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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 Appendix C. Appendix to Section 4 We now prove Proposition 27. Appendix D. Appendix to Section 5 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 Appendix C. Appendix to Section 4 We now prove Proposition 27. Appendix D. Appendix to Section 5 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.

Temporal Hierarchies of Regular Languages: Missing Arguments for the Intermediary Statements

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 6. Conclusion We defined temporal hierarchies using the unary temporal logic operator C 7→ TL(C). We looked at the bases G and G + when G is a group prevariety. In this case, we proved that the corresponding hierarchies are always strict and we compared them with the corresponding concatenation hierarchies. Moreover, we proved that if G has decidable separation, then membership is decidable for levels one and two in both hierarchies. Our main theorem yields a stronger result for the particular bases ST and DD = ST+. Indeed, it implies that level two in both hierarchies has decidable covering. In turn, this implies that level three has decidable membership. References [1] Mustapha Arfi. Polynomial operations on rational languages. In Proceedings of the 4th Annual Symposium on Theoretical Aspects of Computer Science, STACS’87, volume 247 of Lect. Notes Comp. Sci., pages 198–206. Springer, 1987. [2] Mustapha Arfi. Operations polynomiales et hierarchies de concatenation. Theoretical Computer Science, 91(1):71–84, 1991. [3] Christopher J. Ash. Inevitable graphs: a proof of the type II conjecture and some related decision procedures. International Journal of Algebra and Computation, 1(1):127–146, 1991. [4] Janusz A. Brzozowski and Rina S. Cohen. Dot-depth of star-free events. Journal of Computer and System Sciences, 5(1):1–16, 1971. [5] Janusz A. Brzozowski and Robert Knast. The dot-depth hierarchy of star-free languages is infinite. Journal of Computer and System Sciences, 16(1):37–55, 1978. [6] Janusz A. Brzozowski and Imre Simon. Characterizations of locally testable events. Discrete Mathematics, 4(3):243–271, 1973. [7] Laura Chaubard, Jean Eric Pin, and Howard Straubing. First order formulas with modular predicates. In ´ Proceedings of the 21th IEEE Symposium on Logic in Computer Science (LICS’06), pages 211–220, 2006. [8] Luc Dartois and Charles Paperman. Two-variable first order logic with modular predicates over words. In Proceedings of the 30th International Symposium on Theoretical Aspects of Computer Science, STACS’13, volume 20 of Leibniz International Proceedings in Informatics (LIPIcs), pages 329–340. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2013. [9] Manuel Delgado. Abelian poinlikes of a monoid. Semigroup Forum, 56(3):339–361, 1998. [10] Kousha Etessami, Moshe Y. Vardi, and Thomas Wilke. First-order logic with two variables and unary temporal logic. Information and Computation, 179(2):279–295, 2002. [11] Hans W. Kamp. Tense Logic and the Theory of Linear Order. Phd thesis, Computer Science Department, University of California at Los Angeles, USA, 1968. [12] Robert Knast. A semigroup characterization of dot-depth one languages. RAIRO - Theoretical Informatics and Applications, 17(4):321–330, 1983. [13] Andreas Krebs, Kamal Lodaya, Paritosh K. Pandya, and Howard Straubing. Two-variable logic with a between relation. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS’16, pages 106–115, 2016. [14] Andreas Krebs, Kamal Lodaya, Paritosh K. Pandya, and Howard Straubing. An algebraic decision procedure for two-variable logic with a between relation. In 27th EACSL Annual Conference on Computer Science Logic, CSL’18, Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2018. [15] Andreas Krebs, Kamal Lodaya, Paritosh K. Pandya, and Howard Straubing. Two-variable logics with some betweenness relations: Expressiveness, satisfiability and membership. Logical Methods in Computer Science, 16(3), 2020. [16] Manfred Kufleitner and Tobias Walter. One quantifier alternation in first-order logic with modular predicates. RAIRO Theorerical Informatics and Applications, 49(1):1–22, 2015. [17] Alexis Maciel, Pierre P´eladeau, and Denis Th´erien. Programs over semigroups of dot-depth one. Theoretical Computer Science, 245(1):135–148, 2000. [18] Stuart W. Margolis and Jean-Eric Pin. Products of group languages. In ´ FCT’85. Springer, 1985. [19] Robert McNaughton and Seymour A. Papert. Counter-Free Automata. MIT Press, 1971. [20] Jean-Eric Pin. An explicit formula for the intersection of two polynomials of regular languages. In Proceedings of the 17th International Conference on Developments in Language Theory, DLT’13, volume 7907 of Lect. Notes Comp. Sci., pages 31–45. Springer, 2013. [21] Jean-Eric Pin and Pascal Weil. Polynomial closure and unambiguous product. Theory of Computing Systems, 30(4):383–422, 1997. [22] Jean-Eric Pin and Howard Straubing. Monoids of upper triangular boolean matrices. In ´ Semigroups. Structure and Universal Algebraic Problems, volume 39, pages 259–272. North-Holland, 1985. [23] Thomas Place. The amazing mixed polynomial closure and its applications to two-variable first-order logic. In Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS’22, 2022. [24] Thomas Place and Marc Zeitoun. Separation and the successor relation. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS’15, Leibniz International Proceedings in Informatics (LIPIcs), pages 662–675. