02f6b8e77686be6898088519dba2bc2d1df31c3c
—
Florian Märkl
4 years ago

Initial Public Release

22 files changed,2317insertions(+),0deletions(-) A .gitignore A LICENSE A README.md A Setup.hs A app/Main.hs A doc/crackme/.gitignore A doc/crackme/crackme A doc/crackme/solve.hs A doc/paper.pdf A doc/slides.pdf A shida.cabal A src/BitVectorValue.hs A src/Common.hs A src/Flattening.hs A src/Formula.hs A src/MiniSat.hs A src/Propositional.hs A src/Solve.hs A stack.yaml A test/BitVectorValueTest.hs A test/SolveTest.hs A test/Spec.hs

A => .gitignore +5 -0

@@ 1,5 @@dist* out stack.yaml.lock .stack-work .ccls-cache

A => LICENSE +674 -0

@@ 1,674 @@GNU GENERAL PUBLIC LICENSE Version 3, 29 June 2007 Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/> Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The GNU General Public License is a free, copyleft license for software and other kinds of works. The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program--to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. You can apply it to your programs, too. When we speak of free software, we are referring to freedom, not price. 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A => README.md +64 -0

@@ 1,64 @@# 羊歯 Shida Shida is an experimental SMT solver for Bit Vectors written in Haskell that has been developed alongside a paper on the same topic (available in [doc](doc)) for the seminar [Automated Reasoning](https://www21.in.tum.de/teaching/sar/SS20/) at TUM. **Disclaimer**: If you are looking for an efficient, production quality solver, do not use this. Use something like [Z3](https://github.com/Z3Prover/z3) instead. ## Usage Build: ``` stack build ``` Run all tests: ``` stack test ``` The solver can be used conveniently in ghci, for example: ```Haskell $ stack repl λ> f = Atom $ (uVar 8 "a" :-: uConst (13::Word8)) :==: uConst (29::Word8) λ> f ((a - 0b00001101) = 0b00011101) λ> solve f Solution (fromList [("a",0b00101010)]) ``` More examples for formulas can be found in [test/SolveTest.hs](test/SolveTest.hs). The most important solving functions are: ```Haskell -- |Solve to a single solution solve :: Formula -> SolveResult Solution -- |Solve to all solutions as a lazy list solveAll :: Formula -> SolveResult [Solution] -- |Solve incrementally to a single solution -- This can be significantly faster or slower than solve depending on the -- formula, see the paper for more info. solveIncremental :: Int -> Formula -> SolveResult Solution ``` ## About Created by Florian Märkl. This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see <https://www.gnu.org/licenses/>.

A => Setup.hs +2 -0

@@ 1,2 @@import Distribution.Simple main = defaultMain

A => app/Main.hs +117 -0

@@ 1,117 @@module Main where import Data.Word import Data.Int --import Data.Maybe import Data.Bits import Data.Either import qualified BitVectorValue as BV import Common import Formula import Flattening import Solve example0 = (Atom $ uConst (42::Word8) :==: uVar 8 "a") :&&: (Atom $ uVar 8 "b" :==: (uVar 8 "a" :^: uConst (123::Word8))) example1 = Atom $ uVar 8 "a" :==: (uConst (13::Word8) :+: uConst (29::Word8)) bconjunction :: [Formula] -> Formula bconjunction [x, y] = And x y bconjunction (x : xs) = And x $ bconjunction xs bconjunction [] = undefined example2 :: Word8 -> Formula example2 x = bconjunction $ map (\i -> if ((x `shiftR` i) .&. 1) /= 0 then Atom $ Pick (fromIntegral i) (uVar 8 "a") else Not $ Atom $ Pick (fromIntegral i) (uVar 8 "a") ) [0..7] example3 :: Int8 -> Formula example3 x = bconjunction $ (if x == -128 then [] else [ Atom $ sConst (x-1) :<: sVar 8 "a" ]) ++ (if x == 127 then [] else [ Atom $ sVar 8 "a" :<: sConst (x+1) ]) exampleShift :: Word8 -> Word8 -> Formula exampleShift x s = Atom $ uVar 8 "a" :==: (uConst x :>>: Const Unsigned (BV.slice (BV.toBitVector s) 0 3)) exampleMult :: Word8 -> Word8 -> Formula exampleMult l r = Atom $ uVar 8 "a" :==: (uConst l :*: uConst r) exampleDiv :: Int8 -> Int8 -> Formula exampleDiv l r = Atom $ sVar 8 "a" :==: (sConst l :/: sConst r) exampleMod :: Int8 -> Int8 -> Formula exampleMod l r = Atom $ sVar 8 "a" :==: (sConst l :%: sConst r) exampleConcat :: Word8 -> Word8 -> Formula exampleConcat l r = Atom $ sVar 16 "a" :==: Concat Signed (uConst l) (uConst r) exampleTernary :: Bool -> Word8 -> Word8 -> Formula exampleTernary c a b = Atom $ uVar 8 "a" :==: Ternary (BConst c) (uConst a) (uConst b) exampleHardUnsat0 :: Size -> Formula exampleHardUnsat0 sz = Atom ((uVar sz "a" :*: uVar sz "b") :==: uVar sz "c") :&&: Not (Atom ((uVar sz "b" :*: uVar sz "a") :==: uVar sz "c")) :&&: Atom (uVar sz "x" :<: uVar sz "y") :&&: Atom (uVar sz "y" :<: uVar sz "x") exampleHardUnsat1 :: Size -> Formula exampleHardUnsat1 sz = Atom ((uVar sz "a" :*: uVar sz "b") :==: uVar sz "c") :&&: Atom ((uVar sz "b" :*: uVar sz "a") :==: uVar sz "c") :&&: Atom (uVar sz "x" :<: uVar sz "y") :&&: Atom (uVar sz "y" :<: uVar sz "x") exampleSat1 :: Size -> Formula exampleSat1 sz = Atom (uVar sz "a" :==: Const Unsigned (BV.replicate sz False)) :&&: Not ( Not (Atom $ (uVar sz "b" :*: uVar sz "c") :<: uVar sz "d") :&&: Not (Atom $ uVar sz "d" :<: (uVar sz "b" :*: uVar sz "c"))) main :: IO () main = do let x = 1::Int8 let y = -1::Int8 let f = Atom $ sVar 8 "a" :==: (sConst x :*: sConst y) putStrLn $ "Solve: " ++ show f putStrLn "-- Flattened:" let flat = fromRight undefined $ flatten f print flat putStrLn "-- Final Result:" print $ solve f --main :: IO () --main = do -- let x = -1::Int8 -- let y = 2::Int8 -- --print $ flatten $ exampleDiv x y -- putStrLn $ "Solve: " ++ show x ++ " / " ++ show y -- putStrLn $ "Result: " ++ show (solve $ exampleDiv x y) -- putStrLn $ "Remain: " ++ show (solve $ exampleMod x y) --main :: IO () --main = do -- let x = 1 -- let f = exampleHardUnsat1 64 -- example1 -- putStrLn $ "Solve: " ++ show f -- --putStrLn "" -- --putStrLn "-- Flattened:" -- --let flat = fromRight undefined $ flatten f -- --print flat -- --putStrLn "-- Propositional:" -- --print $ propositional flat -- putStrLn "" -- putStrLn "-- Final Result:" -- print $ solveIncremental 1 f -- --print $ solve f

