----------------------------------------------------------------------------- -- | -- 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 #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE TupleSections #-} {-# OPTIONS_GHC -Wall -Werror #-} module Data.SBV.Core.Operations ( -- ** Basic constructors svTrue, svFalse, svBool , svInteger, svFloat, svDouble, svFloatingPoint, svReal, svEnumFromThenTo, svString, svChar -- ** Basic destructors , svAsBool, svAsInteger, svNumerator, svDenominator -- ** Basic operations , svPlus, svTimes, svMinus, svUNeg, svAbs, svSignum , svDivide, svQuot, svRem, svQuotRem, svDivides , svEqual, svNotEqual, svStrongEqual, svImplies , svLessThan, svGreaterThan, svLessEq, svGreaterEq, svStructuralLessThan , svAnd, svOr, svXOr, svNot , svShl, svShr, svRol, svRor , svExtract, svJoin, svZeroExtend, svSignExtend , svIte, svLazyIte, svSymbolicMerge , svSelect , svSign, svUnsign, svSetBit, svWordFromBE, svWordFromLE , svExp, svFromIntegral -- ** Overflows , svMkOverflow1, svMkOverflow2 -- ** Derived operations , svToWord1, svFromWord1, svTestBit , svShiftLeft, svShiftRight , svRotateLeft, svRotateRight , svBarrelRotateLeft, svBarrelRotateRight , svBlastLE, svBlastBE , svAddConstant, svIncrement, svDecrement , svFloatAsSWord32, svDoubleAsSWord64, svFloatingPointAsSWord -- Utils , mkSymOp ) where import Prelude hiding (Foldable(..)) import Data.Bits (Bits(..)) import Data.List (genericIndex, genericLength, genericTake, foldr, length, foldl', elem, nub, sort, null, elemIndex) import Data.Maybe (isNothing) import Data.SBV.Core.AlgReals import Data.SBV.Core.Kind import Data.SBV.Core.Concrete import Data.SBV.Core.Symbolic import Data.SBV.Core.SizedFloats import Data.Ratio import Data.SBV.Utils.Numeric (fpIsEqualObjectH, floatToWord, doubleToWord) import LibBF -------------------------------------------------------------------------------- -- 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 Double svDouble :: Double -> SVal svDouble d = SVal KDouble (Left $! CV KDouble (CDouble d)) -- | Convert from a generalized floating point svFloatingPoint :: FP -> SVal svFloatingPoint f@(FP eb sb _) = SVal k (Left $! CV k (CFP f)) where k = KFP eb sb -- | 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. We handle arbitrary-FP's specially here, just for the negated literals. svUNeg :: SVal -> SVal svUNeg = liftSym1 (mkSymOp1 UNeg) negate negate negate negate negate negate -- | Absolute value. svAbs :: SVal -> SVal svAbs = liftSym1 (mkSymOp1 Abs) abs abs abs abs abs abs -- | Signum. -- -- NB. The following "carefully" tests the number for == 0, as Float/Double's NaN and +/-0 -- cases would cause trouble with explicit equality tests. svSignum :: SVal -> SVal svSignum a | hasSign a = svIte (a `svGreaterThan` z) i $ svIte (a `svLessThan` z) (svUNeg i) a | True = svIte (a `svGreaterThan` z) i a where k = kindOf a z = SVal k $ Left $ mkConstCV k (0 :: Integer) i = SVal k $ Left $ mkConstCV k (1 :: Integer) -- | Division. svDivide :: SVal -> SVal -> SVal svDivide = liftSym2 (mkSymOp Quot) [rationalCheck] (/) idiv (/) (/) (/) (/) where idiv x 0 = x idiv x y = x `div` y -- | Divides predicate svDivides :: Integer -> SVal -> SVal svDivides n v | n <= 0 = error $ "svDivides: The first argument must be a strictly positive number, received: " ++ show n | True = case v of SVal KUnbounded (Left (CV KUnbounded (CInteger val))) -> svBool (val `mod` n == 0) _ -> SVal KBool $ Right $ cache c where c st = do sva <- svToSV st v newExpr st KBool (SBVApp (Divides n) [sva]) -- | 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 or a floating point type. 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 or a floating point type. 