sbv-1.0: Data/SBV/BitVectors/Data.hs
-----------------------------------------------------------------------------
-- |
-- Module : Data.SBV.BitVectors.Data
-- Copyright : (c) Levent Erkok
-- License : BSD3
-- Maintainer : erkokl@gmail.com
-- Stability : experimental
-- Portability : portable
--
-- Internal data-structures for the sbv library
-----------------------------------------------------------------------------
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE TypeSynonymInstances #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE PatternGuards #-}
module Data.SBV.BitVectors.Data
( SBool, SWord8, SWord16, SWord32, SWord64
, SInt8, SInt16, SInt32, SInt64, SInteger
, SymWord(..)
, CW(..), cwSameType, cwIsBit, cwToBool
, mkConstCW ,liftCW2, mapCW, mapCW2
, SW(..), trueSW, falseSW, trueCW, falseCW
, SBV(..), NodeId(..), mkSymSBV
, ArrayContext(..), ArrayInfo, SymArray(..), SFunArray(..), mkSFunArray, SArray(..), arrayUIKind
, sbvToSW, sbvToSymSW
, SBVExpr(..), newExpr
, cache, uncache, uncacheAI, HasSignAndSize(..)
, Op(..), NamedSymVar, UnintKind(..), getTableIndex, Pgm, Symbolic, runSymbolic, runSymbolic', State, inProofMode, SBVRunMode(..), Size(..), Outputtable(..), Result(..)
, getTraceInfo, getConstraints, addConstraint
, SBVType(..), newUninterpreted, unintFnUIKind, addAxiom
, Quantifier(..), needsExistentials
, SMTLibPgm(..), SMTLibVersion(..)
) where
import Control.DeepSeq (NFData(..))
import Control.Monad (when)
import Control.Monad.Reader (MonadReader, ReaderT, ask, runReaderT)
import Control.Monad.Trans (MonadIO, liftIO)
import Data.Char (isAlpha, isAlphaNum)
import Data.Int (Int8, Int16, Int32, Int64)
import Data.Word (Word8, Word16, Word32, Word64)
import Data.IORef (IORef, newIORef, modifyIORef, readIORef, writeIORef)
import Data.List (intercalate, sortBy)
import Data.Maybe (isJust, fromJust, fromMaybe)
import qualified Data.IntMap as IMap (IntMap, empty, size, toAscList, lookup, insert, insertWith)
import qualified Data.Map as Map (Map, empty, toList, size, insert, lookup)
import qualified Data.Foldable as F (toList)
import qualified Data.Sequence as S (Seq, empty, (|>))
import System.Mem.StableName
import System.Random
import Data.SBV.Utils.Lib
-- | 'CW' represents a concrete word of a fixed size:
-- Endianness is mostly irrelevant (see the 'FromBits' class).
-- For signed words, the most significant digit is considered to be the sign.
data CW = CW { cwSigned :: !Bool -- ^ Is the word signed?
, cwSize :: !Size -- ^ Size of the word (unbounded if Nothing)
, cwVal :: !Integer -- ^ The underlying value, represented as a Haskell 'Integer'
}
deriving (Eq, Ord)
cwSameType :: CW -> CW -> Bool
cwSameType x y = cwSigned x == cwSigned y && cwSize x == cwSize y
cwIsBit :: CW -> Bool
cwIsBit x = not (hasSign x) && not (isInfPrec x) && intSizeOf x == 1
-- | Convert a CW to a Haskell boolean
cwToBool :: CW -> Bool
cwToBool x = cwVal x /= 0
normCW :: CW -> CW
normCW x
| isInfPrec x = x
| True = x { cwVal = norm }
where sz = intSizeOf x
norm | sz == 0 = 0
| cwSigned x = let rg = 2 ^ (sz - 1)
in case divMod (cwVal x) rg of
(a, b) | even a -> b
(_, b) -> b - rg
| True = cwVal x `mod` (2 ^ sz)
newtype Size = Size { unSize :: Maybe Int }
deriving (Eq, Ord)
newtype NodeId = NodeId Int deriving (Eq, Ord)
data SW = SW (Bool, Size) NodeId deriving (Eq, Ord)
data Quantifier = ALL | EX deriving Eq
needsExistentials :: [Quantifier] -> Bool
needsExistentials = (EX `elem`)
falseSW, trueSW :: SW
falseSW = SW (False, Size (Just 1)) $ NodeId (-2)
trueSW = SW (False, Size (Just 1)) $ NodeId (-1)
falseCW, trueCW :: CW
falseCW = CW False (Size (Just 1)) 0
trueCW = CW False (Size (Just 1)) 1
newtype SBVType = SBVType [(Bool, Size)]
deriving (Eq, Ord)
-- how many arguments does the type take?
