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语用信息论的数学理论：统一通信、控制与决策

A Mathematical Theory of Pragmatic Information

arXiv:2609.10986 [cs.IT]
Subjects: Information Theory (cs.IT); Artificial Intelligence (cs.AI); Robotics (cs.RO); Systems and Control (eess.SY)
Authors: Kai Niu, Ping Zhang
Submitted: 10 September 2026
Length: 152 pages, 18 figures
arXiv ID: arXiv:2609.10986

文章背景与核心概要

1948 年 Claude Shannon 创立了经典信息论，奠定了现代数字通信的基石，但他明确将信息的语义与语用（即信息产生的作用和价值）排除在理论之外。如今随着具身智能 (Embodied AI)、网络化控制与自主系统的快速发展，智能系统不再仅仅追求比特的无损传输，更追求“信息能否有效指导正确的行动”。北京邮电大学 Kai Niu 与 Ping Zhang 教授发表了长达 152 页的重磅长文，提出了统一通信、控制与决策的“语用信息论”。该理论以“异途同归映射 (Isoteleia Mapping)”为数学核心，建立了语法-语义-语用三层信息体系，证明了推广 Shannon 经典理论的三大语用编码定理，确立了智能系统在资源约束下的语用效率极限与行为信道容量，为下一代任务导向通信与机器智能构筑了严谨坚实的数学根基。

📌 核心概要

Executive Summary

本文提出了一套完备且严谨的语用信息论 (Pragmatic Information Theory)，成功架起了沟通通信、控制理论与决策科学之间的统一桥梁。该理论的核心支柱是异途同归映射 (Isoteleia Mapping)，它在数学上形式化了“殊途同归 (Equifinality)”原则——即引导系统走向完全相同的最优行动的各种不同语义路径，在语用层面是完全等价的。通过彻底剔除与任务无关的冗余区分，该框架确立了“语法、语义、语用”的三层信息层次体系，通过证明全新的编码定理全面推广了经典香农信息论，为下一代智能系统、目标导向决策与具身智能 (Embodied AI) 提供了坚实可靠的数学根基。

This paper introduces a comprehensive pragmatic information theory that bridges communication, control, and decision-making. At its heart is the isoteleia mapping, which formalizes equifinality—the principle that distinct semantic paths leading to the same optimal action are pragmatically equivalent. By discarding task-irrelevant distinctions, the framework establishes a three-tier hierarchy (syntactic, semantic, and pragmatic information), extends classical information theory through new coding theorems, and offers a rigorous foundation for next-generation intelligent systems, goal-directed action, and embodied AI.

📑 论文摘要

Abstract

我们提出了一套统一通信、控制与决策过程的语用信息理论。其数学核心是异途同归映射 (Isoteleia Mapping)，形式化表征了“殊途同归”特性：能够达成相同最优行动的不同语义路径，在语用层面上皆属等价。这自然诱导出了一个由语法信息、语义信息与语用信息构成的三层抽象体系，每一层抽象都精准过滤掉与当前目标任务无关的冗余差异。

We propose a pragmatic information theory unifying communication, control, and decision-making. Its core is the isoteleia mapping, formalizing equifinality: distinct semantic paths leading to the same optimal action are pragmatically equivalent. This induces a three-tier hierarchy of syntactic, semantic, and pragmatic information, each abstraction discarding task-irrelevant distinctions.

我们系统建立了语用熵、上/下互信息、信道容量以及率失真理论，并证明了推广 Shannon 经典理论成果的三大语用编码定理。我们将信息的语用价值 (Value of Information, VoI) 与信息成本 (Cost of Information, CoI) ，分别作为率失真理论与信道容量理论在决策论层面的对偶概念引入，并构建了用于跨层联合优化的 Lagrange 对偶框架。

We develop pragmatic entropy, up/down mutual information, channel capacity, and rate-distortion, and prove three coding theorems generalizing Shannon's classical results. We introduce pragmatic value (VoI) and cost (CoI) of information as decision-theoretic duals to rate-distortion and capacity, respectively, and formulate a Lagrangian dual framework for cross-layer optimization.

语用效率上界：

\[\mathcal{E}_p(\lambda) = \sup_R [\Phi_p(R) - \lambda\,\mathrm{CoI}_p(R)]\]

量化了任何受限于资源约束的智能系统所能汲取的最大净效用，从而确立了一项根本性的“行为容量极限 (Behavioral Capacity Limit)”——将 Shannon 符号级的传输容量极限成功推广到了目标导向的具身行动层面。对于连续变量消息，该理论推广给出了闭式的高斯分布解析表达式；而在时序动态环境中，则通过结合序贯决策的 Bellman 方程予以拓展。该理论框架为任务导向通信、网络化控制、自主无人系统以及具身智能构筑了严密的数学基础，推动信息科学从追求“符号级的高保真度传输”迈向追求“信息指导行动的真实效能”，为下一代机器智能体系提供了统一的数学语言。

The pragmatic efficiency bound

\[\mathcal{E}_p(\lambda) = \sup_R [\Phi_p(R) - \lambda\,\mathrm{CoI}_p(R)]\]

quantifies the maximum net utility any resource-constrained intelligent system can extract, thereby establishing a fundamental behavioral capacity limit—generalizing Shannon's symbol-level capacity to goal-directed action. Extensions to continuous messages yield closed-form Gaussian expressions, while dynamic settings are addressed via a Bellman equation for sequential decision-making. This framework provides a rigorous foundation for task-oriented communication, networked control, autonomous systems, and embodied AI, shifting focus from symbol fidelity to the effectiveness of information in guiding actions, and offers a unified mathematical language for next-generation intelligent systems.

🧠 理论框架的核心支柱

Key Framework Elements

  • 异途同归映射 (The Isoteleia Mapping)：在数学上严格刻画了殊途同归现象，即多条不同的语义发展轨迹最终可收敛于唯一的语用最优行动输出。
  • 三层信息层次架构：涵盖语法、语义和语用三级渐进抽象，专为深度剔除与任务无关的冗余干扰而设计。
  • 推广的广义信息度量：正式确立了语用熵、上下互信息变体、语用信道容量以及语用率失真函数边界。
  • 信息的语用价值与语用成本 (VoI & CoI)：构建了与信道容量和率失真理论紧密对偶的决策论数学工具。
  • 动态时序决策扩展：引入动态规划与 Bellman 方程，实现从静态单步优化向长周期、时间依赖型序贯决策场景的无缝泛化。
  • The Isoteleia Mapping: Formalizes equifinality where multiple semantic trajectories converge on a singular optimal pragmatic outcome.
  • Three-Tier Information Hierarchy: Syntactic, semantic, and pragmatic abstractions optimized to filter out task-irrelevant noise.
  • Generalized Information Measures: Establishes pragmatic entropy, mutual information variations, channel capacity, and rate-distortion bounds.
  • Value and Cost of Information (VoI & CoI): Decision-theoretic duals aligned with capacity and rate-distortion theories.
  • Dynamic Decision-Making: Incorporates Bellman equations to seamlessly scale from static optimization to sequential, time-dependent scenarios.