A mathematical concept is identified and analyzed that is implicit in the 2012 paper Turing Incomputable Computation, presented at the Alan Turing Centenary Conference (Turing-100, Manchester). The concept, called dynamic level sets, is distinct from mathematical concepts in the standard literature on dynamical systems, topology, and computability theory. A new mathematical object is explained and why it may have escaped prior characterizations, including the classical result of de Leeuw, Moore, Shannon, and Shapiro that probabilistic Turing machines (with bias $p$ where $p$ is Turing computable) compute no more than deterministic ones. A key mechanism underlying the concept is the Principle of Self-Modifiability, whereby the physical realization of an invariant logical level set is reconfigured at each computational step by an incomputable physical process.

翻译：本文识别并分析了一个隐含于2012年图灵百年纪念会议（Turing-100，曼彻斯特）论文《图灵不可计算的计算》中的数学概念。该概念被称为动态层级集合，区别于动力系统、拓扑学和可计算性理论标准文献中的数学概念。本文阐释了一种新的数学对象，并解释了为何其可能规避了先前的特征化描述，包括de Leeuw、Moore、Shannon和Shapiro的经典结果——即概率图灵机（具有图灵可计算的偏置$p$）的计算能力不超过确定性图灵机。支撑该概念的一个关键机制是自修改原理：通过不可计算的物理过程，不变逻辑层级集的物理实现会在每个计算步骤中被重新配置。