Universal Functions and KΣ-Structures


We introduce the concept of KΣ-structure and prove the existence of a universal Σ-function in the hereditarily finite superstructure over this structure. We exhibit some examples of families of KΣ-structures of the theory of trees.

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  1. 1.

    Ershov Yu. L., Definability and Computability [Russian], Nauchnaya Kniga, Novosibirsk; Èkonomika, Moscow (2000) (Siberian School of Algebra and Logic).

    Google Scholar 

  2. 2.

    Rogers H., Theory of Recursive Functions and Effective Computability, McGraw-Hill Book Comp., New York, St. Louis, San Francisco, Toronto, London, and Sydney (1967).

    Google Scholar 

  3. 3.

    Rudnev V. A., “A universal recursive function on admissible sets,” Algebra and Logic, vol. 25, no. 4, 267–273 (1986).

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ershov Yu. L., Puzarenko V. G., and Stukachev A. I., “\(\mathbb{HF}\)-computability,” in: Computability in Context, Imp. Coll. Press, London, 2011, 169–242 (Computation and Logic in the Real World).

    Google Scholar 

  5. 5.

    Morozov A. S. and Puzarenko V. G., “Σ-Subsets of natural numbers,” Algebra and Logic, vol. 43, no. 3, 162–178 (2004).

    MathSciNet  Article  Google Scholar 

  6. 6.

    Kalimullin I. Sh. and Puzarenko V. G., “Computable principles on admissible sets,” Siberian Adv. in Math., vol. 15, no. 4, 1–33 (2005).

    MathSciNet  Google Scholar 

  7. 7.

    Puzarenko V. G., “Computability in special models,” Sib. Math. J., vol. 46, no. 1, 148–165 (2005).

    MathSciNet  Article  Google Scholar 

  8. 8.

    Aleksandrova S. A., “The uniformization problem for Σ-predicates in a hereditarily finite list superstructure over the real exponential field,” Algebra and Logic, vol. 53, no. 1, 1–8 (2014).

    MathSciNet  Article  Google Scholar 

  9. 9.

    Korovina M. V., “On a universal recursive function and abstract machines on reals with list superstructure,” in: Structural Algorithmic Properties of Computability (Vychisl. Sistemy; No. 156) [Russian], Novosibirsk, 1996, 24–43.

  10. 10.

    Stukachev A. I., “The uniformization theorem in hereditary finite superstructures,” in: Generalized Computability and Definability (Vychisl. Sistemy; No. 161) [Russian], Izdat. IM SO RAN, Novosibirsk, 1998, 3–14.

    Google Scholar 

  11. 11.

    Khisamiev A. N., “On Σ-Subsets of naturals over abelian groups,” Sib. Math. J., vol. 47, no. 3, 574–583 (2006).

    MathSciNet  Article  Google Scholar 

  12. 12.

    Khisamiev A. N., “Σ-Bounded algebraic systems and universal functions. I,” Sib. Math. J., vol. 51, no. 1, 178–192 (2010).

    MathSciNet  Article  Google Scholar 

  13. 13.

    Khisamiev A. N., “Σ-Bounded algebraic systems and universal functions. II,” Sib. Math. J., vol. 51, no. 3, 537–551 (2010).

    MathSciNet  Article  Google Scholar 

  14. 14.

    Khisamiev A. N., “Σ-Uniform structures and Σ-functions. I,” Algebra and Logic, vol. 50, no. 5, 447–465 (2011).

    MathSciNet  Article  Google Scholar 

  15. 15.

    Khisamiev A. N., “Σ-Uniform structures and Σ-functions. II,” Algebra and Logic, vol. 51, no. 1, 89–102 (2012).

    MathSciNet  Article  Google Scholar 

  16. 16.

    Khisamiev A. N., “On a universal Σ-function over a tree,” Sib. Math. J., vol. 53, no. 3, 551–553 (2012).

    MathSciNet  Article  Google Scholar 

  17. 17.

    Khisamiev A. N., “Universal functions and almost c-simple models,” Sib. Math. J., vol. 56, no. 3, 526–540 (2015).

    MathSciNet  Article  Google Scholar 

  18. 18.

    Khisamiev A. N., “A class of almost c-simple rings,” Sib. Math. J., vol. 56, no. 6, 1133–1141 (2015).

    MathSciNet  Article  Google Scholar 

  19. 19.

    Khisamiev A. N., “Universal functions and unboundedly branching trees,” Algebra and Logic, vol. 57, no. 4, 309–319 (2018).

    MathSciNet  Article  Google Scholar 

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The author is grateful to S. S. Goncharov for stating the problem and giving useful advice.

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Correspondence to A. N. Khisamiev.

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Dedicated to the outstanding mathematician Yuri Leonidovich Ershov on the occasion of his 80th birthday.

Russian Text © The Author(s), 2020, published in Sibirskii Matematicheskii Zhurnal, 2020, Vol. 61, No. 3, pp. 703–716.

The author was funded within the Government Task to the Sobolev Institute of Mathematics (Project 0314-2019-0003).

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Khisamiev, A.N. Universal Functions and KΣ-Structures. Sib Math J 61, 552–562 (2020). https://doi.org/10.1134/S0037446620030192

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  • hereditarily finite superstructure
  • universal Σ-function
  • KΣ-structure
  • tree