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A syntactic approach to Borel functions: some extensions of Louveau’s theorem

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Abstract

Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class \(\Gamma \), then its \(\Gamma \)-code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau’s theorem to Borel functions: If a Borel function on a Polish space happens to be a \( \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t\)-function, then one can find its \( \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t\)-code hyperarithmetically relative to its Borel code. More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau’s theorem for Borel functions.

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References

  1. Block, A.C.: Operations on a Wadge-type hierarchy of ordinal-valued functions. Master’s Thesis, Universiteit van Amsterdam (2014)

  2. Brattka, V., Hertling, P. (eds.): Handbook of Computability and Complexity in Analysis. Theory and Applications of Computability, Springer, Cham (2021)

    MATH  Google Scholar 

  3. Callard, A., Hoyrup, M.: Descriptive complexity on non-polish spaces. In: Paul, C., Bläser, M., (eds.) 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020), volume 154 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 8:1–8:16, Dagstuhl, Germany, 2020. Schloss Dagstuhl–Leibniz-Zentrum für Informatik

  4. Camerlo, R.: Continuous reducibility: functions versus relations. Rep. Math. Logic 54, 45–63 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. de Brecht, M.: Quasi-Polish spaces. Ann. Pure Appl. Logic 164(3), 356–381 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. de Brecht, M., Pauly, A., Schröder, M.: Overt choice. Computability 9(3–4), 169–191 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Duparc, J.: Wadge hierarchy and Veblen hierarchy, part II: Borel sets of infinite rank. unpublished

  8. Duparc, J.: Wadge hierarchy and Veblen hierarchy. I. Borel sets of finite rank. J. Symb. Logic 66(1), 56–86 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duparc, J.: The Steel hierarchy of ordinal valued Borel mappings. J. Symb. Logic 68(1), 187–234 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fujita, H., Mátrai, T.: On the difference property of Borel measurable functions. Fundam. Math. 208(1), 57–73 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gregoriades, V., Kihara, T., Ng, K.M.: Turing degrees in Polish spaces and decomposability of Borel functions. J. Math. Log. 21(1), 2050021, 41 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hertling, P.: Topologische Komplexitätsgrade von Funktionen mit endlichem Bild. PhD Thesis, Fernuniversität Hagen (1993)

  13. Hertling, P.: Forests describing Wadge degrees and topological Weihrauch degrees of certain classes of functions and relations. Computability 9(3–4), 249–307 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hoyrup, M., Rojas, C., Selivanov, V., Stull, D.M.: Computability on quasi-Polish spaces. In: Descriptional complexity of formal systems, volume 11612 of Lecture Notes in Comput. Sci., pp. 171–183. Springer, Cham (2019)

  15. Ikegami, D., Schlicht, P., Tanaka, H.: Borel subsets of the real line and continuous reducibility. Fundam. Math. 244(3), 209–241 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kechris, A.S.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer, New York (1995)

    MATH  Google Scholar 

  17. Kihara, T.: Decomposing Borel functions using the Shore-Slaman join theorem. Fundam. Math. 230(1), 1–13 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kihara, T.: Borel-piecewise continuous reducibility for uniformization problems. Log. Methods Comput. Sci. 12(4), 35 (2016). (Paper No. 4)

    MathSciNet  MATH  Google Scholar 

  19. Kihara, T., Montalbán, A.: On the structure of the Wadge degrees of BQO-valued Borel functions. Trans. Am. Math. Soc. 371(11), 7885–7923 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kihara, T., Selivanov, V.: Wadge-like degrees of Borel BQO-valued functions. Proc. Am. Math. Soc. 150(9), 3989–4003 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  21. Louveau, A.: A separation theorem for \(\Sigma ^{1}_{1}\) sets. Trans. Am. Math. Soc. 260(2), 363–378 (1980)

    MATH  Google Scholar 

  22. Louveau, A.: Some results in the Wadge hierarchy of Borel sets. In: Cabal seminar 79–81, volume 1019 of Lecture Notes in Math., pp. 28–55. Springer, Berlin (1983)

  23. Louveau, A.: Saint-Raymond Jean. The strength of Borel Wadge determinacy. In: Cabal Seminar 81–85, volume 1333 of Lecture Notes in Math., pp. 1–30. Springer, Berlin (1988)

  24. Marks, A., Slaman, T.A., Steel, J.R.: Martin’s conjecture, arithmetic equivalence, and countable Borel equivalence relations. In: Ordinal definability and recursion theory: The Cabal Seminar. Vol. III, volume 43 of Lect. Notes Log., pp. 493–519. Assoc. Symbol. Logic, Ithaca (2016)

  25. Moschovakis, Y.N.: Descriptive set theory, volume 155 of Mathematical Surveys and Monographs, 2nd edn. American Mathematical Society, Providence (2009)

  26. Odifreddi, P.: Theory, Classical Recursion, volume 125 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam. The theory of functions and sets of natural numbers. With a foreword by G. E. Sacks (1989)

  27. Pequignot, Y.: A Wadge hierarchy for second countable spaces. Arch. Math. Logic 54(5–6), 659–683 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sacks, G.E.: Higher recursion theory, Perspectives in Mathematical Logic. Springer, Berlin (1990)

  29. Saint Raymond, J.: Preservation of the Borel class under countable-compact-covering mappings. Topol. Appl. 154(8), 1714–1725 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Schlicht, P.: Continuous reducibility and dimension of metric spaces. Arch. Math. Logic 57(3–4), 329–359 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Schröder, M.: Extended admissibility. Theoret. Comput. Sci. 284(2), 519–538 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Selivanov, V.: A \(Q\)-Wadge hierarchy in quasi-Polish spaces. J. Symb. Log. 87(2), 732–757 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  33. Selivanov, V.L.: Fine hierarchies and Boolean terms. J. Symb. Logic 60(1), 289–317 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Selivanov, V.L.: Hierarchies of \(\Delta ^0_2\)-measurable \(k\)-partitions. MLQ Math. Log. Q. 53(4–5), 446–461 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Selivanov, V.L.: \(Q\)-Wadge degrees as free structures. Computability 9(3–4), 327–341 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  36. Solecki, S.: Decomposing Borel sets and functions and the structure of Baire class \(1\) functions. J. Am. Math. Soc. 11(3), 521–550 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  37. van Engelen, F., Miller A.W., Steel, J.: Rigid Borel sets and better quasi-order theory. In: Logic and combinatorics (Arcata, Calif., 1985), volume 65 of Contemp. Math., pp. 199–222. Amer. Math. Soc., Providence (1987)

  38. Wadge, W.W.: Reducibility and Determinateness on the Baire Space. ProQuest LLC, Ann Arbor, MI, (1983). Thesis (Ph.D.), University of California, Berkeley

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Acknowledgements

The authors are grateful to the anonymous referee for carefully reading the manuscript and providing valuable comments. Kihara’s research was partially supported by JSPS KAKENHI (Grant Numbers 22K03401 and 21H03392).

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  1. Department of Mathematical Informatics, Graduate School of Informatics, Nagoya University, Nagoya, 464-8601, Japan

    Takayuki Kihara & Kenta Sasaki

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  1. Takayuki Kihara
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  2. Kenta Sasaki
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Correspondence to Takayuki Kihara.

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Kihara, T., Sasaki, K. A syntactic approach to Borel functions: some extensions of Louveau’s theorem. Arch. Math. Logic 62, 1041–1082 (2023). https://doi.org/10.1007/s00153-023-00880-8

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