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Young Won Lim12/23/16
A Sudoku Solver – Rules (2A)
● Richard Bird Implementation
Young Won Lim12/23/16
Copyright (c) 2016 Young W. Lim.
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Bird’s Sudoku Rules (2A) 3 Young Won Lim
12/23/16
Based on
Thinking Functionally with Haskell, R. Bird
https://wiki.haskell.org/Sudoku
http://cdsoft.fr/haskell/sudoku.html
https://gist.github.com/wvandyk/3638996
http://www.cse.chalmers.se/edu/year/2015/course/TDA555/lab3.html
Bird’s Sudoku Rules (2A) 4 Young Won Lim
12/23/16
Lawful laws (1)
rows . rows = idcols . cols = idboxs . boxs = id
ungroup. group = idgroup . ungroup = id
map ungroup . ungroup . map cols . group . ungroup . map cols . group . map group
map ungroup . ungroup . map cols .map cols . group . map group
map ungroup . ungroup . group . map group
Bird’s Sudoku Rules (2A) 5 Young Won Lim
12/23/16
Lawful laws (2)
map rows . expand = expand . rowsmap cols . expand = expand . colsmap boxs . expand = expand . boxes
map (map f) . cp = cp . map (map f)filter (all p) . cp = cp . map (filter p)
Bird’s Sudoku Rules (2A) 6 Young Won Lim
12/23/16
Lawful laws (1)
rows . rows = idcols . cols = idboxs . boxs = id
rows :: Matrix a -> [Row a]rows = id
cols :: Matrix a -> [Row a]cols [xs] = [[x] | x <- xs]cols (xs:xss) = zipWith (:) xs (cols xss)
boxs :: Matrix a -> [Row a]boxs = map ungroup . ungroup .
map cols . group . map group
Bird’s Sudoku Rules (2A) 7 Young Won Lim
12/23/16
rows.rows, cols.cols, boxs.boxs
a b c d
e f g h
i j k l
m n o p
a b e f
c d g h
i j m n
k l o p
boxs
colsa b c d
e f g h
i j k l
m n o p
a e i m
b f j n
c g k o
d h l p
a b c d
e f g h
i j k l
m n o p
a b c d
e f g h
i j k l
m n o p
rows
boxs
cols
a b c d
e f g h
i j k l
m n o p
rows
a b c d
e f g h
i j k l
m n o p
a b c d
e f g h
i j k l
m n o p
Bird’s Sudoku Rules (2A) 8 Young Won Lim
12/23/16
group and ungroup
ungroup. group = idgroup . ungroup = id
