## 4 Auto-HPIF Modeling Approaches
### The Implicit Heterogeneous Intent-Slot Interaction Module
Then, the attention mechanism is employed to compute the influence of each neighbor node \(j\) to node \(i\), and they are weighted and summed to obtain the final node representation \(h_{i}^{\prime}\), where \(N_{i}\) denotes the set of neighbors of node \(i\), the variables \(q_{i}\) and \(k_{i}\) are the learnable parameters used for generating the query vector and the key vector, respectively, \(\beta_{i,j}\) denotes the weight obtained from the attention mechanism score normalized by the softmax function, and finally they are weighted and summed to obtain the feature representation \(h_{i}^{\prime\prime}\) of position \(i\), as shown in Equations (11)-(14):  
\[\alpha_{i,j}=\textit{softmax}_{j}(e_{i,j})=\frac{\textit{exp}(e_{i,j})}{\sum \limits_{k\in N_{i}}\textit{exp}(e_{i,k})} \tag{11}\]  
\[\beta_{i,j}=\textit{softmax}_{j}(e_{i,j}^{\prime})=\frac{\textit{exp}(f(W\;q_ {i},Wk_{j}))}{\sum\limits_{k=1}^{n}\textit{exp}(f(W\;q_{i},Wk_{k}))} \tag{12}\]  
\[h_{i}^{\prime}=\sigma(\sum\limits_{j\in N_{i}}\alpha_{i,j}h_{j}) \tag{13}\]  
\[h_{i}^{\prime\prime}=\sigma(\sum\limits_{j\in N_{i}}\beta_{i,j}h_{j}) \tag{14}\]  
As a result, the feature representation of each word node in the input sentence can be extracted, and the information of slot nodes and intention nodes is continuously updated by the heterogeneous graphical layer interaction mechanism, as shown in Equations (15) and (16):  
\[\widetilde{S}=\big{|}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.} \big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.} \big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.} \big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.} \big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.} \big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.} \big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.} \big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.} \big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.} \big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.}\big{.} \big{.  
* **Rational Multiple-Intent Decoder.** Through the heterogeneous graphical layer interaction mechanism, the updated intent node containing the slot information is obtained \(\widetilde{L}=[l_{1}\prime,l_{2}\prime,\ldots,l_{t}\prime]\); to consider both discourse representation and slot information, this paper continues to use bidirectional LSTM to implement slot information to guide the intent decoder, where it is the encoded hidden state after alignment. \(d_{t-1}\) is the hidden state of the decoder at the previous sequential phase, \(l_{t}\) is the slot label vector of the current sequential phase, and the LSTM hidden state is updated by computing the input vector \(x_{t}\) of the current time step as follows: \[x_{t}=\left[e_{t};d_{t-1};L_{t}{}^{\prime}\right]\] (17) \[c_{t}\mathit{=BiLSTM}\left(x_{t},c_{t-1},\,h_{t-1}\right)\] (18) \[h_{t}\mathit{=tanh}(c_{t})\] (19) \[y_{t}^{\mathit{RI}}\mathit{=softmax}\big{(}W_{s}{}^{\prime}h_{t}+b_{s}\big{)}\] (20)  
The hidden state \(h_{t}\) of the current sequential phase is passed to the fully connected layer, and the value of each slot is used as an additional input to generate the intent score \(y_{t}^{\mathit{RI}}\) for the current sequential phase. * **Rational Slot Decoder. ** To enhance the final slot filling task, the predicted multi-intent information is further interacted with the slot information by concatenating the slot node containing the characteristics corresponding to each predicted intent label \(\widetilde{S}=\left[s_{1}{}^{\prime},s_{2}{}^{\prime},\cdots,s_{n}{}^{\prime}\right]\) and the aligned encoder hidden state e as input units to obtain a new sequence of slot labels \(y_{t}^{\mathit{KS}}\) using a method similar to that of the rational multiple-intent decoder.