A => doc/crackme/.gitignore +2 -0

@@ 1,2 @@*.c Makefile

A => doc/crackme/crackme +0 -0

A => doc/crackme/solve.hs +54 -0

@@ 1,54 @@import Common import Formula import Solve import qualified BitVectorValue as BV import Data.Maybe import Data.Word import qualified Data.Map as Map import qualified Data.ByteString as B import Control.Monad hash :: [Word8] hash = [0xa2, 0x35, 0xa3, 0x0f, 0x1c, 0xd0, 0x0e, 0x9e] -- |Combine a list of formulas into a single formula one using And conjunction :: [Formula] -> Formula conjunction [a] = a conjunction (a : as) = And a (conjunction as) conjunction _ = undefined -- |Variable name for the i-th character in the password pwCharVarName :: Int -> String pwCharVarName i = "pw[" ++ show i ++ "]" -- |Term for the i-th character in the password pwChar :: Int -> Term pwChar = uVar 8 . pwCharVarName formula :: Formula formula = conjunction $ map (\i -> let a = pwChar $ i b = pwChar $ (i + 1) `rem` 8 c = pwChar $ (i + 2) `rem` 8 d = pwChar $ (i + 3) `rem` 8 shiftPlus = Slice Unsigned 0 3 (b :+: c) shiftMinus = Slice Unsigned 0 3 (b :-: c) eight = Const Unsigned $ BV.pack [False, False, False] -- Only 3 bits, overflow on purpose for 8 - x term = ((a :<<: shiftPlus) :|: (a :>>: (eight :-: shiftMinus))) :-: d in (Atom $ term :==: uConst (hash!!i)) :&&: (Not $ Atom $ Pick 7 a) ) [0..7] main :: IO () main = case solveAll formula of Solution s -> forM_ s (\s -> putStrLn $ toEnum <$> fromIntegral <$> map (\i -> let bv = s Map.! (pwCharVarName i) in (fromJust (BV.fromBitVector bv))::Word8) [0..7] ) res -> print res

A => doc/paper.pdf +0 -0

A => doc/slides.pdf +0 -0

A => shida.cabal +62 -0

@@ 1,62 @@cabal-version: 1.12 name: shida version: 0.1.0.0 author: Florian Märkl license: GPL-3 license-file: LICENSE build-type: Simple extra-source-files: README.md source-repository head type: git location: https://github.com/thestr4ng3r/shida library exposed-modules: Common, BitVectorValue Formula Propositional Flattening Solve MiniSat hs-source-dirs: src build-depends: base >=4.7 && <5, bytestring, containers, mtl, transformers, minisat-solver default-language: Haskell2010 ghc-options: -W executable shida-exe main-is: Main.hs hs-source-dirs: app ghc-options: -threaded -rtsopts -with-rtsopts=-N build-depends: base >=4.7 && <5, shida, minisat-solver default-language: Haskell2010 test-suite shida-test type: exitcode-stdio-1.0 main-is: Spec.hs other-modules: BitVectorValueTest SolveTest hs-source-dirs: test ghc-options: -threaded -rtsopts -with-rtsopts=-N build-depends: base >=4.7 && <5, QuickCheck, shida, containers default-language: Haskell2010

A => src/BitVectorValue.hs +135 -0

@@ 1,135 @@module BitVectorValue ( Size, BoundsException, BitVectorValue, index, replicate, length, unpack, pack, ToBitVector, toBitVector, FromBitVector, fromBitVector, slice ) where import qualified Prelude as P import Prelude hiding (length,replicate) import Data.Word import Data.Int import Data.Bits import qualified Data.ByteString as B import Control.Exception import Data.Function ((&)) import Common data BoundsException = BoundsException deriving (Show) instance Exception BoundsException data BitVectorValue = BitVectorValue Size B.ByteString instance Eq BitVectorValue where (==) (BitVectorValue lsz ldat) (BitVectorValue rsz rdat) = lsz == rsz && foldl (\acc i -> acc && (index (BitVectorValue lsz ldat) i == index (BitVectorValue rsz rdat) i)) True [0..lsz-1] compareBits :: Size -> BitVectorValue -> BitVectorValue -> Ordering compareBits 0 a b = compare (index a 0) (index b 0) compareBits i a b = let r = compare (index a i) (index b i) in if r == EQ then compareBits (i-1) a b else r instance Ord BitVectorValue where compare a b = let szc = compare (length a) (length b) in if szc == EQ then compareBits (length a - 1) a b else szc instance Show BitVectorValue where show (BitVectorValue sz dat) = foldl (\acc i -> acc ++ (if index (BitVectorValue sz dat) i then "1" else "0")) "0b" $ reverse [0..sz-1] index :: BitVectorValue -> Size -> Bool index (BitVectorValue sz dat) i = if i >= sz then throw BoundsException else let byte = B.index dat $ fromIntegral $ i `shiftR` 3 biti = i .&. 0x7 in ((byte `shiftR` fromIntegral biti) .&. 1) == 1 replicate :: Size -> Bool -> BitVectorValue replicate sz v = BitVectorValue sz $ B.replicate (fromIntegral ((sz + 7) `shiftR` 3)) $ if v then 0xff else 0 length :: BitVectorValue -> Size length (BitVectorValue sz _) = sz unpack :: BitVectorValue -> [Bool] unpack bv = map (index bv) [0..length bv - 1] packByte :: Bool -> Bool -> Bool -> Bool -> Bool -> Bool -> Bool -> Bool -> Word8 packByte b0 b1 b2 b3 b4 b5 b6 b7 = foldr (\b acc -> (acc `shiftL` 1) .|. (if b then 1 else 0)) 0 [b0, b1, b2, b3, b4, b5, b6, b7] packBytes :: [Bool] -> [Word8] packBytes (b0 : b1 : b2 : b3 : b4 : b5 : b6 : b7 : bs) = packByte b0 b1 b2 b3 b4 b5 b6 b7 : packBytes bs packBytes [b0, b1, b2, b3, b4, b5, b6] = [packByte b0 b1 b2 b3 b4 b5 b6 False] packBytes [b0, b1, b2, b3, b4, b5] = [packByte b0 b1 b2 b3 b4 b5 False False] packBytes [b0, b1, b2, b3, b4] = [packByte b0 b1 b2 b3 b4 False False False] packBytes [b0, b1, b2, b3] = [packByte b0 b1 b2 b3 False False False False] packBytes [b0, b1, b2] = [packByte b0 b1 b2 False False False False False] packBytes [b0, b1] = [packByte b0 b1 False False False False False False] packBytes [b0] = [packByte b0 False False False False False False False] packBytes [] = [] pack :: [Bool] -> BitVectorValue pack bits = BitVectorValue (fromIntegral $ P.length bits) $ B.pack $ packBytes bits class ToBitVector a where toBitVector :: a -> BitVectorValue instance ToBitVector Word8 where toBitVector w = BitVectorValue 8 $ B.singleton w instance ToBitVector Word16 where toBitVector w = BitVectorValue 16 $ B.pack $ map fromIntegral [w .&. 0xff, (w `shiftR` 8) .&. 0xff] instance ToBitVector Word32 where toBitVector w = BitVectorValue 32 $ B.pack $ map fromIntegral [w .&. 0xff, (w `shiftR` 8) .&. 0xff, (w `shiftR` 0x10) .&. 0xff, (w `shiftR` 0x18) .&. 0xff] instance ToBitVector Word64 where toBitVector w = BitVectorValue 64 $ B.pack $ map fromIntegral [w .&. 0xff, (w `shiftR` 8) .&. 0xff, (w `shiftR` 0x10) .&. 0xff, (w `shiftR` 0x18) .&. 0xff, (w `shiftR` 0x20) .&. 0xff, (w `shiftR` 0x28) .&. 0xff, (w `shiftR` 0x30) .&. 0xff, (w `shiftR` 0x38) .&. 0xff] instance ToBitVector Int8 where toBitVector v = toBitVector (fromIntegral v :: Word8) instance ToBitVector Int16 where toBitVector v = toBitVector (fromIntegral v :: Word16) instance ToBitVector Int32 where toBitVector v = toBitVector (fromIntegral v :: Word32) instance ToBitVector Int64 where toBitVector v = toBitVector (fromIntegral v :: Word64) class FromBitVector a where fromBitVector :: BitVectorValue -> Maybe a instance FromBitVector Word8 where fromBitVector (BitVectorValue 8 v) = Just $ B.head v fromBitVector _ = Nothing slice :: BitVectorValue -> Size -> Size -> BitVectorValue slice v start sz = unpack v & drop (fromIntegral start) & take (fromIntegral sz) & pack \ No newline at end of file

A => src/Common.hs +13 -0

@@ 1,13 @@module Common where type Size = Word data BitVectorSign = Unsigned | Signed deriving (Eq, Ord) data BitVectorType = BitVectorType BitVectorSign Size deriving (Eq, Ord) instance Show BitVectorType where show (BitVectorType sign sz) = (if sign == Signed then "s" else "u") ++ show sz fixedPoint :: Eq a => (a -> a) -> a -> a fixedPoint f v = if r == v then v else fixedPoint f r where r = f v \ No newline at end of file

