Abstract
An 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(t 1+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:
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∘ we assume addressing operations are performed for free,
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∘ the set of allowed LTTC operations is changed ({+}, {Concatenation}, {+, −}, for example),
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∘ we allow multidimensional arrays (without omitting the time of addressing),
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∘ the integers (respectively addresses) used by the RAM are required to be bounded by a polynomial in timet, orO(t 1+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.
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