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cryptol/sbv/Data/SBV/BitVectors/Data.hs

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Initial import from internal repo
2014-04-18 02:34:25 +04:00
-----------------------------------------------------------------------------
-- |
-- Module : Data.SBV.BitVectors.Data
-- Copyright : (c) Levent Erkok
-- License : BSD3
-- Maintainer : erkokl@gmail.com
-- Stability : experimental
--
-- Internal data-structures for the sbv library
-----------------------------------------------------------------------------
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE TypeSynonymInstances #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE PatternGuards #-}
{-# LANGUAGE DefaultSignatures #-}
module Data.SBV.BitVectors.Data
( SBool, SWord8, SWord16, SWord32, SWord64
, SInt8, SInt16, SInt32, SInt64, SInteger, SReal, SFloat, SDouble
, nan, infinity, sNaN, sInfinity, RoundingMode(..), smtLibSquareRoot, smtLibFusedMA
, SymWord(..)
, CW(..), CWVal(..), cwSameType, cwIsBit, cwToBool, normCW
, mkConstCW ,liftCW2, mapCW, mapCW2
, SW(..), trueSW, falseSW, trueCW, falseCW
, SBV(..), NodeId(..), mkSymSBV, mkSymSBVWithRandom
, ArrayContext(..), ArrayInfo, SymArray(..), SFunArray(..), mkSFunArray, SArray(..), arrayUIKind
, sbvToSW, sbvToSymSW
, SBVExpr(..), newExpr
, cache, Cached, uncache, uncacheAI, HasKind(..)
, Op(..), NamedSymVar, UnintKind(..), getTableIndex, SBVPgm(..), Symbolic, runSymbolic, runSymbolic', State, inProofMode, SBVRunMode(..), Kind(..), Outputtable(..), Result(..)
, readResult, getResult
, getTraceInfo, getConstraints, addConstraint
, SBVType(..), newUninterpreted, unintFnUIKind, addAxiom
, Quantifier(..), needsExistentials
, SMTLibPgm(..), SMTLibVersion(..)
, SolverCapabilities(..)
) where
import Control.Applicative (Applicative)
import Control.DeepSeq (NFData(..))
import Control.Monad (when)
import Control.Monad.Fix (MonadFix)
import Control.Monad.Reader (MonadReader, ReaderT, ask, runReaderT)
import Control.Monad.Trans (MonadIO, liftIO)
import Data.Char (isAlpha, isAlphaNum)
import Data.Generics (Data(..), dataTypeName, dataTypeOf, tyconUQname)
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)
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.Set as Set (Set, empty, toList, insert)
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.BitVectors.AlgReals
import Data.SBV.Utils.Lib
-- | A constant value
data CWVal = CWAlgReal AlgReal -- ^ algebraic real
| CWInteger Integer -- ^ bit-vector/unbounded integer
| CWFloat Float -- ^ float
| CWDouble Double -- ^ double
| CWUninterpreted String -- ^ value of an uninterpreted kind
-- We cannot simply derive Eq/Ord for CWVal, since CWAlgReal doesn't have proper
-- instances for these when values are infinitely precise reals. However, we do
-- need a structural eq/ord for Map indexes; so define custom ones here:
instance Eq CWVal where
CWAlgReal a == CWAlgReal b = a `algRealStructuralEqual` b
CWInteger a == CWInteger b = a == b
CWUninterpreted a == CWUninterpreted b = a == b
CWFloat a == CWFloat b = a == b
CWDouble a == CWDouble b = a == b
_ == _ = False
instance Ord CWVal where
CWAlgReal a `compare` CWAlgReal b = a `algRealStructuralCompare` b
CWAlgReal _ `compare` CWInteger _ = LT
CWAlgReal _ `compare` CWFloat _ = LT
CWAlgReal _ `compare` CWDouble _ = LT
CWAlgReal _ `compare` CWUninterpreted _ = LT
CWInteger _ `compare` CWAlgReal _ = GT
CWInteger a `compare` CWInteger b = a `compare` b
CWInteger _ `compare` CWFloat _ = LT
CWInteger _ `compare` CWDouble _ = LT
CWInteger _ `compare` CWUninterpreted _ = LT
CWFloat _ `compare` CWAlgReal _ = GT
CWFloat _ `compare` CWInteger _ = GT
CWFloat a `compare` CWFloat b = a `compare` b
CWFloat _ `compare` CWDouble _ = LT
CWFloat _ `compare` CWUninterpreted _ = LT
CWDouble _ `compare` CWAlgReal _ = GT
CWDouble _ `compare` CWInteger _ = GT
CWDouble _ `compare` CWFloat _ = GT
CWDouble a `compare` CWDouble b = a `compare` b
CWDouble _ `compare` CWUninterpreted _ = LT
CWUninterpreted _ `compare` CWAlgReal _ = GT
CWUninterpreted _ `compare` CWInteger _ = GT
CWUninterpreted _ `compare` CWFloat _ = GT
CWUninterpreted _ `compare` CWDouble _ = GT
CWUninterpreted a `compare` CWUninterpreted b = a `compare` b
-- | '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 { cwKind :: !Kind
, cwVal :: !CWVal
}
deriving (Eq, Ord)
-- | Are two CW's of the same type?