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. -- Note that this variant does not respect the division/reminder by 0. That's handled at the SBV level. svQuot :: SVal -> SVal -> SVal svQuot x y | not isInteger && isConcreteZero x = x | not isInteger && isConcreteZero y = svInteger (kindOf x) 0 | not isInteger && isConcreteOne y = x | True = liftSym2 (mkSymOp Quot) [nonzeroCheck] (noReal "quot") quot' (noFloat "quot") (noDouble "quot") (noFP "quot") (noRat "quot") x y where isInteger = kindOf x == KUnbounded quot' a b | isInteger = 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 | not isInteger && isConcreteZero x = x | not isInteger && isConcreteZero y = x | not isInteger && isConcreteOne y = svInteger (kindOf x) 0 | True = liftSym2 (mkSymOp Rem) [nonzeroCheck] (noReal "rem") rem' (noFloat "rem") (noDouble "rem") (noFP "rem") (noRat "rem") x y where isInteger = kindOf x == KUnbounded rem' a b | isInteger = 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) -- | Implication. Only for booleans. svImplies :: SVal -> SVal -> SVal svImplies a b | any (\x -> kindOf x /= KBool) [a, b] = error $ "Data.SBV.svImplies: Unexpected arguments: " ++ show (a, kindOf a, b, kindOf b) | isConcreteZero a = svTrue -- F -> _ = T | isConcreteOne b = svTrue -- _ -> T = T | isConcreteOne a && isConcreteZero b = svFalse -- T -> F = F | isConcreteOne a && isConcreteOne b = svTrue -- T -> T = T | True = SVal KBool $ Right $ cache c where c st = do sva <- svToSV st a svb <- svToSV st b -- One final optimization, equal args is just True! if sva == svb then pure trueSV else newExpr st KBool (SBVApp Implies [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 | isFP x, Just f1 <- getFP x, Just f2 <- getFP y = svBool $ f1 `fpIsEqualObjectH` f2 | isFloat x || isDouble x || isFP x = SVal KBool $ Right $ cache r | True = compareSV (Equal True) x y where getF (SVal _ (Left (CV _ (CFloat f)))) = Just f getF _ = Nothing getD (SVal _ (Left (CV _ (CDouble d)))) = Just d getD _ = Nothing getFP (SVal _ (Left (CV _ (CFP f)))) = Just f getFP _ = Nothing r st = do sx <- svToSV st x sy <- svToSV st y newExpr st KBool (SBVApp (IEEEFP FP_ObjEqual) [sx, sy]) -- Comparisons have to be careful in making sure we don't rely on CVal ord/eq instance. compareSV :: Op -> SVal -> SVal -> SVal compareSV op x y -- Make sure we don't get anything we can't handle or expect | op `notElem` [Equal True, Equal False, NotEqual, LessThan, GreaterThan, LessEq, GreaterEq] = error $ "Unexpected call to compareSV: " ++ show (op, x, y) | kx /= ky = error $ "Mismatched kinds in call to compareSV:" ++ show (op, x, kindOf x, kindOf y) | (isSet kx || isArray ky) && op `notElem` [Equal True, Equal False, NotEqual] = error $ "Unexpected Set/Array not-equal comparison: " ++ show (op, x, k) -- Boolean equality optimizations | k == KBool, Equal{} <- op, SVal _ (Left xv) <- x, xv == trueCV = y -- true .== y --> y | k == KBool, Equal{} <- op, SVal _ (Left yv) <- y, yv == trueCV = x -- x .== true --> x | k == KBool, Equal{} <- op, SVal _ (Left xv) <- x, xv == falseCV = svNot y -- false .== y --> svNot y | k == KBool, Equal{} <- op, SVal _ (Left yv) <- y, yv == falseCV = svNot x -- x .== false --> svNot x | k == KBool, op == NotEqual, SVal _ (Left xv) <- x, xv == trueCV = svNot y -- true ./= y --> svNot y | k == KBool, op == NotEqual, SVal _ (Left yv) <- y, yv == trueCV = svNot x -- x ./= true --> svNot x | k == KBool, op == NotEqual, SVal _ (Left xv) <- x, xv == falseCV = y -- false ./= y --> y | k == KBool, op == NotEqual, SVal _ (Left yv) <- y, yv == falseCV = x -- x ./= false --> x -- Comparison optimizations if one operand is min/max bit-vector | op == LessThan, isConcreteMax x = svFalse -- MAX < _ | op == LessThan, isConcreteMin y = svFalse -- _ < MIN | op == GreaterThan, isConcreteMin x = svFalse -- MIN > _ | op == GreaterThan, isConcreteMax y = svFalse -- _ > MAX | op == LessEq, isConcreteMin x = svTrue -- MIN <= _ | op == LessEq, isConcreteMax y = svTrue -- _ <= MAX | op == GreaterEq, isConcreteMax x = svTrue -- MAX >= _ | op == GreaterEq, isConcreteMin y = svTrue -- _ >= MIN -- General constant folding, but be careful not to be too smart here. | SVal _ (Left xv) <- x, SVal _ (Left yv) <- y = case cCompare k op (cvVal xv) (cvVal yv) of Nothing -> -- cCompare is conservative on floats. Give those one more chance, only at the top-level. -- (i.e., if stored under a Maybe/Either/List etc., we'll resort to a symbolic result.) case (k, cvVal xv, cvVal yv) of (KFloat, CFloat a, CFloat b) -> svBool (a `cFPOp` b) (KDouble, CDouble a, CDouble b) -> svBool (a `cFPOp` b) (KFP{} , CFP a, CFP b) -> svBool (a `cFPOp` b) _ -> symResult Just r -> svBool $ case op of Equal _ -> r == EQ NotEqual -> r /= EQ LessThan -> r == LT GreaterThan -> r == GT LessEq -> r `elem` [EQ, LT] GreaterEq -> r `elem` [EQ, GT] _ -> error $ "Unexpected call to compareSV: " ++ show (op, x, y) -- No constant folding opportunities, turn symbolic | True = symResult where kx = kindOf x ky = kindOf y k = kx -- only used after we ensured kx == ky -- Are there any floats embedded down from here? if so, we have to be careful due to presence of NaN safeEq = op == Equal True -- strong equality ok || isSomeKindOfFloat k -- top level OK || not (containsFloats k) -- has floats somewhere: not ok symResult | safeEq = symResultSafe | True = symResultFP -- This will go down to SMTLib's =. So only use it if we're safe to do so! symResultSafe = SVal KBool $ Right $ cache res where res st = do svx :: SV <- svToSV st x svy :: SV <- svToSV st y if svx == svy && eqCheckIsObjectEq k then case op of Equal{} -> pure trueSV LessEq -> pure trueSV GreaterEq -> pure trueSV NotEqual -> pure falseSV LessThan -> pure falseSV GreaterThan -> pure falseSV _ -> error $ "Unexpected call to compareSV, equal SV case: " ++ show (op, svx) else newExpr st KBool (SBVApp op [svx, svy]) a `cFPOp` b = case op of Equal False -> a == b Equal True -> a `fpIsEqualObjectH` b NotEqual -> a /= b LessThan -> a < b GreaterThan -> a > b LessEq -> a <= b GreaterEq -> a >= b _ -> error $ "Unexpected call to cFPOp: " ++ show op -- OK, we have a result that has floats embedded in it. So comparison is problematic. -- Certain subsets of this is supported elsewhere. Here, we simply bail out. symResultFP = error $ unlines $ [ "" , "*** Data.SBV: Unsupported complicated comparison:" , "***" , "*** Op : " ++ show op , "*** Type: " ++ show k , "***" , "*** Due to the presence of NaN, comparisons over this type require" , "*** special support in SMTLib. And in general this can lead to" , "*** performance issues since the comparison is no longer a natively" , "*** supported operation in the logic." , "***" , "*** NB. If you want the semantics NaN == NaN, and +0 /= -0, then you can use .=== instead." , "***" ] ++ case alternative of Nothing -> ["*** Please report this as a feature request."] Just a -> [ "*** For this case, please use: " ++ a , "*** but beware of performance/decidability implications." ] where alternative = case (op, k) of (Equal False, KList f) | isFloat f || isDouble f || isFP f -> Just "Data.SBV.List.listEq" _ -> Nothing -- Compare two CVals; if we can. We're being conservative here and deferring to a symbolic result if we get something complicated. cCompare :: Kind -> Op -> CVal -> CVal -> Maybe Ordering cCompare k op x y = case (x, y) of -- The presence of NaN's throw this off. Why? Because @NaN `compare` x = GT@ in Haskell. But that's just the wrong thing to do here. -- So protect against NaN's. And a similar story for -0/0. (CFloat a, CFloat b) | any (nanOrZero k) [x, y] -> Nothing | True -> Just $ a `compare` b (CDouble a, CDouble