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2015. [25] Thomas Place and Marc Zeitoun. The covering problem. Logical Methods in Computer Science, 14(3), 2018. [26] Thomas Place and Marc Zeitoun. Generic results for concatenation hierarchies. Theory of Computing Systems (ToCS), 63(4):849–901, 2019. Selected papers from CSR’17. [27] Thomas Place and Marc Zeitoun. Going higher in first-order quantifier alternation hierarchies on words. Journal of the ACM, 66(2):12:1–12:65, 2019. [28] Thomas Place and Marc Zeitoun. Separation and covering for group based concatenation hierarchies. In Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS’19, pages 1–13. IEEE Computer Society, 2019. [29] Thomas Place and Marc Zeitoun. Adding successor: A transfer theorem for separation and covering. ACM Transactions on Computational Logic, 21(2):9:1–9:45, 2020. [30] Thomas Place and Marc Zeitoun. A generic polynomial time approach to separation by first-order logic without quantifier alternation. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2022, December 18-20, 2022, IIT Madras, Chennai, India, volume 250 of LIPIcs, pages 43:1–43:22. Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2022. [31] Thomas Place and Marc Zeitoun. All about unambiguous polynomial closure. TheoretiCS, 2(11):1–74, 2023. [32] Thomas Place and Marc Zeitoun. Group separation strikes back. In 38th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS’23, pages 1–13. IEEE Computer Society, 2023. [33] Thomas Place and Marc Zeitoun. On all things star-free. Journal, to appear, 2023. [34] Thomas Place and Marc Zeitoun. A generic characterization of generalized unary temporal logic and two-variable first-order logic. In Proceedings of the 32nd EACSL Annual Conference on Computer Science Logic, CSL’24, 2024. [35] Marcel Paul Sch¨utzenberger. On finite monoids having only trivial subgroups. Information and Control, 8(2):190– 194, 1965. [36] Marcel Paul Sch¨utzenberger. Sur le produit de concat´enation non ambigu. Semigroup Forum, 13:47–75, 1976. [37] Imre Simon. Piecewise testable events. In Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages, pages 214–222. Springer, 1975. [38] Howard Straubing. A generalization of the sch¨utzenberger product of finite monoids. Theoretical Computer Science, 13(2):137–150, 1981. [39] Howard Straubing. Finite semigroup varieties of the form V ∗ D. Journal of Pure and Applied Algebra, 36:53–94, 1985. [40] Denis Th´erien. Classification of finite monoids: The language approach. Theoretical Computer Science, 14(2):195– 208, 1981. [41] Gabriel Thierrin. Permutation automata. Theory of Computing Systems, 2(1):83–90, 1968. [42] Wolfgang Thomas. Classifying regular events in symbolic logic. Journal of Computer and System Sciences, 25(3):360–376, 1982. [43] Denis Therien and Thomas Wilke. Over words, two variables are as powerful as one quantifier alternation. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, STOC’98, pages 234–240, New York, NY, USA, 1998. ACM. [44] Yechezkel Zalcstein. Locally testable languages. Journal of Computer and System Sciences, 6(2):151–167, 1972. 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 6. Conclusion We defined temporal hierarchies using the unary temporal logic operator C 7→ TL(C). We looked at the bases G and G + when G is a group prevariety. In this case, we proved that the corresponding hierarchies are always strict and we compared them with the corresponding concatenation hierarchies. Moreover, we proved that if G has decidable separation, then membership is decidable for levels one and two in both hierarchies. Our main theorem yields a stronger result for the particular bases ST and DD = ST+. Indeed, it implies that level two in both hierarchies has decidable covering. In turn, this implies that level three has decidable membership. References [1] Mustapha Arfi. Polynomial operations on rational languages. In Proceedings of the 4th Annual Symposium on Theoretical Aspects of Computer Science, STACS’87, volume 247 of Lect. Notes Comp. Sci., pages 198–206. Springer, 1987. [2] Mustapha Arfi. Operations polynomiales et hierarchies de concatenation. Theoretical Computer Science, 91(1):71–84, 1991. [3] Christopher J. Ash. Inevitable graphs: a proof of the type II conjecture and some related decision procedures. International Journal of Algebra and Computation, 1(1):127–146, 1991. [4] Janusz A. Brzozowski and Rina S. Cohen. Dot-depth of star-free events. Journal of Computer and System Sciences, 5(1):1–16, 1971. [5] Janusz A. Brzozowski and Robert Knast. The dot-depth hierarchy of star-free languages is infinite. Journal of Computer and System Sciences, 16(1):37–55, 1978. [6] Janusz A. Brzozowski and Imre Simon. Characterizations of locally testable events. Discrete Mathematics, 4(3):243–271, 1973. [7] Laura Chaubard, Jean Eric Pin, and Howard Straubing. First order formulas with modular predicates. In ´ Proceedings of the 21th IEEE Symposium on Logic in Computer Science (LICS’06), pages 211–220, 2006. [8] Luc Dartois and Charles Paperman. Two-variable first order logic with modular predicates over words. In Proceedings of the 30th International Symposium on Theoretical Aspects of Computer Science, STACS’13, volume 20 of Leibniz International Proceedings in Informatics (LIPIcs), pages 329–340. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2013. [9] Manuel Delgado. Abelian poinlikes of a monoid. Semigroup Forum, 56(3):339–361, 1998. [10] Kousha Etessami, Moshe Y. Vardi, and Thomas Wilke. First-order logic with two variables and unary temporal logic. Information and Computation, 179(2):279–295, 2002. [11] Hans W. Kamp. Tense Logic and the Theory of Linear Order. Phd thesis, Computer Science Department, University of California at Los Angeles, USA, 1968. [12] Robert Knast. A semigroup characterization of dot-depth one languages. RAIRO - Theoretical Informatics and Applications, 17(4):321–330, 1983. [13] Andreas Krebs, Kamal Lodaya, Paritosh K. Pandya, and Howard Straubing. Two-variable logic with a between relation. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS’16, pages 106–115, 2016. [14] Andreas Krebs, Kamal Lodaya, Paritosh K. Pandya, and Howard Straubing. An algebraic decision procedure for two-variable logic with a between relation. In 27th EACSL Annual Conference on Computer Science Logic, CSL’18, Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2018. [15] Andreas Krebs, Kamal Lodaya, Paritosh K. Pandya, and Howard Straubing. Two-variable logics with some betweenness relations: Expressiveness, satisfiability and membership. Logical Methods in Computer Science, 16(3), 2020. [16] Manfred Kufleitner and Tobias Walter. One quantifier alternation in first-order logic with modular predicates. RAIRO Theorerical Informatics and Applications, 49(1):1–22, 2015. [17] Alexis Maciel, Pierre P´eladeau, and Denis Th´erien. Programs over semigroups of dot-depth one. Theoretical Computer Science, 245(1):135–148, 2000. [18] Stuart W. Margolis and Jean-Eric Pin. Products of group languages. In ´ FCT’85. Springer, 1985. [19] Robert McNaughton and Seymour A. Papert. Counter-Free Automata. MIT Press, 1971. [20] Jean-Eric Pin. An explicit formula for the intersection of two polynomials of regular languages. In Proceedings of the 17th International Conference on Developments in Language Theory, DLT’13, volume 7907 of Lect. Notes Comp. Sci., pages 31–45. Springer, 2013. [21] Jean-Eric Pin and Pascal Weil. Polynomial closure and unambiguous product. Theory of Computing Systems, 30(4):383–422, 1997. [22] Jean-Eric Pin and Howard Straubing. Monoids of upper triangular boolean matrices. In ´ Semigroups. Structure and Universal Algebraic Problems, volume 39, pages 259–272. North-Holland, 1985. [23] Thomas Place. The amazing mixed polynomial closure and its applications to two-variable first-order logic. In Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS’22, 2022. [24] Thomas Place and Marc Zeitoun. Separation and the successor relation. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS’15, Leibniz International Proceedings in Informatics (LIPIcs), pages 662–675. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2015. [25] Thomas Place and Marc Zeitoun. The covering problem. Logical Methods in Computer Science, 14(3), 2018. [26] Thomas Place and Marc Zeitoun. Generic results for concatenation hierarchies. Theory of Computing Systems (ToCS), 63(4):849–901, 2019. Selected papers from CSR’17. [27] Thomas Place and Marc Zeitoun. Going higher in first-order quantifier alternation hierarchies on words. Journal of the ACM, 66(2):12:1–12:65, 2019. [28] Thomas Place and Marc Zeitoun. Separation and covering for group based concatenation hierarchies. In Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS’19, pages 1–13. IEEE Computer Society, 2019. [29] Thomas Place and Marc Zeitoun. Adding successor: A transfer theorem for separation and covering. ACM Transactions on Computational Logic, 21(2):9:1–9:45, 2020. [30] Thomas Place and Marc Zeitoun. A generic polynomial time approach to separation by first-order logic without quantifier alternation. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2022, December 18-20, 2022, IIT Madras, Chennai, India, volume 250 of LIPIcs, pages 43:1–43:22. Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2022. [31] Thomas Place and Marc Zeitoun. All about unambiguous polynomial closure. TheoretiCS, 2(11):1–74, 2023. [32] Thomas Place and Marc Zeitoun. Group separation strikes back. In 38th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS’23, pages 1–13. IEEE Computer Society, 2023. [33] Thomas Place and Marc Zeitoun. On all things star-free. Journal, to appear, 2023. [34] Thomas Place and Marc Zeitoun. A generic characterization of generalized unary temporal logic and two-variable first-order logic. In Proceedings of the 32nd EACSL Annual Conference on Computer Science Logic, CSL’24, 2024. [35] Marcel Paul Sch¨utzenberger. On finite monoids having only trivial subgroups. Information and Control, 8(2):190– 194, 1965. [36] Marcel Paul Sch¨utzenberger. Sur le produit de concat´enation non ambigu. Semigroup Forum, 13:47–75, 1976. [37] Imre Simon. Piecewise testable events. In Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages, pages 214–222. Springer, 1975. [38] Howard Straubing. A generalization of the sch¨utzenberger product of finite monoids. Theoretical Computer Science, 13(2):137–150, 1981. [39] Howard Straubing. Finite semigroup varieties of the form V ∗ D. Journal of Pure and Applied Algebra, 36:53–94, 1985. [40] Denis Th´erien. Classification of finite monoids: The language approach. Theoretical Computer Science, 14(2):195– 208, 1981. [41] Gabriel Thierrin. Permutation automata. Theory of Computing Systems, 2(1):83–90, 1968. [42] Wolfgang Thomas. Classifying regular events in symbolic logic. Journal of Computer and System Sciences, 25(3):360–376, 1982. [43] Denis Therien and Thomas Wilke. Over words, two variables are as powerful as one quantifier alternation. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, STOC’98, pages 234–240, New York, NY, USA, 1998. ACM. [44] Yechezkel Zalcstein. Locally testable languages. Journal of Computer and System Sciences, 6(2):151–167, 1972. 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.