A => src/Flattening.hs +418 -0

@@ 1,418 @@module Flattening where import Text.Printf import Data.Map (Map) import qualified Data.Map as Map import qualified Data.Set as Set import Control.Monad import Data.Maybe import Control.Monad.Except import Control.Monad.State import qualified BitVectorValue as BV import Common import Formula import qualified Propositional as P import Propositional ((<->), (&&&), (|||), (^^^)) newtype FlattenState = FlattenState P.Identifier data FlattenError = TermTypeError Term | AtomTypeMismatch Atom | AtomPickBoundsError Atom deriving (Eq, Show) type Flattening a = StateT FlattenState (Except FlattenError) a runFlattening :: Flattening a -> FlattenState -> Either FlattenError a runFlattening m = runExcept . evalStateT m type PropVector = Size -> P.Formula data PropVectorVariable = PropVectorVariable P.Identifier Size instance Show PropVectorVariable where show (PropVectorVariable base sz) = "(" ++ show base ++ ":" ++ show sz ++ ")" variableVector :: PropVectorVariable -> PropVector variableVector (PropVectorVariable base sz) i = if i < sz then P.Var $ base + fromIntegral i else error "overflow in reserved prop vector" reserveProps :: Size -> Flattening PropVectorVariable reserveProps sz = do FlattenState nextPropId <- get put $ FlattenState (nextPropId + fromIntegral sz) return $ PropVectorVariable nextPropId sz class Reservable a where requiredProps :: a -> Flattening Size reserveVarFor :: (Reservable a) => a -> Flattening PropVectorVariable reserveVarFor x = requiredProps x >>= reserveProps reserveVarsForAll :: (Reservable a, Ord a, Foldable t) => t a -> Flattening (Map a PropVectorVariable) reserveVarsForAll = foldM (\map x -> do var <- reserveVarFor x return $ Map.insert x var map ) Map.empty curPropsCount :: Flattening P.Identifier curPropsCount = do FlattenState nextPropId <- get return nextPropId maybeToFlattening :: FlattenError -> Maybe a -> Flattening a maybeToFlattening _ (Just v) = return v maybeToFlattening e Nothing = throwError e getTermType :: Term -> Flattening BitVectorType getTermType t = maybeToFlattening (TermTypeError t) $ termType t skeleton :: (Atom -> PropVectorVariable) -> Formula -> P.Formula skeleton atomProps (Atom atom) = variableVector (atomProps atom) 0 -- This is simply using the atom's reserved variable skeleton atomProps (Not f) = P.Not $ skeleton atomProps f skeleton atomProps (And l r) = skeleton atomProps l &&& skeleton atomProps r bitwiseConstraintIff :: (Size -> P.Formula) -> (Term -> PropVectorVariable) -> Term -> Flattening P.Formula bitwiseConstraintIff op termProps t = do (BitVectorType _ sz) <- getTermType t let tProp = variableVector $ termProps t return $ P.conjunction $ map (\i -> P.Iff (tProp i) $ op i) [0..sz - 1] bitwiseConstraintBinary :: (P.Formula -> P.Formula -> P.Formula) -> (Term -> PropVectorVariable) -> Term -> Term -> Term -> Flattening P.Formula bitwiseConstraintBinary op termProps l r = let lProp = variableVector $ termProps l rProp = variableVector $ termProps r in bitwiseConstraintIff (\i -> op (lProp i) (rProp i)) termProps fullAdderSum :: P.Formula -> P.Formula -> P.Formula -> P.Formula fullAdderSum a b cin = (a ^^^ b) ^^^ cin -- (6.37) fullAdderCarry :: P.Formula -> P.Formula -> P.Formula -> P.Formula fullAdderCarry a b cin = (a &&& b) ||| ((a ^^^ b) &&& cin) -- (6.38) -- (6.39) and (6.42) adderCarry :: PropVector -> PropVector -> P.Formula -> PropVector adderCarry _ _ cin 0 = cin adderCarry l r cin i = fullAdderCarry (l (i-1)) (r (i-1)) cin adderCarryRec :: PropVector -> PropVector -> Bool -> PropVector adderCarryRec _ _ cin 0 = P.Const cin adderCarryRec l r cin i = adderCarry l r (adderCarryRec l r cin (i - 1)) i adder :: PropVector -> PropVector -> Bool -> Size -> Flattening ([P.Formula], PropVector) adder l r cin sz = do carriesPropVector <- reserveProps (sz-1) let carryProp = (\i -> if i == 0 then P.Const cin else variableVector carriesPropVector (i-1) ) :: PropVector adderCarryConstraints = map (\i -> carryProp i <-> adderCarry l r (carryProp $ i-1) i) [1..sz-1] return (adderCarryConstraints, \i -> fullAdderSum (l i) (r i) (carryProp i)) doubleIf :: P.Formula -> P.Formula -> P.Formula -> P.Formula -> P.Formula -> P.Formula doubleIf ifa resa ifb resb els = (ifa &&& resa) ||| (P.Not ifa &&& ifb &&& resb) ||| (P.Not ifa &&& P.Not ifb &&& els) singleIf :: P.Formula -> P.Formula -> P.Formula -> P.Formula singleIf ifa resa els = (ifa &&& resa) ||| (P.Not ifa &&& els) shift :: (Size -> Size -> Size) -> (Int -> Size -> Bool) -> PropVector -> PropVector -> Int -> PropVector shift _ _ l _ (-1) i = l i -- (6.48) shift dir cond l r s i | cond s i = doubleIf (r $ fromIntegral s) (shift dir cond l r (s - 1) (i `dir` (2^s))) (P.Not (r $ fromIntegral s)) (shift dir cond l r (s - 1) i) -- else (P.Const False) shift dir cond l r s i = -- (6.49) singleIf (P.Not (r $ fromIntegral s)) (shift dir cond l r (s - 1) i) -- else (P.Const False) shiftLStatic :: PropVector -> Size -> PropVector shiftLStatic x s i = if i < s then P.Const False else x (i - s) lessThanUnsigned :: PropVector -> PropVector -> Size -> P.Formula lessThanUnsigned l r sz = P.Not $ adderCarryRec l (P.Not . r) True (fromIntegral sz) -- (6.46) lessThanSigned :: PropVector -> PropVector -> PropVector lessThanSigned l r sz = l (fromIntegral (sz-1)) <-> r (fromIntegral (sz-1)) ^^^ adderCarryRec l (P.Not . r) True (fromIntegral sz) -- (6.47) but there is a mistake in the book, see "Errata For 2nd Edition" mult :: PropVector -> PropVector -> Size -> Int -> Flattening ([P.Formula], PropVector) mult _ _ _ (-1) = return ([], \_ -> P.Const False) -- (6.50) mult l r sz s = do (prevMultConstraints, prevMultBits) <- mult l r sz (s - 1) (adderConstraints, addedBits) <- adder prevMultBits (\i -> r (fromIntegral s) &&& shiftLStatic l (fromIntegral s) i) False sz return (adderConstraints ++ prevMultConstraints, addedBits) -- (6.51) multiplication :: PropVector -> PropVector -> Size -> Flattening ([P.Formula], PropVector) multiplication l r sz = mult l r sz (fromIntegral sz - 1) extendUnsigned :: PropVector -> Size -> PropVector extendUnsigned bv oldsz i | i < oldsz = bv i | otherwise = P.Const False extendSigned :: PropVector -> Size -> PropVector extendSigned bv oldsz i | i < oldsz - 1 = bv i | otherwise = bv (oldsz - 1) extend :: BitVectorSign -> PropVector -> Size -> PropVector extend Signed = extendSigned extend Unsigned = extendUnsigned increment :: PropVector -> Size -> Flattening ([P.Formula], PropVector) increment v sz = do carries <- reserveProps (sz - 1) let carryProp = (\i -> if i == 0 then P.Const True else variableVector carries (i-1) ) :: PropVector carryConstraints = map (\i -> carryProp i <-> if i == 0 then P.Const True else v (i-1) &&& carryProp (i-1)) [1..sz-1] return (carryConstraints, \i -> v i ^^^ carryProp i) absolute :: PropVector -> Size -> Flattening ([P.Formula], PropVector) absolute v sz = do let inverted i = P.Not (v i) (tcConstraints, tcVec) <- increment inverted sz let signProp = v (sz-1) return (tcConstraints, \i -> (signProp &&& tcVec i) ||| (P.Not signProp &&& v i)) lessThanAbsolute :: PropVector -> PropVector -> Size -> Flattening ([P.Formula], P.Formula) lessThanAbsolute l r sz = do (absLConstraints, absLVec) <- absolute l sz (absRConstraints, absRVec) <- absolute r sz return (absLConstraints ++ absRConstraints, lessThanUnsigned absLVec absRVec sz) divisionConstraint :: BitVectorSign -> PropVector -> PropVector -> PropVector -> PropVector -> Size -> Flattening P.Formula divisionConstraint sign res l r rem sz = do let extsz = sz + sz ext = extend sign extl = ext l sz extr = ext r sz extres = ext res sz extrem = ext rem sz (multConstraints, multBit) <- multiplication extres extr extsz -- term * r (addConstraints, addedBit) <- adder multBit extrem False extsz -- (term * r) + rem let multAddConstraints = map (\i -> extl i <-> addedBit i) [0..extsz - 1] -- (6.52) remainderConstraints <- if sign == Signed then do -- extr (extraConstraints, constraint) <- lessThanAbsolute extrem extr extsz let signConstraint = (l (sz - 1) <-> rem (sz - 1)) ||| P.conjunction (map (P.Not . rem) [0..sz-1]) -- (sign l == sign rem) || (rem == 0) return $ signConstraint : constraint : extraConstraints else return [lessThanUnsigned extrem extr extsz] -- (6.53) return $ P.conjunction $ remainderConstraints ++ multAddConstraints ++ addConstraints ++ multConstraints termPropVector :: (Term -> PropVectorVariable) -> Term -> PropVector termPropVector termProps t = variableVector $ termProps t notTermPropVector :: (Term -> PropVectorVariable) -> Term -> PropVector notTermPropVector termProps t i = P.Not $ variableVector (termProps t) i termConstraint :: (Atom -> PropVectorVariable) -> (Term -> PropVectorVariable) -> Term -> Flattening P.Formula termConstraint atomProps termProps term = let termBit = termPropVector termProps notTermBit = notTermPropVector termProps in case term of (Var _ _) -> return $ P.Const True (Const _ bv) -> return $ P.conjunction $ map (\i -> if BV.index bv i then termBit term i else notTermBit term i ) [0..BV.length bv - 1] -- (6.35) (BinAnd l r) -> bitwiseConstraintBinary (&&&) termProps l r term (BinOr l r) -> bitwiseConstraintBinary (|||) termProps l r term --(BinXor l r) -> bitwiseConstraintBinary P.Xor termProps l r term (BinXor l r) -> do (BitVectorType _ sz) <- getTermType term return $ P.And $ map (\i -> -- Constraint for the i-th bit, bringing the output in relation to the inputs: termBit term i <-> (termBit l i ^^^ termBit r i) ) [0..sz - 1] (Complement t) -> bitwiseConstraintIff (notTermBit t) termProps term (Inc t) -> do (BitVectorType _ sz) <- getTermType term (incConstraints, incVec) <- increment (termBit t) sz return $ P.conjunction $ map (\i -> termBit term i <-> incVec i) [0..sz-1] ++ incConstraints (Abs t) -> do (BitVectorType sign sz) <- getTermType t if sign == Signed then do (absConstraints, absVec) <- absolute (termBit t) sz return $ P.conjunction $ map (\i -> termBit term i <-> absVec i) [0..sz-1] ++ absConstraints else return $ P.conjunction $ map (\i -> termBit term i <-> termBit t i) [0..sz-1] (Plus l r) -> do let (BitVectorType _ sz) = fromJust $ termType term (adderConstraints, addedBit) <- adder (termBit l) (termBit r) False sz return $ P.conjunction $ adderConstraints ++ map (\i -> termBit term i <-> addedBit i -- (6.41), (6.43) ) [0..sz - 1] (Minus l r) -> do (BitVectorType _ sz) <- getTermType term (adderConstraints, addedBit) <- adder (termBit l) (notTermBit r) True sz return $ P.conjunction $ adderConstraints ++ map (\i -> termBit term i <-> addedBit i -- (6.44) ) [0..sz - 1] -- (6.41) (ShL l r) -> do (BitVectorType _ sz) <- getTermType l -- Term type checking ensures sz == 2^ssz (BitVectorType _ ssz) <- getTermType r return $ P.conjunction $ map (\i -> termBit term i <-> shift (-) (\s i -> i >= 2^s) (termBit l) (termBit r) (fromIntegral ssz - 1) i ) [0..fromIntegral sz - 1] (ShR l r) -> do (BitVectorType _ sz) <- getTermType l -- Term type checking ensures sz == 2^ssz (BitVectorType _ ssz) <- getTermType r return $ P.conjunction $ map (\i -> termBit term i <-> shift (+) (\s i -> i + (2^s) < sz) (termBit l) (termBit r) (fromIntegral ssz - 1) i ) [0..fromIntegral sz - 1] (Mult l r) -> do let (BitVectorType _ sz) = fromJust $ termType term (multConstraints, multBit) <- multiplication (termBit l) (termBit r) sz return $ P.conjunction $ multConstraints ++ map (\i -> termBit term i <-> multBit i ) [0..fromIntegral sz - 1] (Div l r) -> do BitVectorType sign sz <- getTermType l remPV <- reserveProps sz divisionConstraint sign (termBit term) (termBit l) (termBit r) (variableVector remPV) sz (Remainder l r) -> do BitVectorType sign sz <- getTermType l resPV <- reserveProps sz divisionConstraint sign (variableVector resPV) (termBit l) (termBit r) (termBit term) sz (Concat _ l r) -> do (BitVectorType _ lsz) <- getTermType l bitwiseConstraintIff (\i -> if i < lsz then termBit l i else termBit r (i-lsz)) termProps term (Ext _ t) -> do (BitVectorType sign sz) <- getTermType t bitwiseConstraintIff (extend sign (termBit t) sz) termProps term (Slice _ off _ t) -> bitwiseConstraintIff (\i -> termBit t (off + i)) termProps term (Ternary c a b) -> let atomProp = variableVector (atomProps c) 0 in bitwiseConstraintIff (\i -> (atomProp &&& termBit a i) ||| (P.Not atomProp &&& termBit b i)) termProps term atomConstraint :: (Atom -> PropVectorVariable) -> (Term -> PropVectorVariable) -> Atom -> Flattening P.Formula atomConstraint atomProps termProps atom = let atomProp = variableVector (atomProps atom) 0 termBit = termPropVector termProps in case atom of (BConst True) -> return $ variableVector (atomProps atom) 0 (BConst False) -> return $ P.Not $ variableVector (atomProps atom) 0 (Equals l r) -> do lt <- getTermType l rt <- getTermType r when (lt /= rt) $ throwError $ AtomTypeMismatch atom (atomProp <->) <$> bitwiseConstraintIff (variableVector (termProps r)) termProps l (Pick i t) -> do (BitVectorType _ sz) <- getTermType t when (i >= sz) $ throwError $ AtomPickBoundsError atom return $ atomProp <-> termBit t i (LessThan l r) -> do lt <- getTermType l rt <- getTermType r when (lt /= rt) $ throwError $ AtomTypeMismatch atom let (BitVectorType sign sz) = lt return $ atomProp <-> if sign == Signed then lessThanSigned (termBit l) (termBit r) sz else lessThanUnsigned (termBit l) (termBit r) sz -- |reserves one propositional variable for each bit in each term termVars :: Foldable a => a Term -> Flattening (Map Term PropVectorVariable) termVars = foldM (\map term -> do (BitVectorType _ sz) <- getTermType term baseProp <- reserveProps sz return $ Map.insert term baseProp map ) Map.empty instance Reservable Term where requiredProps term = (\(BitVectorType _ sz) -> sz) <$> getTermType term instance Reservable Atom where requiredProps _ = return 1 data FlattenedFormula = FlattenedFormula { atomProps :: Map Atom PropVectorVariable, termProps :: Map Term PropVectorVariable, skeletonFormula :: P.Formula, termConstraints :: Map Term P.Formula, atomConstraints :: Map Atom P.Formula, propsCount :: Int } -- |Helper function that applies a function in m to each set member and constructs a -- Map containing the results. mapFromSetM :: (Ord k, Monad m) => (k -> m a) -> Set.Set k -> m (Map k a) mapFromSetM f ks = Map.fromList <$> mapM (\a -> f a >>= \c -> return (a, c)) (Set.toList ks) formulaFlattening :: Formula -> Flattening FlattenedFormula formulaFlattening f = do let allTerms = Set.fromList $ terms f allAtoms = Set.fromList $ atoms f atomProps <- reserveVarsForAll allAtoms termProps <- reserveVarsForAll allTerms let skel = skeleton (atomProps Map.!) f termConstraints <- mapFromSetM (termConstraint (atomProps Map.!) (termProps Map.!)) allTerms atomConstraints <- mapFromSetM (atomConstraint (atomProps Map.!) (termProps Map.!)) allAtoms propsCount <- curPropsCount return $ FlattenedFormula { atomProps = atomProps, termProps = termProps, skeletonFormula = skel, termConstraints = termConstraints, atomConstraints = atomConstraints, propsCount = propsCount } -- |Convenience function that calls formulaFlattening and evaluates the returned monadic value flatten :: Formula -> Either FlattenError FlattenedFormula flatten f = runFlattening (formulaFlattening f) (FlattenState 0) instance Show FlattenedFormula where show (FlattenedFormula atomProps termProps skeletonFormula termConstraints atomConstraints _) = printf "skeleton: %s\nterms:\n%satoms:\n%s" (show skeletonFormula) ( concat $ Map.mapWithKey (\term (prop, constraint) -> " " ++ show term ++ " = " ++ show prop ++ " => " ++ show constraint ++ "\n") $ Map.intersectionWith (,) termProps termConstraints ) ( concat $ Map.mapWithKey (\atom (prop, constraint) -> " " ++ show atom ++ " = " ++ show prop ++ " => " ++ show constraint ++ "\n") $ Map.intersectionWith (,) atomProps atomConstraints ) propositional :: FlattenedFormula -> P.Formula propositional (FlattenedFormula _ _ skeletonFormula termConstraints atomConstraints _) = P.conjunction $ skeletonFormula : Map.elems termConstraints ++ Map.elems atomConstraints -- |Maps the result from the SAT-solver back to the original bit vector variables reconstructResult :: Formula -> FlattenedFormula -> Map P.Identifier Bool -> Map Identifier BV.BitVectorValue reconstructResult f flat propResults = let allVars = Set.fromList $ vars f in Map.fromList $ map (\(BitVectorType sign sz, name) -> let (PropVectorVariable baseProp _) = termProps flat Map.! Var (BitVectorType sign sz) name bits = map (\i -> Map.findWithDefault False (baseProp + i) propResults) [0..fromIntegral sz - 1] in (name, BV.pack bits) ) $ Set.toList allVars

