## 3 Methodology
### Discourse Graph Construction and Positional Encoding of EDUs  
Inspired by previous research [16; 18], the text was cut to _EDUs_ using PTDB 2.0, and the sentences were cut in the following way:  
The sentences in the text and in each option were marked by conjunctions and punctuation marks, and each conjunct and punctuation mark were assumed to act as an edge connecting the EDUs before and after it. For example, the options in Figure 1 are separated into two linguistic units, E5 and E6, where E5 is \"Only very careful drivers use headlights\", and E6 is \"their use is not legally required\", and the edge r connecting them is \"when\". For each structure with one piece of context and multiple options, the graph corresponding to different options is constructed separately, the graph corresponding to option k is denoted by \(G_{k}=\left(V^{k},E^{k}\right)\). Since the embedding of an EDU depends on the tokens that make it up, the original text is first input to the pre-trained language model in the form of \([CLS]+Context+[SEP]+Question+Option+[SEP]\), where \([CLS]\) and \([SEP]\) are special tokens for DeBERTa, to obtain the sequence output \(S=\{t_{0},t_{1},\ldots,t_{n-1}t_{n}\}\) of its encoder layer. The embedding sequence of the output is segmented at the corresponding position according to the pre-marked position, and the embedding \(e_{n}\) of the \(n\)-th EDU is obtained by the following way:  
\[e_{n}=\sum_{l\in S_{n}}t_{l}+PE(n), \tag{1}\]  
where \(s_{n}\) is the index set of tokens that make up the \(n\)-th EDU node, \(PE(n)\) is the position embedding of the node, and the meaning of the location information and the way to obtain it are as follows. Graph neural networks usually process data without considering the position information of the data, only the abstract concept of neighbors. In contrast, the example in Figure 2 demonstrates that even if the neighbors are the same, the order of precedence is actually crucial for semantics. The two sentences in Figure 2 represent completely opposite meanings, and the reason is that the EDUs are in opposite sequential order. When performing graph attention computation, since there is no sequential relationship between each node, shuffling all the EDUs in a sentence will not cause any change in the embedding of an EDU. In previous approaches, when faced with the lack of positional information, there was a risk of ambiguity in reasoning. The typical solution was to feed the enhanced token sequence into a recurrent neural network before predicting the answer. However, this method increased computational complexity and reduced efficiency, while also elevating the risk of overfitting. Therefore, position encoding is added to address the lack of positional relationships between nodes during graph computation so that the same data in different positions are distinguished from each other, but the representation of position information needs to satisfy the following conditions:  
First, it can represent the absolute position of a token in the sequence. Second, the relative positions/distances of tokens in different sequences should be consistent in the case of different sequence lengths. Inspired by Transformer [8], regarding the EDU at position \(t\in pos\), suppose its position embedding is \(PE_{t}\in\mathbb{R}^{d}\). \(\mathbb{R}^{d}\) denotes all vectors of dimension d, and \(PE_{t}\) can be expressed in the following way:  
\[PE_{t}{}^{(i)}=\left\{ \begin{array}{ll}\sin\Bigl{(}pos/10000^{2i/d_{model}} \Bigr{)}&\quad i\;mod\;2=0\\ \cos\Bigl{(}pos/10000^{2i/d_{model}}\Bigr{)}&\quad i\;mod\;2=1\end{array} \right., \tag{2}\]  
where \(PE_{t}{}^{(i)}\) is the \(i\)-th dimension in its position vector, and \(d_{model}\) is the dimension of EDUs embedding. The primary purpose of choosing the constant 10,000 as the denominator is to extend the wavelength of the periodic function to a very high level.