Bird’s Sudoku Rules (2A) 9 Young Won Lim
12/23/16
group . ungroup
[ [ [ ‘5’, ‘3’, ‘4’, ‘6’, ‘7’, ‘8’, ‘9’, ‘1’, ‘2’ ],
[ ‘6’, ‘7’, ‘2’, ‘1’, ‘9’, ‘5’, ‘3’, ‘4’, ‘8’ ],
[ ‘1’, ‘9’, ‘8’, ‘3’, ‘4’, ‘2’, ‘5’, ‘6’, ‘7’ ] ],
[ [ ‘8’, ‘5’, ‘9’, ‘7’, ‘6’, ‘1’, ‘4’, ‘2’, ‘3’ ],
[ ‘4’, ‘2’, ‘6’, ‘8’, ‘5’, ‘3’, ‘7’, ‘9’, ‘1’ ],
[ ‘7’, ‘1’, ‘3’, ‘9’, ‘2’, ‘4’, ‘8’, ‘5’, ‘6’ ] ],
[ [ ‘9’, ‘6’, ‘1’, ‘5’, ‘3’, ‘7’, ‘2’, ‘8’, ‘4’ ],
[ ‘2’, ‘8’, ‘7’, ‘4’, ‘1’, ‘9’, ‘6’, ‘3’, ‘5’ ],
[ ‘3’, ‘4’, ‘5’, ‘2’, ‘8’, ‘6’, ‘1’, ‘7’, ‘9’ ] ] ]
[ [ ‘5’, ‘3’, ‘4’, ‘6’, ‘7’, ‘8’, ‘9’, ‘1’, ‘2’ ],
[ ‘6’, ‘7’, ‘2’, ‘1’, ‘9’, ‘5’, ‘3’, ‘4’, ‘8’ ],
[ ‘1’, ‘9’, ‘8’, ‘3’, ‘4’, ‘2’, ‘5’, ‘6’, ‘7’ ],
[ ‘8’, ‘5’, ‘9’, ‘7’, ‘6’, ‘1’, ‘4’, ‘2’, ‘3’ ],
[ ‘4’, ‘2’, ‘6’, ‘8’, ‘5’, ‘3’, ‘7’, ‘9’, ‘1’ ],
[ ‘7’, ‘1’, ‘3’, ‘9’, ‘2’, ‘4’, ‘8’, ‘5’, ‘6’ ],
[ ‘9’, ‘6’, ‘1’, ‘5’, ‘3’, ‘7’, ‘2’, ‘8’, ‘4’ ],
[ ‘2’, ‘8’, ‘7’, ‘4’, ‘1’, ‘9’, ‘6’, ‘3’, ‘5’ ],
[ ‘3’, ‘4’, ‘5’, ‘2’, ‘8’, ‘6’, ‘1’, ‘7’, ‘9’ ] ]
ungroup
group
Bird’s Sudoku Rules (2A) 10 Young Won Lim
12/23/16
map ungroup . map group
[ [ ‘5’, ‘3’, ‘4’, ‘6’, ‘7’, ‘8’, ‘9’, ‘1’, ‘2’ ],
[ ‘6’, ‘7’, ‘2’, ‘1’, ‘9’, ‘5’, ‘3’, ‘4’, ‘8’ ],
[ ‘1’, ‘9’, ‘8’, ‘3’, ‘4’, ‘2’, ‘5’, ‘6’, ‘7’ ],
[ ‘8’, ‘5’, ‘9’, ‘7’, ‘6’, ‘1’, ‘4’, ‘2’, ‘3’ ],
[ ‘4’, ‘2’, ‘6’, ‘8’, ‘5’, ‘3’, ‘7’, ‘9’, ‘1’ ],
[ ‘7’, ‘1’, ‘3’, ‘9’, ‘2’, ‘4’, ‘8’, ‘5’, ‘6’ ],
[ ‘9’, ‘6’, ‘1’, ‘5’, ‘3’, ‘7’, ‘2’, ‘8’, ‘4’ ],
[ ‘2’, ‘8’, ‘7’, ‘4’, ‘1’, ‘9’, ‘6’, ‘3’, ‘5’ ],
[ ‘3’, ‘4’, ‘5’, ‘2’, ‘8’, ‘6’, ‘1’, ‘7’, ‘9’ ] ]
[ [ [‘5’, ‘3’, ‘4]’, [‘6’, ‘7’, ‘8’], [‘9’, ‘1’, ‘2’] ],
[ [‘6’, ‘7’, ‘2’], [‘1’, ‘9’, ‘5’], [‘3’, ‘4’, ‘8’] ],
[ [‘1’, ‘9’, ‘8’], [‘3’, ‘4’, ‘2’], [‘5’, ‘6’, ‘7’] ],
[ [‘8’, ‘5’, ‘9’], [‘7’, ‘6’, ‘1’], [‘4’, ‘2’, ‘3’] ],
[ [‘4’, ‘2’, ‘6’], [‘8’, ‘5’, ‘3’], [‘7’, ‘9’, ‘1’] ],
[ [‘7’, ‘1’, ‘3’], [‘9’, ‘2’, ‘4’], [‘8’, ‘5’, ‘6’] ],
[ [‘9’, ‘6’, ‘1’], [‘5’, ‘3’, ‘7’], [‘2’, ‘8’, ‘4’] ],
[ [‘2’, ‘8’, ‘7’], [‘4’, ‘1’, ‘9’], [‘6’, ‘3’, ‘5’] ],
[ [‘3’, ‘4’, ‘5’], [‘2’, ‘8’, ‘6’], [‘1’, ‘7’, ‘9’] ] ]