A => src/Formula.hs +262 -0

@@ 1,262 @@{-# LANGUAGE PatternSynonyms #-} module Formula where import Text.Printf import Common import qualified BitVectorValue as BV type Identifier = String data Formula = And Formula Formula | Not Formula | Atom Atom deriving (Eq, Ord) data Atom = BConst Bool | LessThan Term Term | Equals Term Term | Pick Size Term deriving (Eq, Ord) data Term = Plus Term Term | Minus Term Term | Mult Term Term | Div Term Term | Remainder Term Term | ShL Term Term | ShR Term Term | BinAnd Term Term | BinOr Term Term | BinXor Term Term | Var BitVectorType Identifier | Complement Term | Inc Term | Abs Term | Const BitVectorSign BV.BitVectorValue | Ternary Atom Term Term | Concat BitVectorSign Term Term | Ext Size Term | Slice BitVectorSign Size Size Term -- new sign -> start offset -> new size -> term deriving (Eq, Ord) pattern (:&&:) a b = And a b pattern (:!!:) a = Not a pattern (:==:) a b = Equals a b pattern (:<:) a b = LessThan a b pattern (:+:) a b = Plus a b pattern (:-:) a b = Minus a b pattern (:*:) a b = Mult a b pattern (:/:) a b = Div a b pattern (:%:) a b = Remainder a b pattern (:<<:) a b = ShL a b pattern (:>>:) a b = ShR a b pattern (:&:) a b = BinAnd a b pattern (:|:) a b = BinOr a b pattern (:^:) a b = BinXor a b uVar :: BV.Size -> Identifier -> Term uVar sz = Var (BitVectorType Unsigned sz) sVar :: BV.Size -> Identifier -> Term sVar sz = Var (BitVectorType Signed sz) uConst :: BV.ToBitVector a => a -> Term uConst v = Const Unsigned $ BV.toBitVector v sConst :: BV.ToBitVector a => a -> Term sConst v = Const Signed $ BV.toBitVector v instance Show Formula where show (And l r) = printf "(%s ∧ %s)" (show l) (show r) show (Not f) = printf "¬%s" (show f) show (Atom a) = show a instance Show Atom where show (BConst True) = printf "True" show (BConst False) = printf "False" show (LessThan l r) = printf "(%s < %s)" (show l) (show r) show (Equals l r) = printf "(%s = %s)" (show l) (show r) show (Pick i t) = printf "%s[%s]" (show t) (show i) instance Show Term where show (Plus l r) = printf "(%s + %s)" (show l) (show r) show (Minus l r) = printf "(%s - %s)" (show l) (show r) show (Mult l r) = printf "(%s * %s)" (show l) (show r) show (Div l r) = printf "(%s / %s)" (show l) (show r) show (Remainder l r) = printf "(%s %% %s)" (show l) (show r) show (ShL l r) = printf "(%s << %s)" (show l) (show r) show (ShR l r) = printf "(%s >> %s)" (show l) (show r) show (BinAnd l r) = printf "(%s & %s)" (show l) (show r) show (BinOr l r) = printf "(%s | %s)" (show l) (show r) show (BinXor l r) = printf "(%s ^ %s)" (show l) (show r) show (Concat _ l r) = printf "(%s ⚬ %s)" (show l) (show r) show (Var _ name) = name show (Complement term) = printf "~%s" (show term) show (Inc term) = printf "%s+1" (show term) show (Abs term) = printf "|%s|" (show term) show (Const _ bv) = show bv show (Ternary c a b) = printf "(%s ? %s : %s)" (show c) (show a) (show b) show (Ext sz term) = printf "ext_%s %s" (show sz) (show term) show (Slice _ off sz t) = printf "%s[%s:%s]" (show t) (show off) (show sz) combinedTermTypes :: Term -> Term -> Maybe BitVectorType combinedTermTypes l r = case (termType l, termType r) of (Just tl, Just tr) | tl == tr -> Just tl _ -> Nothing shiftTermTypes :: Term -> Term -> Maybe BitVectorType shiftTermTypes l r = case (termType l, termType r) of (Just (BitVectorType sign sz), Just (BitVectorType Unsigned ssz)) | sz == 2 ^ ssz -> Just $ BitVectorType sign sz _ -> Nothing -- |returns the term's type or Nothing if it is invalid wrt. typing termType :: Term -> Maybe BitVectorType termType (Plus l r) = combinedTermTypes l r termType (Minus l r) = combinedTermTypes l r termType (Mult l r) = combinedTermTypes l r termType (Div l r) = combinedTermTypes l r termType (Remainder l r) = combinedTermTypes l r termType (ShL l r) = shiftTermTypes l r termType (ShR l r) = shiftTermTypes l r termType (BinAnd l r) = combinedTermTypes l r termType (BinOr l r) = combinedTermTypes l r termType (BinXor l r) = combinedTermTypes l r termType (Concat sign l r) = do (BitVectorType _ lsz) <- termType l (BitVectorType _ rsz) <- termType r return $ BitVectorType sign (lsz + rsz) termType (Var tp _) = Just tp termType (Complement term) = termType term termType (Inc term) = termType term termType (Abs term) = (\(BitVectorType _ sz) -> BitVectorType Unsigned sz) <$> termType term termType (Const sign bv) = Just $ BitVectorType sign (BV.length bv) termType (Ternary _ a b) = combinedTermTypes a b termType (Ext sz term) = case termType term of Just (BitVectorType sign tsz) | tsz <= sz -> Just $ BitVectorType sign sz _ -> Nothing termType (Slice sign off sz term) = case termType term of Just (BitVectorType _ tsz) | off + sz <= tsz -> Just $ BitVectorType sign sz _ -> Nothing subTerms :: Term -> [Term] subTerms (Plus l r) = [l, r] subTerms (Minus l r) = [l, r] subTerms (Mult l r) = [l, r] subTerms (Div l r) = [l, r] subTerms (Remainder l r) = [l, r] subTerms (ShL l r) = [l, r] subTerms (ShR l r) = [l, r] subTerms (BinAnd l r) = [l, r] subTerms (BinOr l r) = [l, r] subTerms (BinXor l r) = [l, r] subTerms (Concat _ l r) = [l, r] subTerms (Var _ _) = [] subTerms (Complement term) = [term] subTerms (Inc term) = [term] subTerms (Abs term) = [term] subTerms (Const _ _) = [] subTerms (Ternary _ a b) = [a, b] subTerms (Ext _ term) = [term] subTerms (Slice _ _ _ t) = [t] class Terms a where terms :: a -> [Term] instance Terms Term where terms term = term : concatMap terms at ++ concatMap terms st where st = subTerms term at = case term of (Ternary c _ _) -> terms c _ -> [] instance Terms Formula where terms (And l r) = terms l ++ terms r terms (Not f) = terms f terms (Atom a) = terms a instance Terms Atom where terms (BConst _) = [] terms (LessThan l r) = terms l ++ terms r terms (Equals l r) = terms l ++ terms r terms (Pick _ t) = terms t class Atoms a where atoms :: a -> [Atom] instance Atoms Formula where atoms (And l r) = atoms l ++ atoms r atoms (Not f) = atoms f atoms (Atom a) = atoms a instance Atoms Atom where atoms (BConst b) = [BConst b] atoms (LessThan l r) = LessThan l r : atoms l ++ atoms r atoms (Equals l r) = Equals l r : atoms l ++ atoms r atoms (Pick i t) = Pick i t : atoms t instance Atoms Term where atoms (Plus l r) = atoms l ++ atoms r atoms (Minus l r) = atoms l ++ atoms r atoms (Mult l r) = atoms l ++ atoms r atoms (Div l r) = atoms l ++ atoms r atoms (Remainder l r) = atoms l ++ atoms r atoms (ShL l r) = atoms l ++ atoms r atoms (ShR l r) = atoms l ++ atoms r atoms (BinAnd l r) = atoms l ++ atoms r atoms (BinOr l r) = atoms l ++ atoms r atoms (BinXor l r) = atoms l ++ atoms r atoms (Concat _ l r) = atoms l ++ atoms r atoms (Var _ _) = [] atoms (Complement t) = atoms t atoms (Inc t) = atoms t atoms (Abs t) = atoms t atoms (Const _ _) = [] atoms (Ternary c a b) = atoms c ++ atoms a ++ atoms b atoms (Ext _ t) = atoms t atoms (Slice _ _ _ t) = atoms t class Vars a where vars :: a -> [(BitVectorType, Identifier)] instance Vars Formula where vars (And l r) = vars l ++ vars r vars (Not f) = vars f vars (Atom a) = vars a instance Vars Atom where vars (BConst _) = [] vars (LessThan l r) = vars l ++ vars r vars (Equals l r) = vars l ++ vars r vars (Pick _ t) = vars t instance Vars Term where vars (Plus l r) = vars l ++ vars r vars (Minus l r) = vars l ++ vars r vars (Mult l r) = vars l ++ vars r vars (Div l r) = vars l ++ vars r vars (Remainder l r) = vars l ++ vars r vars (ShL l r) = vars l ++ vars r vars (ShR l r) = vars l ++ vars r vars (BinAnd l r) = vars l ++ vars r vars (BinOr l r) = vars l ++ vars r vars (BinXor l r) = vars l ++ vars r vars (Concat _ l r) = vars l ++ vars r vars (Var tp name) = [(tp, name)] vars (Complement t) = vars t vars (Inc t) = vars t vars (Abs t) = vars t vars (Const _ _) = [] vars (Ternary c a b) = vars c ++ vars a ++ vars b vars (Ext _ t) = vars t vars (Slice _ _ _ t) = vars t \ No newline at end of file

