Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Berechnung und programm. I

  • 57 Accesses

  • 44 Citations

Summary

Let A be an algebraic structure and assign to each operation of A a nonnegative real number as the performance time of the operation on a given computer. The notion of a computation (or straight line program) in A yields two functions from finite subsets of A to nonnegative real numbers, namely the computational length (or complexity), and the computational depth. We characterize these functions in a quasiaxiomatic way and prove a number of general results, which will be applied to concrete problems elsewhere (see [12]–[15]).

This is a preview of subscription content, log in to check access.

Literatur

  1. 1.

    Belaga, E. G.: Evaluation of polynomials of one variable with preliminary processing of the coefficients. Problemy Kibernetiki 5, 7–15 (1961).

  2. 2.

    Cohn, P. M.: Universal algebra. New York: Harper & Row 1965.

  3. 3.

    Engeler, E.: Formal languages. Chicago: Markham Publ. Co. 1968.

  4. 4.

    Engeler, E.: Introduction to the theory of computation. Part 2: Recursive functions. Lecture Notes 1970/71. Minneapolis: School of Mathematics, University of Minnesota.

  5. 5.

    Kerkhoff, R.: Gleichungsdefinierbare Klassen partieller Algebren. Math. Ann. 185, 112–133 (1970).

  6. 6.

    Knuth, D. E.: The art of computer programming. Vol. II: Seminumerical algorithms. Addison-Wesley 1969.

  7. 7.

    Lupanov, O. B.: Ueber die Synthese einiger Klassen von Steuersystemen. Probleme der Kybernetik 7, 20–63. Berlin: Akademie-Verlag 1966.

  8. 8.

    Mumford, D.: Introduction to algebraic geometry. Harvard Lecture Notes.

  9. 9.

    Neumann, B. H.: Special topics in algebra: Universal algebra. Lectures delivered in the Fall Semester 1961/62. New York: Courant Inst. of Math. Sci. 1962.

  10. 10.

    Ostrowski, A. M.: On two problems in abstract algebra connected with Horner's rule. Studies in Mathematics and Mechanics, 40–48. New York: Academic Press 1954.

  11. 11.

    Scholz, A.: Jahresbericht der deutschen Mathematiker-Vereinigung, class II, 47, 41–42 (1937).

  12. 12.

    Strassen, V.: Evaluation of rational functions. Complexity of Computer Computations, Plenum Press 1972.

  13. 13.

    Strassen, V.: Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten. Numerische Mathematik, Frühjahr 1973 (?).

  14. 14.

    Strassen, V.: Vermeidung von Divisionen. Crelle Journal für die reine und angew. Mathematik 1973 (?).

  15. 15.

    Strassen, V.: Berechnungen in partiellen Algebren endlichen Typs. Computing 1972/73 (?).

  16. 16.

    Winograd, S.: On the number of multiplications necessary to compute certain functions. Com. PA Math. XXIII/2, 165–179 (1970).

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Strassen, V. Berechnung und programm. I. Acta Informatica 1, 320–335 (1972). https://doi.org/10.1007/BF00289512

Download citation