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tock-mirror/checks/ArrayUsageCheckTest.hs

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{-
Tock: a compiler for parallel languages
Copyright (C) 2007 University of Kent
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 2 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/>.
-}
module ArrayUsageCheckTest (ioqcTests) where
import Control.Monad.Identity
import Control.Monad.State
import Data.Array.IArray
import Data.List
import qualified Data.Map as Map
import Data.Maybe
import Data.Ord
import qualified Data.Set as Set
import Prelude hiding ((**),fail)
import Test.HUnit
import Test.QuickCheck hiding (check)
import ArrayUsageCheck
import qualified AST as A
import Metadata
import Omega
import ShowCode
import TestFramework
import TestHarness
import TestUtils
import UsageCheckUtils hiding (Var)
import Utils
instance Show FlattenedExp where
show fexp = runIdentity $ showFlattenedExp (return . showOccam) fexp
testArrayCheck :: Test
testArrayCheck = TestList
[
-- x_1 = 0
pass (0, [], [[0,1]], [])
-- x_1 = 0, 3x_1 >= 0 --> 0 >= 0
,pass (1, [[0,0]], [[0,1]], [[0,3]])
-- -7 + x_1 = 0
,pass (2, [], [[-7,1]], [])
-- x_1 = 9, 3 + 2x_1 >= 0 --> 21 >= 0
,pass (3, [[21,0]], [[-9,1]], [[3,2]])
-- x_1 + x_2 = 0, 4x_1 = 8, 2x_2 = -4
,pass (4, [], [[0,1,1], [-8,4,0], [4,0,2]], [])
-- - x_1 + x_2 = 0, 4x_1 = 8, 2x_2 = 4
,pass (5, [], [[0,-1,1], [-8,4,0], [-4,0,2]], [])
-- -x_1 = -9, 3 + 2x_1 >= 0 --> 21 >= 0
,pass (6, [[21,0]], [[9,-1]], [[3,2]])
-- From the Omega Test paper (x = x_1, y = x_2, z = x_3, sigma = x_1 (effectively)):
,pass (100, [[11,13,0,0], [28,-13,0,0], [47,-5,0,0], [53,5,0,0]], [[-17,7,12,31], [-7,3,5,14]],
[[-1,1,0,0], [40,-1,0,0], [50,0,1,0], [50,0,-1,0]])
-- Impossible/inconsistent equality constraints:
-- -7 = 0
,TestCase $ assertEqual "testArrayCheck 1002" (Nothing) (solveConstraints' [simpleArray [(0,7),(1,0)]] [])
-- x_1 = 3, x_1 = 4
,TestCase $ assertEqual "testArrayCheck 1003" (Nothing) (solveConstraints' [simpleArray [(0,-3),(1,1)], simpleArray [(0,-4),(1,1)]] [])
-- x_1 + x_2 = 0, x_1 + x_2 = -3
,TestCase $ assertEqual "testArrayCheck 1004" (Nothing) (solveConstraints' [simpleArray [(0,0),(1,1),(2,1)], simpleArray [(0,3),(1,1),(2,1)]] [])
-- 4x_1 = 7
,TestCase $ assertEqual "testArrayCheck 1005" (Nothing) (solveConstraints' [simpleArray [(0,-7),(1,4)]] [])
]
where
solveConstraints' = solveConstraints undefined
pass :: (Int, [[Integer]], [[Integer]], [[Integer]]) -> Test
pass (ind, expIneq, inpEq, inpIneq) = TestCase $ assertEqual ("testArrayCheck " ++ show ind)
(Just $ map arrayise expIneq) (transformMaybe snd $ solveConstraints' (map arrayise inpEq) (map arrayise inpIneq))
arrayise :: [Integer] -> Array Int Integer
arrayise = simpleArray . zip [0..]
-- Various helpers for easily creating equations.
-- Rules for writing equations:
-- * You must use the variables i, j, k in that order as you need them.
-- Never write an equation just involving i and k, or j and k. Always
-- use (i), (i and j), or (i and j and k).
-- * Constant scaling must always be on the left, and does not need the con
-- function. con 1 ** i won't compile.
-- Useful to make sure the equation types are not mixed up:
newtype HandyEq = Eq [(Int, Integer)] deriving (Show, Eq)
newtype HandyIneq = Ineq [(Int, Integer)] deriving (Show, Eq)
-- | The constraint for an arbitrary i,j that exist between low and high (inclusive)
-- and where i and j are distinct and i is taken to be the lower index.
i_j_constraint :: Integer -> Integer -> [HandyIneq]
i_j_constraint low high = [con low <== i, i ++ con 1 <== j, j <== con high]
-- The easy way of writing equations is built on the following Haskell magic.
-- Essentially, everything is a list of (index, coefficient). You can scale
-- with the ** operator, and you can form equalities and inequalities with
-- the ===, <== and >== operators. The type system saves you from doing anything
-- nonsensical. The other neat thing is that + is ++. An &&& operator is defined
-- for combining inequality lists.
leq :: [[(Int,Integer)]] -> [HandyIneq]
leq [] = []
leq [_] = []
leq (x:y:zs) = (x <== y) : (leq (y:zs))
(&&&) :: [a] -> [a] -> [a]
(&&&) = (++)
infixr 4 ===
infixr 4 <==
infixr 4 >==
infix 6 **
(===) :: [(Int,Integer)] -> [(Int,Integer)] -> HandyEq
lhs === rhs = Eq $ lhs ++ negateVars rhs
(<==) :: [(Int,Integer)] -> [(Int,Integer)] -> HandyIneq
lhs <== rhs = Ineq $ negateVars lhs ++ rhs
(>==) :: [(Int,Integer)] -> [(Int,Integer)] -> HandyIneq
lhs >== rhs = Ineq $ lhs ++ negateVars rhs
negateVars :: [(Int,Integer)] -> [(Int,Integer)]
negateVars = map (transformPair id negate)
(**) :: Integer -> [(Int,Integer)] -> [(Int,Integer)]
n ** var = map (transformPair id (* n)) var
con :: Integer -> [(Int,Integer)]
con c = [(0,c)]
i,j,k,m,n,p :: [(Int, Integer)]
i = [(1,1)]
j = [(2,1)]
k = [(3,1)]
m = [(4,1)]
n = [(5,1)]
p = [(6,1)]
-- Turns a list like [(i,3),(j,4)] into proper answers
answers :: [([(Int, Integer)],Integer)] -> Map.Map CoeffIndex Integer
answers = Map.fromList . map (transformPair (fst . head) id)
-- Shows the answers in terms of the test variables
showTestAnswers :: VariableMapping -> String
showTestAnswers vm = concat $ intersperse "\n" $ map showAnswer $ Map.assocs vm
where
showAnswer :: (CoeffIndex,EqualityConstraintEquation) -> String
showAnswer (x,eq) = mylookup x ++ " = " ++ showItems eq
showItems :: EqualityConstraintEquation -> String
showItems eq = concat (intersperse " + " (filter (not . null) $ map showItem (assocs eq)))
showItem :: (CoeffIndex,Integer) -> String
showItem (k,a_k) | a_k == 0 = ""
| k == 0 = show a_k
| a_k == 1 = mylookup k
| otherwise = show a_k ++ mylookup k
mylookup :: CoeffIndex -> String
mylookup x = Map.findWithDefault "unknown" x lookupTable
lookupTable :: Map.Map CoeffIndex String
lookupTable = Map.fromList $ zip [1..] ["i","j","k","m","n","p"]
++ [ (n,"y_" ++ show n) | n <- [7..100]] -- needed for showing QuickCheck failures
showInequality :: InequalityConstraintEquation -> String
showInequality ineq = "0 <= " ++ zeroIfBlank (showItems ineq)
showInequalities :: InequalityProblem -> String
showInequalities ineqs = concat $ intersperse "\n" $ map showInequality ineqs
showEquality :: InequalityConstraintEquation -> String
showEquality eq = "0 = " ++ zeroIfBlank (showItems eq)
showEqualities :: InequalityProblem -> String
showEqualities eqs = concat $ intersperse "\n" $ map showEquality eqs
zeroIfBlank :: String -> String
zeroIfBlank s | null s = "0"
| otherwise = s
showProblem :: (EqualityProblem,InequalityProblem) -> String
showProblem (eqs,ineqs) = showEqualities eqs ++ "\n" ++ showInequalities ineqs
makeConsistent :: [HandyEq] -> [HandyIneq] -> (EqualityProblem, InequalityProblem)
makeConsistent eqs ineqs = (map ensure eqs', map ensure ineqs')
where
eqs' = map (\(Eq e) -> e) eqs
ineqs' = map (\(Ineq e) -> e) ineqs
ensure :: [(CoeffIndex, Integer)] -> EqualityConstraintEquation
ensure = accumArray (+) 0 (0, largestIndex)
largestIndex = maximum $ map (maximum . map fst) $ [[(0,0)]] ++ eqs' ++ ineqs'
-- | Returns Nothing if there is definitely no solution, or (Just ineq) if
-- further investigation is needed
solveAndPrune' :: VariableMapping -> EqualityProblem -> InequalityProblem -> Maybe (VariableMapping,InequalityProblem)
solveAndPrune' vm [] ineq = return (vm,ineq)
solveAndPrune' vm eq ineq = solveConstraints vm eq ineq >>= (seqPair . transformPair return pruneIneq) >>= (\(x,(y,z)) -> solveAndPrune' x y z)
solveAndPrune :: EqualityProblem -> InequalityProblem -> Maybe (VariableMapping,InequalityProblem)
solveAndPrune eq ineq = solveAndPrune' (defaultMapping maxVar) eq ineq
where
maxVar = if null eq && null ineq then 0 else
if null eq then snd $ bounds $ head ineq else snd $ bounds $ head eq
-- | A problem's "solveability"; essentially how much of the Omega Test do you have to
-- run before the result is known, and which result is it
data Solveability =
SolveEq (Map.Map CoeffIndex Integer)
-- ^ Solveable just by solving equalities and pruning.