A => src/MiniSat.hs +18 -0

@@ 1,18 @@module MiniSat where import qualified SAT.MiniSat as M import qualified Propositional as P -- |Convert our representation of propositional formulas to the one accepted by SAT.MiniSat miniSat :: P.Formula -> M.Formula P.Identifier miniSat (P.Iff l r) = miniSat l M.:<->: miniSat r miniSat (P.Impl l r) = miniSat l M.:->: miniSat r --miniSat (P.And l r) = miniSat l M.:&&: miniSat r --miniSat (P.Or l r) = miniSat l M.:||: miniSat r miniSat (P.And fs) = M.All $ map miniSat fs miniSat (P.Or fs) = M.Some $ map miniSat fs miniSat (P.Xor l r) = miniSat l M.:++: miniSat r miniSat (P.Not f) = M.Not $ miniSat f miniSat (P.Var id) = M.Var id miniSat (P.Const c) = if c then M.Yes else M.No \ No newline at end of file

A => src/Propositional.hs +98 -0

@@ 1,98 @@{-# LANGUAGE LambdaCase #-} module Propositional ( Identifier, Formula (..), (<->), (-->), (&&&), (|||), (^^^), (Propositional.!!), conjunction, elimAllConsts, eval ) where import Text.Printf import Data.List import Common type Identifier = Int data Formula = Iff Formula Formula | Impl Formula Formula | And [Formula] | Or [Formula] | Xor Formula Formula | Not Formula | Var Identifier | Const Bool deriving (Eq) instance Show Formula where show (Iff l r) = printf "(%s ↔ %s)" (show l) (show r) show (Impl l r) = printf "(%s → %s)" (show l) (show r) show (And fs) = "(" ++ intercalate " ∧ " (map show fs) ++ ")" show (Or fs) = "(" ++ intercalate " ∨ " (map show fs) ++ ")" show (Xor l r) = printf "(%s ⊕ %s)" (show l) (show r) show (Not f) = printf "¬%s" (show f) show (Var a) = show a show (Const b) = show b (<->) = Iff; (-->) = Impl; (&&&) l r = And [l, r]; (|||) l r = Or [l, r] (^^^) = Xor; (!!) = Not conjunction :: [Formula] -> Formula conjunction = And elimConst :: Formula -> Formula elimConst (Iff l (Const True)) = elimConst l elimConst (Iff l (Const False)) = elimConst $ Not l elimConst (Iff (Const True) r) = elimConst r elimConst (Iff (Const False) r) = elimConst $ Not r elimConst (Iff l r) = Iff (elimConst l) (elimConst r) elimConst (Impl (Const True) r) = elimConst r elimConst (Impl (Const False) _) = Const True elimConst (Impl _ (Const True)) = Const True elimConst (Impl l (Const False)) = Not $ elimConst l elimConst (Impl l r) = Impl (elimConst l) (elimConst r) elimConst (And fs) = if any (\case Const False -> True; _ -> False) fs then Const False else And $ filter (\case Const True -> False; _ -> True) $ map elimConst fs elimConst (Or fs) = if any (\case Const True -> True; _ -> False) fs then Const True else And $ filter (\case Const False -> False; _ -> True) $ map elimConst fs elimConst (Xor l (Const True)) = Not $ elimConst l elimConst (Xor l (Const False)) = elimConst l elimConst (Xor (Const True) r) = Not $ elimConst r elimConst (Xor (Const False) r) = elimConst r elimConst (Xor l r) = Xor (elimConst l) (elimConst r) elimConst (Not (Const True)) = Const False elimConst (Not (Const False)) = Const True elimConst (Not f) = Not $ elimConst f elimConst (Var id) = Var id elimConst (Const v) = Const v elimAllConsts :: Formula -> Formula elimAllConsts = fixedPoint elimConst eval :: (Identifier -> Bool) -> Formula -> Bool eval ass (Iff l r) = eval ass l == eval ass r eval ass (Impl l r) = eval ass l || (eval ass l && eval ass r) eval ass (And fs) = foldl (\acc f -> acc && eval ass f) True fs eval ass (Or fs) = foldl (\acc f -> acc || eval ass f) False fs eval ass (Xor l r) = eval ass l /= eval ass r eval ass (Not f) = not $ eval ass f eval ass (Var id) = ass id eval _ (Const c) = c \ No newline at end of file

