## 2 Preliminaries
### Problem Definition  
In the HTMC-PGT problem, a set of PGTs exists. Each text in this set contains a fixed number of words that together constitute the semantics of the text. Based on semantic features, these texts also correspond to multiple categories with a hierarchical structure. Unlike the nonlinear structure of multiple labels in traditional HMTC (hierarchical multi-label text classification) problems [24], the multiple categories corresponding to text in HTMC-PGT are linear structures. To clearly express the focus of this article, this section first defines the hierarchical structure of multiple labels, text semantics, and the linear structure of multiple target labels. **Definition 1** (Hierarchical Label Tree of PGT). : _The hierarchical label tree of PGT (HLT-PGT)\(\mathcal{T}\) introduced in Section 2.1 can be defined as a partial order set \((C_{T},\prec)\) where \(C_{T}=\{c_{1},c_{2},\cdots,c_{N}\}\) is a finite set of all categories of PGT and the size of the set is \(N\). \(\prec\) is a partial order that represents the parent-child relationship between nodes in \(\mathcal{T}\), which is antireflective and transitive [25]. On the other hand, the node elements in the tree \(C_{T}\) can be demarcated into three parts: root, stem, and leaf, corresponding to three non-overlapping subsets \(R\), \(S\), and \(L\); \(C_{T}=R\bigcup S\bigcup L\). Because each tree only contains one root node, \(\left.\right|R\left.\right|=1\):_  
* \(\forall c_{x}\in C_{T},c_{x}\
ot\prec c_{\mathcal{X}}\)_;_
* \(\forall c_{x},c_{y},c_{z}\in C_{T}\)_, if_ \(c_{x}\prec c_{y}\) _and_ \(c_{y}\prec c_{z}\) _then_ \(c_{x}\prec c_{z}\)_;_
* \(\forall c_{x}\in L,\exists c_{y}\in R\bigcup S,c_{x}\prec c_{y}\) _then_ \(c_{y}\
ot\prec c_{x}\)_;_
* \(\forall c_{x}\in S,\exists c_{y}\in R\bigcup S,c_{x}\prec c_{y}\) _then_ \(c_{y}\
ot\prec c_{x}\)_._  
Therefore, compared with the nonlinearity of HMTC, the elements in the classification result set of HTMC-PGT form a directed acyclic graph connected by \(\prec\) relationships, and the arrangement of nodes is a linear sequence. **Definition 2** (Linear Structure of Target Multiple Labels). : _In the HTMC-PGT problem, each document \(D_{i}\) contains multiple labels \(C_{Ti}\). One and only one of these labels \(C_{Ti}\) belongs to the leaf node set \(L\) (denoted as \(l_{i}\)), and \(C_{Ti}\) must contain elements from the set \(R\) (i.e., root nodes, denoted as \(\gamma\)), while the other node sets \(S_{i}=\{c_{1},c_{2},\cdots,c_{k}\}\), are subsets of the set \(S\). If \(S_{i}\) is not an empty set, a sequence exists \(\left\{c^{{}^{\prime}},{}^{\prime},c^{{}^{\prime}},\cdots,c^{{}^{\prime}}\right\}\) such that all elements in \(S_{i}\) satisfy the following arrangement \(l_{i}\prec c^{{}^{\prime}}_{1}\prec c^{{}^{\prime}}_{2}\prec\cdots\prec c^{{}^ {\prime}}_{k}\prec\gamma\). Therefore, all elements of the leaf node setLcorrespond one-to-one to the sequence of the multi-label classification results in the HTMC-PGT problem. That is, the set of classification results for the HTMC-PGT problem is the set of paths from all leaf nodes in set \(L\) to  
Figure 2: Hierarchical label tree of PGTs. the root node. Therefore, in the HTMC-PGT problem, the set of all the classification result sequences can be denoted as follows, where \(i=1,2,\cdots,|L|\):_  
\[C_{S}=\{\left.C_{T|}i=1,2,\cdots,|L|\right\} \tag{1}\]  
The HTMC-PGT problem exhibits three typical characteristics. First, the classified text has multiple labels, which differs from the traditional single-label text classification. Second, there is a hierarchical semantic relationship between labels, which differs from independent labels. Finally, the hierarchical multi-label classification has strict temporal dependencies among label classifiers.