-- In other words, solveAndPrune will give (Just [])
| ImpossibleEq -- ^ Definitely not solveable just from the equalities.
-- In other words, solveAndPrune will give Nothing
| SolveIneq -- ^ Reduceable to inequalities, where the inequalities (therefore) have a solution.
-- In other words, solveAndPrune will give (Just a) (a /= []),
-- and then feeding a through fmElimination will give back an inequality set
-- that can be fed into <SOME FUNCTION TODO> to give a possible solution
| ImpossibleIneq -- ^ The inequalities are impossible to solve.
-- In other words, solveAndPrune will give (Just a) (a /= []),
-- but feeding this through fmElimination will give Nothing.
-- TODO do we need an option where one variable is left in the inequalities?
deriving (Eq,Show)
check :: Solveability -> (Int,[HandyEq], [HandyIneq]) -> Test
check s (ind, eq, ineq) =
case s of
ImpossibleEq -> TestCase $ assertEqual testName Nothing sapped
SolveEq ans -> TestCase $ assertEqual testName (Just (ans,[]))
(transformMaybe (transformPair getCounterEqs id) sapped)
ImpossibleIneq -> TestCase $ assertEqual testName Nothing elimed
SolveIneq -> TestCase $ assertBool testName (isJust elimed) -- TODO check for a solution to the inequality
where problem = makeConsistent eq ineq
sapped = uncurry solveAndPrune problem
elimed = uncurry solveProblem problem
testName = "check " ++ show s ++ " " ++ show ind
++ "(VM after pruning was: " ++ showMaybe showTestAnswers (transformMaybe fst sapped) ++
", ineqs: " ++ showMaybe showInequalities (transformMaybe snd sapped) ++ ")"
testMakeEquations :: Test
testMakeEquations = TestLabel "testMakeEquations" $ TestList
[
test (0,[(Map.empty,[con 0 === con 1],leq [con 0,con 1,con 7] &&& leq [con 0,con 2,con 7])],
[intLiteral 1, intLiteral 2],intLiteral 8)
,test (1,[(i_mapping,[i === con 3],leq [con 0,con 3,con 7] &&& leq [con 0,i,con 7])],
[exprVariable "i",intLiteral 3],intLiteral 8)
,test (2,[(ij_mapping,[i === j],leq [con 0,i,con 7] &&& leq [con 0,j,con 7])],
[exprVariable "i",exprVariable "j"],intLiteral 8)
,test (3,[(ij_mapping,[i ++ con 3 === j],leq [con 0,i ++ con 3,con 7] &&& leq [con 0,j,con 7])],
[buildExpr $ Dy (Var "i") A.Add (Lit $ intLiteral 3),exprVariable "j"],intLiteral 8)
,test (4,[(ij_mapping,[2 ** i === j],leq [con 0,2 ** i,con 7] &&& leq [con 0,j,con 7])],
[buildExpr $ Dy (Var "i") A.Mul (Lit $ intLiteral 2),exprVariable "j"],intLiteral 8)
,test' (5, [(((0,[]),(1,[])), ijk_mapping, [j === k], leq [con 0, j, i ++ con (-1)] &&& leq [con 0, k, i ++ con (-1)])],
[exprVariable "j", exprVariable "k"], exprVariable "i")
-- Testing (i REM 3) vs (4)
,test' (10,[
(( (0,[XZero]), (1,[]) ) ,i_mod_mapping 3,
[con 0 === con 4, i === con 0], leq [con 0,con 0,con 7] &&& leq [con 0,con 4,con 7])
,(( (0,[XPos]), (1,[]) ), i_mod_mapping 3,
[i ++ 3 ** j === con 4], leq [con 0,con 4,con 7] &&& leq [con 0,i ++ 3 ** j,con 7] &&& [i >== con 1] &&& [j <== con 0] &&& leq [con 0, i ++ 3 ** j, con 2])
,(( (0,[XNeg]), (1,[]) ), i_mod_mapping 3,
[i ++ 3 ** j === con 4], leq [con 0,con 4,con 7] &&& leq [con 0,i ++ 3 ** j,con 7] &&& [i <== con (-1)] &&& [j >== con 0] &&& leq [con (-2), i ++ 3 ** j, con 0])
],[buildExpr $ Dy (Var "i") A.Rem (Lit $ intLiteral 3),intLiteral 4],intLiteral 8)
-- Testing ((3*i - 2*j REM 11) - 5) vs (i + j)
-- Expressed as ((2 * (i - j)) + i) REM 11 - 5, and i + j
,test' (11,[
(( (0,[XZero]), (1,[]) ), _3i_2j_mod_mapping 11,
[con (-5) === i ++ j, 3**i ++ (-2)**j === con 0], leq [con 0,con (-5),con 7] &&& leq [con 0,i ++ j,con 7])
,(( (0,[XPos]), (1,[]) ), _3i_2j_mod_mapping 11,
[3**i ++ (-2)**j ++ 11 ** k ++ con (-5) === i ++ j],
leq [con 0,i ++ j,con 7] &&& leq [con 0,3**i ++ (-2)**j ++ 11 ** k ++ con (-5),con 7]
&&& [3**i ++ (-2)**j >== con 1] &&& [k <== con 0] &&& leq [con 0, 3**i ++ (-2)**j ++ 11 ** k, con 10])
,(( (0,[XNeg]), (1,[]) ), _3i_2j_mod_mapping 11,
[3**i ++ (-2)**j ++ 11 ** k ++ con (-5) === i ++ j],
leq [con 0,i ++ j,con 7] &&& leq [con 0,3**i ++ (-2)**j ++ 11 ** k ++ con (-5),con 7]
&&& [3**i ++ (-2)**j <== con (-1)] &&& [k >== con 0] &&& leq [con (-10), 3**i ++ (-2)**j ++ 11 ** k, con 0])
],[buildExpr $
Dy (Dy (Dy (Dy (Lit $ intLiteral 2)
A.Mul (Dy (Var "i") A.Subtr (Var "j"))
)
A.Add (Var "i")
)
A.Rem (Lit $ intLiteral 11)
)
A.Subtr (Lit $ intLiteral 5)
,buildExpr $ Dy (Var "i") A.Add (Var "j")],intLiteral 8)
-- Testing i REM 2 vs (i + 1) REM 4
,test' (12,combine (0,1) (i_ip1_mod_mapping 2 4)
[ ([XZero],[XZero],[([con 0 === con 0],[]),rr_i_zero, rr_ip1_zero])
,([XZero],[XPos],[([con 0 === i ++ con 1 ++ 4**k],[]),rr_i_zero,rr_ip1_pos])
,([XZero],[XNeg],[([con 0 === i ++ con 1 ++ 4**k],[]),rr_i_zero,rr_ip1_neg])
,([XPos],[XZero],[([i ++ 2**j === con 0],[]),rr_i_pos,rr_ip1_zero])
,([XPos],[XPos],[([i ++ 2**j === i ++ con 1 ++ 4**k],[]),rr_i_pos,rr_ip1_pos])
,([XPos],[XNeg],[([i ++ 2**j === i ++ con 1 ++ 4**k],[]),rr_i_pos,rr_ip1_neg])
,([XNeg],[XZero],[([i ++ 2**j === con 0],[]),rr_i_neg,rr_ip1_zero])
,([XNeg],[XPos],[([i ++ 2**j === i ++ con 1 ++ 4**k],[]),rr_i_neg,rr_ip1_pos])
,([XNeg],[XNeg],[([i ++ 2**j === i ++ con 1 ++ 4**k],[]),rr_i_neg,rr_ip1_neg])
], [buildExpr $ Dy (Var "i") A.Rem (Lit $ intLiteral 2)
,buildExpr $ Dy (Dy (Var "i") A.Add (Lit $ intLiteral 1)) A.Rem (Lit $ intLiteral 4)
], intLiteral 8)
-- Testing i REM j vs 3
,test' (100,[
-- i = 0:
(((0,[XZero]),(1,[])),i_mod_j_mapping,
[con 0 === con 3, i === con 0], leq [con 0, con 0, con 7] &&& leq [con 0, con 3, con 7])
-- i positive, j positive, i REM j = i:
,(((0,[XPosYPosAZero]),(1,[])),i_mod_j_mapping,
[i === con 3], [i >== con 1] &&& leq [con 0, i, j ++ con (-1)] &&& leq [con 0, i, con 7] &&& leq [con 0, con 3, con 7])
-- i positive, j positive, i REM j = i + k:
,(((0,[XPosYPosANonZero]),(1,[])),i_mod_j_mapping,
[i ++ k === con 3], [i >== con 1, k <== (-1)**j] &&&
leq [con 0, i ++ k, j ++ con (-1)] &&& leq [con 0, i ++ k, con 7] &&& leq [con 0, con 3, con 7])
-- i positive, j negative, i REM j = i:
,(((0,[XPosYNegAZero]),(1,[])),i_mod_j_mapping,
[i === con 3], [i >== con 1] &&& leq [con 0, i, (-1)**j ++ con (-1)] &&& leq [con 0, i, con 7] &&& leq [con 0, con 3, con 7])
-- i positive, j negative, i REM j = i + k:
,(((0,[XPosYNegANonZero]),(1,[])),i_mod_j_mapping,
[i ++ k === con 3], [i >== con 1, k <== j] &&&
