----------------------------------------------------------------------------- -- | -- 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 #-} {-# LANGUAGE NamedFieldPuns #-} 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(..), AlgReal(..), cwSameType, cwIsBit, cwToBool , mkConstCW ,liftCW2, mapCW, mapCW2 , SW(..), trueSW, falseSW, trueCW, falseCW, normCW , SBV(..), NodeId(..), mkSymSBV, mkSymSBVWithRandom , ArrayContext(..), ArrayInfo, SymArray(..), SFunArray(..), mkSFunArray, SArray(..), arrayUIKind , sbvToSW, sbvToSymSW, forceSWArg , SBVExpr(..), newExpr , cache, Cached, uncache, uncacheAI, HasKind(..) , Op(..), NamedSymVar, UnintKind(..), getTableIndex, SBVPgm(..), Symbolic, runSymbolic, runSymbolic', State, getPathCondition, extendPathCondition , inProofMode, SBVRunMode(..), Kind(..), Outputtable(..), Result(..) , Logic(..), SMTLibLogic(..) , getTraceInfo, getConstraints, addConstraint , SBVType(..), newUninterpreted, unintFnUIKind, addAxiom , Quantifier(..), needsExistentials , SMTLibPgm(..), SMTLibVersion(..) , SolverCapabilities(..) , extractSymbolicSimulationState, getResult , SMTScript(..), Solver(..), SMTSolver(..), SMTResult(..), SMTModel(..), SMTConfig(..), getSBranchRunConfig ) where import Control.DeepSeq (NFData(..)) import Control.Applicative (Applicative) 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.Exit (ExitCode(..)) 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 KBool -> 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 = KBool | KBounded Bool Int | KUnbounded | KReal | KUninterpreted String | KFloat | KDouble deriving (Eq, Ord) instance Show Kind where show KBool = "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) -- | Forcing an argument; this is a necessary evil to make sure all the arguments -- to an uninterpreted function and sBranch test conditions are evaluated before called; -- the semantics of uinterpreted functions is necessarily strict; deviating from Haskell's forceSWArg :: SW -> IO () forceSWArg (SW k n) = k `seq` n `seq` return () -- | 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 KBool $ NodeId (-2) -- | Constant False as a SW. Note that this value always occupies slot -1. trueSW :: SW trueSW = SW KBool $ NodeId (-1) -- | Constant False as a CW. We represent it using the integer value 0. falseCW :: CW falseCW = CW KBool (CWInteger 0) -- | Constant True as a CW. We represent it using the integer value 1. trueCW :: CW trueCW = CW KBool (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 KBool -> False KBounded b _ -> b KUnbounded -> True KReal -> True KFloat -> True KDouble -> True KUninterpreted{} -> False intSizeOf x = case kindOf x of KBool -> error "SBV.HasKind.intSizeOf((S)Bool)" 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 | KBool{} <- 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 _ = KBool 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])] -- uninterpeted 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, Maybe SMTConfig) -- ^ Symbolic simulation mode, for proof purposes. Bool is True if it's a sat instance. SMTConfig is used for 'sBranch' calls. | 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 , pathCond :: SBool , 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) } -- | Get the current path condition getPathCondition :: State -> SBool getPathCondition = pathCond -- | Extend the path condition with the given test value. extendPathCondition :: State -> (SBool -> SBool) -> State extendPathCondition st f = st{pathCond = f (pathCond st)} -- | Are we running in proof mode? inProofMode :: State -> Bool inProofMode s = case runMode s of Proof{} -> True CodeGen -> False Concrete{} -> False -- | If in proof mode, get the underlying configuration (used for 'sBranch') getSBranchRunConfig :: State -> Maybe SMTConfig getSBranchRunConfig st = case runMode st of Proof (_, s) -> s _ -> Nothing -- | 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 _)) = " :: " ++ 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 KBool a = normCW $ CW KBool (CWInteger (toInteger a)) 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 rand 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 rand 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, Maybe SMTConfig) -> 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 , pathCond = SBV KBool (Left trueCW) , 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 falseCW -- s(-2) == falseSW _ <- newConst st trueCW -- s(-1) == trueSW r <- runReaderT c st res <- extractSymbolicSimulationState st return (r, res) -- | Grab the program from a running symbolic simulation state. This is useful for internal purposes, for -- instance when implementing 'sBranch'. extractSymbolicSimulationState :: State -> IO Result extractSymbolicSimulationState st@State{ spgm=pgm, rinps=inps, routs=outs, rtblMap=tables, rArrayMap=arrays, rUIMap=uis, raxioms=axioms , rUsedKinds=usedKinds, rCgMap=cgs, rCInfo=cInfo, rConstraints = cstrs} = do SBVPgm 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 knds <- readIORef usedKinds cgMap <- Map.toList `fmap` readIORef cgs traceVals <- reverse `fmap` readIORef cInfo extraCstrs <- reverse `fmap` readIORef cstrs return $ Result knds traceVals cgMap inpsO cnsts tbls arrs unint axs (SBVPgm rpgm) extraCstrs outsO getResult :: Symbolic Result getResult = do st <- ask liftIO $ extractSymbolicSimulationState 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: -- -- * 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: -- -- 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 instance NFData SMTResult where rnf (Unsatisfiable _) = () rnf (Satisfiable _ xs) = rnf xs `seq` () rnf (Unknown _ xs) = rnf xs `seq` () rnf (ProofError _ xs) = rnf xs `seq` () rnf (TimeOut _) = () instance NFData SMTModel where rnf (SMTModel assocs unints uarrs) = rnf assocs `seq` rnf unints `seq` rnf uarrs `seq` () -- | SMT-Lib logics. If left unspecified SBV will pick the logic based on what it determines is needed. However, the -- user can override this choice using the 'useLogic' parameter to the configuration. This is especially handy if -- one is experimenting with custom logics that might be supported on new solvers. data SMTLibLogic = AUFLIA -- ^ Formulas over the theory of linear integer arithmetic and arrays extended with free sort and function symbols but restricted to arrays with integer indices and values | AUFLIRA -- ^ Linear formulas with free sort and function symbols over one- and two-dimentional arrays of integer index and real value | AUFNIRA -- ^ Formulas with free function and predicate symbols over a theory of arrays of arrays of integer index and real value | LRA -- ^ Linear formulas in linear real arithmetic | UFLRA -- ^ Linear real arithmetic with uninterpreted sort and function symbols. | UFNIA -- ^ Non-linear integer arithmetic with uninterpreted sort and function symbols. | QF_ABV -- ^ Quantifier-free formulas over the theory of bitvectors and bitvector arrays | QF_AUFBV -- ^ Quantifier-free formulas over the theory of bitvectors and bitvector arrays extended with free sort and function symbols | QF_AUFLIA -- ^ Quantifier-free linear formulas over the theory of integer arrays extended with free sort and function symbols | QF_AX -- ^ Quantifier-free formulas over the theory of arrays with extensionality | QF_BV -- ^ Quantifier-free formulas over the theory of fixed-size bitvectors | QF_IDL -- ^ Difference Logic over the integers. Boolean combinations of inequations of the form x - y < b where x and y are integer variables and b is an integer constant | QF_LIA -- ^ Unquantified linear integer arithmetic. In essence, Boolean combinations of inequations between linear polynomials over integer variables | QF_LRA -- ^ Unquantified linear real arithmetic. In essence, Boolean combinations of inequations between linear polynomials over real variables. | QF_NIA -- ^ Quantifier-free integer arithmetic. | QF_NRA -- ^ Quantifier-free real arithmetic. | QF_RDL -- ^ Difference Logic over the reals. In essence, Boolean combinations of inequations of the form x - y < b where x and y are real variables and b is a rational constant. | QF_UF -- ^ Unquantified formulas built over a signature of uninterpreted (i.e., free) sort and function symbols. | QF_UFBV -- ^ Unquantified formulas over bitvectors with uninterpreted sort function and symbols. | QF_UFIDL -- ^ Difference Logic over the integers (in essence) but with uninterpreted sort and function symbols. | QF_UFLIA -- ^ Unquantified linear integer arithmetic with uninterpreted sort and function symbols. | QF_UFLRA -- ^ Unquantified linear real arithmetic with uninterpreted sort and function symbols. | QF_UFNRA -- ^ Unquantified non-linear real arithmetic with uninterpreted sort and function symbols. | QF_FPABV -- ^ Quantifier-free formulas over the theory of floating point numbers, arrays, and bit-vectors | QF_FPA -- ^ Quantifier-free formulas over the theory of floating point numbers deriving Show -- | Chosen logic for the solver data Logic = PredefinedLogic SMTLibLogic -- ^ Use one of the logics as defined by the standard | CustomLogic String -- ^ Use this name for the logic instance Show Logic where show (PredefinedLogic l) = show l show (CustomLogic s) = s -- | 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? } -- | Solver configuration. See also 'z3', 'yices', 'cvc4', 'boolector', 'mathSAT', etc. which are instantiations of this type