语用信息的数学理论OA

We propose a mathematical theory of pragmatic information that unifies communication,control,and decision-making within a coherent framework.At its core is the isoteleia mapping,which formalizes equifinality:distinct semantic paths that lead to the same optimal action are regarded as pragmatically equivalent.This mapping induces a three-tier hi-erarchy of syntactic,semantic,and pragmatic information,in which each successive abstraction discards distinctions that are irrelevant to the task.Building on this hierarchy,we develop a set of information-theoretic measures at the pragmatic level,including pragmatic entropy,up/down pragmatic mutual information,channel capacity,and rate-distortion.In ad-dition,we prove three coding theorems—lossless source coding,channel coding,and rate-distortion—which generalize Shannon's classical results.We then introduce pragmatic value of information(VoI)and pragmatic cost of information(CoI)as decision-theoretic duals to rate-distortion and capacity,respectively,and formulate a Lagrangian dual framework for cross-layer optimization.The resulting pragmatic efficiency bound εp(λ)=supR[Φp(R)-λCoIp(R)]characterizes the maximum net utility attainable by a resource-constrained intelligent system under a given resource price,thereby providing a limit on purposeful behavior—abehavioral capacity.This limit can be viewed as a goal-directed analogue of Shannon's symbol-level capacity under the stated utility and resource assumptions.Extensions to continuous messages yield closed-form expressions for Gaussian channels and sources,while dynamic settings are addressed through a Bellman equation for sequential decision-making.Finally,this framework provides a foundation for task-oriented communication,networked control,autonomous systems,and embodied AI,shifting the focus from symbol fidelity to the effectiveness of information in guiding actions,and offering a mathematical language for the design of next-generation intelligent systems.