map group
map ungroup
Bird’s Sudoku Rules (2A) 11 Young Won Lim
12/23/16
map group and map ungroup
map ungroup . ungroup . group . map group
map ungroup . ungroup . group . map group
Bird’s Sudoku Rules (2A) 12 Young Won Lim
12/23/16
map ungroup . ungroup . group . map group
[ [ ‘5’, ‘3’, ‘4’, ‘6’, ‘7’, ‘8’, ‘9’, ‘1’, ‘2’ ],
[ ‘6’, ‘7’, ‘2’, ‘1’, ‘9’, ‘5’, ‘3’, ‘4’, ‘8’ ],
[ ‘1’, ‘9’, ‘8’, ‘3’, ‘4’, ‘2’, ‘5’, ‘6’, ‘7’ ],
[ ‘8’, ‘5’, ‘9’, ‘7’, ‘6’, ‘1’, ‘4’, ‘2’, ‘3’ ],
[ ‘4’, ‘2’, ‘6’, ‘8’, ‘5’, ‘3’, ‘7’, ‘9’, ‘1’ ],
[ ‘7’, ‘1’, ‘3’, ‘9’, ‘2’, ‘4’, ‘8’, ‘5’, ‘6’ ],
[ ‘9’, ‘6’, ‘1’, ‘5’, ‘3’, ‘7’, ‘2’, ‘8’, ‘4’ ],
[ ‘2’, ‘8’, ‘7’, ‘4’, ‘1’, ‘9’, ‘6’, ‘3’, ‘5’ ],
[ ‘3’, ‘4’, ‘5’, ‘2’, ‘8’, ‘6’, ‘1’, ‘7’, ‘9’ ] ]
[ [ [‘5’, ‘3’, ‘4]’, [‘6’, ‘7’, ‘8’], [‘9’, ‘1’, ‘2’] ],
[ [‘6’, ‘7’, ‘2’], [‘1’, ‘9’, ‘5’], [‘3’, ‘4’, ‘8’] ],
[ [‘1’, ‘9’, ‘8’], [‘3’, ‘4’, ‘2’], [‘5’, ‘6’, ‘7’] ],
[ [‘8’, ‘5’, ‘9’], [‘7’, ‘6’, ‘1’], [‘4’, ‘2’, ‘3’] ],
[ [‘4’, ‘2’, ‘6’], [‘8’, ‘5’, ‘3’], [‘7’, ‘9’, ‘1’] ],
[ [‘7’, ‘1’, ‘3’], [‘9’, ‘2’, ‘4’], [‘8’, ‘5’, ‘6’] ],
[ [‘9’, ‘6’, ‘1’], [‘5’, ‘3’, ‘7’], [‘2’, ‘8’, ‘4’] ],
[ [‘2’, ‘8’, ‘7’], [‘4’, ‘1’, ‘9’], [‘6’, ‘3’, ‘5’] ],
[ [‘3’, ‘4’, ‘5’], [‘2’, ‘8’, ‘6’], [‘1’, ‘7’, ‘9’] ] ]
[ [ [ [‘5’, ‘3’, ‘4]’, [‘6’, ‘7’, ‘8’], [‘9’, ‘1’, ‘2’] ],
[ [‘6’, ‘7’, ‘2’], [‘1’, ‘9’, ‘5’], [‘3’, ‘4’, ‘8’] ],
[ [‘1’, ‘9’, ‘8’], [‘3’, ‘4’, ‘2’], [‘5’, ‘6’, ‘7’] ] ],
[ [ [‘8’, ‘5’, ‘9’], [‘7’, ‘6’, ‘1’], [‘4’, ‘2’, ‘3’] ],
[ [‘4’, ‘2’, ‘6’], [‘8’, ‘5’, ‘3’], [‘7’, ‘9’, ‘1’] ],
[ [‘7’, ‘1’, ‘3’], [‘9’, ‘2’, ‘4’], [‘8’, ‘5’, ‘6’] ] ],
[ [ [‘9’, ‘6’, ‘1’], [‘5’, ‘3’, ‘7’], [‘2’, ‘8’, ‘4’] ],
[ [‘2’, ‘8’, ‘7’], [‘4’, ‘1’, ‘9’], [‘6’, ‘3’, ‘5’] ],
[ [‘3’, ‘4’, ‘5’], [‘2’, ‘8’, ‘6’], [‘1’, ‘7’, ‘9’] ] ] ]
map group group
map ungroup ungroup
group . map group
map ungroup . ungroup .
Bird’s Sudoku Rules (2A) 13 Young Won Lim
12/23/16
boxs.boxs = id
boxs . boxs = id
boxs :: Matrix a -> [Row a]boxs = map ungroup . ungroup .
