| Copyright | (c) Levent Erkok |
|---|---|
| License | BSD3 |
| Maintainer | erkokl@gmail.com |
| Stability | experimental |
| Safe Haskell | Safe-Inferred |
| Language | Haskell2010 |
Data.SBV
Contents
- Symbolic types
- Arrays of symbolic values
- Creating symbolic values
- Symbolic Equality and Comparisons
- Conditionals: Mergeable values
- Symbolic integral numbers
- Division and Modulus
- Bit-vector operations
- IEEE-floating point numbers
- Enumerations
- Uninterpreted sorts, constants, and functions
- Stopping unrolling: Defined functions
- Special relations
- Properties, proofs, and satisfiability
- Constraints and Quantifiers
- Checking safety
- Quick-checking
- Optimization
- Model extraction
- SMT Interface
- Abstract SBV type
- Module exports
Description
(The sbv library is hosted at http://github.com/LeventErkok/sbv. Comments, bug reports, and patches are always welcome.)
SBV: SMT Based Verification
Express properties about Haskell programs and automatically prove them using SMT solvers.
>>>prove $ \x -> x `shiftL` 2 .== 4 * (x :: SWord8)Q.E.D.
>>>prove $ \x -> x `shiftL` 2 .== 2 * (x :: SWord8)Falsifiable. Counter-example: s0 = 64 :: Word8
And similarly, sat finds a satisfying instance. The types involved are:
prove::Provablea => a ->IOThmResultsat::Satisfiablea => a ->IOSatResult
The classes Provable and Satisfiable come with instances for n-ary predicates, for arbitrary n.
The predicates are just regular Haskell functions over symbolic types listed below.
Functions for checking satisfiability (sat and allSat) are also
provided.
The sbv library introduces the following symbolic types:
SBool: Symbolic Booleans (bits).SWord8,SWord16,SWord32,SWord64: Symbolic Words (unsigned).SInt8,SInt16,SInt32,SInt64: Symbolic Ints (signed).SWordn: Generalized symbolic words of arbitrary bit-size.SIntn: Generalized symbolic ints of arbitrary bit-size.SInteger: Unbounded signed integers.SReal: Algebraic-real numbers.SFloat: IEEE-754 single-precision floating point values.SDouble: IEEE-754 double-precision floating point values.SRational: Rationals. (Ratio of two symbolic integers.)SFloatingPoint: Generalized IEEE-754 floating point values, with user specified exponent and mantissa widths.SChar,SString,RegExp: Characters, strings and regular expressions.SList: Symbolic lists (which can be nested)STuple,STuple2,STuple3, ..,STuple8: Symbolic tuples (upto 8-tuples, can be nested)SEither: Symbolic sumsSMaybe: Symbolic optional values.SSet: Symbolic sets.SArray: Arrays of symbolic values.- Symbolic polynomials over GF(2^n), polynomial arithmetic, and CRCs.
- Uninterpreted constants and functions over symbolic values, with user defined SMT-Lib axioms.
- Uninterpreted sorts, and proofs over such sorts, potentially with axioms.
- Ability to define SMTLib functions, generated directly from Haskell versions, including support for recursive and mutually recursive functions.
- Express quantified formulas (both universals and existentials, including alternating quantifiers), covering first-order logic.
- Model validation: SBV can validate models returned by solvers, which allows
for protection against bugs in SMT solvers and SBV itself. (See the
validateModelparameter.)
The user can construct ordinary Haskell programs using these types, which behave
very similar to their concrete counterparts. In particular these types belong to the
standard classes Num, Bits, custom versions of Eq (EqSymbolic)
and Ord (OrdSymbolic), along with several other custom classes for simplifying
programming with symbolic values. The framework takes full advantage of Haskell's type
inference to avoid many common mistakes.
Furthermore, predicates (i.e., functions that return SBool) built out of
these types can also be:
- proven correct via an external SMT solver (the
provefunction) - checked for satisfiability (the
sat,allSatfunctions) - used in synthesis (the
satfunction with existentials) - quick-checked
If a predicate is not valid, prove will return a counterexample: An
assignment to inputs such that the predicate fails. The sat function will
return a satisfying assignment, if there is one. The allSat function returns
all satisfying assignments.