b) | any (nanOrZero k) [x, y] -> Nothing | True -> Just $ a `compare` b (CFP a, CFP b) | any (nanOrZero k) [x, y] -> Nothing | True -> Just $ a `compare` b -- Simple cases (CInteger a, CInteger b) -> Just $ a `compare` b (CRational a, CRational b) -> Just $ a `compare` b (CChar a, CChar b) -> Just $ a `compare` b (CString a, CString b) -> Just $ a `compare` b -- We can handle algreal, so long as they are exact-rationals (CAlgReal a, CAlgReal b) | isExactRational a && isExactRational b -> Just $ a `compare` b | True -> Nothing -- Lists and tuples use lexicographic ordering (CList a, CList b) -> case k of KList ke -> lexCmp (map (ke,) a) (map (ke,) b) _ -> error $ "cCompare: Unexpected kind in cCompare for List: " ++ show k (CTuple a, CTuple b) | length a == length b -> case k of KTuple ks | length ks == length a -> lexCmp (zip ks a) (zip ks b) _ -> error "cCompare: Unexpected kind in cCompare for tuples" | True -> error $ "cCompare: Received tuples of differing size: " ++ show (op, length a, length b, k) -- Arrays and sets only support equality/inequality. And they have object-equality semantics. So -- if there are any floats or non-exact-rationals down in the index or element kinds, we bail (CSet a, CSet b) | op `elem` [Equal True, Equal False, NotEqual] , KSet ke <- k -> case svSetEqual ke a b of Nothing -> Nothing -- We don't know Just True -> Just EQ -- They're equal Just False -> Just GT -- Pick GT; so equality test will fail, inequality will pass | True -> error $ "cCompare: Received unexpected set comparison: " ++ show (op, k) (CArray a, CArray b) | op `elem` [Equal True, Equal False, NotEqual] , KArray k1 k2 <- k -> case svArrEqual k1 k2 a b of Nothing -> Nothing -- We don't know Just True -> Just EQ -- They're equal Just False -> Just GT -- Pick GT; so equality test will fail, inequality will pass | True -> error $ "cCompare: Received unexpected array comparison: " ++ show (op, k) -- ADTs. Only equal/inequal on full ADTs. Compares on enumerations. (CADT (s, fks), CADT (s', fks')) -> case k of -- Enumerations. We do a straight comparison on the constructor index KADT _ _ cstrs | all (null . snd) cstrs -> let cnms = map fst cstrs in case (s `elemIndex` cnms, s' `elemIndex` cnms) of (Just i, Just j) -> Just (i `compare` j) r -> error $ "cCompare: Unable to locate indexes for CADT: " ++ show (k, s, s', r) -- Arbitrary ADTs. Only allow equality/inequality _ | op `notElem` [Equal True, Equal False, NotEqual] -> error $ "cCompare: Received unexpected ADT comparison: " ++ show (op, k) -- Different constructor | s /= s' -> Just GT -- Pick GT; so equality test will fail, inequality will pass -- Same constructor | map fst fks /= map fst fks' -> error $ "cCompare: Mismatching ADT field kinds in comparison: " ++ show (op, k, map fst fks, map fst fks') | True -> let fmatch = zipWith (\(fk, v1) (_, v2) -> cCompare fk op v1 v2) fks fks' undecided = any isNothing fmatch -- Field comparison undecive allEq = all (== Just EQ) fmatch -- All fields Equal in if undecided then Nothing else if allEq then Just EQ else -- all compared fine, but not all equal Just GT -- Pick GT; so equality test will fail, inequality will pass -- Shouldn't happen: _ -> error $ unlines [ "" , "*** Data.SBV.cCompare: Bug in SBV: Unhandled rank in comparison fallthru" , "***" , "*** Ranks Received: " ++ show (cvRank x, cvRank y, op) , "***" , "*** Please report this as a bug!" ] where -- lexicographic lexCmp :: [(Kind, CVal)] -> [(Kind, CVal)] -> Maybe Ordering lexCmp [] [] = Just EQ lexCmp [] (_:_) = Just LT lexCmp (_:_) [] = Just GT lexCmp ((k1, a):as) ((k2, b):bs) | k1 == k2 = case cCompare k1 op a b of Just EQ -> as `lexCmp` bs other -> other | True = error $ "Mismatching kinds in lexicographic comparison: " ++ show (k1, k2) nanOrZero