This paper presents a new foundational approach to information theory based on the concept of the information efficiency of a recursive function, which is defined as the difference between the information in the input and the output. The theory allows us to study planar representations of various infinite domains. Dilation theory studies the information effects of recursive operations in terms of topological deformations of the plane. I show that the well-known class of finite sets of natural numbers behaves erratically under such transformations. It is subject to phase transitions that in some cases have a fractal nature. The class is \emph{semi-countable}: there is no intrinsic information theory for this class and there are no efficient methods for systematic search. There is a relation between the information efficiency of the function and the time needed to compute it: a deterministic computational process can destroy information in linear time, but it can only generate information at logarithmic speed. Checking functions for problems in $NP$ are information discarding. Consequently, when we try to solve a decision problem based on an efficiently computable checking function, we need exponential time to reconstruct the information destroyed by such a function. At the end of the paper I sketch a systematic taxonomy for problems in $NP$.

翻译：本文提出了一种基于递归函数信息效率的信息论新基础方法，该效率定义为输入与输出信息之差。该理论使我们能够研究各种无限域的平面表示。膨胀理论通过平面的拓扑形变来研究递归运算的信息效应。我证明了著名的自然数有限集类在此类变换下表现出不规律行为。它存在相变现象，某些情况下具有分形性质。该类是半可数类：对此类不存在内在信息理论，亦无系统搜索的有效方法。函数的信息效率与其计算所需时间存在关联：确定性计算过程能以线性时间销毁信息，但仅能以对数速度生成信息。NP问题的检验函数属于信息丢弃型。因此，当我们试图基于高效可计算的检验函数求解判定问题时，需要指数级时间重构被该函数销毁的信息。论文末尾，我勾勒了NP问题的系统性分类框架。