typeArity :: SBVType -> Int
typeArity (SBVType xs) = length xs - 1
instance Show SBVType where
show (SBVType []) = error "SBV: internal error, empty SBVType"
show (SBVType xs) = intercalate " -> " $ map sh xs
where sh (_, Size Nothing) = "SInteger"
sh (False, Size (Just 1)) = "SBool"
sh (s, Size (Just sz)) = (if s then "SInt" else "SWord") ++ show sz
data Op = Plus | Times | Minus
| Quot | Rem -- quot and rem are unsigned only
| Equal | NotEqual
| LessThan | GreaterThan | LessEq | GreaterEq
| Ite
| And | Or | XOr | Not
| Shl Int | Shr Int | Rol Int | Ror Int
| Extract Int Int -- Extract i j: extract bits i to j. Least significant bit is 0 (big-endian)
| Join -- Concat two words to form a bigger one, in the order given
| LkUp (Int, (Bool, Size), (Bool, Size), Int) !SW !SW -- (table-index, arg-type, res-type, length of the table) index out-of-bounds-value
| ArrEq Int Int
| ArrRead Int
| Uninterpreted String
deriving (Eq, Ord)
data SBVExpr = SBVApp !Op ![SW]
deriving (Eq, Ord)
-- minimal complete definition: sizeOf, hasSign
class HasSignAndSize a where
sizeOf :: a -> Size
hasSign :: a -> Bool
intSizeOf :: a -> Int
isInfPrec :: a -> Bool
showType :: a -> String
showType a
| isInfPrec a = "SInteger"
| not (hasSign a) && intSizeOf a == 1 = "SBool"
| True = (if hasSign a then "SInt" else "SWord") ++ show (intSizeOf a)
isInfPrec = maybe True (const False) . unSize . sizeOf
intSizeOf = fromMaybe (error "SBV.HasSignAndSize.bitSize((S)Integer)") . unSize . sizeOf
instance HasSignAndSize Bool where {sizeOf _ = Size (Just 1) ; hasSign _ = False}
instance HasSignAndSize Int8 where {sizeOf _ = Size (Just 8) ; hasSign _ = True }
instance HasSignAndSize Word8 where {sizeOf _ = Size (Just 8) ; hasSign _ = False}
instance HasSignAndSize Int16 where {sizeOf _ = Size (Just 16); hasSign _ = True }
instance HasSignAndSize Word16 where {sizeOf _ = Size (Just 16); hasSign _ = False}
instance HasSignAndSize Int32 where {sizeOf _ = Size (Just 32); hasSign _ = True }
instance HasSignAndSize Word32 where {sizeOf _ = Size (Just 32); hasSign _ = False}
instance HasSignAndSize Int64 where {sizeOf _ = Size (Just 64); hasSign _ = True }
instance HasSignAndSize Word64 where {sizeOf _ = Size (Just 64); hasSign _ = False}
instance HasSignAndSize Integer where {sizeOf _ = Size Nothing; hasSign _ = True}
liftCW :: (Integer -> b) -> CW -> b
liftCW f x = f (cwVal x)
liftCW2 :: (Integer -> Integer -> b) -> CW -> CW -> b
liftCW2 f x y | cwSameType x y = f (cwVal x) (cwVal y)
liftCW2 _ a b = error $ "SBV.liftCW2: impossible, incompatible args received: " ++ show (a, b)
mapCW :: (Integer -> Integer) -> CW -> CW
mapCW f x = normCW $ x { cwVal = f (cwVal x) }
mapCW2 :: (Integer -> Integer -> Integer) -> CW -> CW -> CW
mapCW2 f x y
| cwSameType x y = normCW $ CW (cwSigned x) (cwSize y) (f (cwVal x) (cwVal y))
mapCW2 _ a b = error $ "SBV.mapCW2: impossible, incompatible args received: " ++ show (a, b)
instance HasSignAndSize CW where
intSizeOf = maybe (error "attempting to compute size of SInteger") id . unSize . cwSize
sizeOf = cwSize
hasSign = cwSigned
isInfPrec = maybe True (const False) . unSize . cwSize
instance HasSignAndSize SW where
sizeOf (SW (_, s) _) = s
intSizeOf (SW (_, mbs) _) = maybe (error "attempting to compute size of SInteger") id $ unSize mbs
isInfPrec (SW (_, mbs) _) = maybe True (const False) $ unSize mbs
hasSign (SW (b, _) _) = b
instance Show CW where
show w | cwIsBit w = show (cwToBool w)
show w = liftCW show w ++ " :: " ++ showType w
instance Show SW where
show (SW _ (NodeId n))
| n < 0 = "s_" ++ show (abs n)
| True = 's' : show n
instance Show Op where
show (Shl i) = "<<" ++ show i
show (Shr i) = ">>" ++ show i
show (Rol i) = "<<<" ++ show i
show (Ror i) = ">>>" ++ show i
show (Extract i j) = "choose [" ++ show i ++ ":" ++ show j ++ "]"
show (LkUp (ti, at, rt, l) i e)
= "lookup(" ++ tinfo ++ ", " ++ show i ++ ", " ++ show e ++ ")"
where tinfo = "table" ++ show ti ++ "(" ++ mkT at ++ " -> " ++ mkT rt ++ ", " ++ show l ++ ")"
mkT (_, Size Nothing) = "SInteger"
mkT (b, Size (Just s))
| s == 1 = "SBool"
| True = if b then "SInt" else "SWord" ++ show s
show (ArrEq i j) = "array_" ++ show i ++ " == array_" ++ show j
show (ArrRead i) = "select array_" ++ show i
show (Uninterpreted i) = "uninterpreted_" ++ i
show op
| Just s <- op `lookup` syms = s
| True = error "impossible happened; can't find op!"