KFloat (CFloat v) = isNaN v || v == 0 nanOrZero KDouble (CDouble v) = isNaN v || v == 0 nanOrZero (KFP eb sb) (CFP v) = isNaN v || v == fpFromInteger eb sb 0 nanOrZero knd _ = error $ "Unexpected arguments to nanOrZero: " ++ show knd -- | Set equality. We return Nothing if the result is too complicated for us to concretely calculate. -- Why? Because the Eq instance of CVal is a bit iffy; it's designed to work as an index into maps, not as -- a means of checking this sort of equality svSetEqual :: Kind -> RCSet CVal -> RCSet CVal -> Maybe Bool svSetEqual ek sa sb | eqCheckIsObjectEq ek, RegularSet a <- sa, RegularSet b <- sb = Just $ a == b | eqCheckIsObjectEq ek, ComplementSet a <- sa, ComplementSet b <- sb = Just $ a == b | True = Nothing -- | Array equality. See above comments. svArrEqual :: Kind -> Kind -> ArrayModel CVal CVal -> ArrayModel CVal CVal -> Maybe Bool svArrEqual k1 k2 (ArrayModel asc1 def1) (ArrayModel asc2 def2) | not (all eqCheckIsObjectEq [k1, k2]) = Nothing | True = let -- Use of lookup is safe here, because we already made sure equality is *not* problematic above keysMatch = and [key `lookup` asc1 == key `lookup` asc2 | key <- nub (sort (map fst (asc1 ++ asc2)))] defsMatch = def1 == def2 -- Check if keys cover everything. Clearly, we can't do this for all kinds; but only finite ones -- For the time being, we're retricting ourselves to bool only. Might want to extend this later. complete = case k1 of KBool -> let bools = map cvVal [falseCV, trueCV] covered asc = all (`elem` map fst asc) bools in covered asc1 && covered asc2 _ -> False in case (keysMatch, defsMatch, complete) of (False, _ , _) -> Just False -- keys mismatch. Nothing else matters. (True, True, _) -> Just True -- keys match, def matches; so all is good. Complete doesn't matter. (True, False, True) -> Just True -- keys match, but defs don't. But we keys are complete, so def mismatch is OK _ -> Nothing -- otherwise, we don't really know. So, remain symbolic. -- | Equality. This is SMT object equality. svEqual :: SVal -> SVal -> SVal svEqual = compareSV (Equal False) -- | Inequality. svNotEqual :: SVal -> SVal -> SVal svNotEqual = compareSV NotEqual -- | Less than. svLessThan :: SVal -> SVal -> SVal svLessThan = compareSV LessThan -- | Greater than. svGreaterThan :: SVal -> SVal -> SVal svGreaterThan = compareSV GreaterThan -- | Less than or equal to. svLessEq :: SVal -> SVal -> SVal svLessEq = compareSV LessEq -- | Greater than or equal to. svGreaterEq :: SVal -> SVal -> SVal svGreaterEq = compareSV GreaterEq -- | 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) [] (noReal ".&.") (.&.) (noFloat ".&.") (noDouble ".&.") (noFP ".&.") (noRat ".&") x y where opt a b | a == falseSV || b == falseSV = Just falseSV | a == trueSV = Just b | b == trueSV = Just a | a == b = 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) [] (noReal ".|.") (.|.) (noFloat ".|.") (noDouble ".|.") (noFP ".|.") (noRat ".|.") x y where opt a b | a == trueSV || b == trueSV = Just trueSV | a == falseSV = Just b | b == falseSV = Just a | a == b = 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) [] (noReal "xor") xor (noFloat "xor") (noDouble "xor") (noFP "xor") (noRat "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") (noFPUnary "complement") (noRatUnary "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") (noFPUnary "rotateL") (noRatUnary "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") (noFPUnary "rotateR") (noRatUnary "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 i j v@(SVal KFloat _) = svExtract i j (svFloatAsSWord32 v) svExtract i j v@(SVal KDouble _) = svExtract i j (svDoubleAsSWord64 v) svExtract i j v@(SVal KFP{} _) = svExtract i j (svFloatingPointAsSWord v) svExtract _ _ _ = error "extract: non-bitvector/float