Advertisement

An Introduction to Some Basic Concepts and Techniques in the Theory of Computable Functions

  • Angelo Marzollo
  • Walter Ukovich
Part of the International Centre for Mechanical Sciences book series (CISM, volume 256)

Abstract

We will talk about a class of function defined on subsets of the set of nonnegative integers, and assuming nonnegative integer values: the so-called “computable functions”. We define a function to be computable if there exists an “effective description” for it, that is a finite sentence unambiguously indicating (possibly only some explicit way) what is its “behaviour” in correspondence to each natural number. In other words, an effective description must allow us to deduce, for any given argument, whether the described function is defined on it and, if it is so, the value it assumes there. Perhaps the careful reader will observe that this definition is very different from the one usually found in the literature: in fact, it may seem strange to call “computable” those functions for which we didn’t explicitly prescribe the precise means for computing them, for example, some particular procedure or model of an abstract “computing device”. Moreover, even the general concepts of “effective procedure”, “algorithm”, or “abstract computing device” are not needed for the definition we gave of a computable function.

Keywords

Turing Machine Recursive Function Computable Function Elementary Operation Standard Numbering 
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]
    ARBIB, M., Theories of Abstract Automata, Prentice-Hall, Engelwood Cliffs, NY, 1969.Google Scholar
  2. [2]
    CHOMSKY, N.,“On Certain Formal Properties of Grammars”, Inf. and Control, 2.2 (1959), pp 137–167.Google Scholar
  3. [3]
    CHOMSKY, N., “Formal Properties of Grammars”, Handbook of Math Psych, 2, (1963), Wiley, New York, pp 323–418.Google Scholar
  4. [4]
    CHURCH, A., “An Unsolvable Problem of Elementary Number Theory”, American Journal of Mathematics.Google Scholar
  5. [5]
    CURRY, H., and R. FEYS, “Combinatory Logic, Vol. 1” North Holland, Amsterdam, 1968.Google Scholar
  6. [6]
    DAVIS, M., “Computability and Unsolvability”, McGraw-Hill, New York, 1958.MATHGoogle Scholar
  7. [7]
    DAVIS, M., “The Undecidable”, Raven Press, Hawlett, 1965. 8.Google Scholar
  8. [8]
    HOPCROFT, J., and ULLMANN, J., “Formal Languages and Their Relation to Automata”, Addison-Wesley, Reading, 1969.MATHGoogle Scholar
  9. [9]
    KLEENE, S., “General Recursive Functions of Natural Numbers”, Mathematische Aunalen, Vol. 112, (1936), pp 727–742.CrossRefMathSciNetGoogle Scholar
  10. [10]
    KLEENE, S., “Introduction to Metamathematics” Van Nostrand, Princeton NY, 1952.Google Scholar
  11. [11]
    MARKOV, A., “The Theory of Algorithms” (Russian), Trudy Mathematicheskogo Instituta, imeni V.A. Steklova, vol. 38 (1951), pp 176–189; Engl. transi., Jerusalem, 1962.Google Scholar
  12. [12]
    MINSKY, M., “Computation: “Finite and Infinitc Machines”, Prentice-Hall, Englewood Cliffs, NY, 1967.Google Scholar
  13. [13]
    POST, E., “Finite Combinatory Processes-Formulation”, I, “The Journal of Symbolic Logic” Vol. I, (1936), pp 103–105.Google Scholar
  14. [14]
    RABIN, M., and SCOTT, D., “Finite Automata and Their Decision Problems”, IBM Journal of Research and Development, Vol. 3, (1959), pp 114–125.CrossRefMathSciNetGoogle Scholar
  15. [15]
    ROGERS, H., Jr.: “Theory of Recursive Functions and Effective Computability”, McGraw-Hill, New York, 1967.MATHGoogle Scholar
  16. [16]
    SALOMAA, A., “Theory of Automata”, Pergamon Press, Oxford, 1968Google Scholar
  17. [17]
    SCOTT, D., “The Lattice of Flow Diagrams”, in E. Engeler (ed.), “Symposium on Semantics of Algorithmic Languages”, Lecture Notes in Mathematics No. 168, Springer-Verlag, Wien, 1971.Google Scholar
  18. [18]
    TURING, A. “On Computable Numbers, with an Application to the Entscheidungsproblem”, Proceedings of the London Mathematical Society, ser$12, Vol. 42, (1936), pp 230–265; Vol. 43 (1936) pp 544–546.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 1976

Authors and Affiliations

  • Angelo Marzollo
    • 1
    • 2
  • Walter Ukovich
    • 3
  1. 1.Istituto di Elettrotecnica cd ElettronicaUniversity of TriesteItaly
  2. 2.International Centre for Mechanical SciencesUdineItaly
  3. 3.Istituto di Elettrotecnica ed ElettronicaUniversity of TriesteItaly

Personalised recommendations