# Large-Language-Model-Enabled Text Semantic Communication Systems
## 3 System Model and Mechanism  
### Problem Description  
The primary objective of a communication system is to effectively and reliably transmit information-bearing messages to the receiver. Within such systems, the information sequence is subjected to source encoding, channel encoding, and modulation at the transmitter, enabling it to be transmitted through the channel as symbols. At the receiver, inverse operations are performed to reconstruct the transmitted information. Communication systems typically pursue two optimization goals. Firstly, the reliability metric aims to minimize the discrepancy between the estimated sequence at the receiver and the transmitted sequence at the transmitter. In technical communication, this is evaluated using metrics like the BER, while in semantic communication, it considers semantic differences. Secondly, the efficiency metric aims to minimize the length of transmitted symbols. These objectives often conflict with each other. [PERSON]'s separation theorem states that in technical communication systems, these objectives correspond to channel encoding and source encoding, respectively, and are optimized independently. This study analyzes the problem of semantic communication from the perspective of optimal decoding, focusing on the process where an information sequence \(X=(x_{1},x_{2},\cdots,x_{n})\), with \(x_{i}\in\mathbb{X}\) is transmitted. \(\mathbb{X}\) represents the set of transmitted information symbols. Initially, \(X\) is encoded and modulated at the transmitter using the function described in Equation (1), resulting in \(S=(s_{1},s_{2},\cdots,s_{t})\), where \(s_{i}\in\mathbb{S}\) and \(\mathbb{S}\) denotes the set of transmitted symbols. Subsequently, after transmission through the channel, the receiver acquires the sequence \(O=(o_{1},...,o_{t})\). The receiver task is to estimate \(\hat{S}\) from \(O\), and \(\hat{S}\) is then decoded into \(\hat{X}\) using a semantic decoding function specified by Equation (2). The optimization objectives of the communication system are twofold: Firstly, to minimize information errors by aligning the probability distribution of \(\hat{S}\) as close as possible to \(S\). Here, the function \(\varphi\) is bijective and reversible, ensuring that recovering \(S\) is equivalent to recovering \(X\). Secondly, the system aims to minimize the length \(t\) of the sequence \(S\), thereby improving the efficiency. \[S=\varphi(X), \tag{1}\]  
\[\hat{X}=\varphi^{-1}(\hat{S}), \tag{2}\]  
The relationship between \(o_{t}\) and \(s_{t}\) is shown in Equation (3):  
\[o_{t}=h_{t}\otimes s_{t}+n_{t} \tag{3}\]where \(h_{t}\) and \(n_{t}\) denote the channel impulse response (CSI) and noise at time \(t\), and \(\otimes\) signifies the convolution operator. The process is a typical Markov process, and the optimal decoding and demodulating process involves solving the following:  
\[\hat{S}=\underset{(s_{i}\in\mathbb{S})}{\arg\max}\,P(s_{1},s_{2},\cdots,s_{t}| o_{1},o_{2},\cdots,o_{t}), \tag{4}\]  
In fact, \(P(S|O)\) is often unavailable, hence Bayesian equation in Equation (5) is commonly employed to modify it:  
\[P(S|O)=\frac{P(S,O)}{P(O)}=\frac{P(O|S)P(S)}{P(O)}, \tag{5}\]  
where \(P(O|S)\) represents the channel conditional transition probability. The design of the communication system assumes that the channel is memoryless, so the following formula is valid:  
\[P(o_{1},o_{2},\cdots,o_{t}|s_{1},s_{2},\cdots,s_{t})=\prod_{i=1}^{t}P(o_{i}|s_ {i}), \tag{6}\]  
Therefore, the optimization goal of the optimal decoding system is as follows:  
\[\underset{(s_{i}\in\mathbb{S})}{\arg\max}\,P(s_{1},s_{2},\cdots,s_{t}\mid o_ {1},o_{2},\cdots o_{t})=\quad\underset{(s_{i}\in\mathbb{S})}{\arg\max}\,\frac {P(s_{1},s_{2},\cdots,s_{t})\prod_{i=1}^{t}P(o_{i}\mid s_{i})}{P(o_{1},o_{2}, \cdots,o_{t})}, \tag{7}\]  
In Equation (7), \(P(s_{1},s_{2},\cdots,s_{T})\) represents the probability of the transmitted symbol sequence occurring within the entire set of transmitted symbols \(\mathbb{S}\).