where syms = [ (Plus, "+"), (Times, "*"), (Minus, "-")
, (Quot, "quot")
, (Rem, "rem")
, (Equal, "=="), (NotEqual, "/=")
, (LessThan, "<"), (GreaterThan, ">"), (LessEq, "<"), (GreaterEq, ">")
, (Ite, "if_then_else")
, (And, "&"), (Or, "|"), (XOr, "^"), (Not, "~")
, (Join, "#")
]
reorder :: SBVExpr -> SBVExpr
reorder s = case s of
SBVApp op [a, b] | isCommutative op && a > b -> SBVApp op [b, a]
_ -> s
where isCommutative :: Op -> Bool
isCommutative o = o `elem` [Plus, Times, Equal, NotEqual, And, Or, XOr]
instance Show SBVExpr where
show (SBVApp Ite [t, a, b]) = unwords ["if", show t, "then", show a, "else", show b]
show (SBVApp (Shl i) [a]) = unwords [show a, "<<", show i]
show (SBVApp (Shr i) [a]) = unwords [show a, ">>", show i]
show (SBVApp (Rol i) [a]) = unwords [show a, "<<<", show i]
show (SBVApp (Ror i) [a]) = unwords [show a, ">>>", show i]
show (SBVApp op [a, b]) = unwords [show a, show op, show b]
show (SBVApp op args) = unwords (show op : map show args)
-- | A program is a sequence of assignments
type Pgm = S.Seq (SW, SBVExpr)
-- | 'NamedSymVar' pairs symbolic words and user given/automatically generated names
type NamedSymVar = (SW, String)
-- | 'UnintKind' pairs array names and uninterpreted constants with their "kinds"
-- used mainly for printing counterexamples
data UnintKind = UFun Int String | UArr Int String -- in each case, arity and the aliasing name
deriving Show
-- | Result of running a symbolic computation
data Result = Result Bool -- contains unbounded integers
[(String, CW)] -- quick-check counter-example information (if any)
[(String, [String])] -- uninterpeted code segments
[(Quantifier, NamedSymVar)] -- inputs (possibly existential)
[(SW, CW)] -- constants
[((Int, (Bool, Size), (Bool, Size)), [SW])] -- tables (automatically constructed) (tableno, index-type, result-type) elts
[(Int, ArrayInfo)] -- arrays (user specified)
[(String, SBVType)] -- uninterpreted constants
[(String, [String])] -- axioms
Pgm -- assignments
[SW] -- additional constraints (boolean)
[SW] -- outputs
getConstraints :: Result -> [SW]
getConstraints (Result _ _ _ _ _ _ _ _ _ _ cstrs _) = cstrs
getTraceInfo :: Result -> [(String, CW)]
getTraceInfo (Result _ tvals _ _ _ _ _ _ _ _ _ _) = tvals
instance Show Result where
show (Result _ _ _ _ cs _ _ [] [] _ [] [r])
| Just c <- r `lookup` cs
= show c
show (Result _ _ cgs is cs ts as uis axs xs cstrs os) = intercalate "\n" $
["INPUTS"]
++ map shn is
++ ["CONSTANTS"]
++ map shc cs
++ ["TABLES"]
++ map sht ts
++ ["ARRAYS"]
++ map sha as
++ ["UNINTERPRETED CONSTANTS"]
++ map shui uis
++ ["USER GIVEN CODE SEGMENTS"]
++ concatMap shcg cgs
++ ["AXIOMS"]
++ map shax axs
++ ["DEFINE"]
++ map (\(s, e) -> " " ++ shs s ++ " = " ++ show e) (F.toList xs)
++ ["CONSTRAINTS"]
++ map ((" " ++) . show) cstrs
++ ["OUTPUTS"]
++ map ((" " ++) . show) os
where shs sw = show sw ++ " :: " ++ showType sw
sht ((i, at, rt), es) = " Table " ++ show i ++ " : " ++ mkT at ++ "->" ++ mkT rt ++ " = " ++ show es
shc (sw, cw) = " " ++ show sw ++ " = " ++ show cw
shcg (s, ss) = ("Variable: " ++ s) : map (" " ++) ss
shn (q, (sw, nm)) = " " ++ ni ++ " :: " ++ showType sw ++ ex ++ alias
where ni = show sw
ex | q == ALL = ""
| True = ", existential"
alias | ni == nm = ""
| True = ", aliasing " ++ show nm
sha (i, (nm, (ai, bi), ctx)) = " " ++ ni ++ " :: " ++ mkT ai ++ " -> " ++ mkT bi ++ alias
++ "\n Context: " ++ show ctx
where ni = "array_" ++ show i
alias | ni == nm = ""
| True = ", aliasing " ++ show nm
shui (nm, t) = " uninterpreted_" ++ nm ++ " :: " ++ show t
shax (nm, ss) = " -- user defined axiom: " ++ nm ++ "\n " ++ intercalate "\n " ss
mkT (_, Size Nothing) = "SInteger"
mkT (b, Size (Just s))
| s == 1 = "SBool"
| True = if b then "SInt" else "SWord" ++ show s
data ArrayContext = ArrayFree (Maybe SW)
| ArrayReset Int SW
| ArrayMutate Int SW SW
| ArrayMerge SW Int Int
instance Show ArrayContext where
show (ArrayFree Nothing) = " initialized with random elements"
show (ArrayFree (Just s)) = " initialized with " ++ show s ++ " :: " ++ showType s
show (ArrayReset i s) = " reset array_" ++ show i ++ " with " ++ show s ++ " :: " ++ showType s
show (ArrayMutate i a b) = " cloned from array_" ++ show i ++ " with " ++ show a ++ " :: " ++ showType a ++ " |-> " ++ show b ++ " :: " ++ showType b
show (ArrayMerge s i j) = " merged arrays " ++ show i ++ " and " ++ show j ++ " on condition " ++ show s
