Dynamic Level Sets

TL;DR

This paper introduces and analyzes the concept of dynamic level sets, a novel mathematical idea related to self-modifiability and incomputable physical processes, distinct from traditional dynamical systems and computability theory.

Contribution

It presents a new mathematical object called dynamic level sets and explains its significance and why it has been overlooked in prior literature.

Findings

Dynamic level sets are distinct from standard dynamical systems.

The concept involves reconfiguration by incomputable physical processes.

It challenges classical results on probabilistic Turing machines.

Abstract

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.