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There is a deep 1-generic set

Ang Li

Abstract

An infinite binary sequence is Bennett deep if, for any computable time bound, the difference between the time-bounded prefix-free Kolmogorov complexity and the prefix-free Kolmogorov complexity of its initial segments is eventually unbounded. It is known that weakly 2-generic sets are shallow, i.e. not deep. In this paper, we show that there is a deep 1-generic set.

There is a deep 1-generic set

Abstract

An infinite binary sequence is Bennett deep if, for any computable time bound, the difference between the time-bounded prefix-free Kolmogorov complexity and the prefix-free Kolmogorov complexity of its initial segments is eventually unbounded. It is known that weakly 2-generic sets are shallow, i.e. not deep. In this paper, we show that there is a deep 1-generic set.
Paper Structure (2 sections, 10 theorems, 10 equations)

This paper contains 2 sections, 10 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. The Proof of Theorem 1.5

Key Result

Theorem 1.5

There exists a deep 1-generic set.

Theorems & Definitions (25)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Proposition 1.7
  • proof
  • Remark 1.8
  • Lemma 2.1
  • ...and 15 more