is an
n-dimensional Euclidian space. For
,
, if there exists a finite set
of points such that point
connects to
by a set
of broken lines in
A, then
A is called arcwise connected. That is, any two points in
A can be interconnected by a finite set of broken lines in
A. If
x is a starting state,
y is a goal state, and a broken line connected
with
regarded as an operator, then the problem of judging whether
and
belong to the same arcwise connected set is equivalent to finding out whether there is a finite number of operations such that the given starting state can be transformed into the given goal state.