Computational complexity of random-access models

Abstract

The relative power of several computational models is considered. These models are the Turing machine and its multidimensional variant, the random access machine (RAM), the tree machine, and the pointer machine. The basic computational properties of the pointer machine are examined in more detail. For example, time and space hierarchy theorems for pointer machines are presented.

Every Turing machine of time complexity $t$ and space complexity $s$ can be simulated by a pointer machine of time complexity $O$($t$) using $O$($s$/log $s$) nodes. This strengthens a similar result by van Emde Boas (1989). Every alternating pointer machine of time complexity $t$ can be simulated by a deterministic pointer machine using $O$($t$/log $t$) nodes. Other results concerning nondeterministic and alternating pointer machines are presented.

Every tree machine of time complexity $t$ can be simulated on-line by a log-cost RAM of time complexity $O$(($t$log $t$)/log log $t$). This simulation is shown to be optimal using the notion of incompressibility from Kolmogorov complexity (Solomonoff, 1964; Kolmogorov, 1965).

Every $d$-dimensional Turing machine of time complexity $t$ can be simulated on-line by a log-cost RAM running in time $O$($t$(log $t$)$\sp{1-(1/d)}$)(log log $t$)$\sp{(1/d}$). There is a log-cost RAM $R$ running in time $t$ such that every $d$-dimensional Turing machine requires time $\Omega$($t\sp{1+(1/d)}$/(log $t$(log log $t$)$\sp{1+(1/d)}$)) to simulate $R$ on-line. Every unit-cost RAM of time complexity $t$ can be simulated on-line by a $d$-dimensional Turing machine in time $O$($t$($n$)$\sp2$log $t$($n$)).