leq [con 0, i ++ k, (-1)**j ++ con (-1)] &&& leq [con 0, i ++ k, con 7] &&& leq [con 0, con 3, con 7])
-- i negative, j positive, i REM j = i:
,(((0,[XNegYPosAZero]),(1,[])),i_mod_j_mapping,
[i === con 3], [i <== con (-1)] &&& leq [(-1)**j ++ con 1, i, con 0] &&& leq [con 0, i, con 7] &&& leq [con 0, con 3, con 7])
-- i negative, j positive, i REM j = i + k:
,(((0,[XNegYPosANonZero]),(1,[])),i_mod_j_mapping,
[i ++ k === con 3], [i <== con (-1), k >== j] &&&
leq [(-1)**j ++ con 1, i ++ k, con 0] &&& leq [con 0, i ++ k, con 7] &&& leq [con 0, con 3, con 7])
-- i negative, j negative, i REM j = i:
,(((0,[XNegYNegAZero]),(1,[])),i_mod_j_mapping,
[i === con 3], [i <== con (-1)] &&& leq [j ++ con 1, i, con 0] &&& leq [con 0, i, con 7] &&& leq [con 0, con 3, con 7])
-- i negative, j negative, i REM j = i + k:
,(((0,[XNegYNegANonZero]),(1,[])),i_mod_j_mapping,
[i ++ k === con 3], [i <== con (-1), k >== (-1)**j] &&&
leq [j ++ con 1, i ++ k, con 0] &&& leq [con 0, i ++ k, con 7] &&& leq [con 0, con 3, con 7])
], [buildExpr $ Dy (Var "i") A.Rem (Var "j"), intLiteral 3], intLiteral 8)
-- i vs. i'
,testRep' (199,[(((0,[]),(0,[])),rep_i_mapping, [i === j],
leq [con 3, i, con 4] &&& leq [con 3, j, con 4] &&& [i <== j ++ con (-1)]
&&& leq [con 0, i, con 7] &&& leq [con 0, j, con 7])],
("i", intLiteral 3, intLiteral 2),[exprVariable "i"],intLiteral 8)
-- i vs. i'
,testRep' (200,[(((0,[]),(0,[])),rep_i_mapping, [i === j],
ij_16 &&& [i <== j ++ con (-1)]
&&& leq [con 0, i, con 7] &&& leq [con 0, j, con 7])],
("i", intLiteral 1, intLiteral 6),[exprVariable "i"],intLiteral 8)
-- i vs i' vs 3
,testRep' (201,
[(((0,[]),(0,[])),rep_i_mapping, [i === j],
ij_16 &&& [i <== j ++ con (-1)]
&&& leq [con 0, i, con 7] &&& leq [con 0, j, con 7])]
++ replicate 2 (((0,[]),(1,[])),rep_i_mapping,[i === con 3], leq [con 1,i, con 6] &&& leq [con 0, i, con 7] &&& leq [con 0, con 3, con 7])
++ [(((1,[]),(1,[])),rep_i_mapping,[con 3 === con 3],concat $ replicate 2 (leq [con 0, con 3, con 7]))]
,("i", intLiteral 1, intLiteral 6),[exprVariable "i", intLiteral 3],intLiteral 8)
-- i vs i + 1 vs i' vs i' + 1
,testRep' (202,[
(((0,[]),(1,[])),rep_i_mapping,[i === j ++ con 1],ij_16 &&& [i <== j ++ con (-1)] &&& leq [con 0, i, con 7] &&& leq [con 0, j ++ con 1, con 7])
,(((0,[]),(1,[])),rep_i_mapping,[i ++ con 1 === j],ij_16 &&& [i <== j ++ con (-1)] &&& leq [con 0, i ++ con 1, con 7] &&& leq [con 0, j, con 7])
,(((0,[]),(0,[])),rep_i_mapping,[i === j],ij_16 &&& [i <== j ++ con (-1)] &&& leq [con 0, i, con 7] &&& leq [con 0, j, con 7])
,(((1,[]),(1,[])),rep_i_mapping,[i === j],ij_16 &&& [i <== j ++ con (-1)] &&& leq [con 0, i ++ con 1, con 7] &&& leq [con 0, j ++ con 1, con 7])]
++ [(((0,[]),(1,[])),rep_i_mapping, [i === i ++ con 1], leq [con 1, i, con 6] &&& leq [con 1, i, con 6] &&& -- deliberate repeat
leq [con 0, i, con 7] &&& leq [con 0,i ++ con 1, con 7])]
,("i", intLiteral 1, intLiteral 6),[exprVariable "i", buildExpr $ Dy (Var "i") A.Add (Lit $ intLiteral 1)],intLiteral 8)
-- Only a constant:
,testRep' (210,[(((0,[]),(0,[])),rep_i_mapping,[con 4 === con 4],concat $ replicate 2 $ leq [con 0, con 4, con 7])]
,("i", intLiteral 1, intLiteral 6),[intLiteral 4],intLiteral 8)
-- i REM 3 vs i' REM 3
,testRep' (220,[
-- i REM 3 == 0 and i' REM 3 == 0
(((0,[XZero]),(0,[XZero])), rep_i_mod_mapping 3, [con 0 === con 0, i === con 0, j === con 0], ij_16 &&& [i <== j ++ con (-1)] &&& leq [con 0, con 0, con 7] &&& leq [con 0, con 0, con 7])
-- i REM 3 == 0 and i' >= 1
,(((0,[XZero]),(0,[XPos])), rep_i_mod_mapping 3, [con 0 === j ++ 3**m, i === con 0], ij_16 &&& [i <== j ++ con (-1)] &&& leq [con 0, con 0, con 7]
&&& leq [con 0, j ++ 3**m, con 7] &&& [m <== con 0, j >== con 1] &&& leq [con 0, j ++ 3**m, con 2])
-- i REM 3 == 0 and i' <= -1
,(((0,[XZero]),(0,[XNeg])), rep_i_mod_mapping 3, [con 0 === j ++ 3**m, i === con 0], ij_16 &&& [i <== j ++ con (-1)] &&& leq [con 0, con 0, con 7]
&&& leq [con 0, j ++ 3**m, con 7] &&& [m >== con 0, j <== con (-1)] &&& leq [con (-2), j ++ 3**m, con 0])
-- i >= 1 and i' REM 3 == 0
,(((0,[XPos]),(0,[XZero])), rep_i_mod_mapping 3, [i ++ 3**k === con 0, j === con 0], ij_16 &&& [i <== j ++ con (-1)] &&& leq [con 0, con 0, con 7]
&&& leq [con 0, i ++ 3**k, con 7] &&& [k <== con 0, i >== con 1] &&& leq [con 0, i ++ 3**k, con 2])
-- i >= 1 and i' >= 1
,(((0,[XPos]),(0,[XPos])), rep_i_mod_mapping 3, [i ++ 3**k === j ++ 3**m], ij_16 &&& [i <== j ++ con (-1)]
&&& leq [con 0, i ++ 3**k, con 7] &&& leq [con 0, j ++ 3**m, con 7]
&&& [m <== con 0, k <== con 0, i >== con 1, j >== con 1]
&&& leq [con 0, i ++ 3**k, con 2] &&& leq [con 0, j ++ 3**m, con 2])
-- i >= 1 and i' <= -1
,(((0,[XPos]),(0,[XNeg])), rep_i_mod_mapping 3, [i ++ 3**k === j ++ 3**m], ij_16 &&& [i <== j ++ con (-1)]
&&& leq [con 0, i ++ 3**k, con 7] &&& leq [con 0, j ++ 3**m, con 7]
&&& [m >== con 0, k <== con 0, i >== con 1, j <== con (-1)]
&&& leq [con 0, i ++ 3**k, con 2] &&& leq [con (-2), j ++ 3**m, con 0])
-- i <= -1 and i' REM 3 == 0
,(((0,[XNeg]),(0,[XZero])), rep_i_mod_mapping 3, [i ++ 3**k === con 0, j === con 0], ij_16 &&& [i <== j ++ con (-1)] &&& leq [con 0, con 0, con 7]
&&& leq [con 0, i ++ 3**k, con 7] &&& [k >== con 0, i <== con (-1)] &&& leq [con (-2), i ++ 3**k, con 0])
-- i <= - 1 and i' >= 1
,(((0,[XNeg]),(0,[XPos])), rep_i_mod_mapping 3, [i ++ 3**k === j ++ 3**m], ij_16 &&& [i <== j ++ con (-1)]
&&& leq [con 0, i ++ 3**k, con 7] &&& leq [con 0, j ++ 3**m, con 7]
&&& [m <== con 0, k >== con 0, i <== con (-1), j >== con 1]
&&& leq [con (-2), i ++ 3**k, con 0] &&& leq [con 0, j ++ 3**m, con 2])
-- i <= - 1 and i' <= -1
,(((0,[XNeg]),(0,[XNeg])), rep_i_mod_mapping 3, [i ++ 3**k === j ++ 3**m], ij_16 &&& [i <== j ++ con (-1)]
&&& leq [con 0, i ++ 3**k, con 7] &&& leq [con 0, j ++ 3**m, con 7]
&&& [m >== con 0, k >== con 0, i <== con (-1), j <== con (-1)]
&&& leq [con (-2), i ++ 3**k, con 0] &&& leq [con (-2), j ++ 3**m, con 0])
],("i", intLiteral 1, intLiteral 6),[buildExpr $ Dy (Var "i") A.Rem (Lit $ intLiteral 3)], intLiteral 8)
-- TODO test reads and writes are paired properly
-- TODO test background knowledge being used
]
where
-- These functions assume that you pair each list [x,y,z] as (x,y) (x,z) (y,z) in that order.
