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Well-Quasi Orders and Hierarchy Theory

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Book cover Well-Quasi Orders in Computation, Logic, Language and Reasoning

Part of the book series: Trends in Logic ((TREN,volume 53))

Abstract

We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory. While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities. We survey some results obtained so far and discuss open problems and possible research directions.

Supported by Russian Science Foundation, project 18-11-00100.

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References

  1. Abdulla, P. A., et al. (2004). Using forward reachability analysis for verification of lossy channel systems? Formal Methods in System Design, 25(1), 39–65.

    Article  Google Scholar 

  2. Addison, J. W. (1959). Separation principles in the hierarchies of classical and effective set theory. Fundamenta Mathematicae, 46, 123–135.

    Article  Google Scholar 

  3. Addison, J. W. (1962). Some problems in hierarchy theory. In Recursive function theory. Proceedings of symposia in pure mathematics (Vol. 5 , pp. 123–130). AMS.

    Google Scholar 

  4. Addison, J. W. (1962). The theory of hierarchies. In Logic, methodology and philosophy of science. Proceedings of the 1960 international congress, Palo Alto (pp. 26–37).

    Google Scholar 

  5. Addison, J. W. (1965). The method of alternating chains. In: The theory of models (pp. 1–16). Amsterdam, North Holland.

    Google Scholar 

  6. Abramsky, S., & Jung, A. (1994). Domain theory. In Handbook of logic in computer science (Vol. 3, pp. 1–168). Oxford.

    Google Scholar 

  7. Ash, C. J., & Knight, J. (2000). Computable structures and the hyperarithmetical hierarchy. Amsterdam: Elsevier Science.

    Google Scholar 

  8. Adams, S., & Kechris, A. S. (2000). Linear algebraic groups and countable Borel equivalence relations. Journal of the American Mathematical Society, 13(4), 909–943.

    Article  Google Scholar 

  9. Andretta, A. (2006). More on Wadge determinacy. Annals of Pure and Applied Logic, 144(1–3), 2–32.

    Article  Google Scholar 

  10. Andretta, A., & Martin, D. A. (2003). Borel-Wadge degrees. Fundamenta Mathematicae, 177(2), 175–192.

    Article  Google Scholar 

  11. Becker, H. (1988). A characterization of jump operators. Journal of Symbolic Logic, 53(3), 708–728.

    Article  Google Scholar 

  12. Büchi, J. R., & Landweber, L. H. (1969). Solving sequential conditions by finite-state strategies. Transactions of the American Mathematical Society, 138, 295–311.

    Article  Google Scholar 

  13. Brattka, V., & Gherardi, G. (2009). Weihrauch degrees, omniscience principles and weak computability. arXiv:0905.4679.

  14. Brattka, V., & Gherardi, G. (2009). Effective choice and boundedness principles in computable analysis. arXiv:0905.4685.

  15. Carroy, R. (2013). A quasiorder on continuous functions. Journal of Symbolic Logic, 78(2), 633–648.

    Article  Google Scholar 

  16. Cenzer, D. (1991). Polynomial-time versus recursive models. Annals of Pure and Applied Logic, 54(1), 17–58.

    Article  Google Scholar 

  17. de Brecht, M. (2013). Quasi-Polish spaces. Annals of Pure and Applied Logic, 164, 356–381.

    Article  Google Scholar 

  18. Downey, R. G., & Fellows, M. R. (1998). Parameterized complexity. New York: Springer.

    Google Scholar 

  19. Diekert, V., & Kufleitner, M. (2011). Fragments of first-order logic over infinite words. Theory of Computing Systems, 48(3), 486–516.

    Article  Google Scholar 

  20. de Jongh, D. H. J., & Parikh R. (1977). Well-partial orderings and hierarchies. In Indagationes mathematicae (Proceedings) (Vol. 39, No. 3, pp. 95–207).

    Google Scholar 

  21. Davey, B. A., & Priestley, H. A. (1994). Introduction to lattices and order. Cambridge:

    Google Scholar 

  22. Duparc, J. (2001). Wadge hierarchy and Veblen hierarchy, part I. Journal of Symbolic Logic, 66(1), 56–86.

    Article  Google Scholar 

  23. de Luca, A., & Varricchio, S. (1999). Finiteness and regularity in semigroups and formal languages. Berlin: Springer.

