RAM with compact memory: a realistic and robust model of computation

  • Etienne Grandjean
  • J. M. Robson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 533)


An operation op of arity k on ℕ, i.e. a function op: ℕk → ℕ is linear time Turing computable (for short, LTTC) if it is computable in linear time on a Turing machine (for usual binary or dyadic notation of integers). Let + and Conc respectively denote usual addition of integers and concatenation (of their dyadic notations). A RAM which uses only arithmetical operations of a set I is called an I - RAM. An LTTC-RAM is a RAM which only uses LTTC operations.

In the present paper, we use the logarithmic criterion for time measure of RAMs. A RAM works with polynomially (resp. strongly polynomially) compact memory if it only uses addresses (resp. addresses and register contents) ≤ tO(1) where t is the time of the computation.

Theorem 1. A deterministic LTTC-RAM R with polynomially compact memory is simulated in linear time by a deterministic {+}-RAM (resp. {Conc}-RAM) R' with strongly polynomially compact memory.

Theorem 2. A nondeterministic LTTC-RAM R can be simulated in linear time by a nondeterministic {+}-RAM (resp. {Conc}-RAM) R' with strongly poynomially compact memory.

Note that Theorem 2 holds for both weak nondeterministic RAMs and strong nondeterministic RAMs, i.e. in case the RAMs have only nondeterministic goto instructions or in case they have an instruction to guess an integer.

If moreover the RAMs R of Theorem 1–2 are sane (i.e. R does not use noninitialised registers) then the simulating RAMs R' are sane, too.

We also study and discuss more restrictive notions of compact memory (linearly compact memory).


Hash Function Linear Time Turing Machine Hash Table Block Length 
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.


