Journal of Applied and Industrial Mathematics

, Volume 1, Issue 3, pp 351–360 | Cite as

Superpositions of elementary arithmetic functions

  • S. S. Marchenkov


A new concise proof of the following theorem is found: the system of four functions {x + y, xy, [x/y, 2 x } induces the class of Kalmar elementary functions. An elimination mode of bounded summation is used in the proof.


Industrial Mathematic Recursive Function Binary Representation Binary Digit Selector Function 
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  1. 1.
    N. K. Kosovskii, Fundamentals of the Theory of Elementary Algorithms (Izv. Leningrad. Gos. Univ., Leningrad, 1987) [in Russian].Google Scholar
  2. 2.
    A. I. Mal’cev, Algorithms and Recursive Functions (Nauka, Moscow, 1986; Wolters-Noordhoff, Groningen, 1970).Google Scholar
  3. 3.
    S. S. Marchenkov, “A Superposition Basis in the Class of Kalmar Elementary Functions,” Mat. Zametki 27(3), 321–332 (1980) [Math. Notes 27, 161–166 (1980)]Google Scholar
  4. 4.
    S. S. Marchenkov, “Some Simple Examples of Superposition Bases in the Classes of Kalmar Elementary Functions,” in Combinatorics and Graph Theory: Proceedings of the 30th Semester, Vol. 25 (Banach Cent. Publ., Warsaw, 1989), pp. 119–126.Google Scholar
  5. 5.
    S. S. Marchenkov, Superposition Bases in Classes of Recursive Functions (Nauka, Moscow, 1991) [in Russian].Google Scholar
  6. 6.
    S. S. Marchenkov, Elementary Recursive Functions (Moskov. Tsentr Nepreryvnogo Mat. Obrazovaniya, Moscow, 2003) [in Russian].Google Scholar
  7. 7.
    Yu. V. Matiyasevich, “A New Proof of the Theorem on Exponential Diophantine Representation of Enumerable Sets,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 60, 75–92 (1976) [J. Sov. Math. 14, 1475–1486 (1980)].Google Scholar
  8. 8.
    Yu. V. Matiyasevich, “A Class of Criteria for Primality in Terms of Divisibility of Binomial Coefficients,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 67, 167–183 (1977) [J. Sov. Math. 16, 874–885 (1981)].Google Scholar
  9. 9.
    M. Davis, H. Putnam, and J. Robinson, “The Decision Problem for Exponential Diophantine Equations,” Ann. Math. 74, 425–436 (1961).CrossRefGoogle Scholar
  10. 10.
    A. Grzegorczyk, “Some Classes of Recursive Functions,” in Rozprawy Matematiczne, Vol. 4 (1953) [in Problems of Mathematical Logic (Mir, Moscow, 1970), pp. 9–49].Google Scholar
  11. 11.
    L. Kalmar, “Ein einfach Beispiel für unentscheibares Problem,” Matematikai és fizikai lapok 50, 1–23 (1943).zbMATHGoogle Scholar
  12. 12.
    E. Kummer, “Über die Ergänzugssätze zu den allgemeinen Reciprocitätsgesetzen,” J. Reine Angew. Math. 44, 93–146 (1852).Google Scholar
  13. 13.
    S. Mazzanti, “Plain Bases for Classes of Primitive Recursive Functions,” Math. Logic Quart. 48(1), 93–104 (2002).zbMATHCrossRefGoogle Scholar
  14. 14.
    R. W. Ritchie, “Classes of Predictably Computable Functions,” Trans. Amer. Math. Soc. 106, 139–173 (1963) [in Problems Mathematical Logic (Mir, Moscow, 1970), pp. 50–93].zbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  1. 1.Department of Mathematical CyberneticsMoscow State UniversityVorob’ovy Gory, MoscowRussia

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