cwSameType :: CW -> CW -> Bool
cwSameType x y = cwKind x == cwKind y
-- | Is this a bit?
cwIsBit :: CW -> Bool
cwIsBit x = case cwKind x of
KBounded False 1 -> True
_ -> False
-- | Convert a CW to a Haskell boolean (NB. Assumes input is well-kinded)
cwToBool :: CW -> Bool
cwToBool x = cwVal x /= CWInteger 0
-- | Normalize a CW. Essentially performs modular arithmetic to make sure the
-- value can fit in the given bit-size. Note that this is rather tricky for
-- negative values, due to asymmetry. (i.e., an 8-bit negative number represents
-- values in the range -128 to 127; thus we have to be careful on the negative side.)
normCW :: CW -> CW
normCW c@(CW (KBounded signed sz) (CWInteger v)) = c { cwVal = CWInteger norm }
where norm | sz == 0 = 0
| signed = let rg = 2 ^ (sz - 1)
in case divMod v rg of
(a, b) | even a -> b
(_, b) -> b - rg
| True = v `mod` (2 ^ sz)
normCW c = c
-- | Kind of symbolic value
data Kind = KBounded Bool Int
| KUnbounded
| KReal
| KUninterpreted String
| KFloat
| KDouble
deriving (Eq, Ord)
instance Show Kind where
show (KBounded False 1) = "SBool"
show (KBounded False n) = "SWord" ++ show n
show (KBounded True n) = "SInt" ++ show n
show KUnbounded = "SInteger"
show KReal = "SReal"
show (KUninterpreted s) = s
show KFloat = "SFloat"
show KDouble = "SDouble"
-- | A symbolic node id
newtype NodeId = NodeId Int deriving (Eq, Ord)
-- | A symbolic word, tracking it's signedness and size.
data SW = SW Kind NodeId deriving (Eq, Ord)
-- | Quantifiers: forall or exists. Note that we allow
-- arbitrary nestings.
data Quantifier = ALL | EX deriving Eq
-- | Are there any existential quantifiers?
needsExistentials :: [Quantifier] -> Bool
needsExistentials = (EX `elem`)
-- | Constant False as a SW. Note that this value always occupies slot -2.
falseSW :: SW
falseSW = SW (KBounded False 1) $ NodeId (-2)
-- | Constant False as a SW. Note that this value always occupies slot -1.
trueSW :: SW
trueSW = SW (KBounded False 1) $ NodeId (-1)
-- | Constant False as a CW. We represent it using the integer value 0.
falseCW :: CW
falseCW = CW (KBounded False 1) (CWInteger 0)
-- | Constant True as a CW. We represent it using the integer value 1.
trueCW :: CW
trueCW = CW (KBounded False 1) (CWInteger 1)
-- | A simple type for SBV computations, used mainly for uninterpreted constants.