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  1. [AHU]
    A.V. AHO, J.E. HOPCROFT and J.D. ULLMAN, The design and analysis of computer algorithms, Addison-Wesley, Reading, MA, 1974.Google Scholar
  2. [AnVa]
    D. ANGLUIN and L. VALIANT, Fast probabilistic algorithms for hamiltonian circuits and matchings, J. Comput. System Sci. 18 (1979), pp. 155–193.CrossRefGoogle Scholar
  3. [Ar]
    R.G. ARCHIBALD, Introduction to the theory of numbers, Merrill, 1970.Google Scholar
  4. [CoRe]
    S.A. COOK and R.A. RECKHOW, Time bounded random access machines, J. Comput. System Sci. 7 (1973), pp. 354–375.Google Scholar
  5. [DoGa]
    W.F. DOWLING and J.H. GALLIER, Linear-time algorithms for testing the satisfiability of propositional Horn formulas, J. Logic Prog. 3 (1984), pp. 267–284.CrossRefGoogle Scholar
  6. [Di]
    E.W. DIJKSTRA, Guarded commands, nondeterminacy and the formal derivation of programs, Comm. A.C.M. 18 (1975), pp. 453–457.Google Scholar
  7. [vEB]
    P. Van EMDE BOAS, Space measures for storage modification machines, Inf. Proc. Letters, 30 (1989), pp. 103–110.CrossRefGoogle Scholar
  8. [Grae]
    E. GRAEDEL, On the notion of linear time, Proc. 3rd Italian Conf. Theoret. Comput. Sci., Mantova 1989, World Scientific Publ. Co., 323–334 (also to appear in a Special Issue of Internat. Journal of Foundations Comput. Sci.).Google Scholar
  9. [Gr1]
    E. GRANDJEAN, Universal quantifiers and time complexity of random access machines, Math. Syst. Th. 18, (1985), pp. 171–187.CrossRefGoogle Scholar
  10. [Gr2]
    E. GRANDJEAN, A nontrivial lower bound for an NP problem on automata, SIAM J. Comput. 19, (1990), pp. 438–451.CrossRefGoogle Scholar
  11. [Gr3]
    E. GRANDJEAN, RAMs with polynomially compact memory are efficiently simulated by RAMs with almost linearly compact memory, Abstracts of A.M.S. 90T-68-33 Issue 67 vol. 11 n. 2 (March 1990) p. 238.Google Scholar
  12. [Gr4]
    E. GRANDJEAN, RAMs can be simulated in linear time by RAMs with compact memory, Abstracts of A.M.S. 90T-68-146 Issue 70 vol. 11 n. 4 (August 1990) p. 357.Google Scholar
  13. [Gr5]
    E. GRANDJEAN, Invariance properties of RAMs and linear time, Technical Report L.I.U.C. Univ. Caen FRANCE 90-11 (december 1990).Google Scholar
  14. [GrRo]
    E. GRANDJEAN and J.M. ROBSON, RAM with compact memory, Abstracts of A.M.S. 90T-68-34 Issue 67 vol. 11 n. 2 (March 1990) p. 238.Google Scholar
  15. [Gu]
    Y. GUREVICH, Kolmogorov machines and related issues: The column on logic in computer science, Bull. EATCS 35 (1988), pp. 71–82.Google Scholar
  16. [GuSh]
    Y. GUREVICH and S. SHELAH, Nearly linear time, in Meyer Taitslin (Eds.), Springer-Verlag Berlin, 1989, LNCS 363, pp. 108–118Google Scholar
  17. [HoUl]
    J.E. HOPCROFT and J. D. ULLMAN, Introduction to automata theory, languages and computation, Addison-Wesley, Reading, MA, 1979.Google Scholar
  18. [KvLP]
    J. KATAJAINEN, J. van LEUWEN and M. PENTTONEN, Fast simulation of Turing machines by random access machines, SIAM J. Comput. 17 (1988), pp. 77–88.CrossRefGoogle Scholar
  19. [KoUs]
    A.N. KOLMOGOROV and V.A. USPENSKY, On the definition of an algorithm, Uspekhi Mat. Nauk 13:4 (1958), pp. 3–28 (Russian) or AMS translation, ser. 2, vol 21 (1963), pp. 217–245.Google Scholar
  20. [Mi]
    M. MINOUX, LTUR: a simplified linear-time resolution algorithm for Horn formulae and computer implementation, Inf. Proc. Letters 29 (1988) pp. 1–12.CrossRefGoogle Scholar
  21. [Ro]
    J.M. ROBSON, Random access machines with multi-dimensional memories, Inf. Proc. Letters 34 (1990) pp. 265–266.CrossRefGoogle Scholar
  22. [St]
    A. SCHMITT, On the computational power of the floor function, Inf. Proc. Letters 14 (1982), pp 1–3.CrossRefGoogle Scholar
  23. [Sr]
    C.P. SCHNORR, Satisfiability is quasilinear complete in NQL, J. Assoc. Comput. Mach., 25 (1978), pp. 136–145.Google Scholar
  24. [Se1]
    A. SCHÖNHAGE, Storage modification machines, SIAM J. Comput. 9 (1980), pp. 490–508.CrossRefGoogle Scholar
  25. [Se2]
    A. SCHÖNHAGE, A nonlinear lower bound for random access machines under logarithmic cost, J. Assoc. Comput. Mach. 35 (1988), pp. 748–754.Google Scholar
  26. [SlvEB]
    C. SLOT and P. van EMDE BOAS, The problem of space invariance for sequential machines, Inform. and Comput. 77 (1988), pp.93–122.CrossRefGoogle Scholar
  27. [Wi]
    J. WIEDERMANN, Deterministic and nondeterministic simulation of the RAM by the Turing machine, in Proc. IFIP Congress 83, R.E.A. Mason ed., North Holland, Amsterdam 1983, pp. 163–168.Google Scholar
  28. [Wi2]
    J. WIEDERMANN, Normalizing and accelerating RAM computations and the problem of reasonable space measures, TR OPS-3 / 1990 (June 1990), Dpt of Programming Systems, Bratislava, Czechoslovakia.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Etienne Grandjean
    • 1
  • J. M. Robson
    • 2
  1. 1.LIUCUniversité de CaenCaen CedexFrance
  2. 2.Dept of Computer ScienceAustralian National UniversityCanberraAustralia

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