A => src/Solve.hs +96 -0

@@ 1,96 @@module Solve ( SolveResult (..), Solution, solve, solveAll, solveIncremental, costEstimate ) where import Data.Map (Map) import qualified Data.Map as Map import Data.List -- import Debug.Trace import qualified SAT.MiniSat as M import qualified BitVectorValue as BV import qualified Propositional as P import Formula import Flattening import MiniSat data SolveResult a = Solution a | Unsatisfiable | FlattenError FlattenError deriving (Eq, Show) type Solution = Map Identifier BV.BitVectorValue -- |Solve to a single solution solve :: Formula -> SolveResult Solution solve f = case flatten f of Left e -> FlattenError e Right flat -> case M.solve $ miniSat $ propositional flat of Nothing -> Unsatisfiable Just r -> Solution $ reconstructResult f flat r -- |Solve to all solutions as a lazy list solveAll :: Formula -> SolveResult [Solution] solveAll f = case flatten f of Left e -> FlattenError e Right flat -> case M.solve_all $ miniSat $ propositional flat of [] -> Unsatisfiable sols -> Solution $ map (reconstructResult f flat) sols -- |Solve incrementally to a single solution -- This can be significantly faster or slower than solve depending on the -- formula, see the paper for more info. solveIncremental :: Int -> Formula -> SolveResult Solution solveIncremental steps f = case flatten f of Left e -> FlattenError e Right flat -> solveFlattenedIncremental steps f flat costEstimate :: P.Formula -> Word costEstimate (P.Iff l r) = 1 + costEstimate l + costEstimate r costEstimate (P.Impl l r) = 1 + costEstimate l + costEstimate r costEstimate (P.And fs) = foldl (\acc f -> acc + costEstimate f) (1 + fromIntegral (length fs)) fs costEstimate (P.Or fs) = foldl (\acc f -> acc + costEstimate f) (1 + fromIntegral (length fs)) fs costEstimate (P.Xor l r) = 1 + costEstimate l + costEstimate r costEstimate (P.Not f) = costEstimate f costEstimate (P.Var _) = 1 costEstimate (P.Const _) = 0 -- |Solve a formula incrementally, given the number of new constraints to add in each step solveFlattenedIncremental :: Int -> Formula -> FlattenedFormula -> SolveResult Solution solveFlattenedIncremental stepSize f flat = let (FlattenedFormula _ _ skeletonFormula termConstraints atomConstraints _) = flat initialFormulas = [skeletonFormula] incrementalFormulas = sortOn costEstimate $ filter (/= P.Const True) $ Map.elems termConstraints ++ Map.elems atomConstraints fullFormula = propositional flat in case incrementalSAT stepSize fullFormula initialFormulas incrementalFormulas of Nothing -> Unsatisfiable Just r -> Solution $ reconstructResult f flat r -- |Solve a SAT problem incrementally by recursion, given the number of new constraints to add in -- each step, the full formula, constraints to be considered right now and constraints to be -- considered later. incrementalSAT :: Int -> P.Formula -> [P.Formula] -> [P.Formula] -> Maybe (Map P.Identifier Bool) incrementalSAT stepSize full current pending = -- trace ("incrementalSat " ++ show (length current) ++ "/" ++ show (length pending)) $ case M.solve $ miniSat $ P.conjunction current of Nothing -> Nothing -- partial formula unsatisfiable => full formula unsatisfiable Just r -> -- partial formula satisfiable let conflicts = filter (not . P.eval (\id -> Map.findWithDefault False id r)) pending in if null conflicts then Just r -- no conflicts, full formula satisfied! else -- got conflicts, move the easiest ones into the current list let new = take stepSize conflicts -- pick the easiest conflicts nextPending = filter (not . (`elem` new)) pending in -- remove them from pending incrementalSAT stepSize full (new ++ current) nextPending

A => stack.yaml +66 -0

@@ 1,66 @@# This file was automatically generated by 'stack init' # # Some commonly used options have been documented as comments in this file. # For advanced use and comprehensive documentation of the format, please see: # https://docs.haskellstack.org/en/stable/yaml_configuration/ # Resolver to choose a 'specific' stackage snapshot or a compiler version. # A snapshot resolver dictates the compiler version and the set of packages # to be used for project dependencies. For example: # # resolver: lts-3.5 # resolver: nightly-2015-09-21 # resolver: ghc-7.10.2 # # The location of a snapshot can be provided as a file or url. Stack assumes # a snapshot provided as a file might change, whereas a url resource does not. # # resolver: ./custom-snapshot.yaml # resolver: https://example.com/snapshots/2018-01-01.yaml resolver: lts-12.26 # User packages to be built. # Various formats can be used as shown in the example below. # # packages: # - some-directory # - https://example.com/foo/bar/baz-0.0.2.tar.gz # subdirs: # - auto-update # - wai packages: - . # Dependency packages to be pulled from upstream that are not in the resolver. # These entries can reference officially published versions as well as # forks / in-progress versions pinned to a git hash. For example: # # extra-deps: # - acme-missiles-0.3 # - git: https://github.com/commercialhaskell/stack.git # commit: e7b331f14bcffb8367cd58fbfc8b40ec7642100a # # extra-deps: [] # Override default flag values for local packages and extra-deps # flags: {} # Extra package databases containing global packages # extra-package-dbs: [] # Control whether we use the GHC we find on the path # system-ghc: true # # Require a specific version of stack, using version ranges # require-stack-version: -any # Default # require-stack-version: ">=2.3" # # Override the architecture used by stack, especially useful on Windows # arch: i386 # arch: x86_64 # # Extra directories used by stack for building # extra-include-dirs: [/path/to/dir] # extra-lib-dirs: [/path/to/dir] # # Allow a newer minor version of GHC than the snapshot specifies # compiler-check: newer-minor

A => test/BitVectorValueTest.hs +50 -0

@@ 1,50 @@{-# LANGUAGE TemplateHaskell #-} module BitVectorValueTest (runTests) where import Test.QuickCheck (choose, Gen, Arbitrary, arbitrary, sized) import Test.QuickCheck.All import Text.Printf import Data.Word import Data.Bits import Control.Monad import BitVectorValue as BV instance Arbitrary BV.BitVectorValue where arbitrary = choose (1, 128) >>= \n -> BV.pack <$> replicateM n arbitrary prop_replicate :: Bool -> Gen Bool prop_replicate v = do sz <- choose (1, 128) :: Gen Word let bv = BV.replicate sz v return $ foldl (\acc i -> acc && (BV.index bv i == v)) True [0..sz-1] checkBase :: (ToBitVector a, Bits a, Num a) => BV.Size -> a -> Bool checkBase sz v = let bv = BV.toBitVector v in BV.length bv == sz && foldl (\acc i -> acc && (BV.index bv i == (v .&. (1 `shiftL` fromIntegral i) /= 0))) True [0..sz-1] prop_base8 = checkBase 8 :: Word8 -> Bool prop_base16 = checkBase 16 :: Word16 -> Bool prop_base32 = checkBase 32 :: Word32 -> Bool prop_base64 = checkBase 64 :: Word64 -> Bool prop_eq :: BV.BitVectorValue -> Bool prop_eq bv = bv == bv prop_neq :: Gen Bool prop_neq = do bitsa <- choose (1, 128) >>= \n -> replicateM n (arbitrary::Gen Bool) bitsb <- choose (1, 128) >>= \n -> replicateM n (arbitrary::Gen Bool) return $ (bitsa == bitsb) == (BV.pack bitsa == BV.pack bitsb) prop_compare_eq :: BV.BitVectorValue -> Bool prop_compare_eq bv = compare bv bv == EQ prop_compare :: Word64 -> Word64 -> Bool prop_compare a b = compare a b == compare (BV.toBitVector a) (BV.toBitVector b) return [] runTests :: IO Bool runTests = $quickCheckAll \ No newline at end of file

