Normalizing and accelerating RAM computations and the problem of reasonable space measures

Abstract

It is shown that any RAM of logarithmic time complexity T(n) can be simulated in linear logarithmic time by a RAM that uses only its O(T ²(n)/log T(n)) first registers, each of them being of size O(log T(n)). As a consequence such a RAM can also be simulated by a unit cost RAM in time O(U(n)+T(n)/log log T(n)) where U(n) denotes the number of input/output operations of a RAM to be simulated. In general the last simulation cannot be improved. Further the difference between the two RAM-space measures found in the literature is studied — the first one being defined as the maximum size of all configurations entered by RAM during its computation, the second one as the sum of maxima of each register contents. It is shown that any S(n)-space bounded RAM, under the standard logarithmic space measure, can be simulated in $O(\sqrt {S(n)} )$space, under the first space measure mentioned above; it follows that the underlying space measure does not respect the Invariance Thesis as formulated by Slot and van Emde Boas. However, when the size of register addresses is also reflected in both previous space measure definitions these measures are equivalent, i.e., they induce the same complexity classes.