map cols . group . map group
map ungroup . ungroup . map cols . group . map group .map ungroup . ungroup . map cols . group . map group = id
Bird’s Sudoku Rules (2A) 14 Young Won Lim
12/23/16
boxs.boxs = id
map ungroup . ungroup . map cols . group . map group .map ungroup . ungroup . map cols . group . map group =
map ungroup . ungroup . map cols . group . ungroup . map cols . group . map group = .map ungroup . ungroup . map cols . map cols . group . map group =
map ungroup . ungroup . group . map group = id
Bird’s Sudoku Rules (2A) 15 Young Won Lim
12/23/16
Lawful laws (2)
map rows . expand = expand . rowsmap cols . expand = expand . colsmap boxs . expand = expand . boxes
expand :: Matrix Choices -> [Grid]expand = cp . map cp
cp . map cp = [ [[a]] ] -> [ [[a]] ]
Matrix [Digit][Matrix Digit]expand
Matrix Choices[Grid]
Bird’s Sudoku Rules (2A) 16 Young Won Lim
12/23/16
Lawful laws (2)
map rows . expand = expand . rowsmap cols . expand = expand . colsmap boxs . expand = expand . boxes
Matrix [Digit][Matrix Digit]expand
Matrix Choices[Grid]row
expandmap rowMatrix [Digit][Matrix Digit]
Matrix Choices[Grid]
Bird’s Sudoku Rules (2A) 17 Young Won Lim
12/23/16
Lawful laws (2)
[ [ [‘1’..’9’], [‘1’..’9’], [‘4’], [‘1’..’9’], [‘1’..’9’], [‘5’], [‘7’], [‘1’..’9’], [‘1’..’9’] ],
[ [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘9’], [‘4’], [‘1’..’9’], [‘1’..’9’] ],
[ [‘3’], [‘6’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘8’] ],
[ [‘7’], [‘2’], [‘4’], [‘1’..’9’], [‘6’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’] ],
[ [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘4’], [‘1’..’9’], [‘2’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’] ],
[ [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘8’], [‘1’..’9’], [‘1’..’9’], [‘9’], [‘3’] ],
[ [‘4’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘5’], [‘6’] ],
[ [‘1’..’9’], [‘1’..’9’], [‘5’], [‘3’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’] ],
[ [‘1’..’9’], [‘1’..’9’], [‘6’], [‘1’], [‘1’..’9’], [‘1’..’9’], [‘9’], [‘1’..’9’], [‘1’..’9’] ] ]
map cp
cp (
map
cp
)
[ cp [ [‘1’..’9’], [‘1’..’9’], [‘4’], [‘1’..’9’], [‘1’..’9’], [‘5’], [‘7’], [‘1’..’9’], [‘1’..’9’] ],
cp [ [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘9’], [‘4’], [‘1’..’9’], [‘1’..’9’] ],
cp [ [‘3’], [‘6’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘8’] ],
cp [ [‘7’], [‘2’], [‘4’], [‘1’..’9’], [‘6’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’] ],
cp [ [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘4’], [‘1’..’9’], [‘2’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’] ],
cp [ [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘8’], [‘1’..’9’], [‘1’..’9’], [‘9’], [‘3’] ],
cp [ [‘4’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘5’], [‘6’] ],
cp [ [‘1’..’9’], [‘1’..’9’], [‘5’], [‘3’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’], [‘1’..’9’] ],
cp [ [‘1’..’9’], [‘1’..’9’], [‘6’], [‘1’], [‘1’..’9’], [‘1’..’9’], [‘9’], [‘1’..’9’], [‘1’..’9’] ] ]
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Bird’s Sudoku Rules (2A) 18 Young Won Lim
12/23/16
map row . expand
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,
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,
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,map row .
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,
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,
row row row
Bird’s Sudoku Rules (2A) 19 Young Won Lim
12/23/16
Single-Cell Expansion
> prune :: Matrix Choices -> Matrix Choices> prune => pruneBy boxs . pruneBy cols . pruneBy rows> where pruneBy f = f . map pruneRow . f
> pruneRow :: Row Choices -> Row Choices> pruneRow row = map (remove ones) row> where ones = [d | [d] <- row]
Bird’s Sudoku Rules (2A) 20 Young Won Lim
12/23/16
Single-Cell Expansion
expand1 :: Matrix Choices -> [Matrix Choices]expand1 rows = [rows1 ++ [row1 ++ [c]:row2] ++ rows2 | c <- cs] where (rows1,row:rows2) = break (any smallest) rows (row1,cs:row2) = break smallest row smallest cs = length cs == n n = minimum (counts rows)
counts = filter (/=1) . map length . concat
Bird’s Sudoku Rules (2A) 21 Young Won Lim
12/23/16
Single-Cell Expansion
> solve2 :: Grid -> [Grid]> solve2 = search . choices
> search :: Matrix Choices -> [Grid]> search cm> |not (safe pm) = []> |complete pm = [map (map head) pm]> |otherwise = (concat . map search . expand1) pm> where pm = prune cm
> complete :: Matrix Choices -> Bool> complete = all (all single)
> single [_] = True> single _ = False
Bird’s Sudoku Rules (2A) 22 Young Won Lim
12/23/16
Single-Cell Expansion
> safe :: Matrix Choices -> Bool> safe cm = all ok (rows cm) &&> all ok (cols cm) &&> all ok (boxs cm)
> ok row = nodups [d | [d] <- row]
Young Won Lim12/23/16
References
[1] ftp://ftp.geoinfo.tuwien.ac.at/navratil/HaskellTutorial.pdf[2] https://www.umiacs.umd.edu/~hal/docs/daume02yaht.pdf
