sbv-0.9.19: 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
, SymWord(..)
, CW, cwVal, 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, Size, Outputtable(..), Result(..)
, SBVType(..), newUninterpreted, unintFnUIKind, addAxiom
) where
import Control.DeepSeq (NFData(..))
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 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 Test.QuickCheck (Testable(..))
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, cwSize :: !Size, cwVal :: !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 (cwSigned x) && cwSize x == 1
cwToBool :: CW -> Bool
cwToBool x = cwVal x /= 0
normCW :: CW -> CW
normCW x = x { cwVal = norm }
where norm | cwSize x == 0 = 0
| cwSigned x = let rg = 2 ^ (cwSize x - 1)
in case divMod (cwVal x) rg of
(a, b) | even a -> b
(_, b) -> b - rg
| True = cwVal x `mod` (2 ^ cwSize x)
type Size = Int
newtype NodeId = NodeId Int deriving (Eq, Ord)
data SW = SW (Bool, Size) NodeId deriving (Eq, Ord)
falseSW, trueSW :: SW
falseSW = SW (False, 1) $ NodeId (-2)
trueSW = SW (False, 1) $ NodeId (-1)
falseCW, trueCW :: CW
falseCW = CW False 1 0
trueCW = CW False 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 (False, 1) = "SBool"
sh (s, 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, Int), (Bool, Int), 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)
class HasSignAndSize a where
sizeOf :: a -> Size
hasSign :: a -> Bool
showType :: a -> String
showType a
| not (hasSign a) && sizeOf a == 1 = "SBool"
| True = (if hasSign a then "SInt" else "SWord") ++ show (sizeOf a)
instance HasSignAndSize Bool where {sizeOf _ = 1; hasSign _ = False}
instance HasSignAndSize Int8 where {sizeOf _ = 8; hasSign _ = True }
instance HasSignAndSize Word8 where {sizeOf _ = 8; hasSign _ = False}
instance HasSignAndSize Int16 where {sizeOf _ = 16; hasSign _ = True }
instance HasSignAndSize Word16 where {sizeOf _ = 16; hasSign _ = False}
instance HasSignAndSize Int32 where {sizeOf _ = 32; hasSign _ = True }
instance HasSignAndSize Word32 where {sizeOf _ = 32; hasSign _ = False}
instance HasSignAndSize Int64 where {sizeOf _ = 64; hasSign _ = True }
instance HasSignAndSize Word64 where {sizeOf _ = 64; hasSign _ = False}
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
sizeOf = cwSize
hasSign = cwSigned
instance HasSignAndSize SW where
sizeOf (SW (_, s) _) = s
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 ++ "(" ++ show at ++ " -> " ++ show rt ++ ", " ++ show l ++ ")"
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 [NamedSymVar] -- inputs
[(SW, CW)] -- constants
[((Int, (Bool, Int), (Bool, Int)), [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] -- outputs
instance Show Result where
show (Result _ cs _ _ [] [] _ [r])
| Just c <- r `lookup` cs
= show c
show (Result is cs ts as uis axs xs os) = intercalate "\n" $
["INPUTS"]
++ map shn is
++ ["CONSTANTS"]
++ map shc cs
++ ["TABLES"]
++ map sht ts
++ ["ARRAYS"]
++ map sha as
++ ["UNINTERPRETED CONSTANTS"]
++ map shui uis
++ ["AXIOMS"]
++ map shax axs
++ ["DEFINE"]
++ map (\(s, e) -> " " ++ shs s ++ " = " ++ show e) (F.toList xs)
++ ["OUTPUTS"]
++ map ((" " ++) . show) os
where shs sw = show sw ++ " :: " ++ showType sw
sht ((i, at, rt), es) = " Table " ++ show i ++ " : " ++ show at ++ "->" ++ show rt ++ " = " ++ show es
shc (sw, cw) = " " ++ show sw ++ " = " ++ show cw
shn (sw, nm) = " " ++ ni ++ " :: " ++ showType sw ++ alias
where ni = show sw
alias | ni == nm = ""
| True = ", aliasing " ++ show nm
sha (i, (nm, (ai, bi), ctx)) = " " ++ ni ++ " :: " ++ mkT ai ++ " -> " ++ mkT bi ++ alias
++ "\n Context: " ++ show ctx
where mkT (b, s)
| s == 1 = "SBool"
| True = if b then "SInt" else "SWord" ++ show s
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
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, Int), (Bool, Int))
type ArrayInfo = (String, ((Bool, Size), (Bool, Size)), ArrayContext)
type ArrayMap = IMap.IntMap ArrayInfo
type UIMap = Map.Map String SBVType
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
data State = State { rctr :: IORef Int
, rinps :: IORef [NamedSymVar]
, routs :: IORef [SW]
, rtblMap :: IORef TableMap
