVP-Tuning outperforms all other baselines in 4 out of 5 datasetswith less than 2% trainable parameters. MVPTuning even gets a new state-of-the-art resulton OpenBookQA and is number one on theleaderboard.