Feasible Real Random Access Machines
We present a modified real RAM model which is equipped with the usual discrete and real-valued arithmetic operations and with a finite precision test <kwhich allows comparisons of real numbers only up to a variable uncertainty 1/(k+1). Furthermore, ourfeasible RAMhas an extended semantics which a...
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| Vydané v: | Journal of Complexity Ročník 14; číslo 4; s. 490 - 526 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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Elsevier Inc
01.12.1998
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| ISSN: | 0885-064X, 1090-2708 |
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| Abstract | We present a modified real RAM model which is equipped with the usual discrete and real-valued arithmetic operations and with a finite precision test <kwhich allows comparisons of real numbers only up to a variable uncertainty 1/(k+1). Furthermore, ourfeasible RAMhas an extended semantics which allows approximative computations. Using a logarithmic complexity measure we prove that all functions computable on a RAM in time O(t) can be computed on a Turing machine in time O(t2·log(t)·loglog(t)). Vice versa all functions computable on a Turing machine in time O(t) are computable on a RAM in time O(t). Thus, our real RAM model does not only express exactly the computational power of Turing machines on real numbers (in the sense of Grzegorczyk), but it also yields a high-level tool for realistic time complexity estimations on real numbers. |
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| AbstractList | We present a modified real RAM model which is equipped with the usual discrete and real-valued arithmetic operations and with a finite precision test <kwhich allows comparisons of real numbers only up to a variable uncertainty 1/(k+1). Furthermore, ourfeasible RAMhas an extended semantics which allows approximative computations. Using a logarithmic complexity measure we prove that all functions computable on a RAM in time O(t) can be computed on a Turing machine in time O(t2·log(t)·loglog(t)). Vice versa all functions computable on a Turing machine in time O(t) are computable on a RAM in time O(t). Thus, our real RAM model does not only express exactly the computational power of Turing machines on real numbers (in the sense of Grzegorczyk), but it also yields a high-level tool for realistic time complexity estimations on real numbers. |
| Author | Hertling, Peter Brattka, Vasco |
| Author_xml | – sequence: 1 givenname: Vasco surname: Brattka fullname: Brattka, Vasco organization: Theoretische Informatik I, FernUniversität Hagen, Postfach 940, D-58084, Hagen, Germanyf1E-mail: vasco.brattka@fernuni-hagen.def1 – sequence: 2 givenname: Peter surname: Hertling fullname: Hertling, Peter organization: Department of Computer Science, University of Auckland, Private Bag, 92019, Auckland, New Zealandf2E-mail: hertling@cs.auckland.ac.nzf2 |
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| Cites_doi | 10.1006/jcom.1995.1002 10.1006/jcom.1998.0475 10.1090/S0273-0979-1989-15750-9 10.1109/TEC.1961.5219227 10.4064/fm-44-1-61-71 10.1016/0885-064X(91)90026-T 10.1016/0304-3975(91)90168-2 10.1016/0304-3975(95)00249-9 10.1016/0885-064X(85)90024-X 10.1145/321941.321944 |
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| Keywords | computability complexity in analysis computational complexity |
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| References_xml | – year: 1989 ident: CM980488RF17 publication-title: Computability in Analysis and Physics – start-page: 41 year: 1995 end-page: 53 ident: CM980488RF20 article-title: Topological spaces allowing Type 2 complexity theory publication-title: Proc. of the CCA Workshop – volume: 44 start-page: 61 year: 1957 end-page: 71 ident: CM980488RF7 article-title: On the definition of computable real continuous functions publication-title: Fund. Math. – year: 1985 ident: CM980488RF18 publication-title: Computational Geometry – volume: 23 start-page: 242 year: 1976 end-page: 251 ident: CM980488RF5 article-title: Fast multiple-precision evaluation of elementary functions publication-title: J. Assoc. Comput. Mach. – volume: 21 start-page: 1 year: 1989 end-page: 46 ident: CM980488RF2 article-title: On a theory of computation and complexity over the real numbers: publication-title: Bull. of the Amer. Math. 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Comput. Mach. doi: 10.1145/321941.321944 – year: 1994 ident: 10.1006/jcom.1998.0488_CM980488RF6 article-title: On degeneracy in geometric computations – year: 1986 ident: 10.1006/jcom.1998.0488_CM980488RF14 article-title: Computational complexity of real functions and real numbers publication-title: Informatik Berichte |
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| Title | Feasible Real Random Access Machines |
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