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Dynamic Level Sets

Michael Stephen Fiske

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 (1956) that probabilistic Turing machines compute no more than deterministic ones.

Dynamic Level Sets

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 (1956) that probabilistic Turing machines compute no more than deterministic ones.
Paper Structure (10 sections, 1 theorem, 4 equations)

This paper contains 10 sections, 1 theorem, 4 equations.

Table of Contents

  1. Level Sets in Pure Mathematics
  2. Level Sets in the Active Element Machine
  3. A New Mathematical Object
  4. It is not a parametric family.
  5. It is not bifurcation theory.
  6. It is not a probabilistic Turing machine in the sense of de Leeuw et al.
  7. Incomputability
  8. Connection to the Principle of Self-Modifiability
  9. Level sets and Phase spaces
  10. Summary

Key Result

Lemma 4.1

Let $\varphi : \mathbb{N} \to \{0,1\}$ be an incomputable function. For each $k$ with $1 \le k \le m$, let $B_k : \{0,1\}^n \to \{0,1\}^n$ be an invertible boolean function, and let $h : \mathbb{N} \to \{1,\ldots,m\}$ be a computable function. Define $g : \mathbb{N} \to \{0,1\}$ by Then $g$ is incomputable.

Theorems & Definitions (4)

  • Remark 1.1
  • Definition 3.1: Dynamic Level Set
  • Remark 3.1
  • Lemma 4.1: Fiske 2012