Defining Temporal Hierarchies and How We Proved That Corresponding Hierarchies Are Always Strict

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 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. 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. 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 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. operator 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. 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.

What Are Temporal Hierarchies?

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 Abstract We classify the regular languages using an operator C 7→ TL(C). For each input class of languages C, it builds a larger class TL(C) consisting of all languages definable in a variant of unary temporal logic whose future/past modalities depend on C. This defines the temporal hierarchy of basis C: level n is built by applying this operator n times to C. This hierarchy is closely related to another one, the concatenation hierarchy of basis C. In particular, the union of all levels in both hierarchies is the same. We focus on bases G of group languages and natural extensions thereof, denoted G +. We prove that the temporal hierarchies of bases G and G + are strictly intertwined, and we compare them to the corresponding concatenation hierarchies. Furthermore, we look at two standard problems on classes of languages: membership (decide if an input language is in the class) and separation (decide, for two input regular languages L1, L2, if there is a language K in the class with L1 ⊆ K and L2 ∩ K = ∅). We prove that if separation is decidable for G, then so is membership for level two in the temporal hierarchies of bases G and G +. Moreover, we take a closer look at the case where G is the trivial class ST = {∅, A∗}. The levels one in the hierarchies of bases ST and ST+ are the standard variants of unary temporal logic while the levels two were considered recently using alternate definitions. We prove that for these two bases, level two has decidable separation. Combined with earlier results about the operator G 7→ TL(G), this implies that the levels three have decidable membership. 1. Introduction Context. This paper addresses a fundamental question in automata theory: understanding the formalisms used to describe regular languages. A flagship example is the class of star-free languages. These are languages that can be defined by a regular expression without Kleene star, but where complement is allowed. A striking feature of this class is its robustness. McNaughton and Papert [19] proved that star-free languages are those that can be described in first-order logic FO(<) and Kamp [11] proved that FO(<) has the same expressiveness as linear time temporal logic LTL. Also, Sch¨utzenberger [35] proved that membership is decidable for the star-free languages: given as input a regular language, one can decide if it is star-free. This led Brzozowski and Cohen [4] to define a natural classification of the star-free languages according to the difficulty of expressing them: the dot-depth hierarchy. This robustness justifies to investigate membership for individual levels. It is, however, a difficult problem. For example, in the dot-depth and Straubing-Therien hierarchies, it is known to be decidable only at levels one [37, 12] and two [27] despite decades of active research. 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 Abstract We classify the regular languages using an operator C 7→ TL(C). For each input class of languages C, it builds a larger class TL(C) consisting of all languages definable in a variant of unary temporal logic whose future/past modalities depend on C. This defines the temporal hierarchy of basis C: level n is built by applying this operator n times to C. This hierarchy is closely related to another one, the concatenation hierarchy of basis C. In particular, the union of all levels in both hierarchies is the same. We focus on bases G of group languages and natural extensions thereof, denoted G +. We prove that the temporal hierarchies of bases G and G + are strictly intertwined, and we compare them to the corresponding concatenation hierarchies. Furthermore, we look at two standard problems on classes of languages: membership (decide if an input language is in the class) and separation (decide, for two input regular languages L1, L2, if there is a language K in the class with L1 ⊆ K and L2 ∩ K = ∅). We prove that if separation is decidable for G, then so is membership for level two in the temporal hierarchies of bases G and G +. Moreover, we take a closer look at the case where G is the trivial class ST = {∅, A∗}. The levels one in the hierarchies of bases ST and ST+ are the standard variants of unary temporal logic while the levels two were considered recently using alternate definitions. We prove that for these two bases, level two has decidable separation. Combined with earlier results about the operator G 7→ TL(G), this implies that the levels three have decidable membership. group languages 1. Introduction Context . This paper addresses a fundamental question in automata theory: understanding the formalisms used to describe regular languages. A flagship example is the class of star-free languages. These are languages that can be defined by a regular expression without Kleene star, but where complement is allowed. A striking feature of this class is its robustness. McNaughton and Papert [19] proved that star-free languages are those that can be described in first-order logic FO(<) and Kamp [11] proved that FO(<) has the same expressiveness as linear time temporal logic LTL. Also, Sch¨utzenberger [35] proved that membership is decidable for the star-free languages: given as input a regular language, one can decide if it is star-free. This led Brzozowski and Cohen [4] to define a natural classification of the star-free languages according to the difficulty of expressing them: the dot-depth hierarchy. Context This robustness justifies to investigate membership for individual levels. It is, however, a difficult problem. For example, in the dot-depth and Straubing-Therien hierarchies, it is known to be decidable only at levels one [37, 12] and two [27] despite decades of active research. 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.

