----------------------------------------------------------------------------- -- | -- Module : Data.SBV.Core.Operations -- Copyright : (c) Levent Erkok -- License : BSD3 -- Maintainer: erkokl@gmail.com -- Stability : experimental -- -- Constructors and basic operations on symbolic values ----------------------------------------------------------------------------- {-# LANGUAGE BangPatterns #-} {-# OPTIONS_GHC -Wall -Werror #-} module Data.SBV.Core.Operations ( -- ** Basic constructors svTrue, svFalse, svBool , svInteger, svFloat, svDouble, svReal, svEnumFromThenTo, svString, svChar -- ** Basic destructors , svAsBool, svAsInteger, svNumerator, svDenominator -- ** Basic operations , svPlus, svTimes, svMinus, svUNeg, svAbs , svDivide, svQuot, svRem, svQuotRem , svEqual, svNotEqual, svStrongEqual, svSetEqual , svLessThan, svGreaterThan, svLessEq, svGreaterEq, svStructuralLessThan , svAnd, svOr, svXOr, svNot , svShl, svShr, svRol, svRor , svExtract, svJoin , svIte, svLazyIte, svSymbolicMerge , svSelect , svSign, svUnsign, svSetBit, svWordFromBE, svWordFromLE , svExp, svFromIntegral -- ** Overflows , svMkOverflow -- ** Derived operations , svToWord1, svFromWord1, svTestBit , svShiftLeft, svShiftRight , svRotateLeft, svRotateRight , svBarrelRotateLeft, svBarrelRotateRight , svBlastLE, svBlastBE , svAddConstant, svIncrement, svDecrement -- ** Basic array operations , SArr, readSArr, writeSArr, mergeSArr, newSArr, eqSArr , SFunArr(..), readSFunArr, writeSFunArr, mergeSFunArr, newSFunArr -- Utils , mkSymOp ) where import Data.Bits (Bits(..)) import Data.List (genericIndex, genericLength, genericTake) import qualified Data.IORef as R (modifyIORef', newIORef, readIORef) import qualified Data.Map.Strict as Map (toList, fromList, lookup) import qualified Data.IntMap.Strict as IMap (IntMap, empty, toAscList, fromAscList, lookup, size, insert) import Data.SBV.Core.AlgReals import Data.SBV.Core.Kind import Data.SBV.Core.Concrete import Data.SBV.Core.Symbolic import Data.Ratio import Data.SBV.Utils.Numeric (fpIsEqualObjectH) -------------------------------------------------------------------------------- -- Basic constructors -- | Boolean True. svTrue :: SVal svTrue = SVal KBool (Left trueCV) -- | Boolean False. svFalse :: SVal svFalse = SVal KBool (Left falseCV) -- | Convert from a Boolean. svBool :: Bool -> SVal svBool b = if b then svTrue else svFalse -- | Convert from an Integer. svInteger :: Kind -> Integer -> SVal svInteger k n = SVal k (Left $! mkConstCV k n) -- | Convert from a Float svFloat :: Float -> SVal svFloat f = SVal KFloat (Left $! CV KFloat (CFloat f)) -- | Convert from a Float svDouble :: Double -> SVal svDouble d = SVal KDouble (Left $! CV KDouble (CDouble d)) -- | Convert from a String svString :: String -> SVal svString s = SVal KString (Left $! CV KString (CString s)) -- | Convert from a Char svChar :: Char -> SVal svChar c = SVal KChar (Left $! CV KChar (CChar c)) -- | Convert from a Rational svReal :: Rational -> SVal svReal d = SVal KReal (Left $! CV KReal (CAlgReal (fromRational d))) -------------------------------------------------------------------------------- -- Basic destructors -- | Extract a bool, by properly interpreting the integer stored. svAsBool :: SVal -> Maybe Bool svAsBool (SVal _ (Left cv)) = Just (cvToBool cv) svAsBool _ = Nothing -- | Extract an integer from a concrete value. svAsInteger :: SVal -> Maybe Integer svAsInteger (SVal _ (Left (CV _ (CInteger n)))) = Just n svAsInteger _ = Nothing -- | Grab the numerator of an SReal, if available svNumerator :: SVal -> Maybe Integer svNumerator (SVal KReal (Left (CV KReal (CAlgReal (AlgRational True r))))) = Just $ numerator r svNumerator _ = Nothing -- | Grab the denominator of an SReal, if available svDenominator :: SVal -> Maybe Integer svDenominator (SVal KReal (Left (CV KReal (CAlgReal (AlgRational True r))))) = Just $ denominator r svDenominator _ = Nothing ------------------------------------------------------------------------------------- -- | Constructing [x, y, .. z] and [x .. y]. Only works when all arguments are concrete and integral and the result is guaranteed finite -- Note that the it isn't "obviously" clear why the following works; after all we're doing the construction over Integer's and mapping -- it back to other types such as SIntN/SWordN. The reason is that the values we receive are guaranteed to be in their domains; and thus -- the lifting to Integers preserves the bounds; and then going back is just fine. So, things like @[1, 5 .. 200] :: [SInt8]@ work just -- fine (end evaluate to empty list), since we see @[1, 5 .. -56]@ in the @Integer@ domain. Also note the explicit check for @s /= f@ -- below to make sure we don't stutter and produce an infinite list. svEnumFromThenTo :: SVal -> Maybe SVal -> SVal -> Maybe [SVal] svEnumFromThenTo bf mbs bt | Just bs <- mbs, Just f <- svAsInteger bf, Just s <- svAsInteger bs, Just t <- svAsInteger bt, s /= f = Just $ map (svInteger (kindOf bf)) [f, s .. t] | Nothing <- mbs, Just f <- svAsInteger bf, Just t <- svAsInteger bt = Just $ map (svInteger (kindOf bf)) [f .. t] | True = Nothing ------------------------------------------------------------------------------------- -- Basic operations -- | Addition. svPlus :: SVal -> SVal -> SVal svPlus x y | isConcreteZero x = y | isConcreteZero y = x | True = liftSym2 (mkSymOp Plus) rationalCheck (+) (+) (+) (+) x y -- | Multiplication. svTimes :: SVal -> SVal -> SVal svTimes x y | isConcreteZero x = x | isConcreteZero y = y | isConcreteOne x = y | isConcreteOne y = x | True = liftSym2 (mkSymOp Times) rationalCheck (*) (*) (*) (*) x y -- | Subtraction. svMinus :: SVal -> SVal -> SVal svMinus x y | isConcreteZero y = x | True = liftSym2 (mkSymOp Minus) rationalCheck (-) (-) (-) (-) x y -- | Unary minus. svUNeg :: SVal -> SVal svUNeg = liftSym1 (mkSymOp1 UNeg) negate negate negate negate -- | Absolute value. svAbs :: SVal -> SVal svAbs = liftSym1 (mkSymOp1 Abs) abs abs abs abs -- | Division. svDivide :: SVal -> SVal -> SVal svDivide = liftSym2 (mkSymOp Quot) rationalCheck (/) idiv (/) (/) where idiv x 0 = x idiv x y = x `div` y -- | Exponentiation. svExp :: SVal -> SVal -> SVal svExp b e | Just x <- svAsInteger e = if x >= 0 then let go n v | n == 0 = one | even n = go (n `div` 2) (svTimes v v) | True = svTimes v $ go (n `div` 2) (svTimes v v) in go x b else error $ "svExp: exponentiation: negative exponent: " ++ show x | not (isBounded e) || hasSign e = error $ "svExp: exponentiation only works with unsigned bounded symbolic exponents, kind: " ++ show (kindOf e) | True = prod $ zipWith (\use n -> svIte use n one) (svBlastLE e) (iterate (\x -> svTimes x x) b) where prod = foldr svTimes one one = svInteger (kindOf b) 1 -- | Bit-blast: Little-endian. Assumes the input is a bit-vector. svBlastLE :: SVal -> [SVal] svBlastLE x = map (svTestBit x) [0 .. intSizeOf x - 1] -- | Set a given bit at index svSetBit :: SVal -> Int -> SVal svSetBit x i = x `svOr` svInteger (kindOf x) (bit i :: Integer) -- | Bit-blast: Big-endian. Assumes the input is a bit-vector. svBlastBE :: SVal -> [SVal] svBlastBE = reverse . svBlastLE -- | Un-bit-blast from big-endian representation to a word of the right size. -- The input is assumed to be unsigned. svWordFromLE :: [SVal] -> SVal svWordFromLE bs = go zero 0 bs where zero = svInteger (KBounded False (length bs)) 0 go !acc _ [] = acc go !acc !i (x:xs) = go (svIte x (svSetBit acc i) acc) (i+1) xs -- | Un-bit-blast from little-endian representation to a word of the right size. -- The input is assumed to be unsigned. svWordFromBE :: [SVal] -> SVal svWordFromBE = svWordFromLE . reverse -- | Add a constant value: svAddConstant :: Integral a => SVal -> a -> SVal svAddConstant x i = x `svPlus` svInteger (kindOf x) (fromIntegral i) -- | Increment: svIncrement :: SVal -> SVal svIncrement x = svAddConstant x (1::Integer) -- | Decrement: svDecrement :: SVal -> SVal svDecrement x = svAddConstant x (-1 :: Integer) -- | Quotient: Overloaded operation whose meaning depends on the kind at which -- it is used: For unbounded integers, it corresponds to the SMT-Lib -- "div" operator ("Euclidean" division, which always has a -- non-negative remainder). For unsigned bitvectors, it is "bvudiv"; -- and for signed bitvectors it is "bvsdiv", which rounds toward zero. -- Division by 0 is defined s.t. @x/0 = 0@, which holds even when @x@ itself is @0@. svQuot :: SVal -> SVal -> SVal svQuot x y | isConcreteZero x = x | isConcreteZero y = svInteger (kindOf x) 0 | isConcreteOne y = x | True = liftSym2 (mkSymOp Quot) nonzeroCheck (noReal "quot") quot' (noFloat "quot") (noDouble "quot") x y where quot' a b | kindOf x == KUnbounded = div a (abs b) * signum b | otherwise = quot a b -- | Remainder: Overloaded operation whose meaning depends on the kind at which -- it is used: For unbounded integers, it corresponds to the SMT-Lib -- "mod" operator (always non-negative). For unsigned bitvectors, it -- is "bvurem"; and for signed bitvectors it is "bvsrem", which rounds -- toward zero (sign of remainder matches that of @x@). Division by 0 is -- defined s.t. @x/0 = 0@, which holds even when @x@ itself is @0@. svRem :: SVal -> SVal -> SVal svRem x y | isConcreteZero x = x | isConcreteZero y = x | isConcreteOne y = svInteger (kindOf x) 0 | True = liftSym2 (mkSymOp Rem) nonzeroCheck (noReal "rem") rem' (noFloat "rem") (noDouble "rem") x y where rem' a b | kindOf x == KUnbounded = mod a (abs b) | otherwise = rem a b -- | Combination of quot and rem svQuotRem :: SVal -> SVal -> (SVal, SVal) svQuotRem x y = (x `svQuot` y, x `svRem` y) -- | Optimize away x == true and x /= false to x; otherwise just do eqOpt eqOptBool :: Op -> SV -> SV -> SV -> Maybe SV eqOptBool op w x y | k == KBool && op == Equal && x == trueSV = Just y -- true .== y --> y | k == KBool && op == Equal && y == trueSV = Just x -- x .== true --> x | k == KBool && op == NotEqual && x == falseSV = Just y -- false ./= y --> y | k == KBool && op == NotEqual && y == falseSV = Just x -- x ./= false --> x | True = eqOpt w x y -- fallback where k = swKind x -- | Equality. svEqual :: SVal -> SVal -> SVal svEqual a b | isSet a && isSet b = svSetEqual a b | True = liftSym2B (mkSymOpSC (eqOptBool Equal trueSV) Equal) rationalCheck (==) (==) (==) (==) (==) (==) (==) (==) (==) (==) (==) a b -- | Inequality. svNotEqual :: SVal -> SVal -> SVal svNotEqual a b | isSet a && isSet b = svNot $ svEqual a b | True = liftSym2B (mkSymOpSC (eqOptBool NotEqual falseSV) NotEqual) rationalCheck (/=) (/=) (/=) (/=) (/=) (/=) (/=) (/=) (/=) (/=) (/=) a b -- | Set equality. Note that we only do constant folding if we get both a regular or both a -- complement set. Otherwise we get a symbolic value even if they might be completely concrete. svSetEqual :: SVal -> SVal -> SVal svSetEqual sa sb | not (isSet sa && isSet sb && kindOf sa == kindOf sb) = error $ "Data.SBV.svSetEqual: Called on ill-typed args: " ++ show (kindOf sa, kindOf sb) | Just (RegularSet a) <- getSet sa, Just (RegularSet b) <- getSet sb = svBool (a == b) | Just (ComplementSet a) <- getSet sa, Just (ComplementSet b) <- getSet sb = svBool (a == b) | True = SVal KBool $ Right $ cache r where getSet (SVal _ (Left (CV _ (CSet s)))) = Just s getSet _ = Nothing r st = do sva <- svToSV st sa svb <- svToSV st sb newExpr st KBool $ SBVApp (SetOp SetEqual) [sva, svb] -- | Strong equality. Only matters on floats, where it says @NaN@ equals @NaN@ and @+0@ and @-0@ are different. -- Otherwise equivalent to `svEqual`. svStrongEqual :: SVal -> SVal -> SVal svStrongEqual x y | isFloat x, Just f1 <- getF x, Just f2 <- getF y = svBool $ f1 `fpIsEqualObjectH` f2 | isDouble x, Just f1 <- getD x, Just f2 <- getD y = svBool $ f1 `fpIsEqualObjectH` f2 | isFloat x || isDouble x = SVal KBool $ Right $ cache r | True = svEqual x y where getF (SVal _ (Left (CV _ (CFloat f)))) = Just f getF _ = Nothing getD (SVal _ (Left (CV _ (CDouble d)))) = Just d getD _ = Nothing r st = do sx <- svToSV st x sy <- svToSV st y newExpr st KBool (SBVApp (IEEEFP FP_ObjEqual) [sx, sy]) -- | Less than. svLessThan :: SVal -> SVal -> SVal svLessThan x y | isConcreteMax x = svFalse | isConcreteMin y = svFalse | True = liftSym2B (mkSymOpSC (eqOpt falseSV) LessThan) rationalCheck (<) (<) (<) (<) (<) (<) (<) (<) (<) (<) (uiLift "<" (<)) x y -- | Greater than. svGreaterThan :: SVal -> SVal -> SVal svGreaterThan x y | isConcreteMin x = svFalse | isConcreteMax y = svFalse | True = liftSym2B (mkSymOpSC (eqOpt falseSV) GreaterThan) rationalCheck (>) (>) (>) (>) (>) (>) (>) (>) (>) (>) (uiLift ">" (>)) x y -- | Less than or equal to. svLessEq :: SVal -> SVal -> SVal svLessEq x y | isConcreteMin x = svTrue | isConcreteMax y = svTrue | True = liftSym2B (mkSymOpSC (eqOpt trueSV) LessEq) rationalCheck (<=) (<=) (<=) (<=) (<=) (<=) (<=) (<=) (<=) (<=) (uiLift "<=" (<=)) x y -- | Greater than or equal to. svGreaterEq :: SVal -> SVal -> SVal svGreaterEq x y | isConcreteMax x = svTrue | isConcreteMin y = svTrue | True = liftSym2B (mkSymOpSC (eqOpt trueSV) GreaterEq) rationalCheck (>=) (>=) (>=) (>=) (>=) (>=) (>=) (>=) (>=) (>=) (uiLift ">=" (>=)) x y -- | Bitwise and. svAnd :: SVal -> SVal -> SVal svAnd x y | isConcreteZero x = x | isConcreteOnes x = y | isConcreteZero y = y | isConcreteOnes y = x | True = liftSym2 (mkSymOpSC opt And) (const (const True)) (noReal ".&.") (.&.) (noFloat ".&.") (noDouble ".&.") x y where opt a b | a == falseSV || b == falseSV = Just falseSV | a == trueSV = Just b | b == trueSV = Just a | True = Nothing -- | Bitwise or. svOr :: SVal -> SVal -> SVal svOr x y | isConcreteZero x = y | isConcreteOnes x = x | isConcreteZero y = x | isConcreteOnes y = y | True = liftSym2 (mkSymOpSC opt Or) (const (const True)) (noReal ".|.") (.|.) (noFloat ".|.") (noDouble ".|.") x y where opt a b | a == trueSV || b == trueSV = Just trueSV | a == falseSV = Just b | b == falseSV = Just a | True = Nothing -- | Bitwise xor. svXOr :: SVal -> SVal -> SVal svXOr x y | isConcreteZero x = y | isConcreteOnes x = svNot y | isConcreteZero y = x | isConcreteOnes y = svNot x | True = liftSym2 (mkSymOpSC opt XOr) (const (const True)) (noReal "xor") xor (noFloat "xor") (noDouble "xor") x y where opt a b | a == b && swKind a == KBool = Just falseSV | a == falseSV = Just b | b == falseSV = Just a | True = Nothing -- | Bitwise complement. svNot :: SVal -> SVal svNot = liftSym1 (mkSymOp1SC opt Not) (noRealUnary "complement") complement (noFloatUnary "complement") (noDoubleUnary "complement") where opt a | a == falseSV = Just trueSV | a == trueSV = Just falseSV | True = Nothing -- | Shift left by a constant amount. Translates to the "bvshl" -- operation in SMT-Lib. -- -- NB. Haskell spec says the behavior is undefined if the shift amount -- is negative. We arbitrarily return the value unchanged if this is the case. svShl :: SVal -> Int -> SVal svShl x i | i <= 0 = x | isBounded x, i >= intSizeOf x = svInteger k 0 | True = x `svShiftLeft` svInteger k (fromIntegral i) where k = kindOf x -- | Shift right by a constant amount. Translates to either "bvlshr" -- (logical shift right) or "bvashr" (arithmetic shift right) in -- SMT-Lib, depending on whether @x@ is a signed bitvector. -- -- NB. Haskell spec says the behavior is undefined if the shift amount -- is negative. We arbitrarily return the value unchanged if this is the case. svShr :: SVal -> Int -> SVal svShr x i | i <= 0 = x | isBounded x, i >= intSizeOf x = if not (hasSign x) then z else svIte (x `svLessThan` z) neg1 z | True = x `svShiftRight` svInteger k (fromIntegral i) where k = kindOf x z = svInteger k 0 neg1 = svInteger k (-1) -- | Rotate-left, by a constant. -- -- NB. Haskell spec says the behavior is undefined if the shift amount -- is negative. We arbitrarily return the value unchanged if this is the case. svRol :: SVal -> Int -> SVal svRol x i | i <= 0 = x | True = case kindOf x of KBounded _ sz -> liftSym1 (mkSymOp1 (Rol (i `mod` sz))) (noRealUnary "rotateL") (rot True sz i) (noFloatUnary "rotateL") (noDoubleUnary "rotateL") x _ -> svShl x i -- for unbounded Integers, rotateL is the same as shiftL in Haskell -- | Rotate-right, by a constant. -- -- NB. Haskell spec says the behavior is undefined if the shift amount -- is negative. We arbitrarily return the value unchanged if this is the case. svRor :: SVal -> Int -> SVal svRor x i | i <= 0 = x | True = case kindOf x of KBounded _ sz -> liftSym1 (mkSymOp1 (Ror (i `mod` sz))) (noRealUnary "rotateR") (rot False sz i) (noFloatUnary "rotateR") (noDoubleUnary "rotateR") x _ -> svShr x i -- for unbounded integers, rotateR is the same as shiftR in Haskell -- | Generic rotation. Since the underlying representation is just Integers, rotations has to be -- careful on the bit-size. rot :: Bool -> Int -> Int -> Integer -> Integer rot toLeft sz amt x | sz < 2 = x | True = norm x y' `shiftL` y .|. norm (x `shiftR` y') y where (y, y') | toLeft = (amt `mod` sz, sz - y) | True = (sz - y', amt `mod` sz) norm v s = v .&. ((1 `shiftL` s) - 1) -- | Extract bit-sequences. svExtract :: Int -> Int -> SVal -> SVal svExtract i j x@(SVal (KBounded s _) _) | i < j = SVal k (Left $! CV k (CInteger 0)) | SVal _ (Left (CV _ (CInteger v))) <- x = SVal k (Left $! normCV (CV k (CInteger (v `shiftR` j)))) | True = SVal k (Right (cache y)) where k = KBounded s (i - j + 1) y st = do sv <- svToSV st x newExpr st k (SBVApp (Extract i j) [sv]) svExtract _ _ _ = error "extract: non-bitvector type" -- | Join two words, by concataneting svJoin :: SVal -> SVal -> SVal svJoin x@(SVal (KBounded s i) a) y@(SVal (KBounded _ j) b) | i == 0 = y | j == 0 = x | Left (CV _ (CInteger m)) <- a, Left (CV _ (CInteger n)) <- b = SVal k (Left $! CV k (CInteger (m `shiftL` j .|. n))) | True = SVal k (Right (cache z)) where k = KBounded s (i + j) z st = do xsw <- svToSV st x ysw <- svToSV st y newExpr st k (SBVApp Join [xsw, ysw]) svJoin _ _ = error "svJoin: non-bitvector type" -- | If-then-else. This one will force branches. svIte :: SVal -> SVal -> SVal -> SVal svIte t a b = svSymbolicMerge (kindOf a) True t a b -- | Lazy If-then-else. This one will delay forcing the branches unless it's really necessary. svLazyIte :: Kind -> SVal -> SVal -> SVal -> SVal svLazyIte k t a b = svSymbolicMerge k False t a b -- | Merge two symbolic values, at kind @k@, possibly @force@'ing the branches to make -- sure they do not evaluate to the same result. svSymbolicMerge :: Kind -> Bool -> SVal -> SVal -> SVal -> SVal svSymbolicMerge k force t a b | Just r <- svAsBool t = if r then a else b | force, rationalSBVCheck a b, sameResult a b = a | True = SVal k $ Right $ cache c where sameResult (SVal _ (Left c1)) (SVal _ (Left c2)) = c1 == c2 sameResult _ _ = False c st = do swt <- svToSV st t case () of () | swt == trueSV -> svToSV st a -- these two cases should never be needed as we expect symbolicMerge to be () | swt == falseSV -> svToSV st b -- called with symbolic tests, but just in case.. () -> do {- It is tempting to record the choice of the test expression here as we branch down to the 'then' and 'else' branches. That is, when we evaluate 'a', we can make use of the fact that the test expression is True, and similarly we can use the fact that it is False when b is evaluated. In certain cases this can cut down on symbolic simulation significantly, for instance if repetitive decisions are made in a recursive loop. Unfortunately, the implementation of this idea is quite tricky, due to our sharing based implementation. As the 'then' branch is evaluated, we will create many expressions that are likely going to be "reused" when the 'else' branch is executed. But, it would be *dead wrong* to share those values, as they were "cached" under the incorrect assumptions. To wit, consider the following: foo x y = ite (y .== 0) k (k+1) where k = ite (y .== 0) x (x+1) When we reduce the 'then' branch of the first ite, we'd record the assumption that y is 0. But while reducing the 'then' branch, we'd like to share @k@, which would evaluate (correctly) to @x@ under the given assumption. When we backtrack and evaluate the 'else' branch of the first ite, we'd see @k@ is needed again, and we'd look it up from our sharing map to find (incorrectly) that its value is @x@, which was stored there under the assumption that y was 0, which no longer holds. Clearly, this is unsound. A sound implementation would have to precisely track which assumptions were active at the time expressions get shared. That is, in the above example, we should record that the value of @k@ was cached under the assumption that @y@ is 0. While sound, this approach unfortunately leads to significant loss of valid sharing when the value itself had nothing to do with the assumption itself. To wit, consider: foo x y = ite (y .== 0) k (k+1) where k = x+5 If we tracked the assumptions, we would recompute @k@ twice, since the branch assumptions would differ. Clearly, there is no need to re-compute @k@ in this case since its value is independent of @y@. Note that the whole SBV performance story is based on agressive sharing, and losing that would have other significant ramifications. The "proper" solution would be to track, with each shared computation, precisely which assumptions it actually *depends* on, rather than blindly recording all the assumptions present at that time. SBV's symbolic simulation engine clearly has all the info needed to do this properly, but the implementation is not straightforward at all. For each subexpression, we would need to chase down its dependencies transitively, which can require a lot of scanning of the generated program causing major slow-down; thus potentially defeating the whole purpose of sharing in the first place. Design choice: Keep it simple, and simply do not track the assumption at all. This will maximize sharing, at the cost of evaluating unreachable branches. I think the simplicity is more important at this point than efficiency. Also note that the user can avoid most such issues by properly combining if-then-else's with common conditions together. That is, the first program above should be written like this: foo x y = ite (y .