-- We keep track of the signedness/size of the arguments. A non-function will
-- have just one entry in the list.
newtype SBVType = SBVType [Kind]
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 show xs
-- | Symbolic operations
data Op = Plus | Times | Minus
| Quot | Rem
| 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, Kind, Kind, 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)
-- | SMT-Lib's square-root over floats/doubles. We piggy back on to the uninterpreted function mechanism
-- to implement these; which is not a terrible idea; although the use of the constructor 'Uninterpreted'
-- might be confusing. This function will *not* be uninterpreted in reality, as QF_FPA will define it. It's
-- a bit of a shame, but much easier to implement it this way.
smtLibSquareRoot :: Op
smtLibSquareRoot = Uninterpreted "squareRoot"
-- | SMT-Lib's fusedMA over floats/doubles. Similar to the 'smtLibSquareRoot'. Note that we cannot implement
-- this function in Haskell as precision loss would be inevitable. Maybe Haskell will eventually add this op
-- to the Num class.
smtLibFusedMA :: Op
smtLibFusedMA = Uninterpreted "fusedMA"
-- | A symbolic expression
data SBVExpr = SBVApp !Op ![SW]
deriving (Eq, Ord)
-- | A class for capturing values that have a sign and a size (finite or infinite)
-- minimal complete definition: kindOf. This class can be automatically derived
-- for data-types that have a 'Data' instance; this is useful for creating uninterpreted
-- sorts.
class HasKind a where
kindOf :: a -> Kind
hasSign :: a -> Bool
intSizeOf :: a -> Int
isBoolean :: a -> Bool
isBounded :: a -> Bool
isReal :: a -> Bool
isFloat :: a -> Bool
isDouble :: a -> Bool
isInteger :: a -> Bool
isUninterpreted :: a -> Bool
showType :: a -> String
-- defaults
hasSign x = case kindOf x of
KBounded b _ -> b
KUnbounded -> True
KReal -> True
KFloat -> True
KDouble -> True
KUninterpreted{} -> False
intSizeOf x = case kindOf x of
KBounded _ s -> s
KUnbounded -> error "SBV.HasKind.intSizeOf((S)Integer)"
KReal -> error "SBV.HasKind.intSizeOf((S)Real)"
KFloat -> error "SBV.HasKind.intSizeOf((S)Float)"
KDouble -> error "SBV.HasKind.intSizeOf((S)Double)"
KUninterpreted s -> error $ "SBV.HasKind.intSizeOf: Uninterpreted sort: " ++ s
isBoolean x | KBounded False 1 <- kindOf x = True
| True = False
isBounded x | KBounded{} <- kindOf x = True
| True = False
isReal x | KReal{} <- kindOf x = True
| True = False
isFloat x | KFloat{} <- kindOf x = True
| True = False
isDouble x | KDouble{} <- kindOf x = True
| True = False
isInteger x | KUnbounded{} <- kindOf x = True
| True = False
isUninterpreted x | KUninterpreted{} <- kindOf x = True
| True = False
showType = show . kindOf
-- default signature for uninterpreted kinds
default kindOf :: Data a => a -> Kind
kindOf = KUninterpreted . tyconUQname . dataTypeName . dataTypeOf
instance HasKind Bool where kindOf _ = KBounded False 1
instance HasKind Int8 where kindOf _ = KBounded True 8
instance HasKind Word8 where kindOf _ = KBounded False 8
instance HasKind Int16 where kindOf _ = KBounded True 16
instance HasKind Word16 where kindOf _ = KBounded False 16
instance HasKind Int32 where kindOf _ = KBounded True 32
instance HasKind Word32 where kindOf _ = KBounded False 32
instance HasKind Int64 where kindOf _ = KBounded True 64
instance HasKind Word64 where kindOf _ = KBounded False 64
instance HasKind Integer where kindOf _ = KUnbounded
instance HasKind AlgReal where kindOf _ = KReal
instance HasKind Float where kindOf _ = KFloat
instance HasKind Double where kindOf _ = KDouble
-- | Lift a unary function thruough a CW
liftCW :: (AlgReal -> b) -> (Integer -> b) -> (Float -> b) -> (Double -> b) -> (String -> b) -> CW -> b
liftCW f _ _ _ _ (CW _ (CWAlgReal v)) = f v
liftCW _ f _ _ _ (CW _ (CWInteger v)) = f v
liftCW _ _ f _ _ (CW _ (CWFloat v)) = f v
liftCW _ _ _ f _ (CW _ (CWDouble v)) = f v
liftCW _ _ _ _ f (CW _ (CWUninterpreted v)) = f v