Temporal Hierarchies of Regular Languages: How We Classified Languages

Apia, Samoa, Jan. 23 –  Phemex is currently investigating a report involving one of its hot wallets used for daily operations. As part of the platform’s commitment to user security, withdrawals have been temporarily suspended to allow for a thorough inspection and the enhancement of wallet security measures. It is important to emphasize that Phemex’s cold wallets remain fully secure and unaffected. Cold wallets, which hold the majority of users’ funds, are transparent and verifiable. Users can independently confirm their status at: <https://phemex.com/proof-of-reserves](https://phemex.com/proof-of-reserves). The impact is limited and systems remain secure. Trading services continue uninterrupted, and core business operations remain unaffected. As a leading cryptocurrency exchange, Phemex has consistently demonstrated its commitment to security and transparency. By being the first to publish both proof-of-reserves and proof-of-solvency through a groundbreaking self-proving approach, Phemex has solidified its reputation as one of the most secure and trustworthy platforms in the industry. The temporary suspension of withdrawals is a precautionary measure to ensure the continued safety of user funds. Withdrawals will be reinstated promptly following the conclusion of the investigation and the implementation of all necessary security enhancements. To further address any inconvenience caused, Phemex is preparing a compensation plan, which will be announced shortly. Phemex remains steadfast in its mission to provide a seamless, secure, and trusted trading environment. Additional updates will be shared via official communication channels as the situation evolves. For further updates and information, please visit: <https://phemex.com](https://phemex.com). About Phemex Phemex is a leading cryptocurrency exchange that specializes in spot and derivatives trading. It features 350+ USDT-margined contract trading pairs with up to 100x leverage that all support Hedge Mode, alongside 400+ popular spot trading pairs. Individuals from all over the world can instantly buy, sell, and trade blockchain cryptocurrency through a user-friendly and secure platform. Phemex has released transparent Merkle-Tree Proof-of-Reserves so all users can verify on the blockchain that all funds are 100% backed on the platform. As the first exchange to publish both proof-of-reserves and proof-of-solvency through a unique, self-proving approach, Phemex stands out as one of the best and most trustworthy crypto exchanges. This story was distributed as a release by Btcwire under HackerNoon’s Business Blogging Program. Learn more about the program here Apia, Samoa, Jan. 23 –  Phemex is currently investigating a report involving one of its hot wallets used for daily operations. As part of the platform’s commitment to user security, withdrawals have been temporarily suspended to allow for a thorough inspection and the enhancement of wallet security measures. It is important to emphasize that Phemex’s cold wallets remain fully secure and unaffected. Cold wallets, which hold the majority of users’ funds, are transparent and verifiable. Users can independently confirm their status at: <https://phemex.com/proof-of-reserves](https://phemex.com/proof-of-reserves) . <https://phemex.com/proof-of-reserves](https://phemex.com/proof-of-reserves) The impact is limited and systems remain secure. Trading services continue uninterrupted, and core business operations remain unaffected. As a leading cryptocurrency exchange, Phemex has consistently demonstrated its commitment to security and transparency. By being the first to publish both proof-of-reserves and proof-of-solvency through a groundbreaking self-proving approach, Phemex has solidified its reputation as one of the most secure and trustworthy platforms in the industry. The temporary suspension of withdrawals is a precautionary measure to ensure the continued safety of user funds. Withdrawals will be reinstated promptly following the conclusion of the investigation and the implementation of all necessary security enhancements. To further address any inconvenience caused, Phemex is preparing a compensation plan, which will be announced shortly. Phemex remains steadfast in its mission to provide a seamless, secure, and trusted trading environment. Additional updates will be shared via official communication channels as the situation evolves. For further updates and information, please visit: <https://phemex.com](https://phemex.com) . <https://phemex.com](https://phemex.com) About Phemex Phemex is a leading cryptocurrency exchange that specializes in spot and derivatives trading. It features 350+ USDT-margined contract trading pairs with up to 100x leverage that all support Hedge Mode, alongside 400+ popular spot trading pairs. Individuals from all over the world can instantly buy, sell, and trade blockchain cryptocurrency through a user-friendly and secure platform. Phemex has released transparent Merkle-Tree Proof-of-Reserves so all users can verify on the blockchain that all funds are 100% backed on the platform. As the first exchange to publish both proof-of-reserves and proof-of-solvency through a unique, self-proving approach, Phemex stands out as one of the best and most trustworthy crypto exchanges. This story was distributed as a release by Btcwire under HackerNoon’s Business Blogging Program. Learn more about the program here This story was distributed as a release by Btcwire under HackerNoon’s Business Blogging Program. Learn more about the program here here here