type ExprMap = Map.Map SBVExpr SW
type CnstMap = Map.Map CW SW
type TableMap = Map.Map [SW] (Int, (Bool, Size), (Bool, Size))
type ArrayInfo = (String, ((Bool, Size), (Bool, Size)), ArrayContext)
type ArrayMap = IMap.IntMap ArrayInfo
type UIMap = Map.Map String SBVType
type CgMap = Map.Map String [String]
type Cache a = IMap.IntMap [(StableName (State -> IO a), a)]
unintFnUIKind :: (String, SBVType) -> (String, UnintKind)
unintFnUIKind (s, t) = (s, UFun (typeArity t) s)
arrayUIKind :: (Int, ArrayInfo) -> Maybe (String, UnintKind)
arrayUIKind (i, (nm, _, ctx))
| external ctx = Just ("array_" ++ show i, UArr 1 nm) -- arrays are always 1-dimensional in the SMT-land. (Unless encoded explicitly)
| True = Nothing
where external (ArrayFree{}) = True
external (ArrayReset{}) = False
external (ArrayMutate{}) = False
external (ArrayMerge{}) = False
-- | Different means of running a symbolic piece of code
data SBVRunMode = Proof Bool -- ^ Symbolic simulation mode, for proof purposes. Bool is True if it's a sat instance
| CodeGen -- ^ Code generation mode
| Concrete StdGen -- ^ Concrete simulation mode. The StdGen is for the pConstrain acceptance in cross runs
isConcreteMode :: SBVRunMode -> Bool
isConcreteMode (Concrete _) = True
isConcreteMode (Proof{}) = False
isConcreteMode CodeGen = False
data State = State { runMode :: SBVRunMode
, rStdGen :: IORef StdGen
, rCInfo :: IORef [(String, CW)]
, rctr :: IORef Int
, rInfPrec :: IORef Bool
, rinps :: IORef [(Quantifier, NamedSymVar)]
, rConstraints :: IORef [SW]
, routs :: IORef [SW]
, rtblMap :: IORef TableMap
, spgm :: IORef Pgm
, rconstMap :: IORef CnstMap
, rexprMap :: IORef ExprMap
, rArrayMap :: IORef ArrayMap
, rUIMap :: IORef UIMap
, rCgMap :: IORef CgMap
, raxioms :: IORef [(String, [String])]
, rSWCache :: IORef (Cache SW)
, rAICache :: IORef (Cache Int)
}
inProofMode :: State -> Bool
inProofMode s = case runMode s of
Proof{} -> True
CodeGen -> False
Concrete{} -> False
-- | The "Symbolic" value. Either a constant (@Left@) or a symbolic
-- value (@Right Cached@). Note that caching is essential for making
-- sure sharing is preserved. The parameter 'a' is phantom, but is
-- extremely important in keeping the user interface strongly typed.
data SBV a = SBV !(Bool, Size) !(Either CW (Cached SW))
-- | A symbolic boolean/bit
type SBool = SBV Bool
-- | 8-bit unsigned symbolic value
type SWord8 = SBV Word8
-- | 16-bit unsigned symbolic value
type SWord16 = SBV Word16
-- | 32-bit unsigned symbolic value
type SWord32 = SBV Word32
-- | 64-bit unsigned symbolic value
type SWord64 = SBV Word64
-- | 8-bit signed symbolic value, 2's complement representation
type SInt8 = SBV Int8
-- | 16-bit signed symbolic value, 2's complement representation
type SInt16 = SBV Int16
-- | 32-bit signed symbolic value, 2's complement representation
type SInt32 = SBV Int32
-- | 64-bit signed symbolic value, 2's complement representation
type SInt64 = SBV Int64
-- | Infinite precision signed symbolic value
type SInteger = SBV Integer
-- Not particularly "desirable", but will do if needed
instance Show (SBV a) where
show (SBV _ (Left c)) = show c
show (SBV (_ , Size Nothing) (Right _)) = "<symbolic> :: SInteger"
show (SBV (sgn, Size (Just sz)) (Right _)) = "<symbolic> :: " ++ t
where t | not sgn && sz == 1 = "SBool"
| True = (if sgn then "SInt" else "SWord") ++ show sz
instance Eq (SBV a) where
SBV _ (Left a) == SBV _ (Left b) = a == b
a == b = error $ "Comparing symbolic bit-vectors; Use (.==) instead. Received: " ++ show (a, b)
SBV _ (Left a) /= SBV _ (Left b) = a /= b
a /= b = error $ "Comparing symbolic bit-vectors; Use (./=) instead. Received: " ++ show (a, b)
instance HasSignAndSize a => HasSignAndSize (SBV a) where
sizeOf _ = sizeOf (undefined :: a)
hasSign _ = hasSign (undefined :: a)
incCtr :: State -> IO Int
incCtr s = do ctr <- readIORef (rctr s)
let i = ctr + 1
i `seq` writeIORef (rctr s) i
return ctr
throwDice :: State -> IO Double
throwDice st = do g <- readIORef (rStdGen st)
let (r, g') = randomR (0, 1) g
writeIORef (rStdGen st) g'
return r
newUninterpreted :: State -> String -> SBVType -> Maybe [String] -> IO ()
newUninterpreted st nm t mbCode
| null nm || not (isAlpha (head nm)) || not (all validChar (tail nm))
= error $ "Bad uninterpreted constant name: " ++ show nm ++ ". Must be a valid identifier."