, spgm :: IORef Pgm
, rconstMap :: IORef CnstMap
, rexprMap :: IORef ExprMap
, rArrayMap :: IORef ArrayMap
, rUIMap :: IORef UIMap
, raxioms :: IORef [(String, [String])]
, rSWCache :: IORef (Cache SW)
, rAICache :: IORef (Cache Int)
}
-- | 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
-- Needed to satisfy the Num hierarchy
instance Show (SBV a) where
show (SBV _ (Left c)) = show c
show (SBV (sgn, 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 (SBV a) where
sizeOf (SBV (_, s) _) = s
hasSign (SBV (b, _) _) = b
incCtr :: State -> IO Int
incCtr s = do ctr <- readIORef (rctr s)
let i = ctr + 1
i `seq` writeIORef (rctr s) i
return ctr
newUninterpreted :: State -> String -> SBVType -> IO ()
newUninterpreted st nm t
| null nm || not (isAlpha (head nm)) || not (all isAlphaNum (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 -> modifyIORef (rUIMap st) (Map.insert nm t)
-- 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)
modifyIORef (rconstMap st) (Map.insert c sw)
return sw
-- Create a new table; hash-cons as necessary
getTableIndex :: State -> (Bool, Int) -> (Bool, Int) -> [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)
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 (Monad, MonadIO, MonadReader State)
mkSymSBV :: (Bool, Size) -> Maybe String -> Symbolic (SBV a)
mkSymSBV sgnsz mbNm = do
st <- ask
ctr <- liftIO $ incCtr st
let nm = maybe ('s':show ctr) id mbNm
sw = SW sgnsz (NodeId ctr)
liftIO $ modifyIORef (rinps st) ((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 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 and return a 'Result'
runSymbolic :: Symbolic a -> IO Result
runSymbolic c = do (_, r) <- runSymbolic' c
return r
-- | Run a symbolic computation, and return a extra value paired up with the 'Result'
runSymbolic' :: Symbolic a -> IO (a, Result)
runSymbolic' (Symbolic c) = do
ctr <- newIORef (-2) -- start from -2; False and True will always occupy the first two elements
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
axioms <- newIORef []
swCache <- newIORef IMap.empty
aiCache <- newIORef IMap.empty
let st = State { rctr = ctr
, rinps = inps
, routs = outs
, rtblMap = tables
, spgm = pgm
, rconstMap = cmap
, rArrayMap = arrays
, rexprMap = emap
, rUIMap = uis
, raxioms = axioms
, rSWCache = swCache
, rAICache = aiCache
}
_ <- newConst st (mkConstCW (False,1) (0::Integer)) -- s(-2) == falseSW
_ <- newConst st (mkConstCW (False,1) (1::Integer)) -- s(-1) == trueSW
r <- runReaderT c st
rpgm <- readIORef pgm
inpsR <- readIORef inps
outsR <- 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
return $ (r, Result (reverse inpsR) cnsts tbls arrs unint axs rpgm (reverse outsR))
-------------------------------------------------------------------------------
-- * 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: free, free_, literal, fromCW
class (Bounded a, Ord a) => SymWord a where
-- | Create a user named input
free :: String -> Symbolic (SBV a)
-- | Create an automatically named input
free_ :: Symbolic (SBV a)
-- | Get a bunch of new words
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
-- minimal complete definiton: free, free_, literal, fromCW
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
---------------------------------------------------------------------------------
-- * 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
---------------------------------------------------------------------------------
-- * 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
-- Technicalities..
instance NFData CW where
rnf (CW x y z) = x `seq` y `seq` z `seq` ()
instance NFData Result where
rnf (Result inps consts tbls arrs uis axs pgm outs)
= rnf inps `seq` rnf consts `seq` rnf tbls `seq` rnf arrs `seq` rnf uis `seq` rnf axs `seq` rnf pgm `seq` rnf outs
instance NFData ArrayContext
instance NFData Pgm
instance NFData SW
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` ()
-- Quickcheck interface on symbolic-booleans..
instance Testable SBool where
property (SBV _ (Left b)) = property (cwToBool b)
property s = error $ "Cannot quick-check in the presence of uninterpreted constants! (" ++ show s ++ ")"