Phemex Update On wallet Security And Temporary Suspension Of Withdrawals

Actualització de Phemex sobre la seguretat de la cartera i la suspensió temporal de retirades

Phemex-opdatering om pungsikkerhed og midlertidig suspendering af udbetalinger

Phemex-Update zur Wallet-Sicherheit und zur vorübergehenden Aussetzung von Abhebungen

Actualización de Phemex sobre la seguridad de la billetera y la suspensión temporal de los retiros

Phhemex eguneratzea zorroaren segurtasunari eta erretiratzeak aldi baterako etetea

Phemex-ի թարմացում դրամապանակի անվտանգության և կանխիկացման ժամանակավոր կասեցման վերաբերյալ

Phemex ウォレットのセキュリティと出金の一時停止に関する最新情報

Atualização da Phemex sobre segurança de carteira e suspensão temporária de retiradas

Gapjyk howpsuzlygy we yzyna alynmagy wagtlaýyn togtatmak barada Phemex täzelenmesi

Update ng Phemex Sa Seguridad ng pitaka At Pansamantalang Pagsuspinde Ng Mga Pag-withdraw

Phemex Cüzdan Güvenliği ve Para Çekme İşlemlerinin Geçici Olarak Askıya Alınması Hakkında Güncelleme

پرس کی حفاظت اور واپسی کی عارضی معطلی پر Phemex اپ ڈیٹ

DUBAI, United Arab Emirates, December 13th, 2024/Chainwire/--Bybit, the world’s second-largest cryptocurrency exchange by trading volume, continues to reinforce its commitment to transparency and regulatory compliance. In response to evolving regulations, Bybit has made the difficult but necessary decision to temporarily adjust the availability of its products and services within the European Economic Area (EEA). Paving the Way with MiCAR Compliance Bybit is actively pursuing a Markets in Crypto-Assets Regulation (MiCAR) license in Austria, a cornerstone of its compliance-first approach. This effort underscores Bybit’s dedication to aligning with stringent European regulatory standards, enhancing user protection, and delivering a secure trading environment. “Obtaining the MiCAR license will be a strategic milestone for Bybit,” said Ben Zhou, Co-founder and CEO of Bybit. “Our proactive stance ensures that we meet and exceed the expectations of both regulators and our users, as we work toward a future where innovation and compliance coexist seamlessly.” Temporary Adjustments to EEA Operations To ensure ongoing compliance with applicable regulatory laws, particularly regarding reverse solicitation, Bybit has made the difficult decision to generally cease all communication with the EEA region. This measure is intended to avoid any potential breach of the strict reverse solicitation principle. Existing customers' access to their crypto assets remains uninterrupted. Although this was a challenging decision, it was necessary for Bybit to maintain its compliance-first approach. Bybit is actively working towards obtaining a MiCAR license in Austria to become one of the first players in the EEA. Once the appropriate licensing is secured, Bybit will start engagement with its EEA clients in accordance with applicable laws. Balancing Innovation with Compliance “As cryptocurrency adoption continues to grow globally, it is Bybit's mission to deliver secure and reliable trading experiences underpinned by robust regulatory framework to all crypto communities," said Ben Zhou, Co-founder and CEO of Bybit. "To ensure we meet the highest standards and pave the way for a safer and more sustainable future for the industry, we've made the proactive decision to temporarily adjust our operations in the EEA. This allows us to focus on obtaining the necessary licenses and ensuring a fully compliant platform focused on serving EEA clients." Commitment to the EEA Market Bybit will be deeply committed to serving its EEA clients once it receives a MiCAR license. The company is actively engaging with regulatory authorities to expedite the licensing process and start full operations in the region. Bybit appreciates the support of its users as it navigates this pivotal regulatory journey. For questions or assistance, users are encouraged to contact Bybit’s Customer Support team via Live Chat. #Bybit / #TheCryptoArk About Bybit Bybit is the world’s second-largest cryptocurrency exchange by trading volume, serving over 50 million users. Established in 2018, Bybit provides a professional platform where crypto investors and traders can find an ultra-fast matching engine, 24/7 customer service, and multilingual community support. Bybit is a proud partner of Formula One’s reigning Constructors’ and Drivers’ champions: the Oracle Red Bull Racing team. For media inquiries, please contact: media@bybit.com Contact Head of PR Tony Au Bybit tony.au@bybit.com This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here DUBAI, United Arab Emirates, December 13th, 2024/Chainwire/--Bybit, the world’s second-largest