8puzzle in CP

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initialstate

goal state

A trivial example

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Representation

1 4 72 5 83 6 9

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Representation

1 4 62 73 5 81 4 62 73 5 81 42 7 63 5 81 42 7 63 5 8

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Representationof states

To get from initial state to a goal state we make a number of movesand moves take us to new states

If we have a plan that takes us from I to G in n stepswe 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 havestates 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)

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Representationof moves

There are 24 possible moves from a state

position of blank (zero) possible moves (swaps) move numbers 1 (1,2),(1,4) 1,2 2 (2,1),(2,3),(2,5) 3,4,5 3 (3,2),(3,6) 6,7 4 (4,1),(4,5),(4,7) 8,9,10 5 (5,2),(5,4),(5,6),(5,8) 11,12,13,14, 6 (6,3),(6,5),(6,9) 15,16,17 7 (7,4),(7,8) 18,19 8 (8,5),(8,7),(8,9) 20,21,22 9 (9,6),(9,8) 23,24

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We have n-1 move variables, where move[i] has a domain [1..24]

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Representationof moves & states

1 4 62 73 5 8

1 4 62 73 5 8

1 42 7 63 5 8

1 42 7 63 5 8

move[1] = 14

move[2] = 21

move[1] = 18

position of blank (zero) possible moves (swaps) move numbers 1 (1,2),(1,4) 1,2 2 (2,1),(2,3),(2,5) 3,4,5 3 (3,2),(3,6) 6,7 4 (4,1),(4,5),(4,7) 8,9,10 5 (5,2),(5,4),(5,6),(5,8) 11,12,13,14, 6 (6,3),(6,5),(6,9) 15,16,17 7 (7,4),(7,8) 18,19 8 (8,5),(8,7),(8,9) 20,21,22 9 (9,6),(9,8) 23,24

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Constraints on a move

]2][1[]1][2[0]2][2[1]1[0]1][1[ SSSmoveS

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

]4][1[]1][2[0]4][2[2]1[0]1][1[ SSSmoveS

]2][1[]1][2[0]2][2[3]1[0]2][1[ SSSmoveS

]3][1[]2][2[0]3][2[4]1[0]2][1[ SSSmoveS

]8][1[]9][2[0]8][2[24]1[0]9][1[ SSSmoveS

...

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Constraints on a move

]2][1[]1][[0]2][[1]1[0]1][1[ nSnSnSnmovenS

]4][1[]1][[0]4][[2]1[0]1][1[ nSnSnSnmovenS

]2][1[]1][[0]2][[3]1[0]2][1[ nSnSnSnmovenS

]3][1[]2][[0]3][[4]1[0]2][1[ nSnSnSnmovenS

]8][1[]9][[0]8][[24]1[0]9][1[ nSnSnSnmovenS

...

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Constraints on a move

But … when we make a move, only the things that we move change!

This is the frame problem (GOF AI)

]8][2[]8][1[]7][2[]7][1[]6][2[]6][1[]5][2[]5][1[]4][2[]4][1[]3][2[]3][1[

]2][1[]1][2[0]2][2[1]1[0]1][1[

SSSSSSSSSSSSSSSmoveS

… and so on

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Constraints on a move

We also need to say that if the blank tile (zero) is notin a specific position we cannot make a specified move

2]1[1]1[0]1][1[ movemoveS

… and so on

This is one of our propagation steps!

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Solving

The move variables are our decision variablesFind values for the move variables that satisfy the constraintsThis will find a plan of n-1 steps if one exists

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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 had before in both directions!• add the following constriants

I call this a bridge between backward & forward moves

We can now search in the following orderfmove[1],bmove[n-1],fmove[2],bmove[n-2] ...

][][][ xinverseibmovexifmove

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Propagation through plans

There is little if any propagation through plans

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Propagation through plans

There is little if any propagation through plans

If in state i I cannot move the blank tile into position xthen in state i+1 I cannot make a move out of position x!

Example: If at time I cannot put the bunch of flowers on the table then at time +1 I cannot move the bunch of flowers from the table

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Propagation through plans

If in state i I cannot move the blank tile into position xthen in state i+1 I cannot make a move out of position x!

2]2[1]2[8]1[3]1[ fmovefmovefmovefmove

position of blank (zero) possible moves (swaps) move numbers 1 (1,2),(1,4) 1,2 2 (2,1),(2,3),(2,5) 3,4,5 3 (3,2),(3,6) 6,7 4 (4,1),(4,5),(4,7) 8,9,10 5 (5,2),(5,4),(5,6),(5,8) 11,12,13,14, 6 (6,3),(6,5),(6,9) 15,16,17 7 (7,4),(7,8) 18,19 8 (8,5),(8,7),(8,9) 20,21,22 9 (9,6),(9,8) 23,24

I call this a rippling constraint

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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

That’s all for now folks