for those solvers, with -- reasonable defaults. In particular, custom configuration can be created by varying those values. (Such as @z3{verbose=True}@.) -- -- Most fields are self explanatory. The notion of precision for printing algebraic reals stems from the fact that such values does -- not necessarily have finite decimal representations, and hence we have to stop printing at some depth. It is important to -- emphasize that such values always have infinite precision internally. The issue is merely with how we print such an infinite -- precision value on the screen. The field 'printRealPrec' controls the printing precision, by specifying the number of digits after -- the decimal point. The default value is 16, but it can be set to any positive integer. -- -- When printing, SBV will add the suffix @...@ at the and of a real-value, if the given bound is not sufficient to represent the real-value -- exactly. Otherwise, the number will be written out in standard decimal notation. Note that SBV will always print the whole value if it -- is precise (i.e., if it fits in a finite number of digits), regardless of the precision limit. The limit only applies if the representation -- of the real value is not finite, i.e., if it is not rational. data SMTConfig = SMTConfig { verbose :: Bool -- ^ Debug mode , timing :: Bool -- ^ Print timing information on how long different phases took (construction, solving, etc.) , sBranchTimeOut :: Maybe Int -- ^ How much time to give to the solver for each call of 'sBranch' check. (In seconds. Default: No limit.) , timeOut :: Maybe Int -- ^ How much time to give to the solver. (In seconds. Default: No limit.) , printBase :: Int -- ^ Print integral literals in this base (2, 8, and 10, and 16 are supported.) , printRealPrec :: Int -- ^ Print algebraic real values with this precision. (SReal, default: 16) , solverTweaks :: [String] -- ^ Additional lines of script to give to the solver (user specified) , satCmd :: String -- ^ Usually "(check-sat)". However, users might tweak it based on solver characteristics. , smtFile :: Maybe FilePath -- ^ If Just, the generated SMT script will be put in this file (for debugging purposes mostly) , useSMTLib2 :: Bool -- ^ If True, we'll treat the solver as using SMTLib2 input format. Otherwise, SMTLib1 , solver :: SMTSolver -- ^ The actual SMT solver. , roundingMode :: RoundingMode -- ^ Rounding mode to use for floating-point conversions , useLogic :: Maybe Logic -- ^ If Nothing, pick automatically. Otherwise, either use the given one, or use the custom string. } instance Show SMTConfig where show = show . solver -- | A model, as returned by a solver data SMTModel = SMTModel { modelAssocs :: [(String, CW)] -- ^ Mapping of symbolic values to constants. , modelArrays :: [(String, [String])] -- ^ Arrays, very crude; only works with Yices. , modelUninterps :: [(String, [String])] -- ^ Uninterpreted funcs; very crude; only works with Yices. } deriving Show -- | The result of an SMT solver call. Each constructor is tagged with -- the 'SMTConfig' that created it so that further tools can inspect it -- and build layers of results, if needed. For ordinary uses of the library, -- this type should not be needed, instead use the accessor functions on -- it. (Custom Show instances and model extractors.) data SMTResult = Unsatisfiable SMTConfig -- ^ Unsatisfiable | Satisfiable SMTConfig SMTModel -- ^ Satisfiable with model | Unknown SMTConfig SMTModel -- ^ Prover returned unknown, with a potential (possibly bogus) model | ProofError SMTConfig [String] -- ^ Prover errored out | TimeOut SMTConfig -- ^ Computation timed out (see the 'timeout' combinator) -- | A script, to be passed to the solver. data SMTScript = SMTScript { scriptBody :: String -- ^ Initial feed , scriptModel :: Maybe String -- ^ Optional continuation script, if the result is sat } -- | An SMT engine type SMTEngine = SMTConfig -> Bool -> [(Quantifier, NamedSymVar)] -> [(String, UnintKind)] -> [Either SW (SW, [SW])] -> String -> IO SMTResult -- | Solvers that SBV is aware of data Solver = Z3 | Yices | Boolector | CVC4 | MathSAT deriving (Show, Enum, Bounded) -- | An SMT solver data SMTSolver = SMTSolver { name :: Solver -- ^ The solver in use , executable :: String -- ^ The path to its executable , options :: [String] -- ^ Options to provide to the solver , engine :: SMTEngine -- ^ The solver engine, responsible for interpreting solver output , xformExitCode :: ExitCode -> ExitCode -- ^ Should we re-interpret exit codes. Most solvers behave rationally, i.e., id will do. Some (like CVC4) don't. , capabilities :: SolverCapabilities -- ^ Various capabilities of the solver } instance Show SMTSolver where show = show . name