# FSA Lab Exercises Week 3

> module FSAlab3
> where
> import Data.List
> import System.Random

This page documents the implementation of a Sudoku puzzle solver, starting from a more or less formal specification, using constraint resolution and depth first tree search. Next, a random generator for Sudoku problems is constructed, by randomly deleting values from a randomly generated Sudoku grid.

Your task is to carry out as many of the exercises as you can. Also, do not hesitate to make suggestions for improvement of the code.

Specifying Sudoku Solving

A Sudoku is a $$9 \times 9$$ matrix of numbers in $$\{ 1, \ldots, 9 \}$$, possibly including blanks, satisfying certain constraints. A Sudoku problem is a Sudoku containing blanks, but otherwise satisfying the Sudoku constraints. A Sudoku solver transforms the problem into a solution.

Exercise 1

Give a Hoare triple for a Sudoku solver. If the solver is called $$P$$, the Hoare triple consists of

$\{ \text{precondition} \} \ \ \ P \ \ \ \{ \text{postcondition} \}$

The precondition of the Sudoku solver is that the input is a correct Sudoku problem.

The postcondition of the Sudoku solver is that the transformed input is a solution to the initial problem.

State the pre- and postconditions as clearly and formally as possible.

If declarative specification is to be taken seriously, all there is to solving Sudokus is specifying what a Sudoku problem is. A Sudoku is a $$9 \times 9$$ matrix of numbers in $$\{ 1, \ldots, 9 \}$$ satisfying the following constraints:

• Every row should contain each number in $$\{ 1, \ldots, 9 \}$$

• Every column should contain each number in $$\{ 1, \ldots, 9 \}$$

• Every subgrid $$[i,j]$$ with $$i, j$$ ranging over $$1..3$$, $$4..6$$ and $$7..9$$ should contain each number in $$\{1, \ldots, 9 \}$$.

A Sudoku problem is a partial Sudoku matrix (a list of values in the matrix). A solution to a Sudoku problem is a complete extension of the problem, satisfying the Sudoku constraints.

A partial Sudoku should satisfy the following constraints:

• The values in every row should be in $$\{ 1, \ldots, 9 \}$$, and should all be different.

• The values in every column should be in $$\{ 1, \ldots, 9 \}$$, and should all be different.

• The values in every subgrid $$[i,j]$$ with $$i, j$$ ranging over $$1..3$$, $$4..6$$ and $$7..9$$ should be in $$\{1, \ldots, 9 \}$$, and should all be different.

       +-------+-------+-------+   +-------+-------+-------+
| 5 3   |   7   |       |   | 5 3 4 | 6 7 8 | 9 1 2 |
| 6     | 1 9 5 |       |   | 6 7 2 | 1 9 5 | 3 4 8 |
|   9 8 |       |   6   |   | 1 9 8 | 3 4 2 | 5 6 7 |
+-------+-------+-------+   +-------+-------+-------+
| 8     |   6   |     3 |   | 8 5 9 | 7 6 1 | 4 2 3 |
| 4     | 8   3 |     1 |   | 4 2 6 | 8 5 3 | 7 9 1 |
| 7     |   2   |     6 |   | 7 1 3 | 9 2 4 | 8 5 6 |
+-------+-------+-------+   +-------+-------+-------+
|   6   |       | 2 8   |   | 9 6 1 | 5 3 7 | 2 8 4 |
|       | 4 1 9 |     5 |   | 2 8 7 | 4 1 9 | 6 3 5 |
|       |   8   |   7 9 |   | 3 4 5 | 2 8 6 | 1 7 9 |
+-------+-------+-------+   +-------+-------+-------+

Here is an example problem, with a solution. This is the format we are going to use for display.

Sudoku constraints as injectivity requirements

To express the Sudoku constraints, we have to be able to express the property that a function is injective (or: one-to-one, or: an injection).

A function $$f: X \to Y$$ is an injection if it preserves distinctions: if $$x_1 \neq x_2$$ then $$f(x_1) \neq f(x_2)$$.

Equivalently: a function $$f: X \to Y$$ is injective if $$f (x_1) = f(x_2)$$ implies that $$x_1 = x_2$$.