    Book  Google Scholar 

  24. Ehrenfeucht, A., Haussler, D., & Rozenberg, G. (1983). On regularity of context-free languages. Theoretical Computer Science, 27, 311–332.

    Article  Google Scholar 

  25. Engelking, R. (1989). General topology. Berlin: Heldermann.

    Google Scholar 

  26. van Engelen, F. (1986). Homogeneous zero-dimensional absolute Borel sets. Amsterdam: CWI Tracts.

    Google Scholar 

  27. van Engelen, F., Miller, A., & Steel, J. (1987). Rigid Borel sets and better quasiorder theory. Contemporary Mathematics, 65, 199–222.

    Article  Google Scholar 

  28. Ershov, Y. L. (1968). On a hierarchy of sets 3 (in Russian). Algebra i Logika, 7(4), 15–47.

    Google Scholar 

  29. Ershov, Y. L., & Goncharov, S. S. (1999). Constructive models. Novosibirsk: Scientific Book (in Russian, there is an English Translation).

    Google Scholar 

  30. Ershov, Y. L., Lavrov, I. A., Taimanov, A. D., & Taitslin, M. A. (1965). Elementary theories. Uspekhi Matematicheskikh Nauk, 20(4), 37–108. (in Russian).

    Google Scholar 

  31. Gao, S. (2009). Invariant descriptive set theory. Pure and applied mathematics: A series of monographs and textbooks (p. 293). Taylor & Francis Group.

    Google Scholar 

  32. Glaßer amd, C. & Schmitz, H., (2001). The Boolean structure of dot-depth one. Journal of Automata, Languages and Combinatorics, 6, 437–452.

    Google Scholar 

  33. Glasser, C., Schmitz, H., & Selivanov, V. (2016). Efficient algorithms for membership in Boolean hierarchies of regular languages. Theoretical Computer Science, 646(C), 86–108. https://doi.org/10.1016/j.tcs.2016.07.017. ISSN 0304-3975.

  34. Hertling, P. (1993, December). Topologische Komplexitätsgrade von Funktionen mit endlichem Bild. Informatik-Berichte 152, 34 pages, Fernuniversität Hagen.

    Google Scholar 

  35. Hertling, P. (1996). Unstetigkeitsgrade von Funktionen in der effektiven Analysis. Ph.D. thesis, Fachbereich Informatik, FernUniversität Hagen.

    Google Scholar 

  36. Hertling, P., & Selivanov, V. (2014). Complexity issues for preorders on finite labeled forests. In V. Brattka, H. Diener, & D. Spreen (Eds.), Logic, computation, hierarchies (pp. 165–190). Boston, Berlin: Ontos Publishing, de Gruiter.

    Google Scholar 

  37. Hertling, P., & Weihrauch, K. (1994). Levels of degeneracy and exact lower complexity bounds for geometric algorithms. In Proceedings of the 6th Canadian Conference on Computational Geometry, Saskatoon (pp. 237–224).

    Google Scholar 

  38. Higman, G. (1952). Ordering by divisibility in abstract algebras. Proceedings of the London Mathematical Society, 2(3), 326–336.

    Article  Google Scholar 

  39. Hirschfeldt, D. R. (2014). Slicing the truth—On the computable and reverse mathematics of combinatorial principles. Lecture notes series (Vol. 28). Institute for Mathematical Sciences, National University of Singapore: World Scientific.

    Google Scholar 

  40. Hodges, W. (1993). Model theory. Cambridge University Press.

    Google Scholar 

  41. Hurewicz, W., & Wallman, H. (1948). Dimension theory. Princeton mathematical series (Vol. 4). Princeton, NJ: Princeton University Press.

    Google Scholar 

  42. Ikegami, D. (2010). Games in set theory and logic. Ph.D. thesis, University of Amsterdam.

    Google Scholar 

  43. Ikegami, D., Schlicht, P., & Tanaka, H. (2012). Continuous reducibility for the real line, preprint, submitted. Appeared as: Borel subsets of the real line and continuous reducibility, Fundamenta Mathematicae, 244(3), 209–241.

