Advertisement

A Précis of Classical Computability Theory

  • Apostolos Syropoulos
Chapter
Part of the IFSR International Series on Systems Science and Engineering book series (IFSR, volume 31)

Abstract

Turing machines form the core of computability theory, or recursion theory as it is also known. This chapter introduces basic notions and results and readers already familiar with them can safely skip it. The exposition is based on standard references [18, 39, 67, 83, 109]. In the discussion that follows, the symbol ℕ will stand for the set of positive integer numbers including zero and will stand for the set of rational numbers.

References

  1. [18]
    Boolos, G.S., Burgess, J.P., Jeffrey, R.C.: Computability and Logic, 4th edn. Cambridge University Press, Cambridge (2002)CrossRefMATHGoogle Scholar
  2. [36]
    Cook, S.A., Aanderaa, S.O.: On the minimum computation time of functions. Trans. Am. Math. Soc. 142, 291–314 (1969)MathSciNetCrossRefMATHGoogle Scholar
  3. [37]
    Copeland, B.J.: The Church-Turing thesis. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Fall 2002. http://plato.stanford.edu/archives/fall2002/entries/church-turing/
  4. [38]
    Cotogno, P.: Hypercomputation and the physical Church-Turing thesis. Brit. J. Philos. Sci. 54(2), 181–223 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. [39]
    Davis, M.: Computability and Unsolvability. Dover Publications, Inc., New York (1982)MATHGoogle Scholar
  6. [42]
    Deutsch, D.: Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. Roy. Soc. London A 400, 97–115 (1985)MathSciNetCrossRefMATHGoogle Scholar
  7. [48]
    Etesi, G., Németi, I.: Non-Turing computations via malament-hogarth space-times. Int. J. Theor. Phys. 41(2), 341–370 (2002)CrossRefMATHGoogle Scholar
  8. [61]
    Grigoriev, D.Y.: Kolmogoroff algorithms are stronger than Turing machines. J. Sov. Math. (now known as Journal of Mathematical Sciences) 14, 1445–1450 (1980)Google Scholar
  9. [66]
    Horowitz, E., Sahni, S.: Fundamentals of Data Structures in Pascal. Computer Science Press, Rockville (1984)Google Scholar
  10. [67]
    Hromkovič, J.: Theoretical Computer Science: Introduction to Automata, Computability, Complexity, Algorithmics, Randomization, Communication, and Cryptography. Springer, Berlin (2004)Google Scholar
  11. [73]
    Kleene, S.C.: General recursive functions of natural numbers. Math. Ann. 112, 727–742 (1936)MathSciNetCrossRefGoogle Scholar
  12. [76]
    Kolmogorov, A.N.: On the notion of algorithm. Uspekhi Mat. Nauk 8, 175–176 (1953) (The paper is part of the Meetings of the Moscow Mathematical Society paper and the English translation used here is 77)Google Scholar
  13. [78]
    Kolmogorov, A.N., Uspensky, V.A.: On the definition of an algorithm. Uspekhi Mat. Nauk 13, 3–28 (1958) (English translation in 79)Google Scholar
  14. [83]
    Lewis, H.R., Papadimitriou, C.H.: Elements of the Theory of Computation, 2nd edn. Pearson Education, Harlow (1998)Google Scholar
  15. [93]
    Minsky, M.: Computation: Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs, NJ, USA, 1967.MATHGoogle Scholar
  16. [97]
    Ord, T., Kieu, T.D:. The diagonal method and hypercomputation. Brit. J. Philos. Sci. 56(1), 147–156 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. [106]
    Pontryagin, L.S.: Foundations of Combinatorial Topology. Graylock Press, Rochester (1952) (Translated from the first (1947) Russian edition by F. Bagemihl, H. Komm, and W. Seidel)MATHGoogle Scholar
  18. [109]
    Rogers, Jr., H.: Theory of Recursive Functions and Effective Computability. The MIT Press, Cambridge (1987).Google Scholar
  19. [112]
    Santos, E.S.: Probabilistic Turing machines and computability. Proc. Am. Math. Soc. 22(3), 704–710 (1969)CrossRefMATHGoogle Scholar
  20. [114]
    Santos, E.S.: Computability by probabilistic Turing machines. Trans. Am. Math. Soc. 159, 165–184 (1971)CrossRefMATHGoogle Scholar
  21. [126]
    Syropoulos, A.: Hypercomputation: Computing Beyond the Church-Turing Barrier. Springer New York, Inc., Secaucus (2008)CrossRefGoogle Scholar
  22. [134]
    Tremblay, J.-P., Sorenson, P.G.: The Theory and Practice of Compiler Writing, p. 16. McGraw-Hill, Singapore (1985)Google Scholar
  23. [135]
    Turing, A.M.: On Computable Numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. 42, 230–265 (1936)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Apostolos Syropoulos
    • 1
  1. 1.XanthiGreece

Personalised recommendations