Translational Lemmas for Alternating TMs and PRAMs

  • Chuzo Iwamoto
  • Yoshiaki Nakashiba
  • Kenichi Morita
  • Katsunobu Imai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3623)


We present translational lemmas for alternating Turing machines (ATMs) and parallel random access machines (PRAMs), and apply them to obtain tight hierarchy results on ATM- and PRAM-based complexity classes. It is shown that, for any small rational constant ε, there is a language which can be accepted by a c(9+ε)log r n-time d(4+ε)log n-space ATM with l worktapes but not by any clog r n-time dlog n-space ATM with the same l worktapes if the number of tape symbols is fixed. Here, c,d>0 and r>1 are arbitrary rational constants, and l≥2 is an arbitrary integer. It is also shown that, for any small rational constant ε, there is a language which can be accepted by a c(1 + ε)log r1 n-time PRAM with n r2 processors but not by any c log r1 n-time PRAM with n r2(1 + ε) processors, where c>0, r 1>1, and r 2≥1 are arbitrary rational constants.


Arbitrary Integer Read State Input String Input Tape Common Memory 
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  1. 1.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)zbMATHGoogle Scholar
  2. 2.
    Chandra, A., Kozen, D., Stockmeyer, L.: Alternation. J. Assoc. Comput. Mach. 28, 114–133 (1981)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  4. 4.
    Ibarra, O.H.: A hierarchy theorem for polynomial-space recognition. SIAM J. Comput. 3(3), 184–187 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ibarra, O.H., Kim, S.M., Moran, S.: Sequential machine characterizations of trellis and cellular automata and applications. SIAM J. Comput. 14(2), 426–447 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Iwama, K., Iwamoto, C.: Parallel complexity hierarchies based on PRAMs and DLOGTIME-uniform circuits. In: Proc. 11th Ann. IEEE Conf. on Computational Complexity, Philadelphia, pp. 24–32 (1996)Google Scholar
  7. 7.
    Iwamoto, C., Hatayama, N., Morita, K., Imai, K., Wakamatsu, D.: Hierarchies of DLOGTIME-uniform circuits. In: Margenstern, M. (ed.) MCU 2004. LNCS, vol. 3354, pp. 211–222. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Karp, R.M., Ramachandran, V.: Parallel algorithms for shared-memory machines. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. A, pp. 869–941. MIT Press, Amsterdam (1990)Google Scholar
  9. 9.
    Ruzzo, W.L.: On uniform circuit complexity. J. Comput. System Sci. 22, 365–383 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Stockmeyer, L., Vishkin, U.: Simulation of parallel random access machines by circuits. SIAM J. Comput. 13(2), 409–422 (1984)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Chuzo Iwamoto
    • 1
  • Yoshiaki Nakashiba
    • 1
  • Kenichi Morita
    • 1
  • Katsunobu Imai
    • 1
  1. 1.Hiroshima University, Graduate School of EngineeringHigashi-HiroshimaJapan

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