== 0) x (x+2) In general, the following transformations should be done whenever possible: ite e1 (ite e1 e2 e3) e4 --> ite e1 e2 e4 ite e1 e2 (ite e1 e3 e4) --> ite e1 e2 e4 This is in accordance with the general rule-of-thumb stating conditionals should be avoided as much as possible. However, we might prefer the following: ite e1 (f e2 e4) (f e3 e5) --> f (ite e1 e2 e3) (ite e1 e4 e5) especially if this expression happens to be inside 'f's body itself (i.e., when f is recursive), since it reduces the number of recursive calls. Clearly, programming with symbolic simulation in mind is another kind of beast alltogether. -} let sta = st `extendSValPathCondition` svAnd t let stb = st `extendSValPathCondition` svAnd (svNot t) swa <- svToSV sta a -- evaluate 'then' branch swb <- svToSV stb b -- evaluate 'else' branch -- merge, but simplify for certain boolean cases: case () of () | swa == swb -> return swa -- if t then a else a ==> a () | swa == trueSV && swb == falseSV -> return swt -- if t then true else false ==> t () | swa == falseSV && swb == trueSV -> newExpr st k (SBVApp Not [swt]) -- if t then false else true ==> not t () | swa == trueSV -> newExpr st k (SBVApp Or [swt, swb]) -- if t then true else b ==> t OR b () | swa == falseSV -> do swt' <- newExpr st KBool (SBVApp Not [swt]) newExpr st k (SBVApp And [swt', swb]) -- if t then false else b ==> t' AND b () | swb == trueSV -> do swt' <- newExpr st KBool (SBVApp Not [swt]) newExpr st k (SBVApp Or [swt', swa]) -- if t then a else true ==> t' OR a () | swb == falseSV -> newExpr st k (SBVApp And [swt, swa]) -- if t then a else false ==> t AND a () -> newExpr st k (SBVApp Ite [swt, swa, swb]) -- | Total indexing operation. @svSelect xs default index@ is -- intuitively the same as @xs !! index@, except it evaluates to -- @default@ if @index@ overflows. Translates to SMT-Lib tables. svSelect :: [SVal] -> SVal -> SVal -> SVal svSelect xs err ind | SVal _ (Left c) <- ind = case cvVal c of CInteger i -> if i < 0 || i >= genericLength xs then err else xs `genericIndex` i _ -> error $ "SBV.select: unsupported " ++ show (kindOf ind) ++ " valued select/index expression" svSelect xsOrig err ind = xs `seq` SVal kElt (Right (cache r)) where kInd = kindOf ind kElt = kindOf err -- Based on the index size, we need to limit the elements. For -- instance if the index is 8 bits, but there are 257 elements, -- that last element will never be used and we can chop it off. xs = case kInd of KBounded False i -> genericTake ((2::Integer) ^ i) xsOrig KBounded True i -> genericTake ((2::Integer) ^ (i-1)) xsOrig KUnbounded -> xsOrig _ -> error $ "SBV.select: unsupported " ++ show kInd ++ " valued select/index expression" r st = do sws <- mapM (svToSV st) xs swe <- svToSV st err if all (== swe) sws -- off-chance that all elts are the same then return swe else do idx <- getTableIndex st kInd kElt sws swi <- svToSV st ind let len = length xs -- NB. No need to worry here that the index -- might be < 0; as the SMTLib translation -- takes care of that automatically newExpr st kElt (SBVApp (LkUp (idx, kInd, kElt, len) swi swe) []) -- Change the sign of a bit-vector quantity. Fails if passed a non-bv svChangeSign :: Bool -> SVal -> SVal svChangeSign s x | not (isBounded x) = error $ "Data.SBV." ++ nm ++ ": Received non bit-vector kind: " ++ show (kindOf x) | Just n <- svAsInteger x = svInteger k n | True = SVal k (Right (cache y)) where nm = if s then "svSign" else "svUnsign" k = KBounded s (intSizeOf x) y st = do xsw <- svToSV st x newExpr st k (SBVApp (Extract (intSizeOf x - 1) 0) [xsw]) -- | Convert a symbolic bitvector from unsigned to signed. svSign :: SVal -> SVal svSign = svChangeSign True -- | Convert a symbolic bitvector from signed to unsigned. svUnsign :: SVal -> SVal svUnsign = svChangeSign False -- | Convert a symbolic bitvector from one integral kind to another. svFromIntegral :: Kind -> SVal -> SVal svFromIntegral kTo x | Just v <- svAsInteger x = svInteger kTo v | True = result where result = SVal kTo (Right (cache y)) kFrom = kindOf x y st = do xsw <- svToSV st x newExpr st kTo (SBVApp (KindCast kFrom kTo) [xsw]) -------------------------------------------------------------------------------- -- Derived operations -- | Convert an SVal from kind Bool to an unsigned bitvector of size 1. svToWord1 :: SVal -> SVal svToWord1 b = svSymbolicMerge k True b (svInteger k 1) (svInteger k 0) where k = KBounded False 1 -- | Convert an SVal from a bitvector of size 1 (signed or unsigned) to kind Bool. svFromWord1 :: SVal -> SVal svFromWord1 x = svNotEqual x (svInteger k 0) where k = kindOf x -- | Test the value of a bit. Note that we do an extract here -- as opposed to masking and checking against zero, as we found -- extraction to be much faster with large bit-vectors. svTestBit :: SVal -> Int -> SVal svTestBit x i | i < intSizeOf x = svFromWord1 (svExtract i i x) | True = svFalse -- | Generalization of 'svShl', where the shift-amount is symbolic. svShiftLeft :: SVal -> SVal -> SVal svShiftLeft = svShift True -- | Generalization of 'svShr', where the shift-amount is symbolic. -- -- NB. If the shiftee is signed, then this is an arithmetic shift; -- otherwise it's logical. svShiftRight :: SVal -> SVal -> SVal svShiftRight = svShift False -- | Generic shifting of bounded quantities. The shift amount must be non-negative and within the bounds of the argument -- for bit vectors. For negative shift amounts, the result is returned unchanged. For overshifts, left-shift produces 0, -- right shift produces 0 or -1 depending on the result being signed. svShift :: Bool -> SVal -> SVal -> SVal svShift toLeft x i | Just r <- constFoldValue = r | cannotOverShift = svIte (i `svLessThan` svInteger ki 0) -- Negative shift, no change x regularShiftValue | True = svIte (i `svLessThan` svInteger ki 0) -- Negative shift, no change x $ svIte (i `svGreaterEq` svInteger ki (fromIntegral (intSizeOf x))) -- Overshift, by at least the bit-width of x overShiftValue regularShiftValue where nm | toLeft = "shiftLeft" | True = "shiftRight" kx = kindOf x ki = kindOf i -- Constant fold the result if possible. If either quantity is unbounded, then we only support constants -- as there's no easy/meaningful way to map this combo to SMTLib. Should be rarely needed, if ever! -- We also perform basic sanity check here so that if we go past here, we know we have bitvectors only. constFoldValue | Just iv <- getConst i, iv == 0 = Just x | Just xv <- getConst x, xv == 0 = Just x | Just xv <- getConst x, Just iv <- getConst i = Just $ SVal kx . Left $! normCV $ CV kx (CInteger (xv `opC` shiftAmount iv)) | isUnbounded x || isUnbounded i = bailOut $ "Not yet implemented unbounded/non-constants shifts for " ++ show (kx, ki) ++ ", please file a request!" | not (isBounded x && isBounded i) = bailOut $ "Unexpected kinds: " ++ show (kx, ki) | True = Nothing where bailOut m = error $ "SBV." ++ nm ++ ": " ++ m getConst (SVal _ (Left (CV _ (CInteger val)))) = Just val getConst _ = Nothing opC | toLeft = shiftL | True = shiftR -- like fromIntegral, but more paranoid shiftAmount :: Integer -> Int shiftAmount iv | iv <= 0 = 0 | isUnbounded i, iv > fromIntegral (maxBound :: Int) = bailOut $ "Unsupported constant unbounded shift with amount: " ++ show iv | isUnbounded x = fromIntegral iv | iv >= fromIntegral ub = ub | not (isBounded x && isBounded i) = bailOut $ "Unsupported kinds: " ++ show (kx, ki) | True = fromIntegral iv where ub = intSizeOf x -- Overshift is not possible if the bit-size of x won't even fit into the bit-vector size -- of i. Note that this is a *necessary* check, Consider for instance if we're shifting a -- 32-bit value using a 1-bit shift amount (which can happen if the value is 1 with minimal -- shift widths). We would compare 1 >= 32, but stuffing 32 into bit-vector of size 1 would -- overflow. See http://github.com/LeventErkok/sbv/issues/323 for this case. Thus, we -- make sure that the bit-vector would fit as a value. cannotOverShift = maxRepresentable <= fromIntegral (intSizeOf x) where maxRepresentable :: Integer maxRepresentable | hasSign i = bit (intSizeOf i - 1) - 1 | True = bit (intSizeOf i ) - 1 -- An overshift occurs if we're shifting by more than or equal to the bit-width of x -- For shift-left: this value is always 0 -- For shift-right: -- If x is unsigned: 0 -- If x is signed and is less than 0, then -1 else 0 overShiftValue | toLeft = zx | hasSign x = svIte (x `svLessThan` zx) neg1 zx | True = zx where zx = svInteger kx 0 neg1 = svInteger kx (-1) -- Regular shift, we know that the shift value fits into the