-- for more control use the test' and testRep' versions:
test :: (Integer,[(VarMap,[HandyEq],[HandyIneq])],[A.Expression],A.Expression) -> Test
test (ind, problems, exprs, upperBound) = test' (ind, zipWith (\n (vm,eq,ineq) -> (n,vm,eq,ineq)) (labelNums 0 $ length exprs) problems, exprs, upperBound)
-- The ordering for the original list [0,1,2] should be [(0,1),(0,2),(1,2)]
-- So take each number, pair it with each remaining number in order, then increase
labelNums :: Int -> Int -> [((Int,[a]),(Int,[a]))]
labelNums m n | m >= n = []
| otherwise = [((m,[]),(n',[])) | n' <- [(m + 1) .. n]] ++ labelNums (m + 1) n
makeParItems :: [A.Expression] -> ParItems ([A.Expression],[A.Expression])
makeParItems es = ParItems $ map (\e -> SeqItems [([e],[])]) es
lookup :: [A.Expression] -> (Int, a) -> (A.Expression, a)
lookup es (n,b) = (es !! n, b)
test' :: (Integer,[(((Int, [ModuloCase]),(Int,[ModuloCase])),VarMap,[HandyEq],[HandyIneq])],[A.Expression],A.Expression) -> Test
test' (ind, problems, exprs, upperBound) =
TestCase $ assertEquivalentProblems ("testMakeEquations " ++ show ind)
(map (\((a0,a1),b,c,d) -> ((lookup exprs a0, lookup exprs a1), b, makeConsistent c d)) problems)
=<< (checkRight $ makeEquations [] (makeParItems exprs) upperBound)
testRep' :: (Integer,[(((Int,[ModuloCase]), (Int,[ModuloCase])), VarMap,[HandyEq],[HandyIneq])],(String, A.Expression, A.Expression),[A.Expression],A.Expression) -> Test
testRep' (ind, problems, (repName, repFrom, repFor), exprs, upperBound) =
TestCase $ assertEquivalentProblems ("testMakeEquations " ++ show ind)
(map (\((a0,a1),b,c,d) -> ((lookup exprs a0, lookup exprs a1), b, makeConsistent c d)) problems)
=<< (checkRight $ makeEquations [] (RepParItem (A.For emptyMeta (simpleName repName) repFrom repFor) $ makeParItems exprs) upperBound)
pairLatterTwo (l,a,b,c) = (l,a,(b,c))
joinMapping :: [VarMap] -> ([HandyEq],[HandyIneq]) -> [(VarMap,[HandyEq],[HandyIneq])]
joinMapping vms (eq,ineq) = map (\vm -> (vm,eq,ineq)) vms
i_mapping :: VarMap
i_mapping = Map.singleton (Scale 1 $ (exprVariable "i",0)) 1
ij_mapping :: VarMap
ij_mapping = Map.fromList [(Scale 1 $ (exprVariable "i",0),1),(Scale 1 $ (exprVariable "j",0),2)]
ijk_mapping :: VarMap
ijk_mapping = Map.fromList [(Scale 1 $ (exprVariable "i",0),1),(Scale 1 $ (exprVariable "j",0),2),(Scale 1 $ (exprVariable "k",0),3)]
i_mod_mapping :: Integer -> VarMap
i_mod_mapping n = Map.fromList [(Scale 1 $ (exprVariable "i",0),1),(Modulo 1 (Set.singleton $ Scale 1 $ (exprVariable "i",0)) (Set.singleton $ Const n),2)]
i_mod_j_mapping :: VarMap
i_mod_j_mapping = Map.fromList [(Scale 1 $ (exprVariable "i",0),1),(Scale 1 $ (exprVariable "j",0),2),
(Modulo 1 (Set.singleton $ Scale 1 $ (exprVariable "i",0)) (Set.singleton $ Scale 1 $ (exprVariable "j",0)),3)]
_3i_2j_mod_mapping n = Map.fromList [(Scale 1 $ (exprVariable "i",0),1),(Scale 1 $ (exprVariable "j",0),2),
(Modulo 1 (Set.fromList [(Scale 3 $ (exprVariable "i",0)),(Scale (-2) $ (exprVariable "j",0))]) (Set.singleton $ Const n),3)]
-- i REM m, i + 1 REM n
i_ip1_mod_mapping m n = Map.fromList [(Scale 1 $ (exprVariable "i",0),1)
,(Modulo 1 (Set.singleton $ Scale 1 $ (exprVariable "i",0)) (Set.singleton $ Const m),2)
,(Modulo 1 (Set.fromList [Scale 1 $ (exprVariable "i",0), Const 1]) (Set.singleton $ Const n),3)
]
rep_i_mapping :: VarMap
rep_i_mapping = Map.fromList [((Scale 1 (exprVariable "i",0)),1), ((Scale 1 (exprVariable "i",1)),2)]
rep_i_mapping' :: VarMap
rep_i_mapping' = Map.fromList [((Scale 1 (exprVariable "i",0)),2), ((Scale 1 (exprVariable "i",1)),1)]
both_rep_i = joinMapping [rep_i_mapping, rep_i_mapping']
rep_i_mod_mapping :: Integer -> VarMap
rep_i_mod_mapping n = Map.fromList [((Scale 1 (exprVariable "i",0)),1), ((Scale 1 (exprVariable "i",1)),2)
,(Modulo 1 (Set.singleton $ Scale 1 $ (exprVariable "i",0)) (Set.singleton $ Const n),3)
,(Modulo 1 (Set.singleton $ Scale 1 $ (exprVariable "i",1)) (Set.singleton $ Const n),4)]
-- Helper functions for i REM 2 vs (i + 1) REM 4. Each one is a pair of equalities, inequalities
rr_i_zero = ([i === con 0], leq [con 0,con 0,con 7])
rr_ip1_zero = ([i ++ con 1 === con 0], leq [con 0,con 0,con 7])
rr_i_pos = ([], leq [con 0, i ++ 2**j, con 7] &&& [i >== con 1, j <== con 0] &&& leq [con 0, i ++ 2**j, con 1])
rr_ip1_pos = ([], leq [con 0, i ++ con 1 ++ 4**k, con 7] &&& [i ++ con 1 >== con 1, k <== con 0] &&& leq [con 0, i ++ con 1 ++ 4**k, con 3])
rr_i_neg = ([], leq [con 0, i ++ 2**j, con 7] &&& [i <== con (-1), j >== con 0] &&& leq [con (-1), i ++ 2**j, con 0])
rr_ip1_neg = ([], leq [con 0, i ++ con 1 ++ 4**k, con 7] &&& [i ++ con 1 <== con (-1), k >== con 0] &&& leq [con (-3), i ++ con 1 ++ 4**k, con 0])
combine :: (Int,Int) -> VarMap -> [([ModuloCase],[ModuloCase],[([HandyEq],[HandyIneq])])] -> [(((Int,[ModuloCase]),(Int,[ModuloCase])),VarMap,[HandyEq],[HandyIneq])]
combine (l0,l1) vm eq_ineqs = [(((l0,m0),(l1,m1)),vm,e,i) | (m0,m1,(e,i)) <- map (\(a,b,c) -> (a,b,transformPair concat concat $ unzip c)) eq_ineqs]
-- Helper functions for the replication:
ij_16 = leq [con 1, i, con 6] &&& leq [con 1, j, con 6]
testMakeEquation :: TestMonad m r => ([(((A.Expression, [ModuloCase]), (A.Expression, [ModuloCase])), VarMap,[HandyEq],[HandyIneq])],ParItems [A.Expression],A.Expression) -> m ()
testMakeEquation (problems, exprs, upperBound) =
assertEquivalentProblems ""
(map (\(x,y,z) -> (x, y, uncurry makeConsistent z)) $ map pairLatterTwo problems) =<< (checkRight $ makeEquations [] (transformParItems pairWithEmpty exprs) upperBound)
where
pairWithEmpty a = (a,[])
pairLatterTwo (l,a,b,c) = (l,a,(b,c))
-- TODO add background knowledge
-- TODO add replicators
newtype MakeEquationInput = MEI ([(((A.Expression, [ModuloCase]), (A.Expression, [ModuloCase])), VarMap,[HandyEq],[HandyIneq])],ParItems [A.Expression],A.Expression)
-- Show isn't very useful on QuickCheck failure in this case and just spams the screen:
instance Show MakeEquationInput where
show = const ""
instance Arbitrary MakeEquationInput where
arbitrary = generateEquationInput >>* MEI
frequency' :: [(Int, StateT s Gen a)] -> StateT s Gen a
frequency' items = do dist <- lift $ choose (0, (sum $ map fst items) - 1)
findDist dist items
where
findDist n ((sz, x):sxs)
| n < sz = x
| otherwise = findDist (n - sz) sxs
-- | The item corresponding to the
type GenEqItems = (A.Expression, [(CoeffIndex, Integer)])
-- exprDepth is only really used to stop the possible infinite recursion in the multiplied variable * expression.
