## 4 Approach
### Decomposition and Reasoning  
A core challenge in multi-hop KGQA lies in semantically parsing complex natural language queries. Similarly to human reasoning processes, this requires decomposing intricate questions into simpler sub-questions and resolving them sequentially [36]. Fundamentally, multi-hop QA can be conceptualized as a structured composition of single-hop QA units, where each unit addresses a specific aspect of the query. Crucially, the identification and formulation of sub-questions depend on two critical components: topic words and relation words. Topic words act as starting entities for knowledge graph traversal, while relation words define traversal paths from these entities, guiding the search and reasoning along specific relational chains in the knowledge graph. Figure 2: The overall framework of decomposition and reasoning on the knowledge graph (DRKG). The model first identifies topic entities from the natural language query and subsequently generates structured reasoning plans. This process can be formally represented as follows:  
\[P(\mathcal{T}|Q;\theta)=\prod_{i=1}^{m}P(y_{i}|q_{1},q_{2},\cdots,q_{i-1};\theta) \tag{1}\]  
\[\mathcal{P}\sim LLM(Q)=\prod_{t=1}^{n}P(p_{t}\mid Q,p_{<t};\theta) \tag{2}\]  
\(\mathcal{Q}\in\mathbb{R}^{d\times n}\) is the question embedding matrix, and each \(q_{i}\in\mathbb{R}^{d}\) is the embedding vector for the \(i-th\) token. \(\theta\in\mathbb{R}^{D}\) is the parameter vector. \(P(\mathcal{T}|Q;\theta)\) denotes the probability of identifying \(\mathcal{T}\) from the given \(\mathcal{Q}\). \(\mathcal{P}\in\mathbb{R}^{d\times n}\) is the reasoning plan generated by the model, and \(p\in\mathbb{R}^{d\times n}\) is an independent reasoning sub-plan. \(P(p_{t}\mid Q,p_{<t};\theta)\) represents the probability of generating a reasoning plan \(p_{t}\) conditioned on \(\mathcal{Q}\) and previous plans \(p_{<t}\). For each token \(y_{i}\), the probability of a specific label (topic or non-topic) is computed using the softmax function:  
\[P(y_{i}\mid q_{1},q_{2},\cdots,q_{i-1};\theta)=\frac{\exp\left(\mathbf{e}_{y_{ i}}^{T}\mathbf{W}_{tag}\mathbf{h}_{i}\right)}{\sum_{y\in\{0,1\}}\exp\left( \mathbf{e}_{y}^{T}\mathbf{W}_{tag}\mathbf{h}_{i}\right)} \tag{3}\]  
where \(h_{i}\in\mathbb{R}^{k}\) is the hidden state of the model decoder at the \(i-th\) token and \(\mathbf{W}_{tag}\in\mathbb{R}^{R\times k}\) is the weight matrix mapping hidden states to labels. The label space projection matrix maps the k-dimensional hidden state to the m-dimensional label space, and \(e_{y}\in\mathbb{R}^{m}\) denotes the embedding vector of label \(y\). The decoder hidden state \(h_{i}\) is projected into the label space, and the matching degree between the label and the current context is measured through the vector inner product. Given the input question \(\mathcal{Q}\) and the previously generated plans \(p_{<t}\), the probability distribution of reasoning steps \(p_{t}\) generated at each time step \(t\) is calculated through the softmax function:  
\[P(p_{t}\mid Q,p_{<t};\theta)=\frac{\exp\bigl{(}\mathbf{W}_{pt}\mathbf{h}_{t} \bigr{)}}{\sum_{j=1}^{V}\exp\left(\mathbf{W}_{j}\mathbf{h}_{t}\right)} \tag{4}\]  
where \(\mathbf{W}_{pt}\in\mathbb{R}^{k}\) and \(\mathbf{W}_{j}\in\mathbb{R}^{k}\) are the weight matrices mapping the vocab, and \(\mathbf{h}_{t}\in\mathbb{R}^{k}\) represents the model's hidden layer state. The context state is then linearly mapped to the vocabulary space to generate normalized vocabulary scores. Retrieval is subsequently performed using the identified topic entities and reasoning plans, formalized as follows:  
\[P(r_{i}\;\text{matches}\;p|p,R;\theta)=\frac{\exp(s(r_{i},p;\theta))}{\sum_{j =1}^{n}\exp\left(s(r_{j},p;\theta)\right)} \tag{5}\]  
where \(\mathcal{R}=(r_{1},r_{2},\ldots,r_{n}|r_{n}\in\mathbb{R}^{d})\) represents the set of all relations associated with the topic entity.