bit-width of x, since it's between 0 and sizeOf x. So, we can just -- turn it into a properly sized argument and ship it to SMTLib regularShiftValue = SVal kx $ Right $ cache result where result st = do sw1 <- svToSV st x sw2 <- svToSV st i let op | toLeft = Shl | True = Shr adjustedShift <- if kx == ki then return sw2 else newExpr st kx (SBVApp (KindCast ki kx) [sw2]) newExpr st kx (SBVApp op [sw1, adjustedShift]) -- | Generalization of 'svRol', where the rotation amount is symbolic. -- If the first argument is not bounded, then the this is the same as shift. svRotateLeft :: SVal -> SVal -> SVal svRotateLeft x i | not (isBounded x) = svShiftLeft x i | isBounded i && bit si <= toInteger sx -- wrap-around not possible = svIte (svLessThan i zi) (svSelect [x `svRor` k | k <- [0 .. bit si - 1]] z (svUNeg i)) (svSelect [x `svRol` k | k <- [0 .. bit si - 1]] z i) | True = svIte (svLessThan i zi) (svSelect [x `svRor` k | k <- [0 .. sx - 1]] z (svUNeg i `svRem` n)) (svSelect [x `svRol` k | k <- [0 .. sx - 1]] z ( i `svRem` n)) where sx = intSizeOf x si = intSizeOf i z = svInteger (kindOf x) 0 zi = svInteger (kindOf i) 0 n = svInteger (kindOf i) (toInteger sx) -- | A variant of 'svRotateLeft' that uses a barrel-rotate design, which can lead to -- better verification code. Only works when both arguments are finite and the second -- argument is unsigned. svBarrelRotateLeft :: SVal -> SVal -> SVal svBarrelRotateLeft x i | not (isBounded x && isBounded i && not (hasSign i)) = error $ "Data.SBV.Dynamic.svBarrelRotateLeft: Arguments must be bounded with second argument unsigned. Received: " ++ show (x, i) | Just iv <- svAsInteger i = svRol x $ fromIntegral (iv `rem` fromIntegral (intSizeOf x)) | True = barrelRotate svRol x i -- | A variant of 'svRotateLeft' that uses a barrel-rotate design, which can lead to -- better verification code. Only works when both arguments are finite and the second -- argument is unsigned. svBarrelRotateRight :: SVal -> SVal -> SVal svBarrelRotateRight x i | not (isBounded x && isBounded i && not (hasSign i)) = error $ "Data.SBV.Dynamic.svBarrelRotateRight: Arguments must be bounded with second argument unsigned. Received: " ++ show (x, i) | Just iv <- svAsInteger i = svRor x $ fromIntegral (iv `rem` fromIntegral (intSizeOf x)) | True = barrelRotate svRor x i -- Barrel rotation, by bit-blasting the argument: barrelRotate :: (SVal -> Int -> SVal) -> SVal -> SVal -> SVal barrelRotate f a c = loop blasted a where loop :: [(SVal, Integer)] -> SVal -> SVal loop [] acc = acc loop ((b, v) : rest) acc = loop rest (svIte b (f acc (fromInteger v)) acc) sa = toInteger $ intSizeOf a n = svInteger (kindOf c) sa -- Reduce by the modulus amount, we need not care about the -- any part larger than the value of the bit-size of the -- argument as it is identity for rotations reducedC = c `svRem` n -- blast little-endian, and zip with bit-position blasted = takeWhile significant $ zip (svBlastLE reducedC) [2^i | i <- [(0::Integer)..]] -- Any term whose bit-position is larger than our input size -- is insignificant, since the reduction would've put 0's in those -- bits. For instance, if a is 32 bits, and c is 5 bits, then we -- need not look at any position i s.t. 2^i > 32 significant (_, pos) = pos < sa -- | Generalization of 'svRor', where the rotation amount is symbolic. -- If the first argument is not bounded, then the this is the same as shift. svRotateRight :: SVal -> SVal -> SVal svRotateRight x i | not (isBounded x) = svShiftRight x i | isBounded i && bit si <= toInteger sx -- wrap-around not possible = svIte (svLessThan i zi) (svSelect [x `svRol` k | k <- [0 .. bit si - 1]] z (svUNeg i)) (svSelect [x `svRor` k | k <- [0 .. bit si - 1]] z i) | True = svIte (svLessThan i zi) (svSelect [x `svRol` k | k <- [0 .. sx - 1]] z (svUNeg i `svRem` n)) (svSelect [x `svRor` k | k <- [0 .. sx - 1]] z ( i `svRem` n)) where sx = intSizeOf x si = intSizeOf i z = svInteger (kindOf x) 0 zi = svInteger (kindOf i) 0 n = svInteger (kindOf i) (toInteger sx) -------------------------------------------------------------------------------- -- | Overflow detection. svMkOverflow :: OvOp -> SVal -> SVal -> SVal svMkOverflow o x y = SVal KBool (Right (cache r)) where r st = do sx <- svToSV st x sy <- svToSV st y newExpr st KBool $ SBVApp (OverflowOp o) [sx, sy] --------------------------------------------------------------------------------- -- * Symbolic Arrays --------------------------------------------------------------------------------- -- | Arrays in terms of SMT-Lib arrays data SArr = SArr (Kind, Kind) (Cached ArrayIndex) -- | Read the array element at @a@ readSArr :: SArr -> SVal -> SVal readSArr (SArr (_, bk) f) a = SVal bk $ Right $ cache r where r st = do arr <- uncacheAI f st i <- svToSV st a newExpr st bk (SBVApp (ArrRead arr) [i]) -- | Update the element at @a@ to be @b@ writeSArr :: SArr -> SVal -> SVal -> SArr writeSArr (SArr ainfo f) a b = SArr ainfo $ cache g where g st = do arr <- uncacheAI f st addr <- svToSV st a val <- svToSV st b amap <- R.readIORef (rArrayMap st) let j = ArrayIndex $ IMap.size amap upd = IMap.insert (unArrayIndex j) ("array_" ++ show j, ainfo, ArrayMutate arr addr val) j `seq` modifyState st rArrayMap upd $ modifyIncState st rNewArrs upd return j -- | Merge two given arrays on the symbolic condition -- Intuitively: @mergeArrays cond a b = if cond then a else b@. -- Merging pushes the if-then-else choice down on to elements mergeSArr :: SVal -> SArr -> SArr -> SArr mergeSArr t (SArr ainfo a) (SArr _ b) = SArr ainfo $ cache h where h st = do ai <- uncacheAI a st bi <- uncacheAI b st ts <- svToSV st t amap <- R.readIORef (rArrayMap st) let k = ArrayIndex $ IMap.size amap upd = IMap.insert (unArrayIndex k) ("array_" ++ show k, ainfo, ArrayMerge ts ai bi) k `seq` modifyState st rArrayMap upd $ modifyIncState st rNewArrs upd return k -- | Create a named new array newSArr :: State -> (Kind, Kind) -> (Int -> String) -> Maybe SVal -> IO SArr newSArr st ainfo mkNm mbDef = do amap <- R.readIORef $ rArrayMap st mbSWDef <- case mbDef of Nothing -> return Nothing Just sv -> Just <$> svToSV st sv let i = ArrayIndex $ IMap.size amap nm = mkNm (unArrayIndex i) upd = IMap.insert (unArrayIndex i) (nm, ainfo, ArrayFree mbSWDef) registerLabel "SArray declaration" st nm modifyState st rArrayMap upd $ modifyIncState st rNewArrs upd return $ SArr ainfo $ cache $ const $ return i -- | Compare two arrays for equality eqSArr :: SArr -> SArr -> SVal eqSArr (SArr _ a) (SArr _ b) = SVal KBool $ Right $ cache c where c st = do ai <- uncacheAI a st bi <- uncacheAI b st newExpr st KBool (SBVApp (ArrEq ai bi) []) -- | Arrays managed internally data SFunArr = SFunArr (Kind, Kind) (Cached FArrayIndex) -- | Convert a node-id to an SVal nodeIdToSVal :: Kind -> Int -> SVal nodeIdToSVal k i = swToSVal $ SV k (NodeId i) -- | Convert an 'SV' to an 'SVal' swToSVal :: SV -> SVal swToSVal sv@(SV k _) = SVal k $ Right $ cache $ const $ return sv -- | A variant of SVal equality, but taking into account of constants -- NB. The rationalCheck is paranid perhaps, but is necessary in case -- we have some funky polynomial roots in there. We do allow for -- floating-points here though. Why? Because the Eq instance of 'CV' -- does the right thing by using object equality. (i.e., it does -- the right thing for NaN/+0/-0 etc.) A straightforward equality -- here would be wrong for floats! svEqualWithConsts :: (SVal, Maybe CV) -> (SVal, Maybe CV) -> SVal svEqualWithConsts sv1 sv2 = case (grabCV sv1, grabCV sv2) of (Just cv, Just cv') | rationalCheck cv cv' -> if cv == cv' then svTrue else svFalse _ -> fst sv1 `svEqual` fst sv2 where grabCV (_, Just