cryptocurrency exchange by trading volume, continues to reinforce its commitment to transparency and regulatory compliance. In response to evolving regulations, Bybit has made the difficult but necessary decision to temporarily adjust the availability of its products and services within the European Economic Area (EEA). Paving the Way with MiCAR Compliance Bybit is actively pursuing a Markets in Crypto-Assets Regulation (MiCAR) license in Austria, a cornerstone of its compliance-first approach. This effort underscores Bybit’s dedication to aligning with stringent European regulatory standards, enhancing user protection, and delivering a secure trading environment. “Obtaining the MiCAR license will be a strategic milestone for Bybit,” said Ben Zhou, Co-founder and CEO of Bybit. “Our proactive stance ensures that we meet and exceed the expectations of both regulators and our users, as we work toward a future where innovation and compliance coexist seamlessly.” “Obtaining the MiCAR license will be a strategic milestone for Bybit,” said Ben Zhou, Co-founder and CEO of Bybit. “Our proactive stance ensures that we meet and exceed the expectations of both regulators and our users, as we work toward a future where innovation and compliance coexist seamlessly.” Temporary Adjustments to EEA Operations To ensure ongoing compliance with applicable regulatory laws, particularly regarding reverse solicitation, Bybit has made the difficult decision to generally cease all communication with the EEA region. This measure is intended to avoid any potential breach of the strict reverse solicitation principle. Existing customers' access to their crypto assets remains uninterrupted. Although this was a challenging decision, it was necessary for Bybit to maintain its compliance-first approach. Bybit is actively working towards obtaining a MiCAR license in Austria to become one of the first players in the EEA. Once the appropriate licensing is secured, Bybit will start engagement with its EEA clients in accordance with applicable laws. Balancing Innovation with Compliance “As cryptocurrency adoption continues to grow globally, it is Bybit's mission to deliver secure and reliable trading experiences underpinned by robust regulatory framework to all crypto communities," said Ben Zhou, Co-founder and CEO of Bybit. “As cryptocurrency adoption continues to grow globally, it is Bybit's mission to deliver secure and reliable trading experiences underpinned by robust regulatory framework to all crypto communities," said Ben Zhou, Co-founder and CEO of Bybit. "To ensure we meet the highest standards and pave the way for a safer and more sustainable future for the industry, we've made the proactive decision to temporarily adjust our operations in the EEA. This allows us to focus on obtaining the necessary licenses and ensuring a fully compliant platform focused on serving EEA clients." "To ensure we meet the highest standards and pave the way for a safer and more sustainable future for the industry, we've made the proactive decision to temporarily adjust our operations in the EEA. This allows us to focus on obtaining the necessary licenses and ensuring a fully compliant platform focused on serving EEA clients." Commitment to the EEA Market Bybit will be deeply committed to serving its EEA clients once it receives a MiCAR license. The company is actively engaging with regulatory authorities to expedite the licensing process and start full operations in the region. Bybit appreciates the support of its users as it navigates this pivotal regulatory journey. For questions or assistance, users are encouraged to contact Bybit’s Customer Support team via Live Chat. #Bybit / #TheCryptoArk About Bybit Bybit is the world’s second-largest cryptocurrency exchange by trading volume, serving over 50 million users. Established in 2018, Bybit provides a professional platform where crypto investors and traders can find an ultra-fast matching engine, 24/7 customer service, and multilingual community support. Bybit is a proud partner of Formula One’s reigning Constructors’ and Drivers’ champions: the Oracle Red Bull Racing team. For media inquiries, please contact: media@bybit.com Contact Head of PR Tony Au Bybit tony.au@bybit.com This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here here here

Bybit Advances Regulatory Compliance, Temporarily Adjusts EEA Operations

Bybit прасоўвае адпаведнасць нарматыўным патрабаванням, часова карэктуе дзейнасць ЕЭЗ

Bybit avanza en el cumplimiento normativo y ajusta temporalmente sus operaciones en el EEE

Bybit-ը բարելավում է կանոնակարգման համապատասխանությունը, ժամանակավորապես կարգավորում է ԵՏՀ գործառնությունները

