The paper presents a survey on the speedup phenomenon in the machine-independent theory of recursive functions, the techniques used to prove its existence, its non-effectiveness, its generalizations, and the relations between the speedup in recursion theory, and similar phenomena in logic.


Decidable Theory Turing Machine Regular Expression Recursive Function Complexity Sequence 
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  1. 1.
    ALTON, D., Diversity of Speedups and embeddability in computational complexity, Rep. TR 73-01, Dept. Comp. Sci., Univ. of Iowa (Iowa City, 1973).Google Scholar
  2. 2.
    ALTON, D., Non-existence of program optimizers in an abstract setting, Rep. TR 73-08, Dept. Comp. Sci., Univ. of Iowa (Iowa City, 1973).Google Scholar
  3. 3.
    ARBIB, M.A., Speedup theorems and Incompleteness theorems, in Automata theory, (ed. E.R. Caianello), pp. 6–24, Academic Press, New York, 1966.Google Scholar
  4. 4.
    ARBIB, M.A., Speedup and incompleteness results, in Theories of abstract automata, pp. 261–267, Prentice Hall, New Jersey, 1969.Google Scholar
  5. 5.
    BLUM, M., A machine-independent theory of the complexity of recursive functions, J. Assoc. Comput. Mach., 14 (1967), 322–336.Google Scholar
  6. 6.
    BLUM, M., On effective procedures for speeding up algorithms, J. Assoc. Comput. Mach. 18 (1971), 290–305.Google Scholar
  7. 7.
    BLUM, M. & I. MARQUES, On complexity properties of recursively enumerable sets, J. Symbolic Logic, 38 (1973), 579–593.Google Scholar
  8. 8.
    CONSTABLE, R.L; & J. HARTMANIS, Complexity of formal translations and speedup results, in proc. 3rd ACM Symp. on theory of Computing (1971), pp. 244–250.Google Scholar
  9. 9.
    EHRENFEUCHT, A. & J. MYCIELSKI, Abbreviating proofs by adding new axioms, Bull. Amer. Math. Soc, 77 (1971), 366–367.Google Scholar
  10. 10.
    FISCHER, M.J. & M.O. RABIN, Super-exponential complexity of Pressburger arithmetic, MAC techn. memo 43, Project MAC. MIT Cambridge Mass. Feb. 1974.Google Scholar
  11. 11.
    GÖDEL, K., Über die Länge der Beweise. Ergebnisse eines Math. Kolloquiums, 7, 23–24 (1936). See also in On the lengths of proofs in The Undecidable, (ed. M. Davis), pp. 82–83, Raven Press, NY, 1965.Google Scholar
  12. 12.
    HARTMANIS, J., Oral communication, Dec. 1974.Google Scholar
  13. 13.
    HARTMANIS, J. & J.E. HOPCROFT, An overview of the theory of computational complexity, J. Assoc. Comput. Mach. 18 (1971), 444–475.Google Scholar
  14. 14.
    HELM, J.P. & P.R. YOUNG, On size versus efficiency for programs admitting speedup, J. Symbolic Logic, 36 (1971), 21–27.Google Scholar
  15. 15.
    LANDWEBER, L.H. & E.L. ROBERTSON, Recursive properties of abstract complexity classes, J. Assoc. Comput. Mach., 19 (1972), 296–308.Google Scholar
  16. 16.
    MEYER, A.R., Weak monadic second order theory of successor is not elementary recursive, MAC techn. memo 38. Project MAC, MIT, Cambridge Mass. 1973.Google Scholar
  17. 17.
    MEYER, A.R. & P.C. FISCHER, Computational speedup by effective operators, J. Symbolic Logic, 37 (1972), 55–68.Google Scholar
  18. 18.
    MOSTOWSKI, A., Sentences undecidable in formalized arithmetic, North Holland, Amsterdam, 1957.Google Scholar
  19. 19.
    ROGERS, H., The theory of recursive functions and effective computability, Mc Graw Hill, New York, 1967.Google Scholar
  20. 20.
    SCHNORR, C.P., Does computational speedup concern programming?, in Automata, Languages and Programming, (M. Nivat ed.), pp. 585–592, North Holland/Elsevier, Amsterdam, 1973.Google Scholar
  21. 21.
    SCHNORR, C.P. & G. STUMPF, A characterization of complexity sequences, Tagungsbericht 46/1972, Algorithmen und Komplexitätstheorie, Math. Forschungs-institut Oberwolfach, Nov. 1972.Google Scholar
  22. 22.
    SCHNORR, C.P., Rekursive Funktionen und ihre Komplexität, Teubner, Stuttgart, 1974.Google Scholar
  23. 23.
    STOCKMEYER, L.J., The complexity of decision problems in Automata theory and Logic, Report MAC TR-133, Project MAC. MIT, Cambridge Mass., July 1974.Google Scholar
  24. 24.
    YOUNG, P., Speedups by changing the order in which sets are enumerated, Math. Systems Theory, 5 (1971), 148–156.Google Scholar
  25. 25.
    YOUNG, P., Easy constructions in complexity theory: Gap and Speedup theorems, Proc. Amer. Math. Soc., 37 (1973), 555–563.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1975

Authors and Affiliations

  • Peter van Emde Boas
    • 1
  1. 1.University of Amsterdam/Mathematical Centre Amsterdam

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