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What Are Temporal Hierarchies?by@hierarchy

What Are Temporal Hierarchies?

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January 30th, 2025
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We introduce the operator C 7→ TL(C) and use it to define temporal hierarchies. We compare them to the classic concatenation hierarchies.
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Hierarchy

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Academic Research Paper

Academic Research Paper

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

Abstract and 1 Introduction

2 Preliminaries

3 Temporal Hierarchies

4 Rating Maps

5 Optimal Imprints for TL(AT)

6 Conclusion and References


Appendix A. Appendix to Section 2 & Appendix B. Appendix to Section 3

Appendix C. Appendix to Section 4 & Appendix D. Appendix to Section 5

3. Temporal hierarchies

We introduce the operator C 7→ TL(C) and use it to define temporal hierarchies. We compare them to the classic concatenation hierarchies. We focus on the standard bases which are made of group languages. We prove that for such a basis, the corresponding temporal hierarchy is always strict and that membership for the level two boils down to separation for the basis.


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Proposition 21 can now be combined with Corollary 4. We obtain that for every group prevariety G, if C ∈ {G, G +}, then TL2(C)-membership reduces to BPol(C)-separation. The latter problem has been extensively studied for group classes. If C ∈ {G, G +}, it is shown in [28, 30] that BPol(C)- separation reduces to C-separation. This yields the following result.


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This paper is available on arxiv under CC BY 4.0 DEED license.

Authors:

(1) Thomas Place;

(2) Marc Zaitoun.


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Hierarchy@hierarchy
Hierarchy's nested framework organizes and allocates, channeling power and responsibility with clarity and purpose.

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