Milking the Aanderaa Argument

In the early 1970’s, by an intricate and involved counting argument, Stal Aanderaa managed to show for the first time that every additional tape adds to the power of a Turing machine that operates in real time (Aanderaa, 1974). This was a full decade after Rabin (1963) had shown that the second tape adds power. With time, especially following the introduction of the information-theoretic approach in (Paul et al., 1981), Aanderaa’s argument has become better understood and even extended to new results. In this paper, we present a clean version of the argument and some new extensions. The following four theorems summarize all the extensions we know about. All of them are lower bounds on the time required for on-line