Invariance properties of RAMS and linear time

AbstractAn operation on integers isLTTC if it is computable in linear time on a Turing machine (using the dyadic or binary representation of integers). AnLTTC-RAM (respectivelyI-RAM) is a RAM which only uses LTTC operations (respectively operations in the setI).The address-free time complexity measure of a RAM evaluates execution times using the logarithmic cost criterion but assumes that addressing operations are performed for free.Using the above notions, the present paper proves some invariance properties of RAMs with respect to time complexity.We show that an LTTC-RAM which works within address-free timet may be simulated on a {+}-RAM (respectively, a {Concatenation}-RAM) using only integersO(t1+ε) (for any fixed ε>0) within timeO(t), using the logarithmic cost criterion.It follows that the class of functions computable in linear time on an LTTC-RAM (using the logarithmic cost criterion) is not modified if we make one or several of the following changes:∘ we assume addressing operations are performed for free,∘ the set of allowed LTTC operations is changed ({+}, {Concatenation}, {+, −}, for example),∘ we allow multidimensional arrays (without omitting the time of addressing),∘ the integers (respectively addresses) used by the RAM are required to be bounded by a polynomial in timet, orO(t1+ε) for a fixed ε. As an application, we define and study two robust RAM complexity classes, denoted LINEAR and LARGELINEAR, and discuss their ability to represent linear-time computability.

[1]  Jan van Leeuwen,et al.  Fast Simulation of Turing Machines by Random Access Machines , 1988, SIAM J. Comput..

[2]  Stephen A. Cook,et al.  Time Bounded Random Access Machines , 1973, J. Comput. Syst. Sci..

[3]  Neil Immerman,et al.  Relational Queries Computable in Polynomial Time , 1986, Inf. Control..

[4]  Etienne Grandjean,et al.  Linear Time Algorithms and NP-Complete Problems , 1992, SIAM J. Comput..

[5]  P. Boas.  Machine models and simulations , 1991 .

[6]  Etienne Grandjean.  First-Order Spectra with One Variable , 1990, J. Comput. Syst. Sci..

[7]  Etienne Grandjean.  Universal quantifiers and time complexity of random access machines , 2005, Mathematical systems theory.

[8]  Jirí Wiedermann.  Deterministic and Nondeterministic Simulation of the RAM by the Turing Machine , 1983, IFIP Congress.

[9]  Peter van Emde Boas.  Space Measures for Storage Modification Machines , 1989, Inf. Process. Lett..

[10]  Arnold Schönhage.  Storage Modification Machines , 1980, SIAM J. Comput..

[11]  Leslie G. Valiant,et al.  Fast probabilistic algorithms for hamiltonian circuits and matchings , 1977, STOC '77.

[12]  John Michael Robson.  Random Access Machines with Multi-Dimensional Memories , 1990, Inf. Process. Lett..

[13]  守屋 悦朗,et al.  J.E.Hopcroft, J.D. Ullman 著, "Introduction to Automata Theory, Languages, and Computation", Addison-Wesley, A5変形版, X+418, \6,670, 1979 , 1980 .

[14]  Erich Grädel.  Capturing Complexity Classes by Fragments of Second Order Logic , 1991, Computational Complexity Conference.

[15]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[16]  Leslie G. Valiant,et al.  On Time Versus Space , 1977, JACM.

[17]  Saharon Shelah,et al.  Nearly Linear Time , 1989, Logic at Botik.

[18]  Peter van Emde Boas,et al.  The Problem of Space Invariance for Sequential Machines , 1988, Inf. Comput..

[19]  Arnold L. Rosenberg,et al.  Real-Time Simulation of Multihead Tape Units , 1972, JACM.

[20]  Claus-Peter Schnorr.  Satisfiability Is Quasilinear Complete in NQL , 1978, JACM.

[21]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[22]  Arnold Schönhage.  A nonlinear lower bound for random-access machines under logarithmic cost , 1988, JACM.

[23]  Leslie G. Valiant,et al.  On time versus space and related problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).