-- | Lift a binary function through a CW
liftCW2 :: (AlgReal -> AlgReal -> b) -> (Integer -> Integer -> b) -> (Float -> Float -> b) -> (Double -> Double -> b) -> (String -> String -> b) -> CW -> CW -> b
liftCW2 r i f d u x y = case (cwVal x, cwVal y) of
(CWAlgReal a, CWAlgReal b) -> r a b
(CWInteger a, CWInteger b) -> i a b
(CWFloat a, CWFloat b) -> f a b
(CWDouble a, CWDouble b) -> d a b
(CWUninterpreted a, CWUninterpreted b) -> u a b
_ -> error $ "SBV.liftCW2: impossible, incompatible args received: " ++ show (x, y)
-- | Map a unary function through a CW
mapCW :: (AlgReal -> AlgReal) -> (Integer -> Integer) -> (Float -> Float) -> (Double -> Double) -> (String -> String) -> CW -> CW
mapCW r i f d u x = normCW $ CW (cwKind x) $ case cwVal x of
CWAlgReal a -> CWAlgReal (r a)
CWInteger a -> CWInteger (i a)
CWFloat a -> CWFloat (f a)
CWDouble a -> CWDouble (d a)
CWUninterpreted a -> CWUninterpreted (u a)
-- | Map a binary function through a CW
mapCW2 :: (AlgReal -> AlgReal -> AlgReal) -> (Integer -> Integer -> Integer) -> (Float -> Float -> Float) -> (Double -> Double -> Double) -> (String -> String -> String) -> CW -> CW -> CW
mapCW2 r i f d u x y = case (cwSameType x y, cwVal x, cwVal y) of
(True, CWAlgReal a, CWAlgReal b) -> normCW $ CW (cwKind x) (CWAlgReal (r a b))
(True, CWInteger a, CWInteger b) -> normCW $ CW (cwKind x) (CWInteger (i a b))
(True, CWFloat a, CWFloat b) -> normCW $ CW (cwKind x) (CWFloat (f a b))
(True, CWDouble a, CWDouble b) -> normCW $ CW (cwKind x) (CWDouble (d a b))
(True, CWUninterpreted a, CWUninterpreted b) -> normCW $ CW (cwKind x) (CWUninterpreted (u a b))
_ -> error $ "SBV.mapCW2: impossible, incompatible args received: " ++ show (x, y)
instance HasKind CW where
kindOf = cwKind
instance HasKind SW where
kindOf (SW k _) = k
instance Show CW where
show w | cwIsBit w = show (cwToBool w)
show w = liftCW show show show show id 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, "#")
]
-- | To improve hash-consing, take advantage of commutative operators by
-- reordering their arguments.
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
newtype SBVPgm = SBVPgm {pgmAssignments :: (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 (Set.Set Kind) -- kinds used in the program
[(String, CW)] -- quick-check counter-example information (if any)
[(String, [String])] -- uninterpreted code segments
[(Quantifier, NamedSymVar)] -- inputs (possibly existential)
[(SW, CW)] -- constants
[((Int, Kind, Kind), [SW])] -- tables (automatically constructed) (tableno, index-type, result-type) elts
[(Int, ArrayInfo)] -- arrays (user specified)
[(String, SBVType)] -- uninterpreted constants
[(String, [String])] -- axioms
SBVPgm -- assignments
[SW] -- additional constraints (boolean)
[SW] -- outputs
-- | Extract the constraints from a result
getConstraints :: Result -> [SW]
getConstraints (Result _ _ _ _ _ _ _ _ _ _ cstrs _) = cstrs
-- | Extract the traced-values from a result (quick-check)
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 kinds _ cgs is cs ts as uis axs xs cstrs os) = intercalate "\n" $
(if null usorts then [] else "SORTS" : map (" " ++) usorts)
++ ["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 (pgmAssignments xs))
++ ["CONSTRAINTS"]
++ map ((" " ++) . show) cstrs
++ ["OUTPUTS"]
++ map ((" " ++) . show) os
where usorts = [s | KUninterpreted s <- Set.toList kinds]
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
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 ++ " :: " ++ show ai ++ " -> " ++ show 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
-- | The context of a symbolic array as created
data ArrayContext = ArrayFree (Maybe SW) -- ^ A new array, with potential initializer for each cell
| ArrayReset Int SW -- ^ An array created from another array by fixing each element to another value
| ArrayMutate Int SW SW -- ^ An array created by mutating another array at a given cell
