## 3 Methods
### Co-Interactive Cognitive Inference Layer
#### 3.3.2 Fine-Grained Evidence Aggregating Module  
For the mentioned input, we feed the claim \(\hat{c}\) and the retrieved evidence \(\hat{c_{i}}\in\mathcal{E}\) into mBERT to achieve the corresponding representation. This is shown as follows:  
\[\mathbf{h_{e_{i}}} =\mathbf{mBERT}(\hat{c_{i}}), \tag{9}\] \[\mathbf{h_{c}} =\mathbf{mBERT}(\hat{c}).\]  
Note that the embedding vectors of the claim and evidences are generated from the final hidden state of the special token [CLS]. Next, we extract the potential graph representations of the evidence nodes in the fine-gained evidence aggregating network, where information passing is guided by the noteworthy elements in the claim. Semi-supervised graph networks have been used in the past to capture hidden information for a classification task [42; 43]. Inspired by the previous studies, we propose a Recursive Graph Attention Network (RGAT) for evidence aggregating and reasoning. In order to bridge information gaps between different pieces of evidence, a fully connected graph neural network is firstly constructed, where each network node represents one piece of evidence. In addition, every node is on a self-loop, and mainly considers the mining of fine-grained information and the flowing interaction of information from itself. Then, the three-layer recurrent network that deductively generates connected evidence chains is formulated for evidence reasoning. Specifically, the hidden states of nodes at layer \(l\) in the network are obtained as learning feature representations, as shown in Equation (10):  
\[\mathbf{h}^{l}=\left\{\mathbf{h}_{1}^{l},\mathbf{h}_{2}^{l},\ldots,\mathbf{h}_ {N}^{l}\right\}, \tag{10}\]  
where \(\mathbf{h}_{i}^{l}\in\mathcal{R}^{F\times 1}\) and \(F\) are the number of node features. The initial hidden state of each evidence node \(\mathbf{h}_{i}^{0}\) is initialized by the evidence presentation: \(\mathbf{h}_{i}^{0}=\mathbf{h}_{\mathbf{e}_{i}}\). As far as the correlation coefficients between a node \(i\) and its neighbor \(j(j\in\mathcal{N}_{\backslash})\) is concerned, we compute the coefficients as Equation (11):  
\[c_{ij}=\mathbf{W}_{1}^{l-1}\Big{(}\text{ReLU}\Big{(}\mathbf{W}_{0}^{l-1} \Big{(}\mathbf{h}_{i}^{l-1}\|\mathbf{h}_{j}^{l-1}\Big{)}\Big{)}\Big{)}, \tag{11}\]  
where \(\mathcal{N}_{\backslash}\) denotes the neighbor set of node \(i\), \(\mathbf{W}_{0}^{l-1}\in\mathcal{R}^{H\times 2F}\) and \(\mathbf{W}_{1}^{l-1}\in\mathcal{R}^{1\times H}\) are the weight matrix, and the operation \(\cdot\|\): indicates the concatenation of different variables. Next, the coefficients \(c_{ij}\) are normalized with the softmax function, as shown in Equation (12):  
\[\alpha_{ij}=\text{softmax}_{j}\big{(}c_{ij}\big{)}=\frac{\exp(c_{ij})}{\sum_{k \in\mathcal{N}_{i}}\exp(c_{ik})}, \tag{12}\]  
Finally, through the recursive interactive fusion of evidence nodes layer by layer, the feature representation vector for node \(i\) at layer \(l\) is estimated as Equation (13):  
\[\mathbf{h}_{i}^{l}=\sum_{j\in\mathcal{N}_{i}}\alpha_{ij}\mathbf{h}_{j}^{l-1}. \tag{13}\]  
Following this, the final hidden states of evidence nodes from stacking \(T\) layers are extracted as the component embedding representation of the evidence reasoning, which is seen in Equation (14):  
\[\mathbf{H}^{\text{T}}=\text{CONCAT}\Big{(}\mathbf{h}_{1}^{\text{T}},\mathbf{h }_{2}^{\text{T}},\cdots,\mathbf{h}_{N}^{\text{T}}\Big{)}. \tag{14}\]  
As the next important step, we construct the evidence attention aggregator to gather the fine-grained information from the final embedding representation of the evidence reasoning. Specifically, what information is aggregated preferentially depends on how important it is to the claim, which is measured based on the multi-head attention mechanism [44], as shown in Equation (15):  
\[ \begin{split}\mathbf{H}^{O}&=\text{ MultiHead }(\mathbf{H}^{\text{T}},\mathbf{h}_{\text{c}})\\ &=\text{CONCAT}_{i\in[N_{head}]}\Big{[}\mathbf{Head}^{(i)}\Big{]} \mathbf{W}_{\text{O}}.\end{split}