    Google Scholar 

  44. Jezek, J., & McKenzie, R. (2009). Definability in substructure orderings, IV: Finite lattices. Algebra Universalis, 61(3–4), 301–312.

    Article  Google Scholar 

  45. Jezek, J., & McKenzie, R. (2009). Definability in substructure orderings, I: Fifinite semilattices. Algebra Universalis, 61(1), 59–75.

    Article  Google Scholar 

  46. Jezek, J., & McKenzie, R. (2009). Definability in substructure orderings, III: Finite distributive lattices. Algebra Universalis, 61(3–4), 283–300.

    Article  Google Scholar 

  47. Jezek, J., & McKenzie, R. (2010). Definability in substructure orderings, II: Finite ordered sets. Order, 27(2), 115–145.

    Article  Google Scholar 

  48. Kunos, A. (2015). Definability in the embeddability ordering of finite directed graphs. Order, 32(1), 117–133.

    Article  Google Scholar 

  49. Kruskal, J. B. (1960). Well-quasi-ordering, the tree theorem, and Vazsonyis conjecture. Transactions of the American Mathematical Society, 95, 210–225.

    Google Scholar 

  50. Karandikar, P., Niewerth, M., & Schnoebelen, Ph. (2016). On the state complexity of closures and interiors of regular languages with subwords and superwords. Theoretical Computer Science, 610, 91–107.

    Article  Google Scholar 

  51. Karandikar, P., & Schnoebelen, Ph. (2015). Decidability in the logic of subsequences and supersequences. In Proceedings of FST&TCS 2015, LIPIcs 45 (pp. 84–97). Leibniz-Zentrum für Informatik.

    Google Scholar 

  52. Karandikar, P., & Schnoebelen, Ph. (2015). Generalized post embedding problems. Theory of Computing Systems, 56(4), 697–716.

    Article  Google Scholar 

  53. Karandikar, P., & Schnoebelen, Ph. (2016). The height of piecewise-testable languages with applications in logical complexity. In Proceeding of CSL 2016, LIPIcs62 (37:1–37:22). Leibniz-Zentrum für Informatik.

    Google Scholar 

  54. Kudinov, O. V., & Selivanov, V. L. (2006). Undecidability in the homomorphic quasiorder of finite labeled forests. In Proceedings of CiE 2006. Lecture notes in computer science (Vol. 3988, pp. 289–296). Berlin: Springer.

    Google Scholar 

  55. Kudinov, O. V., & Selivanov, V. L. (2007). Definability in the homomorphic quasiorder of finite labeled forests. In Proceedings of CiE-2007. Lecture notes in computer science (Vol. 4497, 436–445). Berlin: Springer.

    Google Scholar 

  56. Kudinov, O. V., & Selivanov, V. L. (2007). Undecidability in the homomorphic quasiorder of finite labelled forests. Journal of Logic and Computation, 17, 113–1151.

    Article  Google Scholar 

  57. Kudinov, O. V., Selivanov, V. L. (2009). A Gandy theorem for abstract structures and applications to first order definability. In Proceeding of CiE-2009. LNCS 5635 (pp. 290–299). Berlin: Springer.

    Google Scholar 

  58. Kudinov, O., & Selivanov, V. (2009). Definability in the infix order on words. In V. Diekert & D. Nowotka (Eds.), Proceeding of DLT-2009 (Vol. 5583, pp. 454–465). Lecture notes in computer science. Berlin: Springer.

    Google Scholar 

  59. Kudinov, O. V., Selivanov, V. L., & Yartseva, L. V. (2010). Definability in the subwordx order. In F. Ferreira, B. Loewe, E. Mayordomo and L. M. Gomes (Eds.), Proceeding of CiE-2010. Lecture notes in computer science (Vol. 6158, pp. 246–255). Berlin: Springer.

    Google Scholar 

  60. Kudinov, O. V., Selivanov, V. L., & Zhukov, A. V. (2009). Definability in the h-quasiorder of labeled forests. Annals of Pure and Applied Logic, 159(3), 318–332.

