# 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.

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