Thus, we can represent a Sudoku as a matrix $$f[i,j]$$, satisfying:

• The members of each row should be all different. I.e., for every $$i$$, the function $$j \mapsto f[i,j]$$ should be injective (one to one). I.e., the following list of values should not have duplicates.
     [f [i,j] | j <- [1..9] ]
• The members of every column should be all different. I.e., for every $$j$$, the function $$i \mapsto f[i,j]$$ should be injective (one to one). I.e., the following list of values should not have duplicates.
      [f [i,j] | i <- [1..9] ]
• The members of every subgrid $$[i,j]$$ with $$i, j$$ ranging over $$1..3$$, $$4..6$$ and $$7..9$$ should be all different. I.e., the following list of values should not have duplicates, and similarly for the other subgrids.
       [f [i,j] | i <- [1..3], j <- [1..3] ]

Implementing a Sudoku Solver

The specification in the previous section suggests the following declarations:

> type Row    = Int
> type Column = Int
> type Value  = Int
> type Grid   = [[Value]]
>
> positions, values :: [Int]
> positions = [1..9]
> values    = [1..9]
>
> blocks :: [[Int]]
> blocks = [[1..3],[4..6],[7..9]]

Showing Sudoku stuff: use $$0$$ for a blank slot, so show $$0$$ as a blank. Showing a value:

> showVal :: Value -> String
> showVal 0 = " "
> showVal d = show d

Showing a row by sending it to the screen; not the type IO() for the result:

> showRow :: [Value] -> IO()
> showRow [a1,a2,a3,a4,a5,a6,a7,a8,a9] =
>  do  putChar '|'         ; putChar ' '
>      putStr (showVal a1) ; putChar ' '
>      putStr (showVal a2) ; putChar ' '
>      putStr (showVal a3) ; putChar ' '
>      putChar '|'         ; putChar ' '
>      putStr (showVal a4) ; putChar ' '
>      putStr (showVal a5) ; putChar ' '
>      putStr (showVal a6) ; putChar ' '
>      putChar '|'         ; putChar ' '
>      putStr (showVal a7) ; putChar ' '
>      putStr (showVal a8) ; putChar ' '
>      putStr (showVal a9) ; putChar ' '
>      putChar '|'         ; putChar '\n'

Showing a grid, i.e., a sequence of rows.

> showGrid :: Grid -> IO()
> showGrid [as,bs,cs,ds,es,fs,gs,hs,is] =
>  do putStrLn ("+-------+-------+-------+")
>     showRow as; showRow bs; showRow cs
>     putStrLn ("+-------+-------+-------+")
>     showRow ds; showRow es; showRow fs
>     putStrLn ("+-------+-------+-------+")
>     showRow gs; showRow hs; showRow is
>     putStrLn ("+-------+-------+-------+")

Sudoku Type

Define a Sudoku as a function from positions to values

> type Sudoku = (Row,Column) -> Value

Useful conversions:

> sud2grid :: Sudoku -> Grid
> sud2grid s =
>   [ [ s (r,c) | c <- [1..9] ] | r <- [1..9] ]
>
> grid2sud :: Grid -> Sudoku
> grid2sud gr = \ (r,c) -> pos gr (r,c)
>   where
>   pos :: [[a]] -> (Row,Column) -> a
>   pos gr (r,c) = (gr !! (r-1)) !! (c-1)

Showing a Sudoku

Show a Sudoku by displaying its grid:

