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In this paper, we extend the Blahut\u2013Arimoto (BA) algorithm, a fundamental method in classical information theory (CIT) for computing the rate-distortion (RD) function, to semantic communication by proposing the extended Blahut\u2013Arimoto (EBA) algorithm, which iteratively updates transition and reconstruction distributions to calculate the semantic RD function based on synonymous mapping in semantic information theory (SIT). To address scenarios where synonymous mappings are unknown, we develop an optimization framework that combines the EBA algorithm with simulated annealing. Initialized with a syntactic mapping, the framework progressively merges syntactic symbols and identifies the mapping with a maximum synonymous number that satisfies objective constraints. Furthermore, by considering the semantic knowledge base (SKB) as a specific instance of synonymous mapping, the EBA algorithm provides a theoretical approach for analyzing and predicting the SKB size. Numerical results validate the effectiveness of the EBA algorithm. For Gaussian sources, the semantic RD function decreases with an increasing synonymous number and becomes significantly lower than its classical counterpart. Additionally, analysis on the CUB dataset demonstrates that larger SKB sizes lead to higher semantic communication compression efficiency.<\/jats:p>","DOI":"10.3390\/e27060651","type":"journal-article","created":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T07:38:15Z","timestamp":1750232295000},"page":"651","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Extended Blahut\u2013Arimoto Algorithm for Semantic Rate-Distortion Function"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-1665-0767","authenticated-orcid":false,"given":"Yuxin","family":"Han","sequence":"first","affiliation":[{"name":"Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China"},{"name":"Department of Broadband Communication, Pengcheng Laboratory, Shenzhen 518055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yang","family":"Liu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China"},{"name":"Department of Broadband Communication, Pengcheng Laboratory, Shenzhen 518055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yaping","family":"Sun","sequence":"additional","affiliation":[{"name":"Department of Broadband Communication, Pengcheng Laboratory, Shenzhen 518055, China"},{"name":"School of Science and Engineering (SSE) and the Future Network of Intelligent Institute (FNii), The Chinese University of Hong Kong (Shenzhen), Shenzhen 518172, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kai","family":"Niu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China"},{"name":"Department of Broadband Communication, Pengcheng Laboratory, Shenzhen 518055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2302-7148","authenticated-orcid":false,"given":"Nan","family":"Ma","sequence":"additional","affiliation":[{"name":"Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China"},{"name":"Department of Broadband Communication, Pengcheng Laboratory, Shenzhen 518055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuguang","family":"Cui","sequence":"additional","affiliation":[{"name":"School of Science and Engineering (SSE) and the Future Network of Intelligent Institute (FNii), The Chinese University of Hong Kong (Shenzhen), Shenzhen 518172, China"},{"name":"Department of Broadband Communication, Pengcheng Laboratory, Shenzhen 518055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0269-104X","authenticated-orcid":false,"given":"Ping","family":"Zhang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China"},{"name":"Department of Broadband Communication, Pengcheng Laboratory, Shenzhen 518055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,18]]},"reference":[{"key":"ref_1","first-page":"5","article-title":"Beyond transmitting bits: Context, semantics, and task-oriented communications","volume":"41","author":"Qin","year":"2022","journal-title":"IEEE J. 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