| True = do
uiMap <- readIORef (rUIMap st)
case nm `Map.lookup` uiMap of
Just t' -> if t /= t'
then error $ "Uninterpreted constant " ++ show nm ++ " used at incompatible types\n"
++ " Current type : " ++ show t ++ "\n"
++ " Previously used at: " ++ show t'
else return ()
Nothing -> do modifyIORef (rUIMap st) (Map.insert nm t)
when (isJust mbCode) $ modifyIORef (rCgMap st) (Map.insert nm (fromJust mbCode))
where validChar x = isAlphaNum x || x `elem` "_"
-- Create a new constant; hash-cons as necessary
newConst :: State -> CW -> IO SW
newConst st c = do
constMap <- readIORef (rconstMap st)
case c `Map.lookup` constMap of
Just sw -> return sw
Nothing -> do ctr <- incCtr st
let sw = SW (hasSign c, sizeOf c) (NodeId ctr)
when (isInfPrec sw) $ writeIORef (rInfPrec st) True
modifyIORef (rconstMap st) (Map.insert c sw)
return sw
-- Create a new table; hash-cons as necessary
getTableIndex :: State -> (Bool, Size) -> (Bool, Size) -> [SW] -> IO Int
getTableIndex st at rt elts = do
tblMap <- readIORef (rtblMap st)
case elts `Map.lookup` tblMap of
Just (i, _, _) -> return i
Nothing -> do let i = Map.size tblMap
modifyIORef (rtblMap st) (Map.insert elts (i, at, rt))
return i
-- Create a constant word
mkConstCW :: Integral a => (Bool, Size) -> a -> CW
mkConstCW (signed, size) a = normCW $ CW signed size (toInteger a)
-- Create a new expression; hash-cons as necessary
newExpr :: State -> (Bool, Size) -> SBVExpr -> IO SW
newExpr st sgnsz app = do
let e = reorder app
exprMap <- readIORef (rexprMap st)
case e `Map.lookup` exprMap of
Just sw -> return sw
Nothing -> do ctr <- incCtr st
let sw = SW sgnsz (NodeId ctr)
when (isInfPrec sw) $ writeIORef (rInfPrec st) True
modifyIORef (spgm st) (flip (S.|>) (sw, e))
modifyIORef (rexprMap st) (Map.insert e sw)
return sw
sbvToSW :: State -> SBV a -> IO SW
sbvToSW st (SBV _ (Left c)) = newConst st c
sbvToSW st (SBV _ (Right f)) = uncache f st
-------------------------------------------------------------------------
-- * Symbolic Computations
-------------------------------------------------------------------------
-- | A Symbolic computation. Represented by a reader monad carrying the
-- state of the computation, layered on top of IO for creating unique
-- references to hold onto intermediate results.
newtype Symbolic a = Symbolic (ReaderT State IO a)
deriving (Functor, Monad, MonadIO, MonadReader State)
mkSymSBV :: forall a. (Random a, SymWord a) => Maybe Quantifier -> (Bool, Size) -> Maybe String -> Symbolic (SBV a)
mkSymSBV mbQ sgnsz mbNm = do
st <- ask
let q = case (mbQ, runMode st) of
(Just x, _) -> x -- user given, just take it
(Nothing, Concrete{}) -> ALL -- concrete simulation, pick universal
(Nothing, Proof True) -> EX -- sat mode, pick existential
(Nothing, Proof False) -> ALL -- proof mode, pick universal
(Nothing, CodeGen) -> ALL -- code generation, pick universal
case runMode st of
Concrete _ | q == EX -> case mbNm of
Nothing -> error $ "Cannot quick-check in the presence of existential variables, type: " ++ showType (undefined :: SBV a)
Just nm -> error $ "Cannot quick-check in the presence of existential variable " ++ nm ++ " :: " ++ showType (undefined :: SBV a)
Concrete _ -> do v@(SBV _ (Left cw)) <- liftIO randomIO
liftIO $ modifyIORef (rCInfo st) ((maybe "_" id mbNm, cw):)
return v
_ -> do ctr <- liftIO $ incCtr st
let nm = maybe ('s':show ctr) id mbNm
sw = SW sgnsz (NodeId ctr)
when (isInfPrec sw) $ liftIO $ writeIORef (rInfPrec st) True
liftIO $ modifyIORef (rinps st) ((q, (sw, nm)):)
return $ SBV sgnsz $ Right $ cache (const (return sw))
sbvToSymSW :: SBV a -> Symbolic SW
sbvToSymSW sbv = do
st <- ask
liftIO $ sbvToSW st sbv
-- | Mark an interim result as an output. Useful when constructing Symbolic programs
-- that return multiple values, or when the result is programmatically computed.
class Outputtable a where
output :: a -> Symbolic a
instance Outputtable (SBV a) where
output i@(SBV _ (Left c)) = do
st <- ask
sw <- liftIO $ newConst st c
liftIO $ modifyIORef (routs st) (sw:)
return i
output i@(SBV _ (Right f)) = do
st <- ask
sw <- liftIO $ uncache f st
liftIO $ modifyIORef (routs st) (sw:)
return i
instance Outputtable a => Outputtable [a] where
output = mapM output
instance Outputtable () where
output = return
instance (Outputtable a, Outputtable b) => Outputtable (a, b) where
output = mlift2 (,) output output
instance (Outputtable a, Outputtable b, Outputtable c) => Outputtable (a, b, c) where
output = mlift3 (,,) output output output
instance (Outputtable a, Outputtable b, Outputtable c, Outputtable d) => Outputtable (a, b, c, d) where
output = mlift4 (,,,) output output output output
instance (Outputtable a, Outputtable b, Outputtable c, Outputtable d, Outputtable e) => Outputtable (a, b, c, d, e) where
output = mlift5 (,,,,) output output output output output
instance (Outputtable a, Outputtable b, Outputtable c, Outputtable d, Outputtable e, Outputtable f) => Outputtable (a, b, c, d, e, f) where
output = mlift6 (,,,,,) output output output output output output
instance (Outputtable a, Outputtable b, Outputtable c, Outputtable d, Outputtable e, Outputtable f, Outputtable g) => Outputtable (a, b, c, d, e, f, g) where
output = mlift7 (,,,,,,) output output output output output output output
instance (Outputtable a, Outputtable b, Outputtable c, Outputtable d, Outputtable e, Outputtable f, Outputtable g, Outputtable h) => Outputtable (a, b, c, d, e, f, g, h) where
output = mlift8 (,,,,,,,) output output output output output output output output
-- | Add a user specified axiom to the generated SMT-Lib file. Note that the input is a
-- mere string; we perform no checking on the input that it's well-formed or is sensical.