    Article  Google Scholar 

  61. Kuske, D. (2006). Theories of orders on the set of words. Theoretical Informatics and Applications, 40, 53–74.

    Article  Google Scholar 

  62. Kuratowski, K., & Mostowski, A. (1967). Set theory. North Holland.

    Google Scholar 

  63. Kihara, T., & Montalban, A. (17 Aug 2016). The uniform Martin’s conjecture for many-one degrees. arXiv:1608.05065 v1 [Math.LO] . Accepted by Transactions of AMS. Appeared in: Transactions of the American Mathematical Social, 370(12), 9025–9044, 2018.

  64. Kihara. T., & Montalban, A. (22 May 2017). On the structure of the Wadge degrees of BQO-valued Borel functions. arXiv:1705.07802 v1 [Math.LO]. Accepted by Transactions of AMS. Appeared in: Transactions of the American Mathematical Social, 371(11):7885–7923, 2019.

  65. Kechris, A. S. (1995). Classical Descriptive Set Theory. New York: Springer.

    Book  Google Scholar 

  66. Laver, R. (1971). On Fraïssés order type conjecture. Ann. of Math., 93(2), 89–111.

    Article  Google Scholar 

  67. Louveau, A. (1983). Some results in the Wadge hierarchy of Borel sets. Lecture Notes in Mathematics (No. 1019, pp. 28–55).

    Google Scholar 

  68. Marks, A. S. (13 Jan 2016). The universality of polynomial time Turing equivalence. arXiv:1601.033431 [math LO].

  69. Marks, A. S., Slaman, T., & Steel, J. Martins conjecture, arithmetic equivalence and countable Borel equivalence relations. Appeared in: (2016) In Ordinal Definability and Recursion Theory: The Cabal Seminar. Lecture notes in logic (Vol. 3, No.43, pp. 493–520). Cambridge University Press, Cambridge.

    Google Scholar 

  70. Montalban, A. (2007). Computable linearizations of well-partial-orderings. Order, 24, 39–48. https://doi.org/10.1007/s11083-007-9058-0.

    Article  Google Scholar 

  71. Motto Ros, L. (2009). Borel-amenable reducibilities for sets of reals. Journal of Symbolic Logic, 74(1), 27–49.

    Article  Google Scholar 

  72. Motto Ros, L., Schlicht, P., & Selivanov, V. (2015). Wadge-like reducibilities on arbitrary quasi-Polish spaces. Mathematical Structures in Computer Science, 25(8), 1705–1754. https://doi.org/10.1017/S0960129513000339.

    Article  Google Scholar 

  73. Nash-Williams, C. St. J. A. (1965). On well-quasi-ordering infinite trees. Mathematical proceedings of the Cambridge philosophical society (Vol. 61, pp. 697–720). Cambridge University Press.

    Google Scholar 

  74. Nash-Williams, C. St. J. A. (1968). On better-quasi-ordering transfinite sequences. In Mathematical proceedings of the Cambridge philosophical society (Vol. 64, p. 273290). Cambridge University Press.

    Google Scholar 

  75. Nies, A. (2000). Definability in the c.e. degrees: Questions and results. Contemporary Mathematics, 257, 207–213.

    Article  Google Scholar 

  76. Ogawa, M. (2004). Well quasiorders and regular \(\omega \)-languages. Theoretical Computer Science, 324(1), 55–60.

    Article  Google Scholar 

  77. Pequignot, Y. (2015). A Wadge hierarchy for second countable spaces. Archive for Mathematical Logic, 54(5–6), 659683. https://doi.org/10.1007/s00153-015-0434-y.

    Article  Google Scholar 

  78. Perrin, D., & Pin, J.-E. (2004). Infinite words. Volume 141 of pure and applied mathematics.

    Google Scholar 

  79. Rogers, H, Jr. (1967). Theory of recursive functions and effective computability. New York: McGraw-Hill.

    Google Scholar 

  80. Robertson, N., Seymour, P. D. (2004). Graph minors. XX. Wagners conjecture. Journal of Combinatorial Theory, Series B, 92(2), 325–357.

    Google Scholar 

  81. Ramanujam, R., & Thinniyam, R. S. (2016). Definability in first order theories of graph orderings. In Logical foundations of computer science (pp. 31–348). Springer.

