In theoretical computer science, a **crossing sequence at boundary ***i*, denoted as
C
i
(
x
)
or sometimes
c
s
(
x
,
i
)
, is the sequence of states
q
i
1
,
q
i
2
,
.
.
.
,
q
i
k
,
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.