The sbv library uses third-party SMT solvers via the standard SMT-Lib interface: http://smtlib.cs.uiowa.edu/
The SBV library is designed to work with any SMT-Lib compliant SMT-solver. Currently, we support the following SMT-Solvers out-of-the box:
- ABC from University of Berkeley: http://www.eecs.berkeley.edu/~alanmi/abc/
- CVC4, and CVC5 from Stanford University and the University of Iowa. https://cvc4.github.io/ and https://cvc5.github.io
- Boolector from Johannes Kepler University: https://boolector.github.io and its successor Bitwuzla from Stanford university: https://bitwuzla.github.io
- MathSAT from Fondazione Bruno Kessler and DISI-University of Trento: http://mathsat.fbk.eu/
- Yices from SRI: http://yices.csl.sri.com/
- DReal from CMU: http://dreal.github.io/
- Z3 from Microsoft: http://github.com/Z3Prover/z3/wiki
SBV requires recent versions of these solvers; please see the file
SMTSolverVersions.md in the source distribution for specifics.
SBV also allows calling these solvers in parallel, either getting results from multiple solvers
or returning the fastest one. (See proveWithAll, proveWithAny, etc.)
Support for other compliant solvers can be added relatively easily, please get in touch if there is a solver you'd like to see included.
Synopsis
- type SBool = SBV Bool
- sTrue :: SBool
- sFalse :: SBool
- sNot :: SBool -> SBool
- (.&&) :: SBool -> SBool -> SBool
- (.||) :: SBool -> SBool -> SBool
- (.<+>) :: SBool -> SBool -> SBool
- (.~&) :: SBool -> SBool -> SBool
- (.~|) :: SBool -> SBool -> SBool
- (.=>) :: SBool -> SBool -> SBool
- (.<=>) :: SBool -> SBool -> SBool
- fromBool :: Bool -> SBool
- oneIf :: (Ord a, Num a, SymVal a) => SBool -> SBV a
- sAnd :: [SBool] -> SBool
- sOr :: [SBool] -> SBool
- sAny :: (a -> SBool) -> [a] -> SBool
- sAll :: (a -> SBool) -> [a] -> SBool
- type SWord8 = SBV Word8
- type SWord16 = SBV Word16
- type SWord32 = SBV Word32
- type SWord64 = SBV Word64
- type SWord (n :: Nat) = SBV (WordN n)
- data WordN (n :: Nat)
- type SInt8 = SBV Int8
- type SInt16 = SBV Int16
- type SInt32 = SBV Int32
- type SInt64 = SBV Int64
- type SInt (n :: Nat) = SBV (IntN n)
- data IntN (n :: Nat)
- type family BVIsNonZero (arg :: Nat) :: Constraint where ...
- type family FromSized (t :: Type) :: Type where ...
- type family ToSized (t :: Type) :: Type where ...
- fromSized :: FromSizedBV a => a -> FromSized a
- toSized :: ToSizedBV a => a -> ToSized a
- type SInteger = SBV Integer
- type family ValidFloat (eb :: Nat) (sb :: Nat) :: Constraint where ...
- type SFloat = SBV Float
- type SDouble = SBV Double
- type SFloatingPoint (eb :: Nat) (sb :: Nat) = SBV (FloatingPoint eb sb)
- data FloatingPoint (eb :: Nat) (sb :: Nat)
- type SFPHalf = SBV FPHalf
- type FPHalf = FloatingPoint 5 11
- type SFPBFloat = SBV FPBFloat
- type FPBFloat = FloatingPoint 8 8
- type SFPSingle = SBV FPSingle
- type FPSingle = FloatingPoint 8 24
- type SFPDouble = SBV FPDouble
- type FPDouble = FloatingPoint 11 53
- type SFPQuad = SBV FPQuad
- type FPQuad = FloatingPoint 15 113
- type SRational = SBV Rational
- type SReal = SBV AlgReal
- data AlgReal
- sRealToSInteger :: SReal -> SInteger
- algRealToRational :: AlgReal -> RationalCV
- data RealPoint a
- = OpenPoint a
- | ClosedPoint a
