Existing MWP solvers employ sequence or binary tree to present the solution expression and decode it from given problem description. However, such structures fail to handle the variants that can be derived via mathematical manipulation, e.g., (a₁+a₂)*a₃ and a₁ * a₃+a₂ * a₃ can both be possible valid solutions for a same problem but formulated as different expression sequences or trees. The multiple solution variants depicting different possible solving procedures for the same input problem would raise two issues: 1) making it hard for the model to learn the mapping function between the input and output spaces effectively, and 2) wrongly indicating wrong when evaluating a valid expression variant. To address these issues, we introduce a unified tree structure to present a solution expression, where the elements are permutable and identical for all the expression variants. We propose a novel non-autoregressive solver, named MWP-NAS, to parse the problem and deduce the solution expression based on the unified tree. For evaluating the possible expression variants, we design a path-based metric to evaluate the partial accuracy of expressions of a unified tree. The results from extensive experiments conducted on Math23K and MAWPS demonstrate the effectiveness of our proposed MWP-NAS. The codes and checkpoints are available at: https://github.com/mengqunhan/MWP-NAS.

Existing sentence ordering approaches generally employ encoder-decoder frameworks with the pointer net to recover the coherence by recurrently predicting each sentence step-by-step. Such an autoregressive manner only leverages unilateral dependencies during decoding and cannot fully explore the semantic dependency between sentences for ordering. To overcome these limitations, in this paper, we propose a novel Non-Autoregressive Ordering Network, dubbed NAON, which explores bilateral dependencies between sentences and predicts the sentence for each position in parallel. We claim that the non-autoregressive manner is not just applicable but also particularly suitable to the sentence ordering task because of two peculiar characteristics of the task: 1) each generation target is in deterministic length, and 2) the sentences and positions should match exclusively. Furthermore, to address the repetition issue of the naive non-autoregressive Transformer, we introduce an exclusive loss to constrain the exclusiveness between positions and sentences. To verify the effectiveness of the proposed model, we conduct extensive experiments on several common-used datasets and the experimental results show that our method outperforms all the autoregressive approaches and yields competitive performance compared with the state-of-the-arts. The codes are available at: https://github.com/steven640pixel/nonautoregressive-sentence-ordering.