| ArrayMerge SW Int Int -- ^ An array created by symbolically merging two other arrays
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
-- | Expression map, used for hash-consing
type ExprMap = Map.Map SBVExpr SW
-- | Constants are stored in a map, for hash-consing
type CnstMap = Map.Map CW SW
-- | Kinds used in the program; used for determining the final SMT-Lib logic to pick
type KindSet = Set.Set Kind
-- | Tables generated during a symbolic run
type TableMap = Map.Map [SW] (Int, Kind, Kind)
-- | Representation for symbolic arrays
type ArrayInfo = (String, (Kind, Kind), ArrayContext)
-- | Arrays generated during a symbolic run
type ArrayMap = IMap.IntMap ArrayInfo
-- | Uninterpreted-constants generated during a symbolic run
type UIMap = Map.Map String SBVType
-- | Code-segments for Uninterpreted-constants, as given by the user
type CgMap = Map.Map String [String]
-- | Cached values, implementing sharing
type Cache a = IMap.IntMap [(StableName (State -> IO a), a)]
-- | Convert an SBV-type to the kind-of uninterpreted value it represents
unintFnUIKind :: (String, SBVType) -> (String, UnintKind)
unintFnUIKind (s, t) = (s, UFun (typeArity t) s)
-- | Convert an array value type to the kind-of uninterpreted value it represents
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
-- | Is this a concrete run? (i.e., quick-check or test-generation like)
isConcreteMode :: SBVRunMode -> Bool
isConcreteMode (Concrete _) = True
isConcreteMode (Proof{}) = False
isConcreteMode CodeGen = False
-- | The state of the symbolic interpreter
data State = State { runMode :: SBVRunMode
, rStdGen :: IORef StdGen
, rCInfo :: IORef [(String, CW)]
, rctr :: IORef Int
, rUsedKinds :: IORef KindSet
, rinps :: IORef [(Quantifier, NamedSymVar)]
, rConstraints :: IORef [SW]
, routs :: IORef [SW]
, rtblMap :: IORef TableMap
, spgm :: IORef SBVPgm
, 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)
}
-- | Are we running in proof mode?
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 !Kind !(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
-- | Infinite precision symbolic algebraic real value
type SReal = SBV AlgReal
-- | IEEE-754 single-precision floating point numbers
type SFloat = SBV Float
-- | IEEE-754 double-precision floating point numbers
type SDouble = SBV Double
-- | Not-A-Number for 'Double' and 'Float'. Surprisingly, Haskell
-- Prelude doesn't have this value defined, so we provide it here.
nan :: Floating a => a
nan = 0/0
-- | Infinity for 'Double' and 'Float'. Surprisingly, Haskell
-- Prelude doesn't have this value defined, so we provide it here.
infinity :: Floating a => a
infinity = 1/0
-- | Symbolic variant of Not-A-Number. This value will inhabit both
-- 'SDouble' and 'SFloat'.
sNaN :: (Floating a, SymWord a) => SBV a
sNaN = literal nan
-- | Symbolic variant of infinity. This value will inhabit both
-- 'SDouble' and 'SFloat'.
sInfinity :: (Floating a, SymWord a) => SBV a
sInfinity = literal infinity
-- | Rounding mode to be used for the IEEE floating-point operations.
-- Note that Haskell's default is 'RoundNearestTiesToEven'. If you use
-- a different rounding mode, then the counter-examples you get may not
-- match what you observe in Haskell.
data RoundingMode = RoundNearestTiesToEven -- ^ Round to nearest representable floating point value.
-- If precisely at half-way, pick the even number.
-- (In this context, /even/ means the lowest-order bit is zero.)
| RoundNearestTiesToAway -- ^ Round to nearest representable floating point value.
-- If precisely at half-way, pick the number further away from 0.
-- (That is, for positive values, pick the greater; for negative values, pick the smaller.)
| RoundTowardPositive -- ^ Round towards positive infinity. (Also known as rounding-up or ceiling.)
| RoundTowardNegative -- ^ Round towards negative infinity. (Also known as rounding-down or floor.)
| RoundTowardZero -- ^ Round towards zero. (Also known as truncation.)