-- A separate formalization of SMT-Lib would be very useful here.
addAxiom :: String -> [String] -> Symbolic ()
addAxiom nm ax = do
st <- ask
liftIO $ modifyIORef (raxioms st) ((nm, ax) :)
-- | Run a symbolic computation in Proof mode and return a 'Result'. The boolean
-- argument indicates if this is a sat instance or not.
runSymbolic :: Bool -> Symbolic a -> IO Result
runSymbolic b c = snd `fmap` runSymbolic' (Proof b) c
-- | Run a symbolic computation, and return a extra value paired up with the 'Result'
runSymbolic' :: SBVRunMode -> Symbolic a -> IO (a, Result)
runSymbolic' currentRunMode (Symbolic c) = do
ctr <- newIORef (-2) -- start from -2; False and True will always occupy the first two elements
cInfo <- newIORef []
pgm <- newIORef S.empty
emap <- newIORef Map.empty
cmap <- newIORef Map.empty
inps <- newIORef []
outs <- newIORef []
tables <- newIORef Map.empty
arrays <- newIORef IMap.empty
uis <- newIORef Map.empty
cgs <- newIORef Map.empty
axioms <- newIORef []
swCache <- newIORef IMap.empty
aiCache <- newIORef IMap.empty
infPrec <- newIORef False
cstrs <- newIORef []
rGen <- case currentRunMode of
Concrete g -> newIORef g
_ -> newStdGen >>= newIORef
let st = State { runMode = currentRunMode
, rStdGen = rGen
, rCInfo = cInfo
, rctr = ctr
, rInfPrec = infPrec
, rinps = inps
, routs = outs
, rtblMap = tables
, spgm = pgm
, rconstMap = cmap
, rArrayMap = arrays
, rexprMap = emap
, rUIMap = uis
, rCgMap = cgs
, raxioms = axioms
, rSWCache = swCache
, rAICache = aiCache
, rConstraints = cstrs
}
_ <- newConst st (mkConstCW (False, Size (Just 1)) (0::Integer)) -- s(-2) == falseSW
_ <- newConst st (mkConstCW (False, Size (Just 1)) (1::Integer)) -- s(-1) == trueSW
r <- runReaderT c st
rpgm <- readIORef pgm
inpsO <- reverse `fmap` readIORef inps
outsO <- reverse `fmap` readIORef outs
let swap (a, b) = (b, a)
cmp (a, _) (b, _) = a `compare` b
cnsts <- (sortBy cmp . map swap . Map.toList) `fmap` readIORef (rconstMap st)
tbls <- (sortBy (\((x, _, _), _) ((y, _, _), _) -> x `compare` y) . map swap . Map.toList) `fmap` readIORef tables
arrs <- IMap.toAscList `fmap` readIORef arrays
unint <- Map.toList `fmap` readIORef uis
axs <- reverse `fmap` readIORef axioms
hasInfPrec <- readIORef infPrec
cgMap <- Map.toList `fmap` readIORef cgs
traceVals <- reverse `fmap` readIORef cInfo
extraCstrs <- reverse `fmap` readIORef cstrs
return $ (r, Result hasInfPrec traceVals cgMap inpsO cnsts tbls arrs unint axs rpgm extraCstrs outsO)
-------------------------------------------------------------------------------
-- * Symbolic Words
-------------------------------------------------------------------------------
-- | A 'SymWord' is a potential symbolic bitvector that can be created instances of
-- to be fed to a symbolic program. Note that these methods are typically not needed
-- in casual uses with 'prove', 'sat', 'allSat' etc, as default instances automatically
-- provide the necessary bits.
--
-- Minimal complete definiton: forall, forall_, exists, exists_, literal, fromCW
class (HasSignAndSize a, Ord a) => SymWord a where
-- | Create a user named input (universal)
forall :: String -> Symbolic (SBV a)
-- | Create an automatically named input
forall_ :: Symbolic (SBV a)
-- | Get a bunch of new words
mkForallVars :: Int -> Symbolic [SBV a]
-- | Create an existential variable
exists :: String -> Symbolic (SBV a)
-- | Create an automatically named existential variable
exists_ :: Symbolic (SBV a)
-- | Create a bunch of existentials
mkExistVars :: Int -> Symbolic [SBV a]
-- | Create a free variable, universal in a proof, existential in sat
free :: String -> Symbolic (SBV a)
-- | Create an unnamed free variable, universal in proof, existential in sat
free_ :: Symbolic (SBV a)
-- | Create a bunch of free vars
mkFreeVars :: Int -> Symbolic [SBV a]
-- | Turn a literal constant to symbolic
literal :: a -> SBV a
-- | Extract a literal, if the value is concrete
unliteral :: SBV a -> Maybe a
-- | Extract a literal, from a CW representation
fromCW :: CW -> a
-- | Is the symbolic word concrete?