    Google Scholar 

  82. Schmidt, D. (1979). Well-partial orderings and their maximal order types. Habilitationsschrift: University of Heidelberg.

    Google Scholar 

  83. Soare, R. I. (1987) Recursively enumerable sets and degrees. Perspectives in mathematical Logic. A study of computable functions and computably generated sets. Berlin: Springer.

    Google Scholar 

  84. Selivanov, V. L. (1982). On the structure of degrees of generalized index sets. Algebra and Logic, 21, 316–330.

    Article  Google Scholar 

  85. Selivanov, V. L. (1983). Hierarchies of hyperarithmetical sets and functions. Algebra and Logic, 22, 473–491.

    Article  Google Scholar 

  86. Selivanov, V. L. (1988). On algorithmic complexity of algebraic systems. Mathematical Notes, 44(5–6), 944–950.

    Article  Google Scholar 

  87. Selivanov, V. L. (1992). Hierarchies, numberings, index sets. Handwritten notes (300 pp.).

    Google Scholar 

  88. Selivanov, V. L. (1995). Fine hierarchies and Boolean terms. Journal of Symbolic Logic, 60(1), 289–317.

    Article  Google Scholar 

  89. Selivanov, V. L. (1998). Fine hierarchy of regular \(\omega \)-languages. Theoretical Computer Science, 191, 37–59.

    Article  Google Scholar 

  90. Selivanov, V. L. (2010). A logical approach to decidability of hierarchies of regular star-free languages. Lecture notes in computer science (Vol. 2010, pp. 539–550). Berlin: Springer.

    Google Scholar 

  91. Selivanov, V. L. (2005). Variations on the Wadge reducibility. Siberian Advances in Mathematics, 15(3), 44–80.

    Google Scholar 

  92. Selivanov, V. (2006). Towards a descriptive set theory for domain-like structures. Theoretical Computer Science, 365, 258–282.

    Article  Google Scholar 

  93. Selivanov, V. L. (2007). The quotient algebra of labeled forests modulo h-equivalence. Algebra and Logic, 46(2), 120–133.

    Article  Google Scholar 

  94. Selivanov, V. L. (2007). Hierarchies of \({\mathbf{\Delta }}^0_2\)-measurable \(k\)-partitions. Mathematical Logic Quarterly, 53, 446–461.

    Article  Google Scholar 

  95. Selivanov, V. L. (2008). Fine hierarchies and \(m\)-reducibilities in theoretical computer science. Theoretical Computer Science, 405, 116–163.

    Article  Google Scholar 

  96. Selivanov, V. L. (2009). Hierarchies and reducibilities on regular languages related to modulo counting. RAIRO Theoretical Informatics and Applications, 41, 95–132.

    Google Scholar 

  97. Selivanov, V. L., (2011). A fine hierarchy of \(\omega \)-regular \(k\)-partitions. In B. Löwe (Eds.), Proceeding of CiE, et al. (2011). LNCS 6735 (pp. 260–269). Heidelberg: Springer.

    Google Scholar 

  98. Selivanov, V. L. (2012). Fine hierarchies via Priestley duality. Annals of Pure and Applied Logic, 163, 1075–1107.

    Article  Google Scholar 

  99. Selivanov, V. (2013). Total representations, logical methods in computer. Science, 9(2), 1–30. https://doi.org/10.2168/LMCS-9(2:5)2013.

    Article  Google Scholar 

  100. Selivanov, V. L. (2017).Towards a descriptive theory of cb0-spaces. Mathematical Structures in Computer Science, 28(8), 1553–1580. https://doi.org/10.1017/S0960129516000177. Earlier version in: ArXiv:1406.3942v1 [Math.GN] 16 June 2014.

  101. Selivanov, V. L., Wadge, Extending, & theory to k-partitions. In J. Kari, F. Manea, & Ion Petre (Eds.), Proceeding of CiE,. (2017). LNCS 10307 (pp. 387–399). Berlin: Springer.

    Google Scholar 

  102. Schlicht, P. (2017). Continuous reducibility and dimension of metric spaces. arXiv:1703.10144.

  103. Simpson, S.G. (1985). Bqo-theory and Fraïssé conjecture. Chapter 9, Mansfield, R., & Weitkamp, G. (1985). Recursive aspects of descriptive set theory. New York: Oxford University Press.

