Foundational Analyses of Computation

  • Yuri Gurevich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)

Abstract

How can one possibly analyze computation in general? The task seems daunting if not impossible. There are too many different kinds of computation, and the notion of general computation seems too amorphous. As in quicksand, one needs a rescue point, a fulcrum. In computation analysis, a fulcrum is a particular viewpoint on computation that clarifies and simplifies things to the point that analysis becomes possible.

We review from that point of view the few foundational analyses of general computation in the literature: Turing’s analysis of human computations, Gandy’s analysis of mechanical computations, Kolmogorov’s analysis of bit-level computation, and our own analysis of computation on the arbitrary abstraction level.

Keywords

Turing Machine Sequential Algorithm Computational Logic Computable Number Abstract State Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blass, A., Gurevich, Y.: Algorithms: A quest for absolute definitions. In: Paun, G., et al. (eds.) Current Trends in Theoretical Computer Science, pp. 283–311. World Scientific (2004); also in: Olszewski, A. (ed.): Church’s Thesis After 70 Years Ontos Verlag, pp. 24–57. Ontos Verlag (2006)Google Scholar
  2. 2.
    Blass, A., Gurevich, Y.: Abstract state machines capture parallel algorithms. ACM Trans. on Computational Logic 4(4), 578–651 (2003); Correction and Extension, Same Journal 9(3), article 19 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blass, A., Gurevich, Y.: Ordinary interactive small-step algorithms. ACM Trans. Computational Logic 7(2), Part  I, 363–419 (2006); plus 8:3 , articles 15 and 16 (Parts II and III) (2007)Google Scholar
  4. 4.
    Blass, A., Gurevich, Y., Rosenzweig, D., Rossman, B.: Interactive small-step algorithms. Logical Methods in Computer Science 3(4) (2007); papers 3 and 4 (Part I and Part II)Google Scholar
  5. 5.
    Dershowitz, N., Gurevich, Y.: A natural axiomatization of computability and proof of Church’s thesis. Bull. of Symbolic Logic 14(3), 299–350 (2008)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Church, A.: An unsolvable problem of elementary number theory. American Journal of Mathematics 58, 345–363 (1936)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gandy, R.: Church’s thesis and principles for mechanisms. In: Barwise, J., et al. (eds.) The Kleene Symposium, pp. 123–148. North-Holland (1980)Google Scholar
  8. 8.
    Gandy, R.O., Yates, C.E.M. (eds.): Collected works of A.M. Turing: Mathematical logic. Elsevier (2001)Google Scholar
  9. 9.
    Göedel, K.: A philosophical error in Turing’s work. In: Feferman, S., et al. (eds.) Kurt Gödel: Collected Works, vol. II, p. 306. Oxford University Press (1990)Google Scholar
  10. 10.
    Gurevich, Y.: On Kolmogorov machines and related issues. Bull. of Euro. Assoc. for Theor. Computer Science 35, 71–82 (1988)Google Scholar
  11. 11.
    Gurevich, Y.: Evolving algebra 1993: Lipari guide. In: Börger, E. (ed.) Specification and Validation Methods, pp. 9–36. Oxford Univ. Press (1995)Google Scholar
  12. 12.
    Gurevich, Y.: Sequential abstract state machines capture sequential algorithms. ACM Trans. on Computational Logic 1(1), 77–111 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gurevich, Y.: What Is an Algorithm? In: Bielikova, M., et al. (eds.) SOFSEM 2012. LNCS, vol. 7147, pp. 31–42. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  14. 14.
    Kolmogorov, A.N.: On the concept of algorithm. Uspekhi Mat. Nauk 8(4), 175–176 (1953) (Russian)MATHGoogle Scholar
  15. 15.
    Kolmogorov, A.N., Uspensky, V.A.: On the definition of algorithm. Uspekhi Mat. Nauk 13(4), 3–28 (1958) (Russian); English translation in AMS Translations 29, 217–245 (1963)MATHGoogle Scholar
  16. 16.
    Levin, L.A.: Private communication (2003)Google Scholar
  17. 17.
    Markov, A.A.: Theory of algorithms. Trans. of the Steklov Institute of Mathematics 42 (1954) (Russian); English translation by the Israel Program for Scientific Translations, 1962; also by Kluwer (2010)Google Scholar
  18. 18.
    Shagrir, O.: Effective computation by humans and machines. Minds and Machines 12, 221–240 (2002)MATHCrossRefGoogle Scholar
  19. 19.
    Shagrir, O.: Göedel on Turing on computability. In: Olszewski, A., et al. (eds.) Church’s Thesis After 70 Years, pp. 393–419. Ontos-Verlag (2006)Google Scholar
  20. 20.
    Sieg, W.: On computability. In: Irvine, A. (ed.) Handbook of the Philosophy of Mathematics, pp. 535–630. Elsevier (2009)Google Scholar
  21. 21.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proceedings of London Mathematical Society 2(42), 230–265 (1936)Google Scholar
  22. 22.
    Uspensky, V.A.: Kolmogorov and mathematical logic. Journal of Symbolic Logic 57(2), 385–412 (1992)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Uspensky, V.A., Semenov, A.L.: Theory of algorithms: main discoveries and applications, Nauka (1987) (Russian), Kluwer (2010) (English)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yuri Gurevich
    • 1
  1. 1.Microsoft Research, One Microsoft WayRedmondUnited States of America

Personalised recommendations