- realPoint :: RealPoint a -> a
- data RationalCV
- type SChar = SBV Char
- type SString = SBV String
- type SList a = SBV [a]
- class SymTuple a
- type STuple a b = SBV (a, b)
- type STuple2 a b = SBV (a, b)
- type STuple3 a b c = SBV (a, b, c)
- type STuple4 a b c d = SBV (a, b, c, d)
- type STuple5 a b c d e = SBV (a, b, c, d, e)
- type STuple6 a b c d e f = SBV (a, b, c, d, e, f)
- type STuple7 a b c d e f g = SBV (a, b, c, d, e, f, g)
- type STuple8 a b c d e f g h = SBV (a, b, c, d, e, f, g, h)
- type SMaybe a = SBV (Maybe a)
- type SEither a b = SBV (Either a b)
- data RCSet a
- = RegularSet (Set a)
- | ComplementSet (Set a)
- type SSet a = SBV (RCSet a)
- class SymArray array where
- newArray_ :: (SymArray array, HasKind a, HasKind b) => Maybe (SBV b) -> Symbolic (array a b)
- newArray :: (SymArray array, HasKind a, HasKind b) => String -> Maybe (SBV b) -> Symbolic (array a b)
- data SArray a b
- lambdaAsArray :: forall a b. (SymVal a, HasKind b) => (SBV a -> SBV b) -> SArray a b
- sBool :: String -> Symbolic SBool
- sBool_ :: Symbolic SBool
- sWord8 :: String -> Symbolic SWord8
- sWord8_ :: Symbolic SWord8
- sWord16 :: String -> Symbolic SWord16
- sWord16_ :: Symbolic SWord16
- sWord32 :: String -> Symbolic SWord32
- sWord32_ :: Symbolic SWord32
- sWord64 :: String -> Symbolic SWord64
- sWord64_ :: Symbolic SWord64
- sWord :: (KnownNat n, BVIsNonZero n) => String -> Symbolic (SWord n)
- sWord_ :: (KnownNat n, BVIsNonZero n) => Symbolic (SWord n)
- sInt8 :: String -> Symbolic SInt8
- sInt8_ :: Symbolic SInt8
- sInt16 :: String -> Symbolic SInt16
- sInt16_ :: Symbolic SInt16
- sInt32 :: String -> Symbolic SInt32
- sInt32_ :: Symbolic SInt32
- sInt64 :: String -> Symbolic SInt64
- sInt64_ :: Symbolic SInt64
- sInt :: (KnownNat n, BVIsNonZero n) => String -> Symbolic (SInt n)
- sInt_ :: (KnownNat n, BVIsNonZero n) => Symbolic (SInt n)
- sInteger :: String -> Symbolic SInteger
- sInteger_ :: Symbolic SInteger
- sReal :: String -> Symbolic SReal
- sReal_ :: Symbolic SReal
- sRational :: String -> Symbolic SRational
- sRational_ :: Symbolic SRational
- sFloat :: String -> Symbolic SFloat
- sFloat_ :: Symbolic SFloat
- sDouble :: String -> Symbolic SDouble
- sDouble_ :: Symbolic SDouble
- sFloatingPoint :: ValidFloat eb sb => String -> Symbolic (SFloatingPoint eb sb)
- sFloatingPoint_ :: ValidFloat eb sb => Symbolic (SFloatingPoint eb sb)
- sFPHalf :: String -> Symbolic SFPHalf
- sFPHalf_ :: Symbolic SFPHalf
- sFPBFloat :: String -> Symbolic SFPBFloat
- sFPBFloat_ :: Symbolic SFPBFloat
- sFPSingle :: String -> Symbolic SFPSingle
- sFPSingle_ :: Symbolic SFPSingle
- sFPDouble :: String -> Symbolic SFPDouble
- sFPDouble_ :: Symbolic SFPDouble
- sFPQuad :: String -> Symbolic SFPQuad
- sFPQuad_ :: Symbolic SFPQuad
- sChar :: String -> Symbolic SChar
- sChar_ :: Symbolic SChar
- sString :: String -> Symbolic SString
- sString_ :: Symbolic SString
- sList :: SymVal a => String -> Symbolic (SList a)
- sList_ :: SymVal a => Symbolic (SList a)
- sTuple :: (SymTuple tup, SymVal tup) => String -> Symbolic (SBV tup)
- sTuple_ :: (SymTuple tup, SymVal tup) => Symbolic (SBV tup)
- sEither :: (SymVal a, SymVal b) => String -> Symbolic (SEither a b)
- sEither_ :: (SymVal a, SymVal b) => Symbolic (SEither a b)
