## 2 Preliminary
### Pfa  
In the automata theory, an automaton does not refer to a concrete machine but rather a dynamic mathematical model of discrete systems that can transform and process information [27; 28]. Probabilistic finite automata (PFA) is a branch of automata that is a structured  
Figure 1: RNN basic unit. representation of discrete mathematical models. The theory of automata studies the mathematical theory of analyzing and synthesizing automata, an essential branch of control theory. Compared with non-probabilistic finite automata, the characteristic of probabilistic finite automata is that its state transition process provides probability values, which can quantitatively express the randomness of the state transition process [29]. The research fields of automata theory are divided into three categories: abstract theory, structural theory, and self-organization theory. In abstract theoretical research on automata, the structure of the automata itself and the specific forms of its input and output signals are not considered, and the automata is treated as a mathematical system to study its general mathematical properties. The study of the structural theory of automata is a further development of the abstract theory of automata. Structural theory research focuses on the synthesis of automata, methods for constructing automata from meta-automata, and techniques for encoding meta-signals transmitted through input and output channels. The self-organization theory of automata is the result of the development of abstract theory and structural theory of automata. Self-organization theory involves many aspects, such as neural networks, pattern recognition, and other artificial intelligence. In 1969, [PERSON] et al. applied the concept of fuzzy neurons to automata theory, leading to new developments in the research of neural network theory and the application of automata theory to study the behavior of complex systems, such as RNNs [18; 25; 30; 31]. Below is the mathematical definition of PFA. **Definition 1**. : _PFA is a tuple of five elements, defined as the alphabet \(<\Sigma\), \(Q\), \(\delta\), \(Q_{0}\), \(Q_{f}>\), where \(Q\) refers to the possible finite states of the initial, intermediate, and final states observed by the RNN. \(\delta:\ \mathcal{Q}\times\Sigma\to Q\) is the labeled transition function. The transition probability of each edge in the automaton is recorded in \(\delta\). \(Q_{0}\) and \(Q_{f}\) are the finite starting state set and the accepting state set, respectively. Below, we will use a simplified statement to classify positive and negative labels as an example._  
**Example 1**. : _Case of Converting the sentiment classification process of \"This movie is so beautiful!\" in an RNN to PFA._  
Assume that the PFA shown in Figure 2 is the state transition process of the sentiment classification involved in this input. Firstly, vectorize the input texts and convert them into PFA input vectors. We set all the words input into PFA as follows: \"This\", and its corresponding state name is vectorized to \(w_{v0}\), \"film\" to \(w_{v1}\), \"is\" to \(w_{v2}\), \"nice\" to \(w_{v3}\), and \"not\" to \(w_{v4}\). Then, the current string is turned into the vector sequence \(w_{v0}w_{v1}w_{v2}w_{v3}\). If the start word of the string \(w_{v0}\) was read by an RNN, the state of the RNN will transit to state 3, with a transition probability of \(p_{0}\ =\ 0.9\). Next, there is a corresponding transition of \(w_{v1}\), so its transition probability to itself is \(p_{2}\ =\ 0.3\). Then, read \(w_{v2}\), and transition to state 2 with \(p_{2}\) = 0.3. Finally, read \(w_{v3}\); the RNN transitions to an acceptable state 4 with a probability of \(p_{4}\ =\ 0.7\).