The Preliminaries We Used to Classify Languages: Fixing a Finite Alphabet

Table of Links 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 2. Preliminaries Remark 1. The only infinite monoids that we consider are of the form A∗ . From now on, we implicitly assume that every other monoid M, N, . . . appearing in the paper is finite. We shall also consider the class LT of locally testable languages [6, 44]. It consists of all finite Boolean combinations of languages A∗wA∗ , wA∗ and A∗w where w ∈ A∗ is a word. Decision problems. These problems depend on a parameter class C. We use them as tools for analyzing C: intuitively, proving their decidability requires a solid understanding of C. The simplest one is C-membership. Its input is a regular language L. It asks whether L ∈ C. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Thomas Place; (2) Marc Zaitoun. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Preliminaries 2 Preliminaries 3 Temporal Hierarchies 3 Temporal Hierarchies 4 Rating Maps 4 Rating Maps 5 Optimal Imprints for TL(AT) 5 Optimal Imprints for TL(AT) 6 Conclusion and References 6 Conclusion and References Appendix A. Appendix to Section 2 & Appendix B. Appendix to Section 3 Appendix A. Appendix to Section 2 & Appendix B. Appendix to Section 3 Appendix C. Appendix to Section 4 & Appendix D. Appendix to Section 5 Appendix C. Appendix to Section 4 & Appendix D. Appendix to Section 5 2. Preliminaries Remark 1. The only infinite monoids that we consider are of the form A∗ . From now on, we implicitly assume that every other monoid M, N, . . . appearing in the paper is finite. Remark 1. The only infinite monoids that we consider are of the form A∗ . From now on, we implicitly assume that every other monoid M, N, . . . appearing in the paper is finite. We shall also consider the class LT of locally testable languages [6, 44]. It consists of all finite Boolean combinations of languages A∗wA∗ , wA∗ and A∗w where w ∈ A∗ is a word. Decision problems. These problems depend on a parameter class C. We use them as tools for analyzing C: intuitively, proving their decidability requires a solid understanding of C. The simplest one is C-membership. Its input is a regular language L. It asks whether L ∈ C. Decision problems. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Authors: (1) Thomas Place; (2) Marc Zaitoun. Authors: Authors: (1) Thomas Place; (2) Marc Zaitoun.