the tape head

The tape head is the machine's single moving fingertip: the one cell it is touching right now. The tape may be infinitely long, but the machine cannot see all of it at once; it can only read, and write, the one cell under the head. To examine a distant cell it must walk the head over there, one step at a time, like a person reading a scroll through a small window.

On each step the head does exactly three things, dictated by the transition function: it reads the symbol in the current cell, writes a symbol back into that same cell (possibly the same symbol, possibly a different one), and then moves exactly one cell to the left or one cell to the right. It can never stay put (in the basic model), and it can never jump; movement is always by single steps. Because the head reads before it writes, a typical move is 'I see an a here, I'll cross it out by writing an X, and step right to look at the next symbol.'

The head being able to both read and write the cell it visits is the whole difference from earlier machines. A DFA's input pointer can only read, and only moves right, never returning. The Turing head can re-read, rewrite, and shuttle back and forth, which is how a Turing machine compares far-apart parts of its input (for example checking that the first half of a string equals the second). The cost is that the head is a bottleneck: to compare cell 1 with cell 1000 it must physically travel a thousand cells, which is part of why the model is slow even though it is maximally expressive.