> showSudoku :: Sudoku -> IO()
> showSudoku = showGrid . sud2grid

Picking the block of a position

> bl :: Int -> [Int]
> bl x = concat $filter (elem x) blocks  Picking the subgrid of a position in a Sudoku. > subGrid :: Sudoku -> (Row,Column) -> [Value] > subGrid s (r,c) = > [ s (r',c') | r' <- bl r, c' <- bl c ] Free Values Free values are available values at open slot positions. Free in a sequence are all values that have not yet been used. > freeInSeq :: [Value] -> [Value] > freeInSeq seq = values \\ seq  Free in a row are all values not yet used in that row. > freeInRow :: Sudoku -> Row -> [Value] > freeInRow s r = > freeInSeq [ s (r,i) | i <- positions ] Similarly for free in a column. > freeInColumn :: Sudoku -> Column -> [Value] > freeInColumn s c = > freeInSeq [ s (i,c) | i <- positions ] And for free in a subgrid. > freeInSubgrid :: Sudoku -> (Row,Column) -> [Value] > freeInSubgrid s (r,c) = freeInSeq (subGrid s (r,c)) The key notion The available values at a position are the values that are free in the row of that position, free in the column of that position, and free in the subgrid of that position. > freeAtPos :: Sudoku -> (Row,Column) -> [Value] > freeAtPos s (r,c) = > (freeInRow s r) > intersect (freeInColumn s c) > intersect (freeInSubgrid s (r,c))  Injectivity A list of values is injective if each value occurs only once in the list: > injective :: Eq a => [a] -> Bool > injective xs = nub xs == xs Injectivity Checks Check (the non-zero values on) the rows for injectivity. > rowInjective :: Sudoku -> Row -> Bool > rowInjective s r = injective vs where > vs = filter (/= 0) [ s (r,i) | i <- positions ] Check (the non-zero values on) the columns for injectivity. > colInjective :: Sudoku -> Column -> Bool > colInjective s c = injective vs where > vs = filter (/= 0) [ s (i,c) | i <- positions ] Check (the non-zero values on) the subgrids for injectivity. > subgridInjective :: Sudoku -> (Row,Column) -> Bool > subgridInjective s (r,c) = injective vs where > vs = filter (/= 0) (subGrid s (r,c)) Consistency Check Combine the injectivity checks defined above. > consistent :: Sudoku -> Bool > consistent s = and$
>                [ rowInjective s r |  r <- positions ]
>                 ++
>                [ colInjective s c |  c <- positions ]
>                 ++
>                [ subgridInjective s (r,c) |
>                     r <- [1,4,7], c <- [1,4,7]]

Sudoku Extension

Extend a Sudoku by filling in a value in a new position.

> extend :: Sudoku -> ((Row,Column),Value) -> Sudoku
> extend = update

Our well-known update function:

> update :: Eq a => (a -> b) -> (a,b) -> a -> b
> update f (y,z) x = if x == y then z else f x 

Search for a Sudoku Solution

A Sudoku constraint is a list of possible values for a particular position.

> type Constraint = (Row,Column,[Value])

Nodes in the search tree are pairs consisting of a Sudoku and the list of all empty positions in it, together with possible values for those positions, according to the constraints imposed by the Sudoku.

> type Node = (Sudoku,[Constraint])
>
> showNode :: Node -> IO()
> showNode = showSudoku . fst

Solution

A Sudoku is solved if there are no more empty slots.

> solved  :: Node -> Bool
> solved = null . snd

Successors in the Search Tree

The successors of a node are the nodes where the Sudoku gets extended at the next empty slot position on the list, using the values listed in the constraint for that position.

> extendNode :: Node -> Constraint -> [Node]
> extendNode (s,constraints) (r,c,vs) =
>    [(extend s ((r,c),v),
>      sortBy length3rd \$
>          prune (r,c,v) constraints) | v <- vs ]

prune removes the new value $$v$$ from the relevant constraints, given that $$v$$ now occupies position $$(r,c)$$. The definition of prune is given below.

Put constraints that are easiest to solve first

> length3rd :: (a,b,[c]) -> (a,b,[c]) -> Ordering
> length3rd (_,_,zs) (_,_,zs') = compare (length zs) (length zs')

Pruning

Prune values that are no longer possible from constraint list, given a new guess $$(r,c,v)$$ for the value of $$(r,c)$$.

