a stay-put move

A basic Turing machine head is restless: every single step it must shuffle one cell left or one cell right, even when it has nothing useful to do over there. Sometimes that is annoying, you would like the head to write a symbol and just stay where it is, perhaps to read the same cell again next step. A stay-put move is a small variant that grants exactly this: a third head direction, often written S, meaning 'do not move the head this step'.

Concretely, in the basic model the transition function moves the head with a direction from {L, R} (left or right). A stay-put machine allows {L, R, S}, where on an S move the head writes its new symbol and changes state but remains on the same cell. This is purely a convenience for the machine designer, who can now think 'update this cell and reconsider it' as a single step instead of contriving a left-then-right dance.

The point, and the reason this belongs to the robustness story, is that stay-put adds no power whatsoever. Any S step can be simulated by an ordinary machine in two steps: move right (do nothing useful), then move left back to the original cell. So a stay-put Turing machine recognizes exactly the same languages as the basic model, with at most a constant-factor slowdown. It is one of the cleanest illustrations of how a seemingly handy feature leaves the class of computable languages completely unchanged, the whole moral of Turing-machine variants in miniature.