Advertisement

Abstract

We present a generalization of standard Turing machines based on allowing unusual tapes. We present a set of reasonable constraints on tape geometry and conclude that the proper degree of generality is Cayley graphs. Surprisingly, this generalization does not lead to yet another equivalent formulation of the notion of computable function. Rather, it gives an alternative definition of the recursively enumerable Turing degrees that does not rely on oracles.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boone, W.W.: The word problem. Proc. Nat. Acad. Sci. U.S.A. 44, 1061–1065 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Boone, W.W.: Word problems and recursively enumerable degrees of unsolvability. A first paper on Thue systems. Ann. of Math. 83(2), 520–571 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boone, W.W.: Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely presented groups. Ann. of Math. 84(2), 49–84 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Grigorév, D.J.: Time complexity of multidimensional Turing machines. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 88, 47–55 (1979); Studies in constructive mathematics and mathematical logic, VIIIMathSciNetGoogle Scholar
  5. 5.
    Kozen, D.: Automata and Computability. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  6. 6.
    Novikov, P.S.: On algorithmic unsolvability of the problem of identity. Dokl. Akad. Nauk. SSSR 85, 709–719 (1952)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Aubrey da Cunha
    • 1
  1. 1.University of MichiganAnn ArborUSA

Personalised recommendations