> prune :: (Row,Column,Value)
>       -> [Constraint] -> [Constraint]
> prune _ [] = []
> prune (r,c,v) ((x,y,zs):rest)
>   | r == x = (x,y,zs\$v]) : prune (r,c,v) rest > | c == y = (x,y,zs\\[v]) : prune (r,c,v) rest > | sameblock (r,c) (x,y) = > (x,y,zs\\[v]) : prune (r,c,v) rest > | otherwise = (x,y,zs) : prune (r,c,v) rest > > sameblock :: (Row,Column) -> (Row,Column) -> Bool > sameblock (r,c) (x,y) = bl r == bl x && bl c == bl y  Initialisation Success is indicated by return of a unit node [n]. > initNode :: Grid -> [Node] > initNode gr = let s = grid2sud gr in > if (not . consistent) s then [] > else [(s, constraints s)] The open positions of a Sudoku are the positions with value $$0$$. > openPositions :: Sudoku -> [(Row,Column)] > openPositions s = [ (r,c) | r <- positions, > c <- positions, > s (r,c) == 0 ] Sudoku constraints, in a useful order Put the constraints with the shortest lists of possible values first. > constraints :: Sudoku -> [Constraint] > constraints s = sortBy length3rd > [(r,c, freeAtPos s (r,c)) | > (r,c) <- openPositions s ] Depth First Search The depth first search algorithm is completely standard. The goal property is used to end the search. > search :: (node -> [node]) > -> (node -> Bool) -> [node] -> [node] > search children goal [] = [] > search children goal (x:xs) > | goal x = x : search children goal xs > | otherwise = search children goal ((children x) ++ xs) Pursuing the Search > solveNs :: [Node] -> [Node] > solveNs = search succNode solved > > succNode :: Node -> [Node] > succNode (s,[]) = [] > succNode (s,p:ps) = extendNode (s,ps) p  Solving and showing the results This uses some monad operators: fmap and sequence. > solveAndShow :: Grid -> IO[()] > solveAndShow gr = solveShowNs (initNode gr) > > solveShowNs :: [Node] -> IO[()] > solveShowNs = sequence . fmap showNode . solveNs Examples > example1 :: Grid > example1 = [[5,3,0,0,7,0,0,0,0], > [6,0,0,1,9,5,0,0,0], > [0,9,8,0,0,0,0,6,0], > [8,0,0,0,6,0,0,0,3], > [4,0,0,8,0,3,0,0,1], > [7,0,0,0,2,0,0,0,6], > [0,6,0,0,0,0,2,8,0], > [0,0,0,4,1,9,0,0,5], > [0,0,0,0,8,0,0,7,9]] > example2 :: Grid > example2 = [[0,3,0,0,7,0,0,0,0], > [6,0,0,1,9,5,0,0,0], > [0,9,8,0,0,0,0,6,0], > [8,0,0,0,6,0,0,0,3], > [4,0,0,8,0,3,0,0,1], > [7,0,0,0,2,0,0,0,6], > [0,6,0,0,0,0,2,8,0], > [0,0,0,4,1,9,0,0,5], > [0,0,0,0,8,0,0,7,9]] > example3 :: Grid > example3 = [[1,0,0,0,3,0,5,0,4], > [0,0,0,0,0,0,0,0,3], > [0,0,2,0,0,5,0,9,8], > [0,0,9,0,0,0,0,3,0], > [2,0,0,0,0,0,0,0,7], > [8,0,3,0,9,1,0,6,0], > [0,5,1,4,7,0,0,0,0], > [0,0,0,3,0,0,0,0,0], > [0,4,0,0,0,9,7,0,0]] > example4 :: Grid > example4 = [[1,2,3,4,5,6,7,8,9], > [2,0,0,0,0,0,0,0,0], > [3,0,0,0,0,0,0,0,0], > [4,0,0,0,0,0,0,0,0], > [5,0,0,0,0,0,0,0,0], > [6,0,0,0,0,0,0,0,0], > [7,0,0,0,0,0,0,0,0], > [8,0,0,0,0,0,0,0,0], > [9,0,0,0,0,0,0,0,0]] > example5 :: Grid > example5 = [[1,0,0,0,0,0,0,0,0], > [0,2,0,0,0,0,0,0,0], > [0,0,3,0,0,0,0,0,0], > [0,0,0,4,0,0,0,0,0], > [0,0,0,0,5,0,0,0,0], > [0,0,0,0,0,6,0,0,0], > [0,0,0,0,0,0,7,0,0], > [0,0,0,0,0,0,0,8,0], > [0,0,0,0,0,0,0,0,9]] Exercise 2  +---------+---------+---------+ | | 3 | | | +-----|--+ +--|-----+ | | | | 7| | | 3 | | | 2 | | | | | | 8 | +---------+---------+---------+ | | 6 | | |5 | | | | +-----|--+ +--|-----+ | | 9 1 | 6 | | | +-----|--+ +--|-----+ | | 3 | | | 7 |1 | 2 | | +---------+---------+---------+ | | | | | | 3| 1 | | |8 | | 4 | | | | | +-----|--+ +--|-----+ | | 2 | | | +---------+---------+---------+ The goal of this exercise is to extend the Sudoku program described above with functions that can also handle Sudokus of a special kind: the Sudokus that appear in the Dutch evening newspaper