-- Not particularly "desirable", but will do if needed
instance Show (SBV a) where
show (SBV _ (Left c)) = show c
show (SBV k (Right _)) = "<symbolic> :: " ++ show k
-- Equality constraint on SBV values. Not desirable since we can't really compare two
-- symbolic values, but will do.
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 HasKind a => HasKind (SBV a) where
kindOf (SBV k _) = k
-- | Increment the variable counter
incCtr :: State -> IO Int
incCtr s = do ctr <- readIORef (rctr s)
let i = ctr + 1
i `seq` writeIORef (rctr s) i
return ctr
-- | Generate a random value, for quick-check and test-gen purposes
throwDice :: State -> IO Double
throwDice st = do g <- readIORef (rStdGen st)
let (r, g') = randomR (0, 1) g
writeIORef (rStdGen st) g'
return r
-- | Create a new uninterpreted symbol, possibly with user given code
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 SW
newSW :: State -> Kind -> IO (SW, String)
newSW st k = do ctr <- incCtr st
let sw = SW k (NodeId ctr)
registerKind st k
return (sw, 's' : show ctr)
{-# INLINE newSW #-}
registerKind :: State -> Kind -> IO ()
registerKind st k
| KUninterpreted sortName <- k, sortName `elem` reserved
= error $ "SBV: " ++ show sortName ++ " is a reserved sort; please use a different name."
| True
= modifyIORef (rUsedKinds st) (Set.insert k)
where reserved = ["Int", "Real", "List", "Array", "Bool", "NUMERAL", "DECIMAL", "STRING", "FP"] -- Reserved by SMT-Lib
-- | 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 let k = kindOf c
(sw, _) <- newSW st k
modifyIORef (rconstMap st) (Map.insert c sw)
return sw
{-# INLINE newConst #-}
-- | Create a new table; hash-cons as necessary
getTableIndex :: State -> Kind -> Kind -> [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 from an integral
mkConstCW :: Integral a => Kind -> a -> CW
mkConstCW k@(KBounded{}) a = normCW $ CW k (CWInteger (toInteger a))
mkConstCW KUnbounded a = normCW $ CW KUnbounded (CWInteger (toInteger a))
mkConstCW KReal a = normCW $ CW KReal (CWAlgReal (fromInteger (toInteger a)))
mkConstCW KFloat a = normCW $ CW KFloat (CWFloat (fromInteger (toInteger a)))
mkConstCW KDouble a = normCW $ CW KDouble (CWDouble (fromInteger (toInteger a)))
mkConstCW (KUninterpreted s) a = error $ "Unexpected call to mkConstCW with uninterpreted kind: " ++ s ++ " with value: " ++ show (toInteger a)
-- | Create a new expression; hash-cons as necessary
newExpr :: State -> Kind -> SBVExpr -> IO SW
newExpr st k app = do
let e = reorder app
exprMap <- readIORef (rexprMap st)
case e `Map.lookup` exprMap of
Just sw -> return sw
Nothing -> do (sw, _) <- newSW st k
modifyIORef (spgm st) (\(SBVPgm xs) -> SBVPgm (xs S.|> (sw, e)))
modifyIORef (rexprMap st) (Map.insert e sw)
return sw
{-# INLINE newExpr #-}
-- | Convert a symbolic value to a symbolic-word
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, Applicative, Monad, MonadFix, MonadIO, MonadReader State)
-- | Create a symbolic value, based on the quantifier we have. If an explicit quantifier is given, we just use that.
