Dynamic Level Sets
This paper introduces a new mathematical concept that could potentially expand the understanding of computation beyond current theoretical limits, impacting researchers in computability theory and theoretical computer science.
This paper analyzes 'dynamic level sets,' a mathematical concept identified in a 2012 paper, distinct from standard dynamical systems, topology, and computability theory. It explains a new mathematical object and why it may have been overlooked, even in light of the classical result that probabilistic Turing machines (with computable bias) compute no more than deterministic ones.
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.