-- All other recursions are barred by never recursing with specialAllowed = True (outside of the above item)
-- Generates a new variable, or multiplied variable pair
genNewItem :: Int -> Bool -> StateT VarMap Gen (GenEqItems, FlattenedExp)
genNewItem exprDepth specialAllowed
= do (exp, fexp, nextId) <- frequency' $
[(80, do m <- get
let nextId = 1 + maximum (0 : Map.elems m)
let exp = exprVariable $ "x" ++ show nextId
return (exp, Scale 1 (exp,0), nextId))
,(20, if exprDepth <= 1
then
do m <- get
let nextId = 1 + maximum (0 : Map.elems m)
let exp = A.Dyadic emptyMeta A.Mul (exprVariable $ "y" ++ show nextId) (exprVariable $ "y" ++ show nextId)
return (exp,Scale 1 (exp, 0), nextId)
else
do m <- get
((expToMult,_),_) <- genNewItem (exprDepth - 1) True
-- We deliberately overwrite the state here, because we don't need/want the items
-- produced in expToMult to be in the variable map (the real code won't bother
-- inserting them, only the multiplied item
put m
let nextId = 1 + maximum (0 : Map.elems m)
let exp = A.Dyadic emptyMeta A.Mul (exprVariable $ "y" ++ show nextId) expToMult
return (exp, Scale 1 (exp, 0), nextId)
)
] ++ if not specialAllowed then []
else [(10, do ((eT,iT),fT) <- genNewExp (exprDepth - 1) False True
((eB,iB),fB) <- genNewExp (exprDepth - 1) False True
m <- get
let nextId = 1 + maximum (0 : Map.elems m)
return (A.Dyadic emptyMeta A.Rem eT eB, Modulo 1 (errorOrRight $ makeExpSet fT) (errorOrRight $ makeExpSet fB), nextId)
),(10,do ((eT,iT),fT) <- genNewExp (exprDepth - 1) False True
((eB,iB),fB) <- genConst
m <- get
let nextId = 1 + maximum (0 : Map.elems m)
return (A.Dyadic emptyMeta A.Div eT eB, Divide 1 (errorOrRight $ makeExpSet fT) (Set.singleton fB), nextId)
)]
modify (Map.insert fexp nextId)
return ((exp, [(nextId,1)]), fexp)
errorOrRight :: Show a => Either a b -> b
errorOrRight (Left x) = error $ "Not Right: Left " ++ show x
errorOrRight (Right x) = x
genConst :: StateT VarMap Gen (GenEqItems, FlattenedExp)
genConst = do val <- lift $ choose (1, 10)
let exp = intLiteral val
return ((exp, [(0,val)]), Const val)
genNewExp :: Int -> Bool -> Bool -> StateT VarMap Gen (GenEqItems, [FlattenedExp])
genNewExp exprDepth specialAllowed constAllowed
= do num <- lift $ oneof $ map return [1,1,1,1,2,2,3] -- bias towards low numbers of items
items <- replicateM num $ frequency' [(if constAllowed then 20 else 0, maybeMult genConst),
(80, maybeMult $ genNewItem (exprDepth - 1) specialAllowed)]
return $ fromJust $ foldl join Nothing items
where
maybeMult :: StateT VarMap Gen (GenEqItems, FlattenedExp) -> StateT VarMap Gen (GenEqItems, FlattenedExp)
maybeMult x = do multOrNot <- lift $ oneof $ map return [-1,0,0,0,0,1] -- bias towards not multiplying (represented by zero)
unmult <- x
case multOrNot of
0 -> return unmult
sign -> do mult' <- lift $ choose (1 :: Integer,10)
let mult = sign * mult'
return $ transformPair
(transformPair (A.Dyadic emptyMeta A.Mul (intLiteral mult)) (map (transformPair id (* mult))))
(scaleEq mult) unmult
scaleEq :: Integer -> FlattenedExp -> FlattenedExp
scaleEq k (Const n) = Const (k * n)
scaleEq k (Scale n v) = Scale (k * n) v
scaleEq k (Modulo n t b) = Modulo (k * n) t b
scaleEq k (Divide n t b) = Divide (k * n) t b
join :: Maybe (GenEqItems, [FlattenedExp]) -> (GenEqItems,FlattenedExp) -> Maybe (GenEqItems, [FlattenedExp])
join Nothing (e,f) = Just (e,[f])
join (Just ((ex,ix),fxs)) ((ey,iy),fy) = Just ((A.Dyadic emptyMeta A.Add ex ey, ix ++ iy),fxs ++ [fy])
generateEquationInput :: Gen ([(((A.Expression,[ModuloCase]), (A.Expression,[ModuloCase])),VarMap,[HandyEq],[HandyIneq])],ParItems [A.Expression],A.Expression)
generateEquationInput
= do ((items, upper),vm) <- flip runStateT Map.empty
(do upper <- frequency' [(80, genConst >>* fst), (20, genNewExp 4 False True >>* fst)]
itemCount <- lift $ choose (1,5)
items <- replicateM itemCount (genNewExp 2 True True)
return (items, upper)
)
return (makeResults vm items upper, ParItems $ map (\((x,_),_) -> SeqItems [[x]]) items, fst upper)
where
makeResults :: VarMap ->
[(GenEqItems, [FlattenedExp])] ->
GenEqItems ->
[(((A.Expression,[ModuloCase]), (A.Expression,[ModuloCase])),VarMap,[HandyEq],[HandyIneq])]
makeResults vm items upper = concatMap (flip (makeResult vm) upper) (allPairs items)
makeResult :: VarMap -> ((GenEqItems, [FlattenedExp]), (GenEqItems, [FlattenedExp])) -> GenEqItems ->
[(((A.Expression,[ModuloCase]), (A.Expression,[ModuloCase])),VarMap,[HandyEq],[HandyIneq])]
makeResult vm (((ex,x),fx),((ey,y),fy)) (_,u) = mkItem (ex, moduloEq vm fx) (ey, moduloEq vm fy)
where
mkItem :: (A.Expression, [([ModuloCase], [(CoeffIndex, Integer)], [HandyEq], [HandyIneq])]) ->
(A.Expression, [([ModuloCase], [(CoeffIndex, Integer)], [HandyEq], [HandyIneq])]) ->
[(((A.Expression,[ModuloCase]), (A.Expression,[ModuloCase])),VarMap,[HandyEq],[HandyIneq])]
mkItem (ex, xinfo) (ey, yinfo) = map (\(mx,my,eq,ineq) -> (((ex,mx),(ey,my)),vm,eq,ineq)) $ map (uncurry joinItems) (product2 (xinfo, yinfo))
joinItems :: ([ModuloCase],[(CoeffIndex, Integer)], [HandyEq], [HandyIneq]) ->
([ModuloCase],[(CoeffIndex, Integer)], [HandyEq], [HandyIneq]) ->
([ModuloCase], [ModuloCase], [HandyEq],[HandyIneq])
joinItems (mx, x, xEq, xIneq) (my, y, yEq, yIneq) = (mx, my, [x === y] &&& xEq &&& yEq, xIneq &&& yIneq &&& arrayBound x u &&& arrayBound y u)
arrayBound :: [(CoeffIndex, Integer)] -> [(CoeffIndex, Integer)] -> [HandyIneq]
arrayBound x u = leq [con 0, x, u ++ con (-1)]
moduloEq :: VarMap -> [FlattenedExp] -> [([ModuloCase], [(CoeffIndex, Integer)], [HandyEq], [HandyIneq])]
moduloEq vm es = foldl join [([],[],[],[])] (map (moduloEq' vm) es)
where
join :: [([ModuloCase], [(CoeffIndex, Integer)], [HandyEq], [HandyIneq])] ->
