1. 8puzzle in CP

2. 1 2 3 1 2 3 4 5 7 5 6 7 8 4 6 8

3. 1 4 2 7 6 3 5 8 1 4 2 7 6 3 5 8 1 4 6 2 7 3 5 8 1 4 6 2 7 3 5 8

4. Representation 1 4 6 2 7 3 5 8 1 4 7 2 5 8 3 6 9 1 4 7 2 5 8 3 6 9

5. Representation 1 4 6 2 7 3 5 8 1 4 2 7 6 3 5 8 1 4 2 7 6 3 5 8 1 4 6 2 7 3 5 8 1 4 6 2 7 3 5 8

6. 1 4 6 2 7 3 5 8 Representation of states
To get from initial state to a goal state we make a number of moves
and moves take us to new states
If we have a plan that takes us from I to G in n steps
we will have n-1 states between I and G
Represent a state as 9 constrained integer variables
S[i] has a value [0..8], the tile in position i
the value 0 corresponds to the blank tile
Therefore for a plan, let’s say of 3 steps from I to G we will have
states S[1..4][1..9]
S[1][5] is the tile in the 5th position of the 1st state (and is 0 in our example)

7. 1 4 6 2 7 3 5 8 1 4 7 2 5 8 3 6 9 Representation of moves
There are 24 possible moves from a state
position of blank (zero) possible moves (swaps) move numbers
(1,2),(1,4) ,2
(2,1),(2,3),(2,5) ,4,5
(3,2),(3,6) ,7
(4,1),(4,5),(4,7) ,9,10
(5,2),(5,4),(5,6),(5,8) ,12,13,14,
(6,3),(6,5),(6,9) ,16,17
(7,4),(7,8) ,19
(8,5),(8,7),(8,9) ,21,22
(9,6),(9,8) ,24
We have n-1 move variables, where move[i] has a domain [1..24]

8. Representation
of moves & states 1 4 6 2 7 3 5 8 1 4 6 2 7 3 5 8
move[1] = 14 1 4 6 2 7 3 5 8
move[2] = 21 1 4 2 7 6 3 5 8
move[1] = 18 1 4 2 7 6 3 5 8
position of blank (zero) possible moves (swaps) move numbers
(1,2),(1,4) ,2
(2,1),(2,3),(2,5) ,4,5
(3,2),(3,6) ,7
(4,1),(4,5),(4,7) ,9,10
(5,2),(5,4),(5,6),(5,8) ,12,13,14,
(6,3),(6,5),(6,9) ,16,17
(7,4),(7,8) ,19
(8,5),(8,7),(8,9) ,21,22
(9,6),(9,8) ,24

9. Constraints on a move 1 4 6 2 7 3 5 8
If we are in state 1 and the blank tile is in position 1 and we make move 1 (swap(1,2))
then in state 2 the blank is in position 2 and
what is in position 2 of in state 1 is now in position 1 in state 2

10. Constraints on a move 1 4 6 2 7 3 5 8

11. Constraints on a move 1 4 6 2 7 3 5 8
But … when we make a move, only the things that we move change!
This is the frame problem (GOF AI)
… and so on

12. Constraints on a move 1 4 6 2 7 3 5 8
We also need to say that if the blank tile (zero) is not
in a specific position we cannot make a specified move
… and so on

13. 1 4 6 2 7 3 5 8 Solving The move variables are our decision variables
Find values for the move variables that satisfy the constraints
This will find a plan of n-1 steps if one exists

14. 1 4 6 2 7 3 5 8 Bi-directional search
Search from I to G and from G to I simultaneously
Replace the variables move[i]
introduce fmove[i]
the forwards move from state i to state i+1
introduce bmove[i]
the backward move from state i + 1 to I
post all the constraints we hadd before in both directions!
add the following constriants
I call this a bridge between backward & forward moves
We can now search in the following order
fmove[1],bmove[n-1],fmove[2],bmove[n-2] ...

15. 1 4 6 2 7 3 5 8 Propagation through plans
There is little if any propagation through plans

16. 1 4 6 2 7 3 5 8 Propagation through plans
There is little if any propagation through plans
If in state i I cannot move the blank tile into position x
then in state i+1 I cannot make a move out of position x!
Example: If at time T I cannot put the bunch of flowers on the table
then at time T+1 I cannot move the bunch of flowers from the table

17. 1 4 6 2 7 3 5 8 Propagation through plans
If in state i I cannot move the blank tile into position x
then in state i+1 I cannot make a move out of position x!
I call this a rippling constraint
position of blank (zero) possible moves (swaps) move numbers
(1,2),(1,4) ,2
(2,1),(2,3),(2,5) ,4,5
(3,2),(3,6) ,7
(4,1),(4,5),(4,7) ,9,10
(5,2),(5,4),(5,6),(5,8) ,12,13,14,
(6,3),(6,5),(6,9) ,16,17
(7,4),(7,8) ,19
(8,5),(8,7),(8,9) ,21,22
(9,6),(9,8) ,24

18. 1 4 6 2 7 3 5 8 My problems with planning
what happens if we cannot find a plan in n-1 steps?
Build a bigger model … or
add another layer to the model
like iterative dfs
could we have a null move, i.e. a move[i] = 0
yes, but we have to condition the rippling constraint
our rippling constraint can generate a contradiction otherwise!
the model is BIG
I’m working on something neater

19. That’s all for now folks