Consider, for instance, several identical objects, say coins, and a box in which these objects can be placed. There is enough room in the box for 10 coins. We are interested in the effect of the action a of putting a coin in the box on the number c of coins that are currently in the box. This domain can be represented by transition system TS2 (Section 0.1) with N=10.

Prediction. Currently there are 5 coins in the box. If I put one coin in the box twice in a row, how many coins will there be in the box at the end?

In terms of paths in the transition system: consider the path of length 2 that starts at the vertex c=5 and has both edges labeled a=t. Where does it end? (Answer: c=7.)

Postdiction. I just put one coin in the box twice in a row, and now there are 5 coins in it. How many coins were there in the box initially?

In terms of paths: consider the path of length 2 that ends at the vertex c=5 and has both edges labeled a=t. Where does it start? (Answer: c=3.)

Planning. Currently there are 4 coins in the box. Find the shortest sequence of events that will make the box full.

In terms of paths: find the shortest path from the vertex c=4 to the vertex c=10. (Answer: path of length 6 with every edge labeled a=t.)