[([ModuloCase], [(CoeffIndex, Integer)], [HandyEq], [HandyIneq])] ->
[([ModuloCase], [(CoeffIndex, Integer)], [HandyEq], [HandyIneq])]
join xs ys = map (uncurry join') $ product2 (xs,ys)
join' :: ([ModuloCase], [(CoeffIndex, Integer)], [HandyEq], [HandyIneq]) ->
([ModuloCase], [(CoeffIndex, Integer)], [HandyEq], [HandyIneq]) ->
([ModuloCase], [(CoeffIndex, Integer)], [HandyEq], [HandyIneq])
join' (msx, isx, eqsx, ineqsx) (msy, isy, eqsy, ineqsy) = (msx ++ msy, isx ++ isy, eqsx ++ eqsy, ineqsx ++ ineqsy)
moduloEq' :: VarMap -> FlattenedExp -> [([ModuloCase], [(CoeffIndex, Integer)], [HandyEq], [HandyIneq])]
moduloEq' vm m@(Modulo n top bottom) =
let topVar = lookupFS (Set.toList top) vm
botVar = lookupFS (Set.toList bottom) vm
modVar = lookupF m vm
in case onlyConst (Set.toList bottom) of
Just c -> let v = topVar ++ (abs c)**modVar in
[ ([XZero], [(0,0)], [topVar === con 0], [])
, ([XPos], n**v, [], [topVar >== con 1, modVar <== con 0] &&& leq [con 0, v, con (abs c - 1)])
, ([XNeg], n**v, [], [topVar <== con (-1), modVar >== con 0] &&& leq [con (1 - abs c), v, con 0])
]
Nothing -> let v = topVar ++ modVar in
[ ([XZero], [(0,0)], [topVar === con 0], []) -- TODO stop the divisor being zero
, ([XPosYPosAZero], n**topVar, [], [topVar >== con 1] &&& leq [con 0, topVar, botVar ++ con (-1)])
, ([XPosYNegAZero], n**topVar, [], [topVar >== con 1] &&& leq [con 0, topVar, (-1)**botVar ++ con (-1)])
, ([XNegYPosAZero], n**topVar, [], [topVar <== con (-1)] &&& leq [(-1)**botVar ++ con 1, topVar, con 0])
, ([XNegYNegAZero], n**topVar, [], [topVar <== con (-1)] &&& leq [botVar ++ con 1, topVar, con 0])
, ([XPosYPosANonZero], n**v, [], [topVar >== con 1, modVar <== (-1)**botVar] &&& leq [con 0, v, botVar ++ con (-1)])
, ([XPosYNegANonZero], n**v, [], [topVar >== con 1, modVar <== botVar] &&& leq [con 0, v, (-1)**botVar ++ con (-1)])
, ([XNegYPosANonZero], n**v, [], [topVar <== con (-1), modVar >== botVar] &&& leq [(-1)**botVar ++ con 1, v, con 0])
, ([XNegYNegANonZero], n**v, [], [topVar <== con (-1), modVar >== (-1)**botVar] &&& leq [botVar ++ con 1, v, con 0])
]
moduloEq' vm m@(Divide n top bottom) =
[ ([XZero], [(0,0)], [topVar === con 0], [])
, ([XPos], n**divVar, [], [topVar >== con 1] &&& eqs (resultSignum True))
, ([XNeg], n**divVar, [], [topVar <== con (-1)] &&& eqs (resultSignum False))
]
where
topVar = lookupFS (Set.toList top) vm
divVar = lookupF m vm
c = fromJust $ onlyConst (Set.toList bottom)
v = topVar ++ (-c)**divVar
resultSignum xpos = signum c * (if xpos then 1 else -1)
-- TopSign BottomSign Bounds:
-- +++ +++ (0, c - 1)
-- +++ --- (c + 1, 0) (or: 1 - abs c, 0)
-- --- +++ (1 - c, 0)
-- --- --- (0, -1 - c) (or: (0, abs c - 1)
eqs sign = [sign**divVar >== con 0] &&& leq
(if signum c == sign
then [con 0, v, con (abs c - 1)]
else [con (1 - abs c), v, con 0])
moduloEq' vm exp = [([], lookupF exp vm, [], [])]
lookupFS :: [FlattenedExp] -> VarMap -> [(CoeffIndex, Integer)]
lookupFS es vm = concatMap (flip lookupF vm) es
lookupF :: FlattenedExp -> VarMap -> [(CoeffIndex, Integer)]
lookupF (Const c) _ = con c
lookupF f@(Scale a v) vm = [(fromJust $ Map.lookup f vm, a)]
-- We don't scale modulo directly here because the modulo variable is a or m,
-- which shouldn't be scaled
lookupF f@(Modulo a t b) vm = [(fromJust $ Map.lookup f vm, 1)]
lookupF f@(Divide a t b) vm = [(fromJust $ Map.lookup f vm, 1)]
qcTestMakeEquations :: [LabelledQuickCheckTest]
qcTestMakeEquations = [("Turning Code Into Equations", scaleQC (20,100,400,1000) prop)]
where
prop :: MakeEquationInput -> QCProp
prop (MEI mei) = testMakeEquation mei
testIndexes :: Test
testIndexes = TestList
[
check (SolveEq $ answers [(i,7)]) (0, [i === con 7], [])
,check (SolveEq $ answers [(i,6)]) (1, [2 ** i === con 12], [])
,check ImpossibleEq (2, [i === con 7],[i <== con 5])
-- Can i = j?
,check ImpossibleEq (3, [i === j], i_j_constraint 0 9)
-- Can (j + 1 % 10 == i + 1 % 10)?
,check ImpossibleIneq $ withKIsMod (i ++ con 1) 10 $ withNIsMod (j ++ con 1) 10 $ (4, [k === n], i_j_constraint 0 9)
-- Off by one (i + 1 % 9)
,check SolveIneq $ withKIsMod (i ++ con 1) 9 $ withNIsMod (j ++ con 1) 9 $ (5, [k === n], i_j_constraint 0 9)
-- The "nightmare" example from the Omega Test paper:
,check ImpossibleIneq (6,[],leq [con 27, 11 ** i ++ 13 ** j, con 45] &&& leq [con (-10), 7 ** i ++ (-9) ** j, con 4])
-- Solution is: i = 0, j = 0, k = 0
,check (SolveEq $ answers [(i,0),(j,0),(k,0)])
(7, [con 0 === i ++ j ++ k,
con 0 === 5 ** i ++ 4 ** j ++ 3 ** k,
con 0 === i ++ 6 ** j ++ 2 ** k]
, [con 1 >== i ++ 3 ** j ++ k,
con (-4) <== (-5) ** i ++ 2 ** j ++ k,
con 0 >== 4 ** i ++ (-7) ** j ++ (-13) ** k])
-- Solution is i = 0, j = 0, k = 4
,check (SolveEq $ answers [(i,0),(j,0),(k,4)])
(8, [con 4 === i ++ j ++ k,
con 12 === 5 ** i ++ 4 ** j ++ 3 ** k,
con 8 === i ++ 6 ** j ++ 2 ** k]
, [con 5 >== i ++ 3 ** j ++ k,
con 3 <== (-5) ** i ++ 2 ** j ++ k,
con (-52) >== 4 ** i ++ (-7) ** j ++ (-13) ** k])
-- Solution is: i = 0, j = 5, k = 4, but
-- this can't be determined from the equalities alone.
,check SolveIneq (9, [con 32 === 4 ** i ++ 4 ** j ++ 3 ** k,
con 17 === i ++ j ++ 3 ** k,
con 54 === 10 ** i ++ 10 ** j ++ k]
, [3 ** i ++ 8 ** j ++ 5 ** k >== con 60,
i ++ j ++ 3 ** k >== con 17,
5 ** i ++ j ++ 5 ** k >== con 25])
-- If we have (solution: 1,2):
-- 5 <= 5y - 4x <= 7
-- 9 <= 3y + 4x <= 11
-- Bounds on x:
-- Upper: 4x <= 5y - 5, 4x <= 11 - 3y
-- Lower: 5y - 7 <= 4x, 9 - 3y <= 4x
-- Dark shadow of x includes:
-- 4(11 - 3y) - 4(9 - 3y) >= 9, gives 8 >= 9.
-- Bounds on y:
-- Upper: 5y <= 7 + 4x, 3y <= 11 - 4x
-- Lower: 5 + 4x <= 5y, 9 - 4x <= 3y
-- Dark shadow of y includes:
-- 5(7 + 4x) - 5(5 + 4x) >= 16, gives 10 >= 16
-- So no solution to dark shadow, either way!