type" -- | Join two words, by concatenating svJoin :: SVal -> SVal -> SVal svJoin x@(SVal (KBounded s i) a) y@(SVal (KBounded s' j) b) | s /= s' = error $ "svJoin: received differently signed values: " ++ show (x, y) | i == 0 = y | j == 0 = x | Left (CV _ (CInteger m)) <- a, Left (CV _ (CInteger n)) <- b = let val | s -- signed, arithmetic doesn't work; blast and come back = let xbits = [m `testBit` xi | xi <- [0 .. i-1]] ybits = [n `testBit` yi | yi <- [0 .. j-1]] rbits = zip [0..] (ybits ++ xbits) in foldl' (\acc (idx, set) -> if set then setBit acc idx else acc) 0 rbits | True -- unsigned, go fast = m `shiftL` j .|. n in SVal k (Left $! normCV (CV k (CInteger val))) | 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" -- | Zero-extend by given number of bits. svZeroExtend :: Int -> SVal -> SVal svZeroExtend = svExtend True ZeroExtend -- | Sign-extend by given number of bits. svSignExtend :: Int -> SVal -> SVal svSignExtend = svExtend False SignExtend svExtend :: Bool -> (Int -> Op) -> Int -> SVal -> SVal svExtend isZeroExtend extender i x@(SVal (KBounded s sz) a) | i < 0 = error $ "svExtend: Received negative extension amount: " ++ show i | i == 0 = x | Left (CV _ (CInteger cv)) <- a = SVal k' (Left (normCV (CV k' (CInteger (replBit (not isZeroExtend && (cv `testBit` (sz-1))) cv))))) | True = SVal k' (Right (cache z)) where k' = KBounded s (sz+i) z st = do xsw <- svToSV st x newExpr st k' (SBVApp (extender i) [xsw]) replBit :: Bool -> Integer -> Integer replBit b = go sz where stop = sz + i go k v | k == stop = v | b = go (k+1) (v `setBit` k) | True = go (k+1) (v `clearBit` k) svExtend _ _ _ _ = error "svExtend: 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 aggressive 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 altogether. -} 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]) -- | 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 '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 = svRotate svShiftLeft svRor svRol -- | 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 = svRotate svShiftRight svRol svRor -- | Common implementation for rotations. This is more complicated than it might first seem, since SMTLib does -- not allow for non-constant rotation amounts, and only defines rotations for bit-vectors. In SBV, we support -- both finite/infinite combos, and also non-constant (i.e., symbolic) rotations. Furthermore, if the rotation -- amount is negative, then the direction of the rotation is reversed. -- -- Case 1. Infinite x. In this case, we call unbounded-shifter, since you can't rotate an unbounded integer value. -- This is the Haskell semantics for rotates. -- Case 2. Finite x. -- Case 2.1. Infinite i, or finite i but i can contain a value > |x|. In this case, wrap-around can happen, -- so we reduce by the size of |x|. -- Case 2.2. Finite i, and it can't contain a value > |x|. In this case, no reduction is needed. svRotate :: (SVal -> SVal -> SVal) -> (SVal -> Int -> SVal) -> (SVal -> Int -> SVal) -> SVal -> SVal -> SVal svRotate unboundedShifter opRot curRot x i | not (isBounded x) = unboundedShifter x i | True = svSelect table (svInteger (kindOf x) 0) curRotate where sx = intSizeOf x si = intSizeOf i -- Is it the case that this rotation can never "wrap-around?" This happens if -- i is bounded and the max rotation it can represent is less than the bit-size of the input noWrapAround :: Bool noWrapAround = isBounded i && maxRotate <= toInteger sx where maxRotate :: Integer maxRotate | hasSign i = 2^(si-1) | True = 2^si-1 ifNegRotate = svIte (svLessThan i (svInteger (kindOf i) 0)) -- the lookup table has sx entries if index can wrap-around. Otherwise it is just as wide as it needs to be. table :: [SVal] table = map rotK vals where rotK k = ifNegRotate (x `opRot` k) (x `curRot` k) vals | noWrapAround = if hasSign i then -- If signed then bit (si-1) is the max abs value. (consider 3 bits, [-4..3] is the range) [0 .. bit (si - 1)] else [0 .. bit si - 1] | True -- If wrap-around can happen, then compute all rotations up to |x| = [0 .. sx - 1] -- What's the current rotation amount? Here we change the type of the -- index to make it one bit larger if the index is signed, since otherwise -- we run into (-(-1)) = -1 problem. See https://github.com/LeventErkok/sbv/issues/673#issuecomment-1782296700 -- Note that curRotate is always non-negative. curRotate :: SVal curRotate | noWrapAround = ifNegRotate (svUNeg i' ) i' | True = ifNegRotate (svUNeg i' `svRem` n) (i' `svRem` n) where i' | hasSign i && isBounded i = toWord $ svAbs $ enlarge i | True = i -- Make sure sx can fit into this many bits si' = (si + 1) `max` bitsNeeded sx enlarge | isBounded i = svFromIntegral (KBounded True si') -- Increase bit size | True = id toWord | isBounded i = svFromIntegral (KBounded False si') -- Treat as word, after call to svAbs above | True = id n = svInteger (kindOf i') (toInteger sx) bitsNeeded :: Int -> Int bitsNeeded = go 0 where go s 0 = s go s v = let s' = s + 1 in s' `seq` go s' (v `shiftR` 1) -------------------------------------------------------------------------------- -- | Overflow detection. svMkOverflow1 :: OvOp -> SVal -> SVal svMkOverflow1 o x = SVal KBool (Right (cache r)) where r st = do sx <- svToSV st x newExpr st KBool $ SBVApp (OverflowOp o) [sx] svMkOverflow2 :: OvOp -> SVal -> SVal -> SVal svMkOverflow2 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] -------------------------------------------------------------------------------- -- Utility functions liftSym1 :: (State -> Kind -> SV -> IO SV) -> (AlgReal -> AlgReal) -> (Integer -> Integer) -> (Float -> Float) -> (Double -> Double) -> (FP -> FP) -> (Rational -> Rational) -> SVal -> SVal liftSym1 _ opCR opCI opCF opCD opFP opRA (SVal k (Left a)) = SVal k . Left $! mapCV opCR opCI opCF opCD opFP opRA 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 symbolically 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) -> (FP -> FP -> FP) -> (Rational -> Rational-> Rational) -> SVal -> SVal -> SVal liftSym2 _ okCV opCR opCI opCF opCD opFP opRA (SVal k (Left a)) (SVal _ (Left b)) | and [f a b | f <- okCV] = SVal k . Left $! mapCV2 opCR opCI opCF opCD opFP opRA a b liftSym2 opS _ _ _ _ _ _ _ a@(SVal k _) b = SVal k $ Right $ liftSV2 opS k 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) -- | 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 -> a noReal o a b = error $ "SBV.AlgReal." ++ o ++ ": Unexpected arguments: " ++ show (a, b) noFloat :: String -> Float -> Float -> a noFloat o a b = error $ "SBV.Float." ++ o ++ ": Unexpected arguments: " ++ show (a, b) noDouble :: String -> Double -> Double -> a noDouble o a b = error $ "SBV.Double." ++ o ++ ": Unexpected arguments: " ++ show (a, b) noFP :: String -> FP -> FP -> a noFP o a b = error $ "SBV.FPR." ++ o ++ ": Unexpected arguments: " ++ show (a, b) noRat:: String -> Rational -> Rational -> a noRat o a b = error $ "SBV.Rational." ++ o ++ ": Unexpected arguments: " ++ show (a, b) noRealUnary :: String -> AlgReal -> a noRealUnary o a = error $ "SBV.AlgReal." ++ o ++ ": Unexpected argument: " ++ show a noFloatUnary :: String -> Float -> a noFloatUnary o a = error $ "SBV.Float." ++ o ++ ": Unexpected argument: " ++ show a noDoubleUnary :: String -> Double -> a noDoubleUnary o a = error $ "SBV.Double." ++ o ++ ": Unexpected argument: " ++ show a noFPUnary :: String -> FP -> a noFPUnary o a = error $ "SBV.FPR." ++ o ++ ": Unexpected argument: " ++ show a noRatUnary :: String -> Rational -> a noRatUnary o a = error $ "SBV.Rational." ++ 