Turing Computable Embeddings, Computable Infinitary Equivalence, and Linear Orders

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10307)

Abstract

We study Turing computable embeddings for various classes of linear orders. The concept of a Turing computable embedding (or tc-embedding for short) was developed by Calvert, Cummins, Knight, and Miller as an effective counterpart for Borel embeddings. We are focused on tc-embeddings for classes equipped with computable infinitary\(\varSigma _{\alpha }\)equivalence, denoted by \(\sim ^c_{\alpha }\). In this paper, we isolate a natural subclass of linear orders, denoted by WMB, such that \((WMB,\cong )\) is not universal under tc-embeddings, but for any computable ordinal \(\alpha \ge 5\), \((WMB, \sim ^c_{\alpha })\) is universal under tc-embeddings. Informally speaking, WMB is not tc-universal, but it becomes tc-universal if one imposes some natural restrictions on the effective complexity of the syntax. We also give a complete syntactic characterization for classes \((K,\cong )\) that are Turing computably embeddable into some specific classes \((\mathcal {C},\cong )\) of well-orders. This extends the similar result of Knight, Miller, and Vanden Boom for the class of all finite linear orders \(\mathcal {C}_{fin}\).

Keywords

Turing computable embedding Linear order Ordinal Computable infinitary equivalence Computable structure 

References

  1. 1.
    Ash, C.J.: Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees. Trans. Am. Math. Soc. 298(2), 497–514 (1986). doi:10.1090/S0002-9947-1986-0860377-7 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Hirschfeldt, D.R., Khoussainov, B., Shore, R.A., Slinko, A.M.: Degree spectra and computable dimensions in algebraic structures. Ann. Pure Appl. Logic 115(1–3), 71–113 (2002). doi:10.1016/S0168-0072(01)00087-2 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Calvert, W., Cummins, D., Knight, J.F., Miller, S.: Comparing classes of finite structures. Algebra Logic 43(6), 374–392 (2004). doi:10.1023/B:ALLO.0000048827.30718.2c MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Fokina, E.B., Friedman, S.-D.: On \(\Sigma ^1_1\) equivalence relations over the natural numbers. Math. Log. Q. 58(1–2), 113–124 (2012). doi:10.1002/malq.201020063 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Harrison-Trainor, M., Melnikov, A., Miller, R., Montálban, A.: Computable functors and effective interpretability. J. Symbolic Logic (to appear). doi:10.1017/jsl.2016.12
  6. 6.
    Friedman, H., Stanley, L.: A Borel reducibility theory for classes of countable structures. J. Symbolic Logic 54(3), 894–914 (1989). doi:10.2307/2274750 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Knight, J.F., Miller, S., Vanden Boom, M.: Turing computable embeddings. J. Symbolic Logic 72(3), 901–918 (2007). doi:10.2178/jsl/1191333847 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chisholm, J., Knight, J.F., Miller, S.: Computable embeddings and strongly minimal theories. J. Symbolic Logic 72(3), 1031–1040 (2007). doi:10.2178/jsl/1191333854 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fokina, E., Knight, J.F., Melnikov, A., Quinn, S.M., Safranski, C.: Classes of Ulm type and coding rank-homogeneous trees in other structures. J. Symbolic Logic 76(3), 846–869 (2011). doi:10.2178/jsl/1309952523 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ocasio-González, V.A.: Turing computable embeddings and coding families of sets. In: Cooper, S.B., Dawar, A., Löwe, B. (eds.) CiE 2012. LNCS, vol. 7318, pp. 539–548. Springer, Heidelberg (2012). doi:10.1007/978-3-642-30870-3_54 CrossRefGoogle Scholar
  11. 11.
    Andrews, U., Dushenin, D.I., Hill, C., Knight, J.F., Melnikov, A.G.: Comparing classes of finite sums. Algebra Logic 54(6), 489–501 (2016). doi:10.1007/s10469-016-9368-7 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    VanDenDriessche, S.M.: Embedding computable infinitary equivalence into \(p\)-groups. Ph.D. thesis, University of Notre Dame (2013)Google Scholar
  13. 13.
    Wright, M.: Turing computable embeddings of equivalences other than isomorphism. Proc. Am. Math. Soc. 142, 1795–1811 (2014). doi:10.1090/S0002-9939-2014-11878-8 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Goncharov, S.S.: Groups with a finite number of constructivizations. Sov. Math. Dokl. 23, 58–61 (1981)MATHGoogle Scholar
  15. 15.
    Ash, C.J., Knight, J.F.: Computable structures and the hyperarithmetical hierarchy. In: Studies in Logic and the Foundations of Mathematics, vol. 144. Elsevier Science B.V., Amsterdam (2000)Google Scholar
  16. 16.
    Knight, J.F.: Using computability to measure complexity of algebraic structures and classes of structures. Lobachevskii J. Math. 35(4), 304–312 (2014). doi:10.1134/S1995080214040192 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ash, C.J., Knight, J.F.: Pairs of recursive structures. Ann. Pure Appl. Logic 46(3), 211–234 (1990). doi:10.1016/0168-0072(90)90004-L MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations