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Christophe Delord

**Monday 18. july 2016**: Updates on my new simulation framework project in Haskell.

**Friday 25. march 2016**: Dear backers, unfortunately, the FUN project was not successfully funded. I will now focus on FRP (Functional Reactive Programming) applied to real-time critical system specification and simulation.

CDSoft :: CV/Resume :: Free softwares Essays Haskell Handy Calc pp TPG BonaLuna Calculadoira todo pwd w Live :: AI tools in Prolog AI dialog

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.

Once I have been offered a snake puzzle. It’s made of 64 cubes of wood, some of them can turn. The goal is to fold this snake into a 4x4x4 cube.

After a while trying to solve this cube I decided to write a solver in Prolog. I present here an Haskell version of this solver.

The snake is made of 64 cubes. Cubes are joined in a way that the next cube is either in the same direction, either in a perpendicular direction. We will model theses constraints by a list of terms `F`

or `T`

:

`F`

: the next cube goes forward`T`

: the next cube “turns”

```
import Data.Array
import Control.Parallel
import Control.Parallel.Strategies
data SnakeSection = F | T deriving (Eq) -- Forward or Turn
snake :: [SnakeSection]
snake = [ F,F,T,T,F,T,T,T,
F,F,T,T,F,T,T,F,
T,T,F,T,T,T,T,T,
T,T,T,T,F,T,F,T,
T,T,T,T,T,F,T,F,
F,T,T,T,T,F,F,T,
T,F,T,T,T,T,T,T,
T,T,T,T,F,F,T ]
```

The cube is a 4x4x4 array of booleans. `True`

means the cell is occupied by the partial solution and `False`

means the cells is still available.

```
type Cube = Array (Int,Int,Int) Bool
type Position = (Int,Int,Int)
type Direction = (Int,Int,Int)
```

A solution is a list of terms indicating the direction to follow in the cube to fill it while walking throught the snake.

```
data Move = Forward | Backward | Left | Right | Up | Down deriving (Show)
type Solution = [Move]
```

The solver is a brute force backtracking solver. Given a partial solution, a current position and direction it tries all the possibilities and concat them. `solve`

returns a list of all the solutions. Thanks to the lazyness of Haskell we will only compute the first one. There are a lot of solutions because of symetries.

So the solver starts with:

- an empty partial solution
- a cube fill with the first snake cube
- at any positions in the cube
- in any directions

```
solve :: [SnakeSection] -> [Solution]
solve snake = concat [ solve [] (emptyCube//[(p,True)]) p d snake
| p <- r3D, d <- dirs
]
where
```

The size of the cube is $\sqrt[3]{1 + length(snake)}$^{1}. The cube is a 3D array. `i3D`

and `r3D`

are the coordinates of each small cubes.

```
size = round (fromIntegral (length snake + 1) ** (1/3))
i3D = ((1,1,1),(size,size,size))
r3D = range i3D
```

The initial empty cube is filled with `False`

values (no cube occupied yet).

```
emptyCube :: Cube
emptyCube = array i3D [(p,False) | p <- r3D]
```

Here is the real solver. There are two possibilities at each stage.

- if all the snake cubes have been placed in the cube, the partial solution is a complete.
- if a snake cube must still be placed, the solver tries continuing in all the possible directions from the current position and direction. A new position is possible only if it is in the big cube and if it is not yet occupied.

```
solve :: Solution -> Cube -> Position -> Direction -> [SnakeSection] -> [Solution]
solve path cube _ _ [] = [path]
solve path cube p d (s:ss) = concat [
solve (dp p p' : path) (cube//[(p',True)]) p' d' ss
| d' <- turn s d,
let p' = nextPos p d',
inRange i3D p', not (cube!p')
]
```

The recursive search can be performed in parallel on several cores. This is pretty easy in Haskell. `parL`

is a strategy that evaluates items in a list in parallel:

```
solve' :: Solution -> Cube -> Position -> Direction -> [SnakeSection] -> [Solution]
solve' path cube _ _ [] = [path]
solve' path cube p d (s:ss) = concat $ (if s==T then id else parL) [
solve' (dp p p' : path) (cube//[(p',True)]) p' d' ss
| d' <- turn s d,
let p' = nextPos p d',
inRange i3D p', not (cube!p')
]
```

Directions are 3D unit vectors describing the eight possible directions in the cube.

```
dirs :: [Direction]
dirs = [(-1,0,0), (1,0,0), (0,-1,0), (0,1,0), (0,0,-1), (0,0,1)]
```

`turn`

computes the next possible directions from the current position and direction.

- if the snake goes Forward, the only possible direction is the current one
- if the snake turns, the possible directions are perpendicular to the current one

```
turn :: SnakeSection -> Direction -> [Direction]
turn F d = [d]
turn T (_,0,0) = [d | d@(0,_,_) <- dirs]
turn T (0,_,0) = [d | d@(_,0,_) <- dirs]
turn T (0,0,_) = [d | d@(_,_,0) <- dirs]
```

Computing the next position is just a matter of adding vectors.

```
nextPos :: Position -> Direction -> Position
nextPos (x,y,z) (dx,dy,dz) = (x+dx, y+dy, z+dz)
```

A step in the solution is simply the move required to go from one position to the next one.

```
dp :: Position -> Position -> Move
dp (x,y,z) (x',y',z') | dx == 1 = Forward
| dx == -1 = Backward
| dy == 1 = Main.Right
| dy == -1 = Main.Left
| dz == 1 = Up
| dz == -1 = Down
where (dx, dy, dz) = (x'-x, y'-y, z'-z)
```

`parL`

is a strategy that evaluate items of a list in parallel. This fasten significally the search.

`parL = withStrategy (parList rseq)`

There are many solutions because of symetries. Let’s take only the first one. `main`

takes the first solution, enumerates and prints all the steps.

```
main = printSol $ zip [1..] $ reverse $ head $ solve snake
where printSol ((i,d):ds) = do
putStrLn (show i ++ ": " ++ show d)
printSol ds
printSol [] = return ()
```

It’s better to compile the script with ghc. The interpreted version is 20 times slower than the compiled one.

```
$ snake
1: Forward
2: Forward
3: Right
4: Backward
5: Backward
6: Up
7: Left
8: Forward
9: Forward
10: Forward
11: Down
12: Right
13: Right
14: Backward
15: Up
16: Up
17: Backward
18: Down
19: Down
20: Backward
21: Right
22: Forward
23: Up
24: Forward
25: Down
26: Forward
27: Up
28: Left
29: Left
30: Backward
31: Backward
32: Up
33: Backward
34: Right
35: Down
36: Right
37: Up
38: Up
39: Left
40: Left
41: Left
42: Down
43: Forward
44: Up
45: Right
46: Right
47: Right
48: Down
49: Forward
50: Forward
51: Left
52: Up
53: Left
54: Down
55: Backward
56: Left
57: Forward
58: Up
59: Backward
60: Right
61: Right
62: Right
63: Forward
```

The Haskell source code is here: snake.lhs

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