-- If not, then we pick existential for SAT calls and universal for everything else.
mkSymSBV :: forall a. (Random a, SymWord a) => Maybe Quantifier -> Kind -> Maybe String -> Symbolic (SBV a)
mkSymSBV = mkSymSBVWithRandom randomIO
mkSymSBVWithRandom :: forall a. SymWord a =>
IO (SBV a) -> Maybe Quantifier -> Kind -> Maybe String -> Symbolic (SBV a)
mkSymSBVWithRandom random mbQ k 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 random
liftIO $ modifyIORef (rCInfo st) ((maybe "_" id mbNm, cw):)
return v
_ -> do (sw, internalName) <- liftIO $ newSW st k
let nm = maybe internalName id mbNm
liftIO $ modifyIORef (rinps st) ((q, (sw, nm)):)
return $ SBV k $ Right $ cache (const (return sw))
-- | Convert a symbolic value to an SW, inside the Symbolic monad
sbvToSymSW :: SBV a -> Symbolic SW
sbvToSymSW sbv = do
st <- ask
liftIO $ sbvToSW st sbv
-- | A class representing what can be returned from a symbolic computation.
class Outputtable a where
-- | Mark an interim result as an output. Useful when constructing Symbolic programs
-- that return multiple values, or when the result is programmatically computed.
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. The first argument is a mere
-- string, use for commenting purposes. The second argument is intended to hold the multiple-lines
-- of the axiom text as expressed in SMT-Lib notation. Note that we perform no checks on the axiom
-- itself, to see whether it's actually well-formed or is sensical by any means.
-- 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 (SBVPgm 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
usedKinds <- newIORef Set.empty
cstrs <- newIORef []
rGen <- case currentRunMode of
Concrete g -> newIORef g
_ -> newStdGen >>= newIORef
let st = State { runMode = currentRunMode
, rStdGen = rGen
, rCInfo = cInfo
, rctr = ctr
, rUsedKinds = usedKinds
, 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 (KBounded False 1) (0::Integer)) -- s(-2) == falseSW
_ <- newConst st (mkConstCW (KBounded False 1) (1::Integer)) -- s(-1) == trueSW
r <- runReaderT c st
res <- readResult st
return $ (r, res)
readResult :: State -> IO Result
readResult st = do
rpgm <- readIORef (spgm st)
inpsO <- reverse `fmap` readIORef (rinps st)
outsO <- reverse `fmap` readIORef (routs st)
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 (rtblMap st)
arrs <- IMap.toAscList `fmap` readIORef (rArrayMap st)
unint <- Map.toList `fmap` readIORef (rUIMap st)
axs <- reverse `fmap` readIORef (raxioms st)
knds <- readIORef (rUsedKinds st)
cgMap <- Map.toList `fmap` readIORef (rCgMap st)
traceVals <- reverse `fmap` readIORef (rCInfo st)
extraCstrs <- reverse `fmap` readIORef (rConstraints st)
return $ Result knds traceVals cgMap inpsO cnsts tbls arrs unint axs rpgm extraCstrs outsO
getResult :: Symbolic Result
getResult = do
st <- ask
liftIO $ readResult st
-------------------------------------------------------------------------------
-- * 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.
class (HasKind 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]
-- | Similar to free; Just a more convenient name
symbolic :: String -> Symbolic (SBV a)
-- | Similar to mkFreeVars; but automatically gives names based on the strings
symbolics :: [String] -> 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
-- | One stop allocator
mkSymWord :: Maybe Quantifier -> Maybe String -> Symbolic (SBV a)
-- minimal complete definition, Nothing.
-- Giving no instances is ok when defining an uninterpreted sort, but otherwise you really
-- want to define: mbMaxBound, mbMinBound, literal, fromCW, mkSymWord
forall = mkSymWord (Just ALL) . Just
forall_ = mkSymWord (Just ALL) Nothing
exists = mkSymWord (Just EX) . Just
exists_ = mkSymWord (Just EX) Nothing
free = mkSymWord Nothing . Just
free_ = mkSymWord Nothing Nothing
mkForallVars n = mapM (const forall_) [1 .. n]
mkExistVars n = mapM (const exists_) [1 .. n]
mkFreeVars n = mapM (const free_) [1 .. n]
symbolic = free
symbolics = mapM symbolic
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
-- Followings, you really want to define them unless the instance is for an uninterpreted sort
mbMaxBound = Nothing
mbMinBound = Nothing
literal x = error $ "Cannot create symbolic literals for kind: " ++ show (kindOf x)
fromCW cw = error $ "Cannot convert CW " ++ show cw ++ " to kind " ++ show (kindOf (undefined :: a))
default mkSymWord :: Data a => Maybe Quantifier -> Maybe String -> Symbolic (SBV a)
mkSymWord mbQ mbNm = do
let sortName = tyconUQname . dataTypeName . dataTypeOf $ (undefined :: a)
st <- ask
let k = KUninterpreted sortName
liftIO $ registerKind st k
let q = case (mbQ, runMode st) of
(Just x, _) -> x
(Nothing, Proof True) -> EX
(Nothing, Proof False) -> ALL
(Nothing, Concrete{}) -> error $ "SBV: Uninterpreted sort " ++ sortName ++ " can not be used in concrete simulation mode."
