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Multiple access (MA) design is investigated for facilitating the coexistence of the emerging semantic transmission and the conventional bit-based transmission in future networks. The semantic rate is considered for measuring the performance of the semantic transmission. However, a key challenge is that there is a lack of a closed-form expression for a key parameter, namely the semantic similarity, which characterizes the sentence similarity between an original sentence and the corresponding recovered sentence. To overcome this challenge, we propose a data regression method, where the semantic similarity is approximated by a generalized logistic function. Using the obtained tractable function, we propose a heterogeneous semantic and bit communication framework, where an access point simultaneously sends the semantic and bit streams to one semantics-interested user (S-user) and one bit-interested user (B-user). To realize this heterogeneous semantic and bit transmission in multi-user networks, three MA schemes are proposed, namely orthogonal multiple access (OMA), non-orthogonal multiple access (NOMA), and semi-NOMA. More specifically, the bit stream in semi-NOMA is split into two streams, one is transmitted with the semantic stream over the shared frequency sub-band and the other is transmitted over the separate orthogonal frequency sub-band. To study the fundamental performance limits of the three proposed MA schemes, the semantic-versus-bit (SvB) rate region and the power region are defined. An optimal resource allocation procedure is then derived for characterizing the boundary of the SvB rate region and the power region achieved by each MA scheme. The structures of the derived solutions demonstrate that semi-NOMA is superior to both NOMA and OMA given its highly flexible transmission policy. Our numerical results validate the analysis and show the superiority of semi-NOMA.