- sMaybe :: SymVal a => String -> Symbolic (SMaybe a)
- sMaybe_ :: SymVal a => Symbolic (SMaybe a)
- sSet :: (Ord a, SymVal a) => String -> Symbolic (SSet a)
- sSet_ :: (Ord a, SymVal a) => Symbolic (SSet a)
- sBools :: [String] -> Symbolic [SBool]
- sWord8s :: [String] -> Symbolic [SWord8]
- sWord16s :: [String] -> Symbolic [SWord16]
- sWord32s :: [String] -> Symbolic [SWord32]
- sWord64s :: [String] -> Symbolic [SWord64]
- sWords :: (KnownNat n, BVIsNonZero n) => [String] -> Symbolic [SWord n]
- sInt8s :: [String] -> Symbolic [SInt8]
- sInt16s :: [String] -> Symbolic [SInt16]
- sInt32s :: [String] -> Symbolic [SInt32]
- sInt64s :: [String] -> Symbolic [SInt64]
- sInts :: (KnownNat n, BVIsNonZero n) => [String] -> Symbolic [SInt n]
- sIntegers :: [String] -> Symbolic [SInteger]
- sReals :: [String] -> Symbolic [SReal]
- sRationals :: [String] -> Symbolic [SRational]
- sFloats :: [String] -> Symbolic [SFloat]
- sDoubles :: [String] -> Symbolic [SDouble]
- sFloatingPoints :: ValidFloat eb sb => [String] -> Symbolic [SFloatingPoint eb sb]
- sFPHalfs :: [String] -> Symbolic [SFPHalf]
- sFPBFloats :: [String] -> Symbolic [SFPBFloat]
- sFPSingles :: [String] -> Symbolic [SFPSingle]
- sFPDoubles :: [String] -> Symbolic [SFPDouble]
- sFPQuads :: [String] -> Symbolic [SFPQuad]
- sChars :: [String] -> Symbolic [SChar]
- sStrings :: [String] -> Symbolic [SString]
- sLists :: SymVal a => [String] -> Symbolic [SList a]
- sTuples :: (SymTuple tup, SymVal tup) => [String] -> Symbolic [SBV tup]
- sEithers :: (SymVal a, SymVal b) => [String] -> Symbolic [SEither a b]
- sMaybes :: SymVal a => [String] -> Symbolic [SMaybe a]
- sSets :: (Ord a, SymVal a) => [String] -> Symbolic [SSet a]
- class EqSymbolic a where
- class (Mergeable a, EqSymbolic a) => OrdSymbolic a where
- class Equality a where
- class Mergeable a where
- ite :: Mergeable a => SBool -> a -> a -> a
- iteLazy :: Mergeable a => SBool -> a -> a -> a
- class (SymVal a, Num a, Bits a, Integral a) => SIntegral a
- class SDivisible a where
- sEDivMod :: SInteger -> SInteger -> (SInteger, SInteger)
- sEDiv :: SInteger -> SInteger -> SInteger
- sEMod :: SInteger -> SInteger -> SInteger
- sFromIntegral :: forall a b. (Integral a, HasKind a, Num a, SymVal a, HasKind b, Num b, SymVal b) => SBV a -> SBV b
- sShiftLeft :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a
- sShiftRight :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a
- sRotateLeft :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a
- sBarrelRotateLeft :: (SFiniteBits a, SFiniteBits b) => SBV a -> SBV b -> SBV a
- sRotateRight :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a
- sBarrelRotateRight :: (SFiniteBits a, SFiniteBits b) => SBV a -> SBV b -> SBV a
- sSignedShiftArithRight :: (SFiniteBits a, SIntegral b) => SBV a -> SBV b -> SBV a
- class (Ord a, SymVal a, Num a, Bits a) => SFiniteBits a where
- sFiniteBitSize :: SBV a -> Int
- lsb :: SBV a -> SBool
- msb :: SBV a -> SBool
- blastBE :: SBV a -> [SBool]
- blastLE :: SBV a -> [SBool]
- fromBitsBE :: [SBool] -> SBV a
- fromBitsLE :: [SBool] -> SBV a
- sTestBit :: SBV a -> Int -> SBool
- sExtractBits :: SBV a -> [Int] -> [SBool]
- sPopCount :: SBV a -> SWord8