isConcrete :: SBV a -> Bool
-- | Is the symbolic word really symbolic?
isSymbolic :: SBV a -> Bool
-- | Does it concretely satisfy the given predicate?
isConcretely :: SBV a -> (a -> Bool) -> Bool
-- | max/minbounds, if available. Note that we don't want
-- to impose "Bounded" on our class as Integer is not Bounded but it is a SymWord
mbMaxBound, mbMinBound :: Maybe a
-- minimal complete definiton: forall, forall_, exists, exists_, free, free_, literal, fromCW
mkForallVars n = mapM (const forall_) [1 .. n]
mkExistVars n = mapM (const exists_) [1 .. n]
mkFreeVars n = mapM (const free_) [1 .. n]
unliteral (SBV _ (Left c)) = Just $ fromCW c
unliteral _ = Nothing
isConcrete (SBV _ (Left _)) = True
isConcrete _ = False
isSymbolic = not . isConcrete
isConcretely s p
| Just i <- unliteral s = p i
| True = False
instance (Random a, SymWord a) => Random (SBV a) where
randomR (l, h) g = case (unliteral l, unliteral h) of
(Just lb, Just hb) -> let (v, g') = randomR (lb, hb) g in (literal (v :: a), g')
_ -> error $ "SBV.Random: Cannot generate random values with symbolic bounds"
random g = let (v, g') = random g in (literal (v :: a) , g')
---------------------------------------------------------------------------------
-- * Symbolic Arrays
---------------------------------------------------------------------------------
-- | Flat arrays of symbolic values
-- An @array a b@ is an array indexed by the type @'SBV' a@, with elements of type @'SBV' b@
-- If an initial value is not provided in 'newArray_' and 'newArray' methods, then the elements
-- are left unspecified, i.e., the solver is free to choose any value. This is the right thing
-- to do if arrays are used as inputs to functions to be verified, typically.
--
-- While it's certainly possible for user to create instances of 'SymArray', the
-- 'SArray' and 'SFunArray' instances already provided should cover most use cases
-- in practice. (There are some differences between these models, however, see the corresponding
-- declaration.)
--
--
-- Minimal complete definition: All methods are required, no defaults.
class SymArray array where
-- | Create a new array, with an optional initial value
newArray_ :: (HasSignAndSize a, HasSignAndSize b) => Maybe (SBV b) -> Symbolic (array a b)
-- | Create a named new array, with an optional initial value
newArray :: (HasSignAndSize a, HasSignAndSize b) => String -> Maybe (SBV b) -> Symbolic (array a b)
-- | Read the array element at @a@
readArray :: array a b -> SBV a -> SBV b
-- | Reset all the elements of the array to the value @b@
resetArray :: SymWord b => array a b -> SBV b -> array a b
-- | Update the element at @a@ to be @b@
writeArray :: SymWord b => array a b -> SBV a -> SBV b -> array a b
-- | 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
mergeArrays :: SymWord b => SBV Bool -> array a b -> array a b -> array a b
-- | Arrays implemented in terms of SMT-arrays: <http://goedel.cs.uiowa.edu/smtlib/theories/ArraysEx.smt2>
--
-- * Maps directly to SMT-lib arrays
--
-- * Reading from an unintialized value is OK and yields an uninterpreted result
--
-- * Can check for equality of these arrays
--
-- * Cannot quick-check theorems using @SArray@ values
--
-- * Typically slower as it heavily relies on SMT-solving for the array theory
--
data SArray a b = SArray ((Bool, Size), (Bool, Size)) (Cached ArrayIndex)
type ArrayIndex = Int
instance (HasSignAndSize a, HasSignAndSize b) => Show (SArray a b) where
show (SArray{}) = "SArray<" ++ showType (undefined :: a) ++ ":" ++ showType (undefined :: b) ++ ">"
instance SymArray SArray where
newArray_ = declNewSArray (\t -> "array_" ++ show t)
newArray n = declNewSArray (const n)
readArray (SArray (_, bsgnsz) f) a = SBV bsgnsz $ Right $ cache r
where r st = do arr <- uncacheAI f st
i <- sbvToSW st a
newExpr st bsgnsz (SBVApp (ArrRead arr) [i])
resetArray (SArray ainfo f) b = SArray ainfo $ cache g
where g st = do amap <- readIORef (rArrayMap st)
val <- sbvToSW st b
i <- uncacheAI f st
let j = IMap.size amap
j `seq` modifyIORef (rArrayMap st) (IMap.insert j ("array_" ++ show j, ainfo, ArrayReset i val))
return j
writeArray (SArray ainfo f) a b = SArray ainfo $ cache g
where g st = do arr <- uncacheAI f st
addr <- sbvToSW st a
val <- sbvToSW st b
amap <- readIORef (rArrayMap st)
let j = IMap.size amap
j `seq` modifyIORef (rArrayMap st) (IMap.insert j ("array_" ++ show j, ainfo, ArrayMutate arr addr val))
return j
mergeArrays t (SArray ainfo a) (SArray _ b) = SArray ainfo $ cache h
where h st = do ai <- uncacheAI a st
bi <- uncacheAI b st
ts <- sbvToSW st t
amap <- readIORef (rArrayMap st)
let k = IMap.size amap
k `seq` modifyIORef (rArrayMap st) (IMap.insert k ("array_" ++ show k, ainfo, ArrayMerge ts ai bi))
return k
declNewSArray :: forall a b. (HasSignAndSize a, HasSignAndSize b) => (Int -> String) -> Maybe (SBV b) -> Symbolic (SArray a b)
declNewSArray mkNm mbInit = do
let asgnsz = (hasSign (undefined :: a), sizeOf (undefined :: a))
bsgnsz = (hasSign (undefined :: b), sizeOf (undefined :: b))
st <- ask
amap <- liftIO $ readIORef $ rArrayMap st
let i = IMap.size amap
nm = mkNm i
actx <- liftIO $ case mbInit of
Nothing -> return $ ArrayFree Nothing
Just ival -> sbvToSW st ival >>= \sw -> return $ ArrayFree (Just sw)
liftIO $ modifyIORef (rArrayMap st) (IMap.insert i (nm, (asgnsz, bsgnsz), actx))
return $ SArray (asgnsz, bsgnsz) $ cache $ const $ return i
-- | Arrays implemented internally as functions
--
-- * Internally handled by the library and not mapped to SMT-Lib
--
-- * Reading an uninitialized value is considered an error (will throw exception)
--
-- * Cannot check for equality (internally represented as functions)
--
-- * Can quick-check
--
-- * Typically faster as it gets compiled away during translation
--
data SFunArray a b = SFunArray (SBV a -> SBV b)
instance (HasSignAndSize a, HasSignAndSize b) => Show (SFunArray a b) where
show (SFunArray _) = "SFunArray<" ++ showType (undefined :: a) ++ ":" ++ showType (undefined :: b) ++ ">"
-- | Lift a function to an array. Useful for creating arrays in a pure context. (Otherwise use `newArray`.)