cv) = Just cv grabCV (SVal _ (Left cv), _ ) = Just cv grabCV _ = Nothing -- | Read the array element at @a@. For efficiency purposes, we create a memo-table -- as we go along, as otherwise we suffer significant performance penalties. See: -- and -- . readSFunArr :: SFunArr -> SVal -> SVal readSFunArr (SFunArr (ak, bk) f) address | kindOf address /= ak = error $ "Data.SBV.readSFunArr: Impossible happened, accesing with address type: " ++ show (kindOf address) ++ ", expected: " ++ show ak | True = SVal bk $ Right $ cache r where r st = do fArrayIndex <- uncacheFAI f st fArrMap <- R.readIORef (rFArrayMap st) constMap <- R.readIORef (rconstMap st) let consts = Map.fromList [(i, cv) | (cv, SV _ (NodeId i)) <- Map.toList constMap] case unFArrayIndex fArrayIndex `IMap.lookup` fArrMap of Nothing -> error $ "Data.SBV.readSFunArr: Impossible happened while trying to access SFunArray, can't find index: " ++ show fArrayIndex Just (uninitializedRead, rCache) -> do memoTable <- R.readIORef rCache SV _ (NodeId addressNodeId) <- svToSV st address -- If we hit the cache, return the result right away. If we miss, we need to -- walk through each element to "merge" all possible reads as we do not know -- whether the symbolic access may end up the same as one of the earlier ones -- in the cache. case addressNodeId `IMap.lookup` memoTable of Just v -> return v -- cache hit! Nothing -> -- cache miss; walk down the cache items to form the chain of reads: do let aInfo = (address, addressNodeId `Map.lookup` consts) find :: [(Int, SV)] -> SVal find [] = uninitializedRead address find ((i, v) : ivs) = svIte (svEqualWithConsts (nodeIdToSVal ak i, i `Map.lookup` consts) aInfo) (swToSVal v) (find ivs) finalValue = find (IMap.toAscList memoTable) finalSW <- svToSV st finalValue -- Cache the result, so next time we can retrieve it faster if we look it up with the same address! -- The following line is really the whole point of having caching in SFunArray! R.modifyIORef' rCache (IMap.insert addressNodeId finalSW) return finalSW -- | Update the element at @address@ to be @b@ writeSFunArr :: SFunArr -> SVal -> SVal -> SFunArr writeSFunArr (SFunArr (ak, bk) f) address b | kindOf address /= ak = error $ "Data.SBV.writeSFunArr: Impossible happened, accesing with address type: " ++ show (kindOf address) ++ ", expected: " ++ show ak | kindOf b /= bk = error $ "Data.SBV.writeSFunArr: Impossible happened, accesing with address type: " ++ show (kindOf b) ++ ", expected: " ++ show bk | True = SFunArr (ak, bk) $ cache g where g st = do fArrayIndex <- uncacheFAI f st fArrMap <- R.readIORef (rFArrayMap st) constMap <- R.readIORef (rconstMap st) let consts = Map.fromList [(i, cv) | (cv, SV _ (NodeId i)) <- Map.toList constMap] case unFArrayIndex fArrayIndex `IMap.lookup` fArrMap of Nothing -> error $ "Data.SBV.writeSFunArr: Impossible happened while trying to access SFunArray, can't find index: " ++ show fArrayIndex Just (aUi, rCache) -> do memoTable <- R.readIORef rCache SV _ (NodeId addressNodeId) <- svToSV st address val <- svToSV st b -- There are three cases: -- -- (1) We hit the cache, and old value is the same as new: No write necessary, just return the array -- (2) We hit the cache, values are different. Simply insert, overriding the old-memo table location -- (3) We miss the cache: Now we have to walk through all accesses and update the memo table accordingly. -- Why? Just because we missed the cache doesn't mean that it's not there with a different "symbolic" -- address. So, we have to walk through and update each entry in case the address matches. -- -- Below, we determine which case we're in and then insert the value at the end and continue cont <- case addressNodeId `IMap.lookup` memoTable of Just oldVal -- Cache hit | val == oldVal -> return $ Left fArrayIndex -- Case 1 | True -> return $ Right memoTable -- Case 2 Nothing -> do -- Cache miss let aInfo = (address, addressNodeId `Map.lookup` consts) -- NB. The order of modifications here is important as we -- keep the keys in ascending order. (Since we'll call fromAscList later on.) walk :: [(Int, SV)] -> [(Int, SV)] -> IO [(Int, SV)] walk [] sofar = return $ reverse sofar walk ((i, s):iss) sofar = modify i s >>= \s' -> walk iss ((i, s') : sofar) -- At the cached address i, currently storing value s. Conditionally update it to `b` (new value) -- if the addresses match. Otherwise keep it the same. modify :: Int -> SV -> IO SV modify i s = svToSV st $ svIte (svEqualWithConsts (nodeIdToSVal ak i, i `Map.lookup` consts) aInfo) b (swToSVal s) Right . IMap.fromAscList <$> walk (IMap.toAscList memoTable) [] case cont of Left j -> return j -- There was a hit, and value was unchanged, nothing to do Right mt -> do -- There was a hit, and the value was different; or there was a miss. Insert the new value -- and create a new array. Note that we keep the aUi the same: Just because we modified -- an array, it doesn't mean we change the uninitialized reads: they still come from the same place. -- newMemoTable <- R.newIORef $ IMap.insert addressNodeId val mt let j = FArrayIndex $ IMap.size fArrMap upd = IMap.insert (unFArrayIndex j) (aUi, newMemoTable) j `seq` modifyState st rFArrayMap upd (return ()) return j -- | Merge two given arrays on the symbolic condition -- Intuitively: @mergeArrays cond a b = if cond then a else b@. -- Merging pushes the if-then-else choice down on to elements mergeSFunArr :: SVal -> SFunArr -> SFunArr -> SFunArr mergeSFunArr t array1@(SFunArr ainfo@(sourceKind, targetKind) a) array2@(SFunArr binfo b) | ainfo /= binfo = error $ "Data.SBV.mergeSFunArr: Impossible happened, merging incompatbile arrays: " ++ show (ainfo, binfo) | Just test <- svAsBool t = if test then array1 else array2 | True = SFunArr ainfo $ cache c where c st = do ai <- uncacheFAI a st bi <- uncacheFAI b st constMap <- R.readIORef (rconstMap st) let consts = Map.fromList [(i, cv) | (cv, SV _ (NodeId i)) <- Map.toList constMap] -- Catch the degenerate case of merging an array with itself. One -- can argue this is pointless, but actually it comes up when one -- is merging composite structures (through a Mergeable class instance) -- that has an array component that didn't change. So, pays off in practice! if unFArrayIndex ai == unFArrayIndex bi then return ai -- merging with itself, noop else do fArrMap <- R.readIORef (rFArrayMap st) case (unFArrayIndex ai `IMap.lookup` fArrMap, unFArrayIndex bi `IMap.lookup` fArrMap) of (Nothing, _) -> error $ "Data.SBV.mergeSFunArr: Impossible happened while trying to access SFunArray, can't find index: " ++ show ai (_, Nothing) -> error $ "Data.SBV.mergeSFunArr: Impossible happened while trying to access SFunArray, can't find index: " ++ show bi (Just (aUi, raCache), Just (bUi, rbCache)) -> do -- This is where the complication happens. We need to merge the caches. If the same -- key appears in both, then that's the easy case: Just merge the entries. But if -- a key only appears in one but not the other? Just like in the read/write cases, -- we have to consider the possibility that the missing key can be any one of the -- other elements in the cache. So, for each non-matching key in either memo-table, -- we traverse the other and create a chain of look-up values. aMemo <- R.readIORef raCache bMemo <- R.readIORef rbCache let aMemoT = IMap.toAscList aMemo bMemoT = IMap.toAscList bMemo -- gen takes a uninitialized-read creator, a key, and the choices from the "other" -- cache that this key may map to. And creates a new SV that corresponds to the -- merged value: gen :: (SVal -> SVal) -> Int -> [(Int, SV)] -> IO SV gen mk k choices = svToSV st $ walk choices where kInfo = (nodeIdToSVal sourceKind k, k `Map.lookup` consts) walk :: [(Int, SV)] -> SVal walk [] = mk (fst kInfo) walk ((i, v) : ivs) = svIte (svEqualWithConsts (nodeIdToSVal sourceKind i, i `Map.lookup` consts) kInfo) (swToSVal v) (walk ivs) -- Insert into an existing map the new key value by merging according to the test fill :: Int -> SV -> SV -> IMap.IntMap SV -> IO (IMap.IntMap SV) fill k (SV _ (NodeId ni1)) (SV _ (NodeId ni2)) m = do v <- svToSV st (svIte t sval1 sval2) return $ IMap.insert k v m where sval1 = nodeIdToSVal targetKind ni1 sval2 = nodeIdToSVal targetKind ni2 -- Walk down the memo-tables in tandem. If we find a common key: Simply fill it in. If we find -- a key only in one, generate the corresponding read from the other cache, and do the fill. merge [] [] sofar = return sofar merge ((k1, v1) : as) [] sofar = gen bUi k1 bMemoT >>= \v2 -> fill k1 v1 v2 sofar >>= merge as [] merge [] ((k2, v2) : bs) sofar = gen aUi k2 aMemoT >>= \v1 -> fill k2 v1 v2 sofar >>= merge [] bs merge ass@((k1, v1) : as) bss@((k2, v2) : bs) sofar | k1 < k2 = gen bUi k1 bMemoT >>= \nv2 -> fill k1 v1 nv2 sofar >>= merge as bss | k1 > k2 = gen aUi k2 aMemoT >>= \nv1 -> fill k2 nv1 v2 sofar >>= merge ass bs | True = fill k1 v1 v2 sofar >>= merge as bs mergedMap <- merge aMemoT bMemoT IMap.empty memoMerged <- R.newIORef mergedMap -- Craft a new uninitializer. Note that we do *not* create a new initializer here, -- but simply merge the two initializers which will inherit their original unread -- references should the test be a constant. let mkUninitialized i = svIte t (aUi i) (bUi i) -- Add it to our collection: let j = FArrayIndex $ IMap.size fArrMap upd = IMap.insert (unFArrayIndex j) (mkUninitialized, memoMerged) j `seq` modifyState st rFArrayMap upd (return ()) return j -- | Create a named new array newSFunArr :: State -> (Kind, Kind) -> (Int -> String) -> Maybe SVal -> IO SFunArr newSFunArr st (ak, bk) mkNm mbDef = do fArrMap <- R.readIORef (rFArrayMap st) memoTable <- R.newIORef IMap.empty let j = FArrayIndex $ IMap.size fArrMap nm = mkNm (unFArrayIndex j) mkUninitialized i = case mbDef of Just def -> def _ -> svUninterpreted bk (nm ++ "_uninitializedRead") Nothing [i] upd = IMap.insert (unFArrayIndex j) (mkUninitialized, memoTable) registerLabel "SFunArray declaration" st nm j `seq` modifyState st rFArrayMap upd (return ()) return $ SFunArr (ak, bk) $ cache $ const $ return j -------------------------------------------------------------------------------- -- Utility functions noUnint :: (Maybe Int, String) -> a noUnint x = error $ "Unexpected operation called on uninterpreted/enumerated value: " ++ show x noUnint2 :: (Maybe Int, String) -> (Maybe Int, String) -> a noUnint2 x y = error $ "Unexpected binary operation called on uninterpreted/enumerated values: " ++ show (x, y) noCharLift :: Char -> a noCharLift x = error $ "Unexpected operation called on char: " ++ show x noStringLift :: String -> a noStringLift x = error $ "Unexpected operation called on string: " ++ show x noCharLift2 :: Char -> Char -> a noCharLift2 x y = error $ "Unexpected binary operation called on chars: " ++ show (x, y) noStringLift2 :: String -> String -> a noStringLift2 x y = error $ "Unexpected binary operation called on strings: " ++ show (x, y) liftSym1 :: (State -> Kind -> SV -> IO SV) -> (AlgReal -> AlgReal) -> (Integer -> Integer) -> (Float -> Float) -> (Double -> Double) -> SVal -> SVal liftSym1 _ opCR opCI opCF opCD (SVal k (Left a)) = SVal k . Left $! mapCV opCR opCI opCF opCD noCharLift noStringLift noUnint a liftSym1 opS _ _ _ _ a@(SVal k _) = SVal k $ Right $ cache c where c st = do sva <- svToSV st a opS st k sva {- A note on constant folding. There are cases where we miss out on certain constant foldings. On May 8 2018, Matt Peddie pointed this out, as the C code he was getting had redundancies. I was aware that could be missing constant foldings due to missed out optimizations, or some other code snafu, but till Matt pointed it out I haven't realized that we could be hiding constants inside an if-then-else. The example is: proveWith z3{verbose=True} $ \x -> 0 .< ite (x .== (x::SWord8)) 1 (2::SWord8) If you try this, you'll see that it generates (shortened): (define-fun s1 () (_ BitVec 8) #x00) (define-fun s2 () (_ BitVec 8) #x01) (define-fun s3 () Bool (bvult s1 s2)) But clearly we have all the info for s3 to be computed! The issue here is that the reduction of @x .== x@ to @true@ happens after we start computing the if-then-else, hence we are already committed to an SV at that point. The call to ite eventually recognizes this, but at that point it picks up the now constants from SV's, missing the constant folding opportunity. We can fix this, by looking up the constants table in liftSV2, along the lines of: liftSV2 :: (CV -> CV -> Bool) -> (CV -> CV -> CV) -> (State -> Kind -> SV -> SV -> IO SV) -> Kind -> SVal -> SVal -> Cached SV liftSV2 okCV opCV opS k a b = cache c where c st = do sw1 <- svToSV st a sw2 <- svToSV st b cmap <- readIORef (rconstMap st) let cv1 = [cv | ((_, cv), sv) <- M.toList cmap, sv == sv1] cv2 = [cv | ((_, cv), sv) <- M.toList cmap, sv == sv2] case (cv1, cv2) of ([x], [y]) | okCV x y -> newConst st $ opCV x y _ -> opS st k sv1 sv2 (with obvious modifications to call sites to get the proper arguments.) But this means that we have to grab the constant list for every symbolicly lifted operation, also do the same for other places, etc.; for the rare opportunity of catching a @x .== x@ optimization. Even then, the constants for the branches would still be generated. (i.e., in the above example we would still generate @s1@ and @s2@, but would skip @s3@.) It seems to me that the price to pay is rather high, as this is hardly the most common case; so we're opting here to ignore these cases. See http://github.com/LeventErkok/sbv/issues/379 for some further discussion. -} liftSV2 :: (State -> Kind -> SV -> SV -> IO SV) -> Kind -> SVal -> SVal -> Cached SV liftSV2 opS k a b = cache c where c st = do sw1 <- svToSV st a sw2 <- svToSV st b opS st k sw1 sw2 liftSym2 :: (State -> Kind -> SV -> SV -> IO SV) -> (CV -> CV -> Bool) -> (AlgReal -> AlgReal -> AlgReal) -> (Integer -> Integer -> Integer) -> (Float -> Float -> Float) -> (Double -> Double -> Double) -> SVal -> SVal -> SVal liftSym2 _ okCV opCR opCI opCF opCD (SVal k (Left a)) (SVal _ (Left b)) | okCV a b = SVal k . Left $! mapCV2 opCR opCI opCF opCD noCharLift2 noStringLift2 noUnint2 a b liftSym2 opS _ _ _ _ _ a@(SVal k _) b = SVal k $ Right $ liftSV2 opS k a b liftSym2B :: (State -> Kind -> SV -> SV -> IO SV) -> (CV -> CV -> Bool) -> (AlgReal -> AlgReal -> Bool) -> (Integer -> Integer -> Bool) -> (Float -> Float -> Bool) -> (Double -> Double -> Bool) -> (Char -> Char -> Bool) -> (String -> String -> Bool) -> ([CVal] -> [CVal] -> Bool) -> ([CVal] -> [CVal] -> Bool) -> (Maybe CVal -> Maybe CVal -> Bool) -> (Either CVal CVal -> Either CVal CVal -> Bool) -> ((Maybe Int, String) -> (Maybe Int, String) -> Bool) -> SVal -> SVal -> SVal liftSym2B _ okCV opCR opCI opCF opCD opCC opCS opCSeq opCTup opCMaybe opCEither opUI (SVal _ (Left a)) (SVal _ (Left b)) | okCV a b = svBool (liftCV2 opCR opCI opCF opCD opCC opCS opCSeq opCTup opCMaybe opCEither opUI a b) liftSym2B opS _ _ _ _ _ _ _ _ _ _ _ _ a b = SVal KBool $ Right $ liftSV2 opS KBool a b -- | Create a symbolic two argument operation; with shortcut optimizations mkSymOpSC :: (SV -> SV -> Maybe SV) -> Op -> State -> Kind -> SV -> SV -> IO SV mkSymOpSC shortCut op st k a b = maybe (newExpr st k (SBVApp op [a, b])) return (shortCut a b) -- | Create a symbolic two argument operation; no shortcut optimizations mkSymOp :: Op -> State -> Kind -> SV -> SV -> IO SV mkSymOp = mkSymOpSC (const (const Nothing)) mkSymOp1SC :: (SV -> Maybe SV) -> Op -> State -> Kind -> SV -> IO SV mkSymOp1SC shortCut op st k a = maybe (newExpr st k (SBVApp op [a])) return (shortCut a) mkSymOp1 :: Op -> State -> Kind -> SV -> IO SV mkSymOp1 = mkSymOp1SC (const Nothing) -- | eqOpt says the references are to the same SV, thus we can optimize. Note that -- we explicitly disallow KFloat/KDouble here. Why? Because it's *NOT* true that -- NaN == NaN, NaN >= NaN, and so-forth. So, we have to make sure we don't optimize -- floats and doubles, in case the argument turns out to be NaN. eqOpt :: SV -> SV -> SV -> Maybe SV eqOpt w x y = case swKind x of KFloat -> Nothing KDouble -> Nothing _ -> if x == y then Just w else Nothing -- For uninterpreted/enumerated values, we carefully lift through the constructor index for comparisons: uiLift :: String -> (Int -> Int -> Bool) -> (Maybe Int, String) -> (Maybe Int, String) -> Bool uiLift _ cmp (Just i, _) (Just j, _) = i `cmp` j uiLift w _ a b = error $ "Data.SBV.Core.Operations: Impossible happened while trying to lift " ++ w ++ " over " ++ show (a, b) -- | Predicate to check if a value is concrete isConcrete :: SVal -> Bool isConcrete (SVal _ Left{}) = True isConcrete _ = False -- | Predicate for optimizing word operations like (+) and (*). -- NB. We specifically do *not* match for Double/Float; because -- FP-arithmetic doesn't obey traditional rules. For instance, -- 0 * x = 0 fails if x happens to be NaN or +/- Infinity. So, -- we merely return False when given a floating-point value here. isConcreteZero :: SVal -> Bool isConcreteZero (SVal _ (Left (CV _ (CInteger n)))) = n == 0 isConcreteZero (SVal KReal (Left (CV KReal (CAlgReal v)))) = isExactRational v && v == 0 isConcreteZero _ = False -- | Predicate for optimizing word operations like (+) and (*). -- NB. See comment on 'isConcreteZero' for why we don't match -- for Float/Double values here. isConcreteOne :: SVal -> Bool isConcreteOne (SVal _ (Left (CV _ (CInteger 1)))) = True isConcreteOne (SVal KReal (Left (CV KReal (CAlgReal v)))) = isExactRational v && v == 1 isConcreteOne _ = False -- | Predicate for optimizing bitwise operations. The unbounded integer case of checking -- against -1 might look dubious, but that's how Haskell treats 'Integer' as a member -- of the Bits class, try @(-1 :: Integer) `testBit` i@ for any @i@ and you'll get 'True'. isConcreteOnes :: SVal -> Bool isConcreteOnes (SVal _ (Left (CV (KBounded b w) (CInteger n)))) = n == if b then -1 else bit w - 1 isConcreteOnes (SVal _ (Left (CV KUnbounded (CInteger n)))) = n == -1 -- see comment above isConcreteOnes (SVal _ (Left (CV KBool (CInteger n)))) = n == 1 isConcreteOnes _ = False -- | Predicate for optimizing comparisons. isConcreteMax :: SVal -> Bool isConcreteMax (SVal _ (Left (CV (KBounded False w) (CInteger n)))) = n == bit w - 1 isConcreteMax (SVal _ (Left (CV (KBounded True w) (CInteger n)))) = n == bit (w - 1) - 1 isConcreteMax (SVal _ (Left (CV KBool (CInteger n)))) = n == 1 isConcreteMax _ = False -- | Predicate for optimizing comparisons. isConcreteMin :: SVal -> Bool isConcreteMin (SVal _ (Left (CV (KBounded False _) (CInteger n)))) = n == 0 isConcreteMin (SVal _ (Left (CV (KBounded True w) (CInteger n)))) = n == - bit (w - 1) isConcreteMin (SVal _ (Left (CV KBool (CInteger n)))) = n == 0 isConcreteMin _ = False -- | Most operations on concrete rationals require a compatibility check to avoid faulting -- on algebraic reals. rationalCheck :: CV -> CV -> Bool rationalCheck a b = case (cvVal a, cvVal b) of (CAlgReal x, CAlgReal y) -> isExactRational x && isExactRational y _ -> True -- | Quot/Rem operations require a nonzero check on the divisor. -- nonzeroCheck :: CV -> CV -> Bool nonzeroCheck _ b = cvVal b /= CInteger 0 -- | Same as rationalCheck, except for SBV's rationalSBVCheck :: SVal -> SVal -> Bool rationalSBVCheck (SVal KReal (Left a)) (SVal KReal (Left b)) = rationalCheck a b rationalSBVCheck _ _ = True noReal :: String -> AlgReal -> AlgReal -> AlgReal noReal o a b = error $ "SBV.AlgReal." ++ o ++ ": Unexpected arguments: " ++ show (a, b) noFloat :: String -> Float -> Float -> Float noFloat o a b = error $ "SBV.Float." ++ o ++ ": Unexpected arguments: " ++ show (a, b) noDouble :: String -> Double -> Double -> Double noDouble o a b = error $ "SBV.Double." ++ o ++ ": Unexpected arguments: " ++ show (a, b) noRealUnary :: String -> AlgReal -> AlgReal noRealUnary o a = error $ "SBV.AlgReal." ++ o ++ ": Unexpected argument: " ++ show a noFloatUnary :: String -> Float -> Float noFloatUnary o a = error $ "SBV.Float." ++ o ++ ": Unexpected argument: " ++ show a noDoubleUnary :: String -> Double -> Double noDoubleUnary o a = error $ "SBV.Double." ++ o ++ ": Unexpected argument: " ++ show a -- | Given a composite structure, figure out how to compare for less than svStructuralLessThan :: SVal -> SVal -> SVal svStructuralLessThan x y | isConcrete x && isConcrete y = x `svLessThan` y | KTuple{} <- kx = tupleLT x y | KMaybe{} <- kx = maybeLT x y | KEither{} <- kx = eitherLT x y | True = x `svLessThan` y where kx = kindOf x -- | Structural less-than for tuples tupleLT :: SVal -> SVal -> SVal tupleLT x y = SVal KBool $ Right $ cache res where ks = case kindOf x of KTuple xs -> xs k -> error $ "Data.SBV: Impossible happened, tupleLT called with: " ++ show (k, x, y) n = length ks res st = do sx <- svToSV st x sy <- svToSV st y let chkElt i ek = let xi = SVal ek $ Right $ cache $ \_ -> newExpr st ek $ SBVApp (TupleAccess i n) [sx] yi = SVal ek $ Right $ cache $ \_ -> newExpr st ek $ SBVApp (TupleAccess i n) [sy] lt = xi `svStructuralLessThan` yi eq = xi `svEqual` yi in (lt, eq) walk [] = svFalse walk [(lti, _)] = lti walk ((lti, eqi) : rest) = lti `svOr` (eqi `svAnd` walk rest) svToSV st $ walk $ zipWith chkElt [1..] ks -- | Structural less-than for maybes maybeLT :: SVal -> SVal -> SVal maybeLT x y = sMaybeCase ( sMaybeCase svFalse (const svTrue) y) (\jx -> sMaybeCase svFalse (jx `svStructuralLessThan`) y) x where ka = case kindOf x of KMaybe k' -> k' k -> error $ "Data.SBV: Impossible happened, maybeLT called with: " ++ show (k, x, y) sMaybeCase brNothing brJust s = SVal KBool $ Right $ cache res where res st = do sv <- svToSV st s let justVal = SVal ka $ Right $ cache $ \_ -> newExpr st ka $ SBVApp MaybeAccess [sv] justRes = brJust justVal br1 <- svToSV st brNothing br2 <- svToSV st justRes -- Do we have a value? noVal <- newExpr st KBool $ SBVApp (MaybeIs ka False) [sv] newExpr st KBool $ SBVApp Ite [noVal, br1, br2] -- | Structural less-than for either eitherLT :: SVal -> SVal -> SVal eitherLT x y = sEitherCase (\lx -> sEitherCase (lx `svStructuralLessThan`) (const svTrue) y) (\rx -> sEitherCase (const svFalse) (rx `svStructuralLessThan`) y) x where (ka, kb) = case kindOf x of KEither k1 k2 -> (k1, k2) k -> error $ "Data.SBV: Impossible happened, eitherLT called with: " ++ show (k, x, y) sEitherCase brA brB sab = SVal KBool $ Right $ cache res where res st = do abv <- svToSV st sab let leftVal = SVal ka $ Right $ cache $ \_ -> newExpr st ka $ SBVApp (EitherAccess False) [abv] rightVal = SVal kb $ Right $ cache $ \_ -> newExpr st kb $ SBVApp (EitherAccess True) [abv] leftRes = brA leftVal rightRes = brB rightVal br1 <- svToSV st leftRes br2 <- svToSV st rightRes -- Which branch are we in? Return the appropriate value: onLeft <- newExpr st KBool $ SBVApp (EitherIs ka kb False) [abv] newExpr st KBool $ SBVApp Ite [onLeft, br1, br2] {-# ANN svIte ("HLint: ignore Eta reduce" :: String) #-} {-# ANN svLazyIte ("HLint: ignore Eta reduce" :: String) #-} {-# ANN module ("HLint: ignore Reduce duplication" :: String) #-}