- setBitTo :: SBV a -> Int -> SBool -> SBV a
- fullAdder :: SBV a -> SBV a -> (SBool, SBV a)
- fullMultiplier :: SBV a -> SBV a -> (SBV a, SBV a)
- sCountLeadingZeros :: SBV a -> SWord8
- sCountTrailingZeros :: SBV a -> SWord8
- bvExtract :: forall i j n bv proxy. (KnownNat n, BVIsNonZero n, SymVal (bv n), KnownNat i, KnownNat j, (i + 1) <= n, j <= i, BVIsNonZero ((i - j) + 1)) => proxy i -> proxy j -> SBV (bv n) -> SBV (bv ((i - j) + 1))
- (#) :: (KnownNat n, BVIsNonZero n, SymVal (bv n), KnownNat m, BVIsNonZero m, SymVal (bv m)) => SBV (bv n) -> SBV (bv m) -> SBV (bv (n + m))
- zeroExtend :: forall n m bv. (KnownNat n, BVIsNonZero n, SymVal (bv n), KnownNat m, BVIsNonZero m, SymVal (bv m), (n + 1) <= m, SIntegral (bv (m - n)), BVIsNonZero (m - n)) => SBV (bv n) -> SBV (bv m)
- signExtend :: forall n m bv. (KnownNat n, BVIsNonZero n, SymVal (bv n), KnownNat m, BVIsNonZero m, SymVal (bv m), (n + 1) <= m, SFiniteBits (bv n), SIntegral (bv (m - n)), BVIsNonZero (m - n)) => SBV (bv n) -> SBV (bv m)
- bvDrop :: forall i n m bv proxy. (KnownNat n, BVIsNonZero n, KnownNat i, (i + 1) <= n, ((i + m) - n) <= 0, BVIsNonZero (n - i)) => proxy i -> SBV (bv n) -> SBV (bv m)
- bvTake :: forall i n bv proxy. (KnownNat n, BVIsNonZero n, KnownNat i, BVIsNonZero i, i <= n) => proxy i -> SBV (bv n) -> SBV (bv i)
- class ByteConverter a where
- (.^) :: (Mergeable b, Num b, SIntegral e) => b -> SBV e -> b
- class (SymVal a, RealFloat a) => IEEEFloating a where
- fpAbs :: SBV a -> SBV a
- fpNeg :: SBV a -> SBV a
- fpAdd :: SRoundingMode -> SBV a -> SBV a -> SBV a
- fpSub :: SRoundingMode -> SBV a -> SBV a -> SBV a
- fpMul :: SRoundingMode -> SBV a -> SBV a -> SBV a
- fpDiv :: SRoundingMode -> SBV a -> SBV a -> SBV a
- fpFMA :: SRoundingMode -> SBV a -> SBV a -> SBV a -> SBV a
- fpSqrt :: SRoundingMode -> SBV a -> SBV a
- fpRem :: SBV a -> SBV a -> SBV a
- fpRoundToIntegral :: SRoundingMode -> SBV a -> SBV a
- fpMin :: SBV a -> SBV a -> SBV a
- fpMax :: SBV a -> SBV a -> SBV a
- fpIsEqualObject :: SBV a -> SBV a -> SBool
- fpIsNormal :: SBV a -> SBool
- fpIsSubnormal :: SBV a -> SBool
- fpIsZero :: SBV a -> SBool
- fpIsInfinite :: SBV a -> SBool
- fpIsNaN :: SBV a -> SBool
- fpIsNegative :: SBV a -> SBool
- fpIsPositive :: SBV a -> SBool
- fpIsNegativeZero :: SBV a -> SBool
- fpIsPositiveZero :: SBV a -> SBool
- fpIsPoint :: SBV a -> SBool
- data RoundingMode
- type SRoundingMode = SBV RoundingMode
- nan :: Floating a => a
- infinity :: Floating a => a
- sNaN :: (Floating a, SymVal a) => SBV a
- sInfinity :: (Floating a, SymVal a) => SBV a
- sRoundNearestTiesToEven :: SRoundingMode
- sRoundNearestTiesToAway :: SRoundingMode
- sRoundTowardPositive :: SRoundingMode
- sRoundTowardNegative :: SRoundingMode
- sRoundTowardZero :: SRoundingMode
- sRNE :: SRoundingMode
- sRNA :: SRoundingMode
- sRTP :: SRoundingMode
- sRTN :: SRoundingMode
- sRTZ :: SRoundingMode
- class SymVal a => IEEEFloatConvertible a where
- fromSFloat :: SRoundingMode -> SFloat -> SBV a
- toSFloat :: SRoundingMode -> SBV a -> SFloat
- fromSDouble :: SRoundingMode -> SDouble -> SBV a
- toSDouble :: SRoundingMode -> SBV a -> SDouble
- fromSFloatingPoint :: ValidFloat eb sb => SRoundingMode -> SFloatingPoint eb sb -> SBV a