NRC-Handelsblad each week (designed by Peter Ritmeester, from Oct 8, 2005 onward). These NRC Sudokus are special in that they have to satisfy a few extra constraints: in addition to the usual Sudoku constraints, each of the $$3 \times 3$$ subgrids with left-top corner (2,2), (2,6), (6,2), and (6,6) should also yield a surjective function. The above figure gives an example (this is the NRC sudoku that appeared Saturday Nov 26, 2005). Your task is to formalize this extra constraint, and to use your formalization in a program that can solve this Sudoku. See also the webpage of Andries Brouwer. Sudoku Generation An empty node is a Sudoku function that assigns $$0$$ everywhere, together with the trivial constraints that forbid nothing. > emptyN :: Node > emptyN = (\ _ -> 0,constraints (\ _ -> 0)) Get a random integer from the random generator: > getRandomInt :: Int -> IO Int > getRandomInt n = getStdRandom (randomR (0,n)) Pick a random member from a list; the empty list indicates failure. > getRandomItem :: [a] -> IO [a] > getRandomItem [] = return [] > getRandomItem xs = do n <- getRandomInt maxi > return [xs !! n] > where maxi = length xs - 1 Randomize a list. > randomize :: Eq a => [a] -> IO [a] > randomize xs = do y <- getRandomItem xs > if null y > then return [] > else do ys <- randomize (xs\\y) > return (head y:ys) > sameLen :: Constraint -> Constraint -> Bool > sameLen (_,_,xs) (_,_,ys) = length xs == length ys > getRandomCnstr :: [Constraint] -> IO [Constraint] > getRandomCnstr cs = getRandomItem (f cs) > where f [] = [] > f (x:xs) = takeWhile (sameLen x) (x:xs) > rsuccNode :: Node -> IO [Node] > rsuccNode (s,cs) = do xs <- getRandomCnstr cs > if null xs > then return [] > else return > (extendNode (s,cs\\xs) (head xs)) Find a random solution. > rsolveNs :: [Node] -> IO [Node] > rsolveNs ns = rsearch rsuccNode solved (return ns) > rsearch :: (node -> IO [node]) > -> (node -> Bool) -> IO [node] -> IO [node] > rsearch succ goal ionodes = > do xs <- ionodes > if null xs > then return [] > else > if goal (head xs) > then return [head xs] > else do ys <- rsearch succ goal (succ (head xs)) > if (not . null) ys > then return [head ys] > else if null (tail xs) then return [] > else > rsearch > succ goal (return  tail xs) > genRandomSudoku :: IO Node > genRandomSudoku = do [r] <- rsolveNs [emptyN] > return r > randomS = genRandomSudoku >>= showNode > uniqueSol :: Node -> Bool > uniqueSol node = singleton (solveNs [node]) where > singleton [] = False > singleton [x] = True > singleton (x:y:zs) = False Erase a position from a Sudoku. > eraseS :: Sudoku -> (Row,Column) -> Sudoku > eraseS s (r,c) (x,y) | (r,c) == (x,y) = 0 > | otherwise = s (x,y) Erase a position from a Node. > eraseN :: Node -> (Row,Column) -> Node > eraseN n (r,c) = (s, constraints s) > where s = eraseS (fst n) (r,c)  Return a minimal node with a unique solution by erasing positions until the result becomes ambiguous. > minimalize :: Node -> [(Row,Column)] -> Node > minimalize n [] = n > minimalize n ((r,c):rcs) | uniqueSol n' = minimalize n' rcs > | otherwise = minimalize n rcs > where n' = eraseN n (r,c) > filledPositions :: Sudoku -> [(Row,Column)] > filledPositions s = [ (r,c) | r <- positions, > c <- positions, s (r,c) /= 0 ] > genProblem :: Node -> IO Node > genProblem n = do ys <- randomize xs > return (minimalize n ys) > where xs = filledPositions (fst n) > main :: IO () > main = do [r] <- rsolveNs [emptyN] > showNode r > s <- genProblem r > showNode s Example output from this:  *FSAlab3> main +-------+-------+-------+ | 8 9 5 | 6 7 1 | 4 2 3 | | 7 4 2 | 9 5 3 | 8 6 1 | | 6 1 3 | 2 4 8 | 9 5 7 | +-------+-------+-------+ | 9 2 4 | 1 3 7 | 5 8 6 | | 5 8 7 | 4 9 6 | 1 3 2 | | 1 3 6 | 5 8 2 | 7 9 4 | +-------+-------+-------+ | 2 7 1 | 8 6 9 | 3 4 5 | | 4 6 9 | 3 1 5 | 2 7 8 | | 3 5 8 | 7 2 4 | 6 1 9 | +-------+-------+-------+ +-------+-------+-------+ | 5 | | | | 4 | 9 | 1 | | | 2 8 | 5 7 | +-------+-------+-------+ | | 3 | 5 8 6 | | 7 | 9 | 3 | | 3 | | 9 | +-------+-------+-------+ | | 8 | 3 | | 4 6 | | | | 5 8 | 7 | | +-------+-------+-------+ Exercise 3 Extend the code above to create a program that generates NRC Sudoku problems, that is, Sudoku problems satisfying the extra constraint explained in the NRC exercise above. Exercise 4 A Sudoku problem $$P$$ is minimal if it admits a unique solution, and every problem $$P'$$ that you can get from $$P$$ by erasing one of the hints admits more than one solution. How can you test whether the problems generated by the code given above are minimal? Exercise 5 Write a program that generates Sudoku problems with three empty blocks. Is it also possible to generate Sudoku problems with four empty blocks? Five? Exercise 6 Bertram Felgenhauer and Frazer Jarvis (Felgenhauer and Jarvis 2005) describe a computer program to enumerate the number of possible Sudoku grids. Confirm their results with an alternative program. Exercise 7 Can you find a way of classifying the difficulty of a Sudoku problem? Can you modify the Sudoku problem generator so that it can generate problems that are minimal, but easy to solve by hand? Problems that are minimal but hard to solve by hand? How can you test whether the problems your program generates satisfy these properties? Consult (Pelánek 2014). Exercise 8 Minimal problems for NRC Sudokus need fewer hints than standard Sudoku problems. Investigate the difference. What is the average number of hints in a minimal standard Sudoku problem? What is the average number of hints in a minimal NRC Sudoku problem? Exercise 9 Further constraints are possible. Let's investigate some. Let's say that a crossed Sudoku is a Sudoku satisfying the additional constraints that the values on the two diagonals are all different. Write a program that generates minimal problems for crossed Sudokus. How many hints do these problems contain, on average? Exercise 10 How about crossed NRC Sudokus or NRCX Sudokus: Sudokus satisfying both the NRC constraints and the diagonal constraints? Write a program that generates minimal problems for crossed NRC Sudokus. How many hints do these problems contain, on average? Can you generate examples with 9 hints? With less than 9 hints? Exercise 11 Andries Brouwer mentions on his NRC Sudoku webpage that there are \[ 6337174388428800$

different NRC Sudokus. Confirm this number with a program.

Exercise 12

How many NRCX Sudokus are there?

For further background, you might wish to read (Rosenhouse and Taalman 2011). Information about counting methods for Sudokus can be found in (Felgenhauer and Jarvis 2006) and (Russell and Jarvis 2006).

Note These exercises are open-ended. Don't worry if you do not manage to carry out all of them before the deadline.

Submission deadline is Monday evening, September 26th, at midnight.

Felgenhauer, Bertram, and Frazer Jarvis. 2005. “Enumerating Possible Sudoku Grids.” TU Dresden; University of Sheffield.

———. 2006. “Mathematics of Sudoku I.”

Pelánek, Radek. 2014. “Difficulty Rating of Sudoku Puzzles: An Overview and Evaluation.” arXiv:1403.7373.

Rosenhouse, Jason, and Laura Taalman. 2011. Taking Sudoku Seriously: The Math Behind the World’s Most Popular Pencil Puzzle. Oxford University Press.

Russell, Ed, and Frazer Jarvis. 2006. “Mathematics of Sudoku II.”