,check SolveIneq (10, [], leq [con 5, 5 ** i ++ (-4) ** j, con 7] &&& leq [con 9, 3 ** i ++ 4 ** j, con 11])
,safeParTest 100 True (0,10) [i]
,safeParTest 120 False (0,10) [i,i ++ con 1]
,safeParTest 140 True (0,10) [2 ** i, 2 ** i ++ con 1]
]
where
-- Given some indexes using "i", this function checks whether these can
-- ever overlap within the bounds given, and matches this against
-- the expected value; True for safe, False for unsafe.
safeParTest :: Int -> Bool -> (Integer,Integer) -> [[(Int,Integer)]] -> Test
safeParTest ind expSafe (low, high) usesI = TestCase $
(if expSafe
then assertEqual ("testIndexes " ++ show ind ++ " should be safe (unsolveable)") []
else assertNotEqual ("testIndexes " ++ show ind ++ " should be solveable") []
)
$ findSolveable $ zip3 [ind..] (equalityCombinations) (repeat constraint)
where
changeItoJ (1,n) = (2,n)
changeItoJ x = x
usesJ = map (map changeItoJ) usesI
constraint = i_j_constraint low high
equalityCombinations :: [[HandyEq]]
equalityCombinations = map (\(lhs,rhs) -> [lhs === rhs]) $ product2 (usesI,usesJ)
findSolveable :: [(Int, [HandyEq], [HandyIneq])] -> [(Int, [HandyEq], [HandyIneq])]
findSolveable = filter isSolveable
isSolveable :: (Int, [HandyEq], [HandyIneq]) -> Bool
isSolveable (ind, eq, ineq) = isJust $ (uncurry solveProblem) (makeConsistent eq ineq)
isMod :: [(Int,Integer)] -> [(Int,Integer)] -> Integer -> ([HandyEq], [HandyIneq])
isMod var@[(ind,1)] alpha divisor = ([alpha_minus_div_sigma === var], leq [con 0, alpha_minus_div_sigma, con $ divisor - 1])
where
alpha_minus_div_sigma = alpha ++ (negate divisor) ** sigma
sigma :: [(Int, Integer)]
sigma = [(ind+1,1)]
-- | Adds both k and m to the equation!
withKIsMod :: [(Int,Integer)] -> Integer -> (Int, [HandyEq], [HandyIneq]) -> (Int, [HandyEq], [HandyIneq])
withKIsMod alpha divisor (ind,eq,ineq) = let (eq',ineq') = isMod k alpha divisor in (ind,eq ++ eq',ineq ++ ineq')
-- | Adds both n and p to the equation!
withNIsMod :: [(Int,Integer)] -> Integer -> (Int, [HandyEq], [HandyIneq]) -> (Int, [HandyEq], [HandyIneq])
withNIsMod alpha divisor (ind,eq,ineq) = let (eq',ineq') = isMod n alpha divisor in (ind,eq ++ eq',ineq ++ ineq')
-- | Given one mapping and a second mapping, gives a function that converts the indexes
-- from one to the indexes of the next. If any of the keys in the map don't match
-- (i.e. if (keys m0 /= keys m1)) Nothing will be returned
generateMapping :: TestMonad m r => String -> VarMap -> VarMap -> m [(CoeffIndex,CoeffIndex)]
generateMapping msg m0 m1
= do testEqual ("Keys in variable mapping " ++ msg) (Map.keys m0) (Map.keys m1)
return $ Map.elems $ zipMap mergeMaybe m0 m1
-- | Given a forward mapping list, translates equations across
translateEquations :: forall m r. TestMonad m r =>
[(CoeffIndex,CoeffIndex)] -> (EqualityProblem, InequalityProblem) -> m (EqualityProblem, InequalityProblem)
translateEquations mp (eq,ineq)
= do testEqual "translateEquations mapping not one-to-one" (sort $ map fst mp) (sort $ map snd mp)
testCompareCustom "translateEquations input not square" (>=) 1 $ length $ nub $ map (snd . bounds) $ eq ++ ineq
eq' <- mapM swapColumns eq
ineq' <- mapM swapColumns ineq
return (eq', ineq')
where
swapColumns :: Array CoeffIndex Integer -> m (Array CoeffIndex Integer)
swapColumns arr
= do swapped <- mapM swapColumns' $ assocs arr
check arr swapped
return $ simpleArray swapped
where
swapColumns' :: (CoeffIndex,Integer) -> m (CoeffIndex,Integer)
swapColumns' (0,v) = return (0,v) -- Never swap the units column
swapColumns' (x,v)
= case find ((== x) . fst) mp of
Just (_,y) -> return (y,v)
Nothing -> testFailure "Could not find column to swap to" >> return undefined
check :: Show a => a -> [(CoeffIndex,Integer)] -> m ()
check x ies = if length ies == 1 + maximum (map fst ies) then return () else
testFailure $ "Error in translateEquations, not all indexes present after swap: " ++ show ies
++ " value beforehand was: " ++ show x ++ " mapping was: " ++ show mp
instance (ShowOccam a, Show b) => ShowOccam (a,b) where
showOccamM (x,y) = showOccamM x >>* (++ show y)
type Problem = (((A.Expression, [ModuloCase]), (A.Expression, [ModuloCase])), VarMap, (EqualityProblem, InequalityProblem))
-- | Asserts that the two problems are equivalent, once you take into account the potentially different variable mappings
assertEquivalentProblems :: forall m r. (TestMonad m r) => String -> [Problem] -> [Problem] -> m ()
assertEquivalentProblems title exp act
= do testEqualCustomShow (showListCustom $ showPairCustom showLabel showLabel) "Label sets not equal"
(map fst3 $ sortByLabels exp) (map fst3 $ sortByLabels act)
transformed <- mapM (uncurry $ transform $ showPairCustom showLabel showLabel) $ zip (sortByLabels exp) (sortByLabels act)
-- let transformedSortedZipped = map (transformPair id (zip . transformPair sortProblem sortProblem . unzip)) $ transformed
-- To give a more useful error on large problems we compare each item individually:
mapM_ test $ zip [0..] $ map (transformPair (\(e,a) -> showOccam e ++ " = " ++ showOccam a) id) transformed
testEqual (title ++ " Problems were not the same size") (length exp) (length act)
where
test :: (Int, (String, ((EqualityProblem, InequalityProblem), (EqualityProblem, InequalityProblem)))) -> m ()
test (n, (l, (eps, aps))) = testEqualCustomShow showProblem (title ++ " " ++ l ++ " #" ++ show n) eps aps
showLabel :: (A.Expression, [ModuloCase]) -> String
showLabel = showPairCustom showOccam show
showFunc :: (Int, [(EqualityProblem, InequalityProblem)]) -> String
showFunc = showPairCustom show $ showListCustom $ showProblem
fst3 :: (a,b,c) -> a
fst3 (a,_,_) = a
sortByLabels :: [Problem] -> [Problem]
sortByLabels = sortBy (comparing fst3) . map (\(es,b,c) -> (sortPair es, b, c))
sortPair :: Ord a => (a,a) -> (a, a)
sortPair (x,y) | x <= y = (x,y)
| otherwise = (y,x)
sortP :: (EqualityProblem, InequalityProblem) -> (EqualityProblem, InequalityProblem)
sortP (eq,ineq) = (sort $ map normaliseEquality eq, sort ineq)
transform :: Eq label => (label -> String) ->
(label, VarMap, (EqualityProblem, InequalityProblem)) ->
(label, VarMap, (EqualityProblem, InequalityProblem)) ->
m ( label, ((EqualityProblem, InequalityProblem), (EqualityProblem, InequalityProblem)))
transform s (el, vmexp, (e_eq, e_ineq)) (al, vmact, (a_eq, a_ineq))
= do testEqualCustomShow s "Labels did not match" el al
mapping <- generateMapping (showListCustom showOccam $ sort . nub $ map (fst . fst . fst3) exp ++ map (fst . snd . fst3) exp) (vmexp) (vmact)
translatedExp <- translateEquations mapping (resize e_eq, resize e_ineq)
return (el, (sortP translatedExp, sortP (resize a_eq, resize a_ineq)))
where
size = maximum $ map (snd . bounds) $ concat [e_eq, e_ineq, a_eq, a_ineq]
resize :: [Array CoeffIndex Integer] -> [Array CoeffIndex Integer]
resize = map (makeArraySize (0, size) 0)
pairPairs (xa,ya) (xb,yb) = ((xa,xb), (ya,yb))
sortProblem :: [(EqualityProblem, InequalityProblem)] -> [(EqualityProblem, InequalityProblem)]
sortProblem = sort
checkRight :: (Show a, TestMonad m r) => Either a b -> m b
checkRight (Left err) = testFailure ("Not Right: " ++ show err) >> return undefined
checkRight (Right x) = return x
-- QuickCheck tests for Omega Test:
-- The idea is to begin with a random list of integers, representing answers.
-- Combine this with a randomly generated matrix of coefficients for equalities
-- and the similar for inequalities. Correct all the unit coefficients such that
-- the equalities are true, and the inequalities should all resolve such that
-- LHS = RHS (and therefore they will be pruned out)
-- | We want to generate a solveable equation. Expressing our N equations as a matrix A (size: NxN),
-- we get: A . x = b, where b is our solution. The equations are solveable iff x = inv(A) . b
-- Or expressed another way, the equations are solveable iff A is nonsingular;
-- see http://mathworld.wolfram.com/LinearSystemofEquations.html A is singular if it
-- has determinant zero, therefore A is non-singular if the determinant is non-zero.
-- See http://mathworld.wolfram.com/Determinant.html for this.
--
-- At first I tried to simply check the determinant of a randomly generated matrix.
-- I implemented the standard naive algorithm, which is O(N!). Eeek! Reading the maths
-- more, a quicker way to do the determinant of a matrix M is to decompose it into
-- M = LU (where L is lower triangular, and U is upper triangular). Once you have
-- done this, you can use det M = det (LU) = (det L) . (det U) (from the Determinant page)
-- This is easier because det (A) where A is triangular, is simply the product
-- of its diagonal elements (see http://planetmath.org/encyclopedia/TriangularMatrix.html).