- toSFloatingPoint :: ValidFloat eb sb => SRoundingMode -> SBV a -> SFloatingPoint eb sb
- sFloatAsSWord32 :: SFloat -> SWord32
- sWord32AsSFloat :: SWord32 -> SFloat
- sDoubleAsSWord64 :: SDouble -> SWord64
- sWord64AsSDouble :: SWord64 -> SDouble
- sFloatingPointAsSWord :: forall eb sb. (ValidFloat eb sb, KnownNat (eb + sb), BVIsNonZero (eb + sb)) => SFloatingPoint eb sb -> SWord (eb + sb)
- sWordAsSFloatingPoint :: forall eb sb. (KnownNat (eb + sb), BVIsNonZero (eb + sb), ValidFloat eb sb) => SWord (eb + sb) -> SFloatingPoint eb sb
- blastSFloat :: SFloat -> (SBool, [SBool], [SBool])
- blastSDouble :: SDouble -> (SBool, [SBool], [SBool])
- blastSFloatingPoint :: forall eb sb. (ValidFloat eb sb, KnownNat (eb + sb), BVIsNonZero (eb + sb)) => SFloatingPoint eb sb -> (SBool, [SBool], [SBool])
- crack :: SBV a -> String
- mkSymbolicEnumeration :: Name -> Q [Dec]
- mkUninterpretedSort :: Name -> Q [Dec]
- class SMTDefinable a where
- smtFunction :: Lambda Symbolic a => String -> a -> a
- uninterpret :: String -> a
- cgUninterpret :: String -> [String] -> a -> a
- sbvDefineValue :: String -> UIKind a -> a
- sym :: String -> a
- type Relation a = (SBV a, SBV a) -> SBool
- isPartialOrder :: SymVal a => String -> Relation a -> SBool
- isLinearOrder :: SymVal a => String -> Relation a -> SBool
- isTreeOrder :: SymVal a => String -> Relation a -> SBool
- isPiecewiseLinearOrder :: SymVal a => String -> Relation a -> SBool
- mkTransitiveClosure :: forall a. SymVal a => String -> Relation a -> Relation a
- type Predicate = Symbolic SBool
- type ConstraintSet = Symbolic ()
- type Provable = ProvableM IO
- type Satisfiable = SatisfiableM IO
- prove :: Provable a => a -> IO ThmResult
- proveWith :: Provable a => SMTConfig -> a -> IO ThmResult
- dprove :: Provable a => a -> IO ThmResult
- dproveWith :: Provable a => SMTConfig -> a -> IO ThmResult
- sat :: Satisfiable a => a -> IO SatResult
- satWith :: Satisfiable a => SMTConfig -> a -> IO SatResult
- dsat :: Satisfiable a => a -> IO SatResult
- dsatWith :: Satisfiable a => SMTConfig -> a -> IO SatResult
- allSat :: Satisfiable a => a -> IO AllSatResult
- allSatWith :: Satisfiable a => SMTConfig -> a -> IO AllSatResult
- optimize :: Satisfiable a => OptimizeStyle -> a -> IO OptimizeResult
- optimizeWith :: Satisfiable a => SMTConfig -> OptimizeStyle -> a -> IO OptimizeResult
- isVacuousProof :: Provable a => a -> IO Bool
- isVacuousProofWith :: Provable a => SMTConfig -> a -> IO Bool
- isTheorem :: Provable a => a -> IO Bool
- isTheoremWith :: Provable a => SMTConfig -> a -> IO Bool
- isSatisfiable :: Satisfiable a => a -> IO Bool
- isSatisfiableWith :: Satisfiable a => SMTConfig -> a -> IO Bool
- proveWithAny :: Provable a => [SMTConfig] -> a -> IO (Solver, NominalDiffTime, ThmResult)
- proveWithAll :: Provable a => [SMTConfig] -> a -> IO [(Solver, NominalDiffTime, ThmResult)]
- proveConcurrentWithAny :: Provable a => SMTConfig -> [Query b] -> a -> IO (Solver, NominalDiffTime, ThmResult)
- proveConcurrentWithAll :: Provable a => SMTConfig -> [Query b] -> a -> IO [(Solver, NominalDiffTime, ThmResult)]
- satWithAny :: Satisfiable a => [SMTConfig] -> a -> IO (Solver, NominalDiffTime, SatResult)
- satWithAll :: Satisfiable a => [SMTConfig] -> a -> IO [(Solver, NominