The simple roots of real-time computation hierarchies

Preliminary version
  • Paul M. B. Vitányi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 172)


A BLAH machine is any memory device that can be simulated in real-time by a multitape Turing machine and such that a multiBLAH machine can real-time simulate a pushdown store. A multiBLAH machine consists of a finite control connected to an input terminal and an output terminal and one or more copies of the BLAH memory unit. It is shown that a (k+1)-BLAH machine is more powerful in real-time than a k-BLAH machine, for each k. Thus the hierarchies, within the real-time definable computations, are proper and smooth, that is, adding a device always increases power. It also turns out that all real-time hierarchy results in this vein are simple corollaries of a single root: the real-time hierarchy of multipushdown store machines. As examples of such new results we mention that in real-time, k+1 tape-units with a fast rewind square are more powerful than k such units; that (k+1)-head tape-units with fast rewind squares are more powerful than k-head tape-units with fast rewind squares; that (k+1)-dequeue machines are more powerful than k-dequeue machines; and that (k+1)-concatenable-dequeue machines are more powerful than k-concatenable-dequeue machines.


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  1. Aa.
    Aanderaa, S.O., On k-tape versus (k − 1)-tape real-time computation. In: SIAM-AMS Proceedings, Vol. 7 (Complexity of Computation), 1974, 75–96.Google Scholar
  2. FMR.
    Fischer, P.C., A.R. Meyer and A.L. Rosenberg, Real-time simulation of multihead tape-units, J. Ass. Comp. Mach. 19 (1972) 590–607.Google Scholar
  3. Fr.
    Fischer, M.J., and A.L. Rosenberg, Limited random access Turing machines, Proceedings 9-th IEEE Conference on Switching and Automata Theory, 1968, 356–367.Google Scholar
  4. Ko.
    Kosaraju, S.R., Real-time simulation of concatenable double-ended queues by double-ended queues. Proceedings 11-th ACM Symposium on Theory of Computing, 1979, 346–351.Google Scholar
  5. LS.
    Leong, B., and J.I. Seiferas, New real-time simulations of multihead tape units, Proceedings 9-th ACM Symposium on Theory of Computing, 1977, 239–248.Google Scholar
  6. Pa.
    Paul, W.J., On-line simulation of k + 1 tapes by k tapes requires nonlinear time. Proceedings 22-nd IEEE Conference on Foundations of Computer Science, 1982, 53–56.Google Scholar
  7. PSS.
    Paul, W.J., J.I. Seiferas and J. Simon, An information-theoretic approach to time bounds for on-line computation. Proceedings 12-th ACM Symposium on Theory of Computing, 1980, 357–367.Google Scholar
  8. PF.
    Pippenger, N., and M.J. Fischer, Relations among complexity measures, Journal ACM 26 (1979) 361–384.Google Scholar
  9. Vi.
    Vitányi, P.M.B., On the power of real-time Turing machines under varying specifications. Proceedings of the 7-th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 85, Springer Verlag, Berlin, 1980, 658–671.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Paul M. B. Vitányi
    • 1
  1. 1.Centre for Mathematics and Computer Science (C.W.I.)AmsterdamThe Netherlands

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