----------------------------------------------------------------------------- -- | -- Module : Data.SBV.Examples.Existentials.Diophantine -- Copyright : (c) Levent Erkok -- License : BSD3 -- Maintainer : erkokl@gmail.com -- Stability : experimental -- -- Finding minimal natural number solutions to linear Diophantine equations, -- using explicit quantification. ----------------------------------------------------------------------------- module Data.SBV.Examples.Existentials.Diophantine where import Data.SBV -------------------------------------------------------------------------------------------------- -- * Representing solutions -------------------------------------------------------------------------------------------------- -- | For a homogeneous problem, the solution is any linear combination of the resulting vectors. -- For a non-homogeneous problem, the solution is any linear combination of the vectors in the -- second component plus one of the vectors in the first component. data Solution = Homogeneous [[Integer]] | NonHomogeneous [[Integer]] [[Integer]] deriving Show -------------------------------------------------------------------------------------------------- -- * Solving diophantine equations -------------------------------------------------------------------------------------------------- -- | ldn: Solve a (L)inear (D)iophantine equation, returning minimal solutions over (N)aturals. -- The input is given as a rows of equations, with rhs values separated into a tuple. ldn :: [([Integer], Integer)] -> IO Solution ldn problem = do solution <- basis (map (map literal) m) if homogeneous then return $ Homogeneous solution else do let ones = [xs | (1:xs) <- solution] zeros = [xs | (0:xs) <- solution] return $ NonHomogeneous ones zeros where rhs = map snd problem lhs = map fst problem homogeneous = all (== 0) rhs m | homogeneous = lhs | True = zipWith (\x y -> -x : y) rhs lhs -- | Find the basis solution. By definition, the basis has all non-trivial (i.e., non-0) solutions -- that cannot be written as the sum of two other solutions. We use the mathematically equivalent -- statement that a solution is in the basis if it's least according to the lexicographic -- order using the ordinary less-than relation. (NB. We explicitly tell z3 to use the logic -- AUFLIA for this problem, as the BV solver that is chosen automatically has a performance -- issue. See: <https://z3.codeplex.com/workitem/88>.) basis :: [[SInteger]] -> IO [[Integer]] basis m = extractModels `fmap` allSatWith z3{useLogic = Just (PredefinedLogic AUFLIA)} cond where cond = do as <- mkExistVars n bs <- mkForallVars n return $ ok as &&& (ok bs ==> as .== bs ||| bnot (bs `less` as)) n = if null m then 0 else length (head m) ok xs = bAny (.> 0) xs &&& bAll (.>= 0) xs &&& bAnd [sum (zipWith (*) r xs) .== 0 | r <- m] as `less` bs = bAnd (zipWith (.<=) as bs) &&& bOr (zipWith (.<) as bs) -------------------------------------------------------------------------------------------------- -- * Examples -------------------------------------------------------------------------------------------------- -- | Solve the equation: -- -- @2x + y - z = 2@ -- -- We have: -- -- >>> test -- NonHomogeneous [[0,2,0],[1,0,0]] [[0,1,1],[1,0,2]] -- -- which means that the solutions are of the form: -- -- @(1, 0, 0) + k (0, 1, 1) + k' (1, 0, 2) = (1+k', k, k+2k')@ -- -- OR -- -- @(0, 2, 0) + k (0, 1, 1) + k' (1, 0, 2) = (k', 2+k, k+2k')@ -- -- for arbitrary @k@, @k'@. It's easy to see that these are really solutions -- to the equation given. It's harder to see that they cover all possibilities, -- but a moments thought reveals that is indeed the case. test :: IO Solution test = ldn [([2,1,-1], 2)] -- | A puzzle: Five sailors and a monkey escape from a naufrage and reach an island with -- coconuts. Before dawn, they gather a few of them and decide to sleep first and share -- the next day. At night, however, one of them awakes, counts the nuts, makes five parts, -- gives the remaining nut to the monkey, saves his share away, and sleeps. All other -- sailors do the same, one by one. When they all wake up in the morning, they again make 5 shares, -- and give the last remaining nut to the monkey. How many nuts were there at the beginning? -- -- We can model this as a series of diophantine equations: -- -- @ -- x_0 = 5 x_1 + 1 -- 4 x_1 = 5 x_2 + 1 -- 4 x_2 = 5 x_3 + 1 -- 4 x_3 = 5 x_4 + 1 -- 4 x_4 = 5 x_5 + 1 -- 4 x_5 = 5 x_6 + 1 -- @ -- -- We need to solve for x_0, over the naturals. We have: -- -- >>> sailors -- [15621,3124,2499,1999,1599,1279,1023] -- -- That is: -- -- @ -- * There was a total of 15621 coconuts -- * 1st sailor: 15621 = 3124*5+1, leaving 15621-3124-1 = 12496 -- * 2nd sailor: 12496 = 2499*5+1, leaving 12496-2499-1 = 9996 -- * 3rd sailor: 9996 = 1999*5+1, leaving 9996-1999-1 = 7996 -- * 4th sailor: 7996 = 1599*5+1, leaving 7996-1599-1 = 6396 -- * 5th sailor: 6396 = 1279*5+1, leaving 6396-1279-1 = 5116 -- * In the morning, they had: 5116 = 1023*5+1. -- @ -- -- Note that this is the minimum solution, that is, we are guaranteed that there's -- no solution with less number of coconuts. In fact, any member of @[15625*k-4 | k <- [1..]]@ -- is a solution, i.e., so are @31246@, @46871@, @62496@, @78121@, etc. sailors :: IO [Integer] sailors = do NonHomogeneous (xs:_) _ <- ldn [ ([1, -5, 0, 0, 0, 0, 0], 1) , ([0, 4, -5 , 0, 0, 0, 0], 1) , ([0, 0, 4, -5 , 0, 0, 0], 1) , ([0, 0, 0, 4, -5, 0, 0], 1) , ([0, 0, 0, 0, 4, -5, 0], 1) , ([0, 0, 0, 0, 0, 4, -5], 1) ] return xs