An Improved Simulation Result for Ink Bounded Turing Machines

Abstract

A (one tape, deterministic) Turing machine is $f(n)$ ink bounded if the machine changes a symbol of its work tape at most $O(f(n))$ times while processing any input of length $n$. The main result of our paper is the construction of an "ink efficient" universal machine which, for any $f(n)$ ink bounded machine $M$ and input $x$, can simulate the processing of $M$ on $x$ or detect that $M$ is looping infinitely on input $x$. The universal machine requires $O(f(n)^{1+\epsilon)$ ink for this simulation where $\epsilon$ is an arbitrarily small positive number. As a corollary, we establish that the class of all $f(n)$ ink bounded computations is properly contained in the class of all $g(n)$ ink bounded computations assuming $\stackrel{inf}{n \rightarrow \infty} \frac{f(n)^{1+\varepsilon}}{g(n)} = 0$ and a technical condition on g.