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 | 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 -- | Convert an 'Data.SBV.SFloat' to an 'Data.SBV.SWord32', preserving the bit-correspondence. Note that since the -- representation for @NaN@s are not unique, this function will return a symbolic value when given a -- concrete @NaN@. -- -- Implementation note: Since there's no corresponding function in SMTLib for conversion to -- bit-representation due to partiality, we use a translation trick by allocating a new word variable, -- converting it to float, and requiring it to be equivalent to the input. In code-generation mode, we simply map -- it to a simple conversion. svFloatAsSWord32 :: SVal -> SVal svFloatAsSWord32 (SVal KFloat (Left (CV KFloat (CFloat f)))) | not (isNaN f) = let w32 = KBounded False 32 in SVal w32 $ Left $ CV w32 $ CInteger (fromIntegral (floatToWord f)) svFloatAsSWord32 fVal@(SVal KFloat _) = SVal w32 (Right (cache y)) where w32 = KBounded False 32 y st = do cg <- isCodeGenMode st if cg then do f <- svToSV st fVal newExpr st w32 (SBVApp (IEEEFP (FP_Reinterpret KFloat w32)) [f]) else do n <- newInternalVariable st w32 ysw <- newExpr st KFloat (SBVApp (IEEEFP (FP_Reinterpret w32 KFloat)) [n]) internalConstraint st False [] $ fVal `svStrongEqual` SVal KFloat (Right (cache (\_ -> return ysw))) return n svFloatAsSWord32 (SVal k _) = error $ "svFloatAsSWord32: non-float type: " ++ show k -- | Convert an 'Data.SBV.SDouble' to an 'Data.SBV.SWord64', preserving the bit-correspondence. Note that since the -- representation for @NaN@s are not unique, this function will return a symbolic value when given a -- concrete @NaN@. -- -- Implementation note: Since there's no corresponding function in SMTLib for conversion to -- bit-representation due to partiality, we use a translation trick by allocating a new word variable, -- converting it to float, and requiring it to be equivalent to the input. In code-generation mode, we simply map -- it to a simple conversion. svDoubleAsSWord64 :: SVal -> SVal svDoubleAsSWord64 (SVal KDouble (Left (CV KDouble (CDouble f)))) | not (isNaN f) = let w64 = KBounded False 64 in SVal w64 $ Left $ CV w64 $ CInteger (fromIntegral (doubleToWord f)) svDoubleAsSWord64 fVal@(SVal KDouble _) = SVal w64 (Right (cache y)) where w64 = KBounded False 64 y st = do cg <- isCodeGenMode st if cg then do f <- svToSV st fVal newExpr st w64 (SBVApp (IEEEFP (FP_Reinterpret KDouble w64)) [f]) else do n <- newInternalVariable st w64 ysw <- newExpr st KDouble (SBVApp (IEEEFP (FP_Reinterpret w64 KDouble)) [n]) internalConstraint st False [] $ fVal `svStrongEqual` SVal KDouble (Right (cache (\_ -> return ysw))) return n svDoubleAsSWord64 (SVal k _) = error $ "svDoubleAsSWord64: non-float type: " ++ show k -- | Convert a float to the word containing the corresponding bit pattern svFloatingPointAsSWord :: SVal -> SVal svFloatingPointAsSWord (SVal (KFP eb sb) (Left (CV _ (CFP f@(FP _ _ fpV))))) | not (isNaN f) = let wN = KBounded False (eb + sb) in SVal wN $ Left $ CV wN $ CInteger $ bfToBits (mkBFOpts eb sb NearEven) fpV svFloatingPointAsSWord fVal@(SVal kFrom@(KFP eb sb) _) = SVal kTo (Right (cache y)) where kTo = KBounded False (eb + sb) y st = do cg <- isCodeGenMode st if cg then do f <- svToSV st fVal newExpr st kTo (SBVApp (IEEEFP (FP_Reinterpret kFrom kTo)) [f]) else do n <- newInternalVariable st kTo ysw <- newExpr st kFrom (SBVApp (IEEEFP (FP_Reinterpret kTo kFrom)) [n]) internalConstraint st False [] $ fVal `svStrongEqual` SVal kFrom (Right (cache (\_ -> return ysw))) return n svFloatingPointAsSWord (SVal k _) = error $ "svFloatingPointAsSWord: non-float type: " ++ show k {- HLint ignore svIte "Eta reduce" -} {- HLint ignore svLazyIte "Eta reduce" -} {- HLint ignore module "Reduce duplication" -}