Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the future cover of a sofic shift

Klaus Thomsen

TL;DR

This work provides a new, more conceptual proof of Krieger's theorem on the canonicity of the future cover for sofic shifts and introduces a framework to generate new strongly canonical covers from weakly canonical ones. It formalizes a universal-property perspective for the future cover and its labeling map, and develops a subset-construction approach to obtain the future cover from arbitrary presentations. The paper also clarifies the relationship between the future cover and the follower-set graph, and furnishes constructive procedures (merging, subset constructions) with concrete examples illustrating when the future cover coincides with or differs from other canonical covers such as the minimal right-resolving cover. Overall, it expands the taxonomy of canonical covers for sofic shifts and provides tools to derive and compare strongly canonical covers beyond Krieger’s original construction.

Abstract

The paper contains a new proof of the theorem by Krieger which establishes the canonicity of the future cover of a sofic shift. In addition the paper describes a method to produce new canonical covers from a given one, resulting in canonical covers related to, but generally different from the future cover.

On the future cover of a sofic shift

TL;DR

This work provides a new, more conceptual proof of Krieger's theorem on the canonicity of the future cover for sofic shifts and introduces a framework to generate new strongly canonical covers from weakly canonical ones. It formalizes a universal-property perspective for the future cover and its labeling map, and develops a subset-construction approach to obtain the future cover from arbitrary presentations. The paper also clarifies the relationship between the future cover and the follower-set graph, and furnishes constructive procedures (merging, subset constructions) with concrete examples illustrating when the future cover coincides with or differs from other canonical covers such as the minimal right-resolving cover. Overall, it expands the taxonomy of canonical covers for sofic shifts and provides tools to derive and compare strongly canonical covers beyond Krieger’s original construction.

Abstract

The paper contains a new proof of the theorem by Krieger which establishes the canonicity of the future cover of a sofic shift. In addition the paper describes a method to produce new canonical covers from a given one, resulting in canonical covers related to, but generally different from the future cover.

Paper Structure

This paper contains 18 sections, 47 theorems, 84 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Krieger's future cover and its canonicity
  3. Setup and notation
  4. Regular vertexes and rays
  5. The future cover and Krieger's theorem
  6. Proof of Krieger's theorem, Theorem \ref{['Krieger']}
  7. The terminal component in the future cover of an irreducible sofic shift
  8. Additional properties of the future cover
  9. A universal property of the future cover $(\mathbb K(Y),L_{\mathbb K(Y)})$.
  10. A universal property of the future cover $L_G : X_{\mathbb K(Y)} \to Y$.
  11. The relation between the future cover and the follower set graph
  12. An addendum to Krieger's theorem
  13. Subset constructions
  14. Subset constructions for sofic shifts
  15. Constructing the future cover
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $X$ be an SFT and $\pi : X \to Y$ a right-closing factor code. There is a right-resolving labeled graph $(G,L_G)$ and a conjugacy $\psi : X \to X_G$ such that commutes.

Figures (7)

  • Figure 1: A right-resolving follower-separated labeled graph which is not regular.
  • Figure 2: A strongly connected labeled graph which is right-resolving and regular, but not synchronizing.
  • Figure 3: A regular labeled graph with non-regular rays.
  • Figure 4: The future cover of the sofic shift presented by the labeled graph in Figure \ref{['04-09-25gxx']}.
  • Figure 5: A right-resolving, regular and follower-separated presentation of the even shift.
  • ...and 2 more figures

Theorems & Definitions (94)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Example 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Example 2.7
  • ...and 84 more