--
-- However, we don't need to do this the hard way. We just want to generate a matrix M
-- where its determinant is non-zero. If we imagine M = LU, then (det M) is non-zero
-- as long as (det L) is non-zero AND (det U) is non-zero. In turn, det L and det U are
-- non-zero as long as all their diagonal elements are non-zero. Therefore we just
-- need to randomly generate L and U (such that the diagonal elements are all non-zero)
-- and do M = LU.
--
-- Note that we should not take the shortcut of using just L or just U. This would
-- lead to trivially solveable linear equations, which would not test our algorithm well!
generateInvertibleMatrix :: Int -> Gen [[Integer]]
generateInvertibleMatrix size
= do u <- genUpper
l <- genLower
return $ l `multMatrix` u
where
ns = [0 .. size - 1]
-- | From http://mathworld.wolfram.com/MatrixMultiplication.html:
-- To multiply two square matrices of size N:
-- c_ik = sum (j in 0 .. N-1) (a_ij . b_jk)
-- Note that we begin our indexing at zero, because that's how !! works.
multMatrix a b = [[sum [((a !! i) !! j) * ((b !! j) !! k) | j <- ns] | k <- ns] | i <- ns]
genUpper :: Gen [[Integer]]
genUpper = mapM sequence [[
case i `compare` j of
EQ -> oneof [choose (-10,-1),choose (1,10)]
LT -> return 0
GT -> choose (-10,10)
| i <- ns] |j <- ns]
genLower :: Gen [[Integer]]
genLower = mapM sequence [[
case i `compare` j of
EQ -> oneof [choose (-10,-1),choose (1,10)]
GT -> return 0
LT -> choose (-10,10)
| i <- ns] |j <- ns]
-- | Given a solution, and the coefficients, work out the result.
-- Effectively the dot-product of the two lists.
calcUnits :: [Integer] -> [Integer] -> Integer
calcUnits a b = sum $ zipWith (*) a b
-- | Given the solution for an equation (values of x_1 .. x_n), generates
-- equalities and inequalities. The equalities will all be true and consistent,
-- the inequalities will all turn out to be equal. That is, when the inequalities
-- are resolved, the LHS will equal 0. Therefore we can generate the inequalities
-- using the same method as equalities. Also, the equalities are assured to be
-- distinct. If they were not distinct (one could be transformed into another by scaling)
-- then the equation set would be unsolveable.
generateEqualities :: [Integer] -> Gen (EqualityProblem, InequalityProblem)
generateEqualities solution = do eqCoeffs <- generateInvertibleMatrix num_vars
ineqCoeffs <- generateInvertibleMatrix num_vars
return (map mkCoeffArray eqCoeffs, map mkCoeffArray ineqCoeffs)
where
num_vars = length solution
mkCoeffArray coeffs = arrayise $ (negate $ calcUnits solution coeffs) : coeffs
-- | The input to a test that will have an exact solution after the equality problems have been
-- solved. All the inequalities will be simplified to 0 = 0. The answers to the equation are
-- in the map.
newtype OmegaTestInput = OMI (Map.Map CoeffIndex Integer,(EqualityProblem, InequalityProblem)) deriving (Show)
-- | Generates an Omega test problem with between 1 and 10 variables (incl), where the solutions
-- are numbers between -20 and + 20 (incl).
generateProblem :: Gen (Map.Map CoeffIndex Integer,(EqualityProblem, InequalityProblem))
generateProblem = choose (1,10) >>= (\n -> replicateM n $ choose (-20,20)) >>=
(\ans -> seqPair (return $ makeAns (zip [1..] ans),generateEqualities ans))
where
makeAns :: [(Int, Integer)] -> Map.Map CoeffIndex Integer
makeAns = Map.fromList
instance Arbitrary OmegaTestInput where
arbitrary = generateProblem >>* OMI
qcOmegaEquality :: [LabelledQuickCheckTest]
qcOmegaEquality = [("Omega Test Equality Solving", scaleQC (40,200,2000,10000) prop)]
where
prop :: OmegaTestInput -> QCProp
prop (OMI (ans,(eq,ineq))) = omegaCheck actAnswer
where
actAnswer = solveConstraints (defaultMapping $ Map.size ans) eq ineq
-- We use Map.assocs because pshow doesn't work on Maps
omegaCheck (Just (vm,ineqs)) = (True *==* all (all (== 0) . elems) ineqs)
*&&* ((Map.assocs ans) *==* (Map.assocs $ getCounterEqs vm))
omegaCheck Nothing = testFailure ("Found Nothing while expecting answer: " ++ show (eq,ineq))
-- | A randomly mutated problem ready for testing the inequality pruning.
-- The first part is the input to the pruning, and the second part is the expected result;
-- the remaining inequalities, preceding by a list of equalities.
type MutatedProblem =
(InequalityProblem
,Maybe ([EqualityConstraintEquation],InequalityProblem))
-- | The type for inside the function; easier to work with since it can't be
-- inconsistent until the end.
type MutatedProblem' =
(InequalityProblem
,[EqualityConstraintEquation]
,InequalityProblem)
-- | Given a distinct inequality list, mutates each one at random using one of these mutations:
-- * Unchanged
-- * Generates similar but redundant equations
-- * Generates its dual (to be transformed into an equality equation)
-- * Generates an inconsistent partner (rare - 20% chance of existing in the returned problem).
-- The equations passed in do not have to be consistent, merely unique and normalised.
-- Returns the input, and the expected output.
mutateEquations :: InequalityProblem -> Gen MutatedProblem
mutateEquations ineq = do (a,b,c) <- mapM mutate ineq >>*
foldl (\(a,b,c) (x,y,z) -> (a++x,b++y,c++z)) ([],[],[])
frequency
[
(80,return (a,Just (b,c)))
,(20,addInconsistent a >>* (\x -> (x,Nothing)))
]
where
-- We take an equation like 5 + 3x - y >=0 (i.e. 3x - y >= -5)
-- and add -6 -3x + y >= 0 (i.e. -6 >= 3x - y)
-- This works for all cases, even where the unit coeff is zero;
-- 3x - y >= 0 becomes -1 -3x + y >= 0 (i.e. -1 >= 3x - y)
addInconsistent :: InequalityProblem -> Gen InequalityProblem
addInconsistent inpIneq
= do randEq <- oneof (map return inpIneq)
let negEq = amap negate randEq
let modRandEq = (negEq) // [(0, (negEq ! 0) - 1)]
return (modRandEq : inpIneq)
mutate :: InequalityConstraintEquation -> Gen MutatedProblem'
mutate ineq = oneof
[
return ([ineq],[],[ineq])
,addRedundant ineq
,return $ addDual ineq
]
addDual :: InequalityConstraintEquation -> MutatedProblem'
addDual eq = ([eq,neg],[eq],[]) where neg = amap negate eq
addRedundant :: InequalityConstraintEquation -> Gen MutatedProblem'
addRedundant ineq = do i <- choose (1,5) -- number of redundant equations to add
newIneqs <- replicateM i addRedundant'
return (ineq : newIneqs, [], [ineq])
where
-- A redundant equation is one with a bigger unit coefficient:
addRedundant' = do n <- choose (1,100)
return $ ineq // [(0,n + (ineq ! 0))]
-- | Puts an equality into normal form. This is where the first non-zero coefficient is positive.
-- If all coefficients are zero, it doesn't matter (it will be equal to its negation)
normaliseEquality :: EqualityConstraintEquation -> EqualityConstraintEquation
normaliseEquality eq = case listToMaybe $ filter (/= 0) $ elems eq of
Nothing -> eq -- all zeroes
Just x -> amap (* (signum x)) eq
newtype OmegaPruneInput = OPI MutatedProblem deriving (Show)
instance Arbitrary OmegaPruneInput where
arbitrary = ((generateProblem >>* snd) >>= (return . snd) >>= mutateEquations) >>* OPI
qcOmegaPrune :: [LabelledQuickCheckTest]
qcOmegaPrune = [("Omega Test Pruning", scaleQC (100,1000,10000,50000) prop)]
where
--We perform the map assocs because we can't compare arrays using *==*
-- (toConstr fails in the pretty-printing!).
prop (OPI (inp,out)) = True
{-
case out of
Nothing -> Nothing *==* result
Just (expEq,expIneq) ->
case result of
Nothing -> mkFailResult $ "Expected success but got failure: " ++ pshow (inp,out)
Just (actEq,actIneq) ->
(sort (map assocs expIneq) *==* sort (map assocs actIneq))
*&&* ((sort $ map normaliseEquality expEq) *==* (sort $ map normaliseEquality actEq))
where
result = undefined -- TODO replace solveAndPrune: solveProblem [] inp
-}
ioqcTests :: IO (Test, [LabelledQuickCheckTest])
ioqcTests
= seqPair
(liftM (TestLabel "ArrayUsageCheckTest" . TestList) $ sequence
[
return testArrayCheck
,return testIndexes
,return testMakeEquations
,automaticTest "testcases/automatic/usage-check-1.occ.test"
,automaticTest "testcases/automatic/usage-check-2.occ.test"
,automaticTest "testcases/automatic/usage-check-3.occ.test"
,automaticTest "testcases/automatic/usage-check-4.occ.test"
,automaticTest "testcases/automatic/usage-check-5.occ.test"
]
,return $ qcOmegaEquality ++ qcOmegaPrune ++ qcTestMakeEquations)
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