自从1948年经典信息论诞生以来，在其指导下，现代通信技术已经逼近了理论性能极限，例如信息熵

<math id="M1"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592415&type=

2.87866688

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592416&type=

8.55133343

、信道容量

<math id="M2"> <mi>C</mi> <mo>=</mo> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <msub> <mrow> <mi mathvariant="normal">x</mi> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592417&type=

3.89466691

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592418&type=

28.10933495

以及率失真函数

<math id="M3"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <msub> <mrow> <mi mathvariant="normal">n</mi> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mover accent="true"> <mi>x</mi> <mo>^</mo> </mover> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi mathvariant="double-struck">E</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>

</mo> <mover accent="true"> <mi>x</mi> <mo>^</mo> </mover> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>D</mi> </mrow> </msub> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mover accent="true"> <mi>X</mi> <mo>^</mo> </mover> <mo stretchy="false">)</mo> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592396&type=

4.65666676

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592397&type=

46.73600006

。长期以来，由于经典信息论只研究语法信息，限制了通信科学的进一步发展。近年来，研究语义信息处理与传输的通信技术获得了学术界的普遍关注，语义通信开辟了未来通信技术发展的新方向，但还缺乏一般性的数学指导理论。为了解决这一难题，构建了语义信息论的理论框架，对语义信息的度量体系与语义通信的理论极限进行了系统性阐述。首先，通过深入分析各类信源的数据特征，以及各种下游任务的需求，总结归纳出语义信息的普遍属性——同义性。由此指出语义信息是语法信息的上级概念，是许多等效或相似语法信息的抽象特征，表征隐藏在数据或消息背后的含义或内容。将语义信息与语法信息之间的关系命名为同义映射，这是一种“一对多”映射，即一个语义符号可以由许多不同的语法符号表示。基于同义映射

<math id="M4"> <mi>f</mi> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592398&type=

2.96333337

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592399&type=

2.11666679

这一核心概念，引入语义熵

<math id="M5"> <msub> <mrow> <mi>H</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mover> <mrow> <mi>U</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> <mo stretchy="false">)</mo> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592400&type=

4.57200003

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592401&type=

9.22866726

作为语义信息的基本度量指标，表示为信源概率分布与同义映射的泛函。在此基础上，引入上/下语义互信息

<math id="M6"> <msup> <mrow> <mi>I</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mover> <mrow> <mi>X</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> <mo>;</mo> <mover> <mrow> <mi>Y</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mrow> <mi>I</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mover> <mrow> <mi>X</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> <mo>;</mo> <mover> <mrow> <mi>Y</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592402&type=

4.82600021

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592403&type=

28.95599937

，语义

信道容量

<math id="M7"> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <msub> <mrow> <mi mathvariant="normal">x</mi> </mrow> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </msub> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <msub> <mrow> <mi mathvariant="normal">x</mi> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msub> <msup> <mrow> <mi>I</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mover> <mrow> <mi>X</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> <mo>;</mo> <mover> <mrow> <mi>Y</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> <mo stretchy="false">)</mo> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592404&type=

6.01133299

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592405&type=

38.43866730

以及语义率失真函数

<math id="M8"> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <msub> <mrow> <mi mathvariant="normal">n</mi> </mrow> <mrow> <mo stretchy="false">{</mo> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mi>x</mi> </mrow> </msub> <mo>

</mo> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mover accent="true"> <mi>x</mi> <mo>^</mo> </mover> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> </msub> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <msub> <mrow> <mi mathvariant="normal">n</mi> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mover accent="true"> <mi>x</mi> <mo>^</mo> </mover> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi mathvariant="double-struck">E</mi> <msub> <mrow> <mi>d</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mover> <mrow> <mi>x</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> <mo>

</mo> <mover accent="true"> <mrow> <mover> <mrow> <mi>x</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> </mrow> <mo>^</mo> </mover> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>D</mi> </mrow> </msub> <msub> <mrow> <mi>I</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mover> <mrow> <mi>X</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> <mo>;</mo> <mover accent="true"> <mrow> <mover> <mrow> <mi>X</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> </mrow> <mo>^</mo> </mover> <mo stretchy="false">)</mo> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592406&type=

6.85799980

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592419&type=

60.70600128

，从而构建了完整的语义信息度量体系。这些语义信息度量是经典信息度量的自然延伸，都由同义映射约束，如果采用“一对一”映射，则可以退化为传统的信息度量。由此可见，语义信息度量体系包含语法信息度量，前者与后者具有兼容性。其次，证明了3个重要的语义编码定理，以揭示语义通信的性能优势。基于同义映射，引入新的数学工具——语义渐近均分（AEP），详细探讨了同义典型序列的数学性质，并应用随机编码和同义典型序列译码/编码，证明了语义无失真信源编码定理、语义信道编码定理和语义限失真信源编码定理。类似于经典信息论，这些基本编码定理也都是存在性定理，但它们指出了语义通信系统的性能极限，在语义信息论中起着关键作用。由同义映射和这些基本编码定理可以推断，语义通信系统的性能优于经典通信系统，即语义熵小于信息熵

<math id="M9"> <msub> <mrow> <mi>H</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mover> <mrow> <mi>U</mi> </mrow> <mrow> <mi>˜</mi> </mrow> </mover> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592420&type=

4.57200003

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592421&type=

21.33600044

，语义信道容量大于经典信道容量

<math id="M10"> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>≥</mo> <mi>C</mi> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592422&type=

3.21733332

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592423&type=

9.22866726

，以及语义率失真函数小于经典率失真函数

<math id="M11"> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592424&type=

3.89466691

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592425&type=

20.40466690

。最后，讨论了连续条件下的语义信息度量。此时，同义映射转换为连续随机变量分布区间的划分方式。相应地，划分后的子区间被命名为同义区间，其平均长度定义为同义长度

<math id="M12"> <mi>S</mi> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592430&type=

2.28600001

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592431&type=

1.77800000

。特别是对于限带高斯信道，得到了一个新的信道容量公式

<math id="M13"> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mi>B</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">g</mi> <mfenced open="[" close="]" separators="|"> <mrow> <msup> <mrow> <mi>S</mi> </mrow> <mrow> <mn mathvariant="normal">4</mn> </mrow> </msup> <mfenced separators="|"> <mrow> <mn mathvariant="normal">1</mn> <mo>+</mo> <mfrac> <mrow> <mi>P</mi> </mrow> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mn mathvariant="normal">0</mn> </mrow> </msub> <mi>B</mi> </mrow> </mfrac> </mrow> </mfenced> </mrow> </mfenced> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592428&type=

8.97466660

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592429&type=

33.86666870

，其中，平均同义长度

<math id="M14"> <mi>S</mi> </math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592430&type=

2.28600001

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=63592431&type=

1.77800000

表征了信息的辨识能力。这一容量公式是经典信道容量的重要扩展，当

<math id="M15"> <mi>S</mi> <mo>=</mo> <mn mathvariant="normal">1</mn> </math>

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2.28600001

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7.11199999

时，该公式退化为著名的香农信道容量公式。综上所述，语义信息论依据同义映射这一语义信息的本质特征，构建了语义信息的度量体系，引入新的数学工具，证明了语义编码的基本定理，论证了语义通信系统的性能极限，揭示了未来语义通信的巨大性能潜力。