(Nothing, CodeGen) -> error $ "SBV: Uninterpreted sort " ++ sortName ++ " can not be used in code-generation mode."
ctr <- liftIO $ incCtr st
let sw = SW k (NodeId ctr)
nm = maybe ('s':show ctr) id mbNm
liftIO $ modifyIORef (rinps st) ((q, (sw, nm)):)
return $ SBV k $ Right $ cache (const (return sw))
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_ :: (HasKind a, HasKind b) => Maybe (SBV b) -> Symbolic (array a b)
-- | Create a named new array, with an optional initial value
newArray :: (HasKind a, HasKind 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 (Kind, Kind) (Cached ArrayIndex)
-- | An array index is simple an int value
type ArrayIndex = Int
instance (HasKind a, HasKind 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 (_, bk) f) a = SBV bk $ Right $ cache r
where r st = do arr <- uncacheAI f st
i <- sbvToSW st a
newExpr st bk (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
-- | Declare a new symbolic array, with a potential initial value
declNewSArray :: forall a b. (HasKind a, HasKind b) => (Int -> String) -> Maybe (SBV b) -> Symbolic (SArray a b)
declNewSArray mkNm mbInit = do
let aknd = kindOf (undefined :: a)
bknd = kindOf (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, (aknd, bknd), actx))
return $ SArray (aknd, bknd) $ 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 (HasKind a, HasKind 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:)
-- | Add a constraint with a given probability
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.
--
-- See Andy Gill's type-safe obervable sharing trick for the inspiration behind
-- this technique: <http://ittc.ku.edu/~andygill/paper.php?label=DSLExtract09>
--
-- Note that this is *not* a general memo utility!
newtype Cached a = Cached (State -> IO a)
-- | Cache a state-based computation
cache :: (State -> IO a) -> Cached a
cache = Cached
-- | Uncache a previously cached computation
uncache :: Cached SW -> State -> IO SW
uncache = uncacheGen rSWCache
-- | Uncache, retrieving array indexes
uncacheAI :: Cached ArrayIndex -> State -> IO ArrayIndex
uncacheAI = uncacheGen rAICache
-- | Generic uncaching. Note that this is entirely safe, since we do it in the IO monad.
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 Program versions, currently we only know of versions 1 and 2.
-- (NB. Eventually, we should just drop SMTLib1.)
data SMTLibVersion = SMTLib1
| SMTLib2
deriving Eq
-- | Representation of an SMT-Lib program. 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) = x `seq` y `seq` ()
instance NFData Result where
rnf (Result kindInfo qcInfo cgs inps consts tbls arrs uis axs pgm cstr outs)
= rnf kindInfo `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 Kind
instance NFData ArrayContext
instance NFData SW
instance NFData SBVExpr
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` ()
instance NFData SBVPgm
-- | Translation tricks needed for specific capabilities afforded by each solver
data SolverCapabilities = SolverCapabilities {
capSolverName :: String -- ^ Name of the solver
, mbDefaultLogic :: Maybe String -- ^ set-logic string to use in case not automatically determined (if any)
, supportsMacros :: Bool -- ^ Does the solver understand SMT-Lib2 macros?
, supportsProduceModels :: Bool -- ^ Does the solver understand produce-models option setting
, supportsQuantifiers :: Bool -- ^ Does the solver understand SMT-Lib2 style quantifiers?
, supportsUninterpretedSorts :: Bool -- ^ Does the solver understand SMT-Lib2 style uninterpreted-sorts
, supportsUnboundedInts :: Bool -- ^ Does the solver support unbounded integers?
, supportsReals :: Bool -- ^ Does the solver support reals?
, supportsFloats :: Bool -- ^ Does the solver support single-precision floating point numbers?
, supportsDoubles :: Bool -- ^ Does the solver support double-precision floating point numbers?
}
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