Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 161 — #169
6.2. The Invariant Principle 161
y
2
1
0
0 1 2 3
x
Figure 6.2
The Diagonally Moving Robot.
can’t reach .1; 0/. This follows because the evenness of the sum of the coordinates
is a property that is preserved by transitions. This is an example of a preserved
invariant.
This once, let’s go through this preserved invariant argument, carefully highlighting
where induction comes in. Specifically, define the even-sum property of
states to be:
Even-sum..m; n// WWD Œm C n is even:
! r of the diagonally-moving robot, if Even-
Lemma 6.2.1. For any transition q
sum(q), then Even-sum(r).
This lemma follows immediately from the definition of the robot’s transitions:
.m; n/ ! .m ˙ 1; n ˙ 1/. After a transition, the sum of coordinates changes by
.˙1/ C .˙1/, that is, by 0, 2, or -2. Of course, adding 0, 2 or -2 to an even number
gives an even number. So by a trivial induction on the number of transitions, we
can prove:
Theorem 6.2.2. The sum of the coordinates of any state reachable by the diagonallymoving
robot is even.