mkSFunArray :: (SBV a -> SBV b) -> SFunArray a b
mkSFunArray = SFunArray
-- | Handling constraints
imposeConstraint :: SBool -> Symbolic ()
imposeConstraint c = do st <- ask
case runMode st of
CodeGen -> error "SBV: constraints are not allowed in code-generation"
_ -> do liftIO $ do v <- sbvToSW st c
modifyIORef (rConstraints st) (v:)
addConstraint :: Maybe Double -> SBool -> SBool -> Symbolic ()
addConstraint Nothing c _ = imposeConstraint c
addConstraint (Just t) c c'
| t < 0 || t > 1
= error $ "SBV: pConstrain: Invalid probability threshold: " ++ show t ++ ", must be in [0, 1]."
| True
= do st <- ask
when (not (isConcreteMode (runMode st))) $ error "SBV: pConstrain only allowed in 'genTest' or 'quickCheck' contexts."
case () of
() | t > 0 && t < 1 -> liftIO (throwDice st) >>= \d -> imposeConstraint (if d <= t then c else c')
| t > 0 -> imposeConstraint c
| True -> imposeConstraint c'
---------------------------------------------------------------------------------
-- * Cached values
---------------------------------------------------------------------------------
-- We implement a peculiar caching mechanism, applicable to the use case in
-- implementation of SBV's. Whenever we do a state based computation, we do
-- not want to keep on evaluating it in the then-current state. That will
-- produce essentially a semantically equivalent value. Thus, we want to run
-- it only once, and reuse that result, capturing the sharing at the Haskell
-- level. This is similar to the "type-safe observable sharing" work, but also
-- takes into the account of how symbolic simulation executes.
--
-- Note that this is *not* a general memo utility!
newtype Cached a = Cached (State -> IO a)
cache :: (State -> IO a) -> Cached a
cache = Cached
uncache :: Cached SW -> State -> IO SW
uncache = uncacheGen rSWCache
uncacheAI :: Cached ArrayIndex -> State -> IO ArrayIndex
uncacheAI = uncacheGen rAICache
uncacheGen :: (State -> IORef (Cache a)) -> Cached a -> State -> IO a
uncacheGen getCache (Cached f) st = do
let rCache = getCache st
stored <- readIORef rCache
sn <- f `seq` makeStableName f
let h = hashStableName sn
case maybe Nothing (sn `lookup`) (h `IMap.lookup` stored) of
Just r -> return r
Nothing -> do r <- f st
r `seq` modifyIORef rCache (IMap.insertWith (++) h [(sn, r)])
return r
-- Representation of SMTLib Programs
data SMTLibVersion = SMTLib1
| SMTLib2
deriving Eq
-- in between pre and post goes the refuted models
data SMTLibPgm = SMTLibPgm SMTLibVersion ( [(String, SW)] -- alias table
, [String] -- pre: declarations.
, [String]) -- post: formula
instance NFData SMTLibVersion
instance NFData SMTLibPgm
instance Show SMTLibPgm where
show (SMTLibPgm _ (_, pre, post)) = intercalate "\n" $ pre ++ post
-- Other Technicalities..
instance NFData CW where
rnf (CW x y z) = x `seq` y `seq` z `seq` ()
instance NFData Result where
rnf (Result isInf qcInfo cgs inps consts tbls arrs uis axs pgm cstr outs)
= rnf isInf `seq` rnf qcInfo `seq` rnf cgs `seq` rnf inps `seq` rnf consts `seq` rnf tbls `seq` rnf arrs `seq` rnf uis `seq` rnf axs `seq` rnf pgm `seq` rnf cstr `seq` rnf outs
instance NFData Size
instance NFData ArrayContext
instance NFData Pgm
instance NFData SW
instance NFData Quantifier
instance NFData SBVType
instance NFData UnintKind
instance NFData a => NFData (Cached a) where
rnf (Cached f) = f `seq` ()
instance NFData a => NFData (SBV a) where
rnf (SBV x y) = rnf x `seq` rnf y `seq` ()