    Google Scholar 

  104. Slaman, T. A., & Steel, J. R. (1988). Definable functions on degrees. In Cabal seminar 81–85. Lecture notes in mathematics (Vol. 1333, pp. 37–55). Berlin: Springer.

    Google Scholar 

  105. Saint Raymond, J. (2007). Preservation of the Borel class under countable-compact-covering mappings. Topology and Its Applications, 154, 1714–1725.

    Article  Google Scholar 

  106. Stern, J. (1985). Characterizations of some classes of regular events. Theoretical Computer Science, 35, 163–176.

    Article  Google Scholar 

  107. Straubing, H. (1994). Finite automata, formal logic and circuit complexity. Boston: Birkhäuser.

    Book  Google Scholar 

  108. Steel, J. R. (1982). A classification of jump operators. Journal of Symbolic Logic, 47(2), 347358.

    Article  Google Scholar 

  109. Steel, J. (1980). Determinateness and the separation property. Journal of Symbolic Logic, 45, 143–146.

    Google Scholar 

  110. Selivanov, V. L., & Wagner, K. W. (2005). A reducibility for the dot-depth hierarchy. Theoretical Computer Science, 345(2–3), 448–472.

    Article  Google Scholar 

  111. Trakhtenbrot, B. A., & Barzdin, J. M. (1970). Finite automata. Behavior and synthesis (p. 1973). Moscow: Mir (Russian, English translation: North Holland, Amsterdam).

    Google Scholar 

  112. Thomas, W. (1990). Automata on infinite objects. Handbook of theoretical computer science (Vo. B, pp. 133–191).

    Google Scholar 

  113. Thomas, W. (1996). Languages, automata and logic. Handbook of formal language theory (Vol. B, pp. 133–191).

    Google Scholar 

  114. Tarski, A., Mostowski, A., & Robinson, J. (1953). Undecidable Theories. Amsterdam: North Holland.

    Google Scholar 

  115. Van Wesep, R. (1976). Wadge degrees and descriptive set theory. Lecture Notes in Mathematics, 689, 151–170.

    Article  Google Scholar 

  116. Wadge, W. (1984). Reducibility and determinateness in the Baire space. Ph.D. thesis, University of California, Berkely.

    Google Scholar 

  117. Wagner, K. (1979). On \(\omega \)-regular sets. Information and Control, 43, 123–177.

    Article  Google Scholar 

  118. Weihrauch, K. (1992). The degrees of discontinuity of some translators between representations of the real numbers. Technical Report TR-92-050. Berkeley: International Computer Science Institute.

    Google Scholar 

  119. Weihrauch, K. (2000). Computable analysis. Berlin: Springer.

    Book  Google Scholar 

  120. Wires, A. (2012). Definability in the substructure ordering of simple graphs. http://www.math.uwaterloo.ca/awires/SimpleGraphs.pdf.

  121. Wires, A. (2016). Definability in the substructure ordering of simple graphs. Annals of Combinatorics, 20(1), 139–176.

    Article  Google Scholar 

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Acknowledgements

I am grateful to Leibniz-Zentrum für Informatik for accepting and taking care of Dagstuhl Seminars 08271, 11411, 15392, 16031 which were important for promoting the topic of this paper.

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  1. A.P. Ershov Institute of Informatics Systems SB RAS and Kazan Federal University, Lavrentyev av. 6, Novosibirsk, 630090, Russia

    Victor Selivanov

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  1. Dipartimento di Informatica, Università degli Studi di Verona, Verona, Italy

    Peter M. Schuster

  2. Department of Computer Science, Swansea University, Swansea, UK

    Monika Seisenberger

  3. Vakgroep Wiskunde, Ghent University, Ghent, Belgium

    Andreas Weiermann

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Selivanov, V. (2020). Well-Quasi Orders and Hierarchy Theory. In: Schuster, P., Seisenberger, M., Weiermann, A. (eds) Well-Quasi Orders in Computation, Logic, Language and Reasoning. Trends in Logic, vol 53. Springer, Cham. https://doi.org/10.1007/978-3-030-30229-0_10

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