The simple roots of real-time computation hierarchies

Abstract

A BLAH machine is any memory device that can be simulated in real-time by a multitape Turing machine and such that a multiBLAH machine can real-time simulate a pushdown store. A multiBLAH machine consists of a finite control connected to an input terminal and an output terminal and one or more copies of the BLAH memory unit. It is shown that a (k+1)-BLAH machine is more powerful in real-time than a k-BLAH machine, for each k. Thus the hierarchies, within the real-time definable computations, are proper and smooth, that is, adding a device always increases power. It also turns out that all real-time hierarchy results in this vein are simple corollaries of a single root: the real-time hierarchy of multipushdown store machines. As examples of such new results we mention that in real-time, k+1 tape-units with a fast rewind square are more powerful than k such units; that (k+1)-head tape-units with fast rewind squares are more powerful than k-head tape-units with fast rewind squares; that (k+1)-dequeue machines are more powerful than k-dequeue machines; and that (k+1)-concatenable-dequeue machines are more powerful than k-concatenable-dequeue machines.

This work was supported by the Stichting Mathematisch Centrum.