On the speed of addition and multiplication on one-tape, off-line turing machines

Abstract

It is shown that for any ε > 0 and for any sufficiently large l (1 — ε )2 l 2 /log b Q is a lower bound for the average computation time required by any one-tape, off-line Turing machine with Q internal states for implementing addition or multiplication of two consecutively written b -adic numbers ( b ⩾ 2) with l digits each, where the average is taken over all pairs of numbers with l digits. Conversely: For any ε > 0 and for both operations Turing machine constructions are indicated, whose computation times are smaller than (1 + ε )2 l 2 /log b Q for any sufficiently large l and for any pair of numbers with l digits each, where the number Q of internal states of the Turing machine has to be chosen large for small ε .