A => test/SolveTest.hs +170 -0

@@ 1,170 @@{-# LANGUAGE TemplateHaskell #-} module SolveTest (runTests) where import Test.QuickCheck (choose, Gen, Arbitrary, arbitrary, sized, (===), Property, property) import Test.QuickCheck.All import Data.Word import Data.Int import Data.Bits import Data.Maybe import Data.Map (Map) import qualified Data.Map as Map import Common import Formula import Solve import qualified BitVectorValue as BV example0 x y = (Atom $ uConst x :==: uVar 8 "a") :&&: (Atom $ uVar 8 "b" :==: (uVar 8 "a" :^: uConst y)) prop_example0 x y = let f = example0 (x::Word8) (y::Word8) in solve f == Solution (Map.fromList [("a", BV.toBitVector x), ("b", BV.toBitVector (x `xor` y))]) prop_example0Incremental x y = let f = example0 (x::Word8) (y::Word8) in solveIncremental 3 f === Solution (Map.fromList [("a", BV.toBitVector x), ("b", BV.toBitVector (x `xor` y))]) prop_inc x = let f = Atom $ uVar 8 "a" :==: Inc (uConst (x::Word8)) in solve f == Solution (Map.fromList [("a", BV.toBitVector (x+1))]) prop_abs x = let f = Atom $ uVar 8 "a" :==: Abs (sConst (x :: Int8)) in solve f == Solution (Map.fromList [("a", BV.toBitVector (fromIntegral (abs x) :: Word8))]) checkOperator :: BV.ToBitVector a => (Term -> Term -> Term) -> (a -> a -> a) -> BitVectorSign -> a -> a -> Bool checkOperator top op sign x y = let (mVar, mConst) = if sign == Signed then (sVar, sConst) else (uVar, uConst) f = Atom $ mVar 8 "a" :==: (mConst x `top` mConst y) in solve f == Solution (Map.fromList [("a", BV.toBitVector (x `op` y))]) prop_plus x y = checkOperator (:+:) (+) Unsigned (x::Word8) (y::Word8) prop_plusS x y = checkOperator (:+:) (+) Signed (x::Int8) (y::Int8) prop_minus x y = checkOperator (:-:) (-) Unsigned (x::Word8) (y::Word8) prop_minusS x y = checkOperator (:-:) (-) Signed (x::Int8) (y::Int8) checkDiv :: (BV.ToBitVector a, Integral a) => BitVectorSign -> a -> a -> Bool checkDiv sign x y = let (mVar, mConst) = if sign == Signed then (sVar, sConst) else (uVar, uConst) f = Atom $ mVar 8 "a" :==: (mConst x :/: mConst y) in solveAll f == if y == 0 then Unsatisfiable else Solution [Map.fromList [("a", BV.toBitVector (x `quot` y))]] prop_concat :: Word8 -> Word8 -> Bool prop_concat x y = let f = Atom $ sVar 16 "a" :==: Concat Signed (uConst x) (uConst y) v = fromIntegral ((fromIntegral x::Word16) .|. ((fromIntegral y::Word16) `shiftL` 8))::Word16 in solve f == Solution (Map.fromList [("a", BV.toBitVector v)]) prop_extz :: Word8 -> Property prop_extz x = let f = Atom $ uVar 16 "a" :==: Ext 16 (uConst x) in solve f === Solution (Map.fromList [("a", BV.toBitVector (fromIntegral x :: Word16))]) prop_exts :: Int8 -> Property prop_exts x = let f = Atom $ sVar 16 "a" :==: Ext 16 (sConst x) in solve f === Solution (Map.fromList [("a", BV.toBitVector (fromIntegral x :: Int16))]) prop_slice :: Word64 -> Gen Property prop_slice x = do let bv = BV.toBitVector x off <- choose (0, 62) sz <- choose (1, 64 - off) let f = Atom $ uVar sz "a" :==: Slice Unsigned off sz (Const Signed bv) return $ solve f === Solution (Map.fromList [("a", BV.slice bv off sz)]) prop_complement :: Word8 -> Bool prop_complement x = let f = Atom $ uVar 8 "a" :==: Complement (uConst x) in solve f == Solution (Map.fromList [("a", BV.toBitVector (x `xor` 0xff))]) prop_ternary :: Bool -> Word8 -> Word8 -> Property prop_ternary c a b = let f = Atom $ uVar 8 "a" :==: Ternary (BConst c) (uConst a) (uConst b) in solve f === Solution (Map.fromList [("a", BV.toBitVector $ if c then a else b)]) prop_ternary1 :: Word8 -> Word8 -> Word8 -> Word8 -> Property prop_ternary1 ca cb a b = let f = Atom $ uVar 8 "a" :==: Ternary (uConst ca :==: uConst cb) (uConst a) (uConst b) in solve f === Solution (Map.fromList [("a", BV.toBitVector $ if ca == cb then a else b)]) prop_atomEquals x = let f = Atom $ uVar 8 "a" :==: uConst (x::Word8) in solve f == Solution (Map.fromList [("a", BV.toBitVector x)]) bconjunction :: [Formula] -> Formula bconjunction [x] = x bconjunction [x, y] = And x y bconjunction (x : xs) = And x $ bconjunction xs bconjunction [] = undefined prop_atomPick :: Word8 -> Bool prop_atomPick x = let f = bconjunction $ map (\i -> if ((x `shiftR` i) .&. 1) /= 0 then Atom $ Pick (fromIntegral i) (uVar 8 "a") else Not $ Atom $ Pick (fromIntegral i) (uVar 8 "a") ) [0..7] in solve f == Solution (Map.fromList [("a", BV.toBitVector x)]) prop_atomLessThanUnsigned :: Word8 -> Bool prop_atomLessThanUnsigned x = let f = bconjunction $ (if x == 0 then [] else [ Atom $ uConst (x-1) :<: uVar 8 "a" ]) ++ (if x == 255 then [] else [ Atom $ uVar 8 "a" :<: uConst (x+1) ]) in solve f == Solution (Map.fromList [("a", BV.toBitVector x)]) prop_atomLessThanSigned :: Int8 -> Bool prop_atomLessThanSigned x = let f = bconjunction $ (if x == -128 then [] else [ Atom $ sConst (x-1) :<: sVar 8 "a" ]) ++ (if x == 127 then [] else [ Atom $ sVar 8 "a" :<: sConst (x+1) ]) in solve f == Solution (Map.fromList [("a", BV.toBitVector x)]) prop_shiftL :: Word8 -> Word8 -> Bool prop_shiftL x s = let f = Atom $ uVar 8 "a" :==: (uConst x :<<: Const Unsigned (BV.slice (BV.toBitVector s) 0 3)) in solve f == Solution (Map.fromList [("a", BV.toBitVector (x `shiftL` fromIntegral (s .&. 0x7)))]) prop_shiftR :: Word8 -> Word8 -> Bool prop_shiftR x s = let f = Atom $ uVar 8 "a" :==: (uConst x :>>: Const Unsigned (BV.slice (BV.toBitVector s) 0 3)) in solve f == Solution (Map.fromList [("a", BV.toBitVector (x `shiftR` fromIntegral (s .&. 0x7)))]) prop_incrementalUnsatisfiable :: Property prop_incrementalUnsatisfiable = let f = Atom ((uVar 64 "a" :*: uVar 64 "b") :==: uVar 64 "c") :&&: Not (Atom ((uVar 64 "b" :*: uVar 64 "a") :==: uVar 64 "c")) :&&: Atom (uVar 64 "x" :<: uVar 64 "y") :&&: Atom (uVar 64 "y" :<: uVar 64 "x") in solveIncremental 1 f === Unsatisfiable --prop_incrementalSatisfiable :: Word64 -> Property --prop_incrementalSatisfiable x = -- let f = Atom (uVar 64 "a" :==: uConst x) -- :&&: Not ( Not (Atom $ (uVar 64 "b" :*: uVar 64 "c") :<: uVar 64 "d") -- :&&: Not (Atom $ uVar 64 "d" :<: (uVar 64 "b" :*: uVar 64 "c"))) -- sol = solveIncremental 1 f in -- case sol of -- Solution m -> m Map.! "a" === BV.toBitVector x -- _ -> property False prop_mult x y = checkOperator (:*:) (*) Unsigned (x::Word8) (y::Word8) prop_multS x y = checkOperator (:*:) (*) Signed (x::Int8) (y::Int8) prop_div x y = checkDiv Unsigned (x::Word8) (y::Word8) prop_divS x y = checkDiv Signed (x::Int8) (y::Int8) return [] runTests :: IO Bool runTests = $quickCheckAll \ No newline at end of file

A => test/Spec.hs +11 -0

@@ 1,11 @@import System.Exit import qualified BitVectorValueTest import qualified SolveTest main :: IO () main = do good <- and <$> sequence [BitVectorValueTest.runTests, SolveTest.runTests] if good then exitSuccess else exitFailure