实用信息论的数学理论
A Mathematical Theory of Pragmatic Information
- Beijing University of Posts and Telecommunications（北京邮电大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种统一通信、控制与决策的实用信息论，通过同终映射形式化等终性，建立三层信息层级，并证明三个编码定理及实用效率界，为任务导向智能系统奠定数学基础。
AI中文摘要：
我们提出了一种统一的实用信息论，将通信、控制和决策制定融为一体。其核心是同终映射（isoteleia mapping），它形式化了等终性（equifinality）：通向同一最优行动的不同语义路径在实用上是等价的。这引出了一个由句法信息、语义信息和实用信息组成的三层层级结构，每一层抽象都丢弃了与任务无关的区分。我们发展了实用熵、上/下互信息、信道容量和率失真理论，并证明了三个推广了香农经典结果的编码定理。我们引入了信息的实用价值（VoI）和实用成本（CoI），分别作为率失真和容量的决策论对偶，并构建了一个用于跨层优化的拉格朗日对偶框架。实用效率界 $\mathcal{E}_p(\lambda)=\sup_R[\Phi_p(R)-\lambda\\,\mathrm{CoI}_p(R)]$ 量化了任何资源受限的智能系统所能提取的最大净效用，从而确立了一个基本的行为能力极限——将香农的符号级容量推广到目标导向的行动。对连续消息的扩展产生了闭式高斯表达式，而动态场景则通过用于序贯决策的贝尔曼方程加以处理。该框架为面向任务的通信、网络化控制、自主系统和具身人工智能提供了严格的基础，将焦点从符号保真度转移到信息在指导行动中的有效性，并为下一代智能系统提供了一种统一的数学语言。
英文摘要：
We propose a mathematical theory of pragmatic information that connects communication, control, and decision-making. Its central notion is the isoteleia mapping, which formalizes equifinality: distinct semantic paths that lead to the same optimal action are treated as pragmatically equivalent. This mapping yields a three-tier hierarchy of syntactic, semantic, and pragmatic information, in which each successive abstraction removes distinctions that are irrelevant to the task. We then define pragmatic entropy, up/down pragmatic mutual information, channel capacity, and rate-distortion, and prove lossless source coding, channel coding, and rate-distortion theorems that extend Shannon's results. These measures quantify decision uncertainty, reliable transmission, and task-oriented compression at the level of terminal actions. We further introduce pragmatic value of information (VoI) and pragmatic cost of information (CoI) as decision-theoretic duals to rate-distortion and capacity, and develop a Lagrangian dual framework for cross-layer optimization. The resulting pragmatic efficiency bound $\mathcal{E}_p(λ)=\sup_R[Φ_p(R)-λ\mathrm{CoI}_p(R)]$ characterizes the maximum net utility attainable by a resource-constrained intelligent system under a given resource price, yielding a behavioral capacity that extends Shannon's symbol-level capacity to goal-directed action. Extensions to continuous messages provide closed-form expressions for Gaussian channels and sources, while dynamic settings are addressed through a Bellman equation for sequential decision-making. The framework supports task-oriented communication, networked control, autonomous systems, and embodied AI by shifting emphasis from symbol fidelity to the effectiveness of information in guiding actions. In this way, it offers a common language for systems that extract value from information under resource constraints.