Bybit、規制遵守を推進し、EEAの業務を一時的に調整

„Bybit“ pagerina teisės aktų laikymąsi, laikinai koreguoja EEE operacijas

Bybit uzlabo atbilstību normatīvajiem aktiem, īslaicīgi pielāgo EEZ darbības

Bybit зохицуулалтын нийцлийг сайжруулж, EEA-ийн үйл ажиллагааг түр зохицуулдаг

Bybit e Tšwetša Pele Kobamelo ya Taolo, e Beakanya Mešomo ya EEA ka Nakwana

Bybit zwiększa zgodność z przepisami, tymczasowo dostosowuje operacje w EOG

Bybit zlepšuje súlad s predpismi, dočasne upravuje operácie v EHP

Bybit Advances Regulatory Compliance, Nenguva Inogadzirisa EEA Operations

Bybit ส่งเสริมการปฏิบัติตามกฎระเบียบ ปรับการดำเนินงาน EEA ชั่วคราว

Author:
(1) David Staines. Table of Links Abstract 1 Introduction 2 Mathematical Arguments 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 5 Existing Solutions 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.3 Literature Comparison 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.3 Discussion 9 Bifurcation Analysis 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 11 Policy Rule 12 Conclusions and References Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.3 De Rham Cohomology F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.3 Trend Inflation Volatility C.2 Extending No Persistence This subsection generalizes lack of persistence in two directions. The first is to show that in the absence of price dispersion, it extends to all approximations from ZINSS. Similar results are well-known for Rotemberg. The second shows that even when the output gap is adjusted to allow technology shocks, inflation still responds to demand shocks only on impact. C.2.1 Higher Order Dynamics C.2.2 Dynamics with Unobserved Natural Rate Although no persistence breaks down, propagation problems still arise, in particular, with respect to inflation and aggregate demand shocks. The destruction of endogenous persistence when shock persistence takes limiting values is a theme common to the whole paper. This analysis has appeared before. The task at hand is to diagonalize (240) and substitute into the forward solution of (239) and then compute the persistence. First solve (241)-(243), this yields The relevant inverse matrix is using the forward solution and several rounds of substitution yields the following expression for inflation Deploying the persistent properties of the errors, communicated in Subsection 4.7, this yields It is intuitive that a positive supply shock drives down inflation. Away from the two limiting cases where it vanishes, as with Divine Coincidence, there is positive persistence, measured by the product moment correlation coefficient It is still pathological to Keynesian intuition that inflation does not respond to demand shocks. For output the expression is Again, the desired limiting cases arise. There can be overshooting or undershooting with respect to the efficient benchmark.110 However, the underlying message is that this system cannot generate persistent responses from inflation to demand shocks. This paper is available on arxiv under CC 4.0 license. Author: (1) David Staines. Author: Author: (1) David Staines. Table of Links Abstract Abstract 1 Introduction 1 Introduction 2 Mathematical Arguments 2 Mathematical Arguments 3 Outline and Preview 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.2 Preferences 4.3 Household Equilibrium Conditions 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 4.8 Recursive Equilibrium 5 Existing Solutions 5 Existing Solutions 5.1 Singular Phillips Curve 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.3 Two Comparison Models 5.4 Lucas Critique 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.2 Equilibrium Construction 6.3 Literature Comparison 6.3 Literature Comparison 6.4 Equilibrium Analysis 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.1 Slope Coefficients 7.2 Error Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.2 Key Proofs 8.3 Discussion 8.3 Discussion 9 Bifurcation Analysis 9 Bifurcation Analysis 9.1 Analytic Aspects 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.2 Econometric Duality 10.3 Coefficient Properties 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 10.4 Microeconomic Interpretation 11 Policy Rule 11 Policy Rule 12 Conclusions and References 12 Conclusions and References Appendices Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.2 Behaviour of ∆ A.3 Proof Part (iii) A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.2 Basic Categories F.3 De Rham Cohomology F.3 De Rham Cohomology F.4 Marginal Costs and Inflation F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.2 Lucas Critique I.3 Trend Inflation Volatility I.3 Trend Inflation Volatility C.2 Extending No Persistence This subsection generalizes lack of persistence in two directions. The first is to show that in the absence of price dispersion, it extends to all approximations from ZINSS. Similar results are well-known for Rotemberg. The second shows that even when the output gap is adjusted to allow technology shocks, inflation still responds to demand shocks only on impact. C.2.1 Higher Order Dynamics C.2.1 Higher Order Dynamics C.2.2 Dynamics with Unobserved Natural Rate C.2.2 Dynamics with Unobserved Natural Rate Although no persistence breaks down, propagation problems still arise, in particular, with respect to inflation and aggregate demand shocks. The destruction of endogenous persistence when shock persistence takes limiting values is a theme common to the whole paper. This analysis has appeared before. The task at hand is to diagonalize (240) and substitute into the forward solution of (239) and then compute the persistence. First solve (241)-(243), this yields The relevant inverse matrix is using the forward solution and several rounds of substitution yields the following expression for inflation Deploying the persistent properties of the errors, communicated in Subsection 4.7, this yields It is intuitive that a positive supply shock drives down inflation. Away from the two limiting cases where it vanishes, as with Divine Coincidence, there is positive persistence, measured by the product moment correlation coefficient It is still pathological to Keynesian intuition that inflation does not respond to demand shocks. For output the expression is Again, the desired limiting cases arise. There can be overshooting or undershooting with respect to the efficient benchmark.110 However, the underlying message is that this system cannot generate persistent responses from inflation to demand shocks. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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