Crossing sequence (Turing machines)

In theoretical computer science, a crossing sequence at boundary "i", denoted as formula_1 or sometimes formula_2, is the sequence of states formula_3 of a Turing machine on input x, such that in this sequence of states, the head crosses between cell "i" and "i" + 1 (note that the first crossing is always a right crossing, and the next left, and so on...)
Sometimes, crossing sequence is considered as the sequence of configurations, which represent the three elements: the states, the contents of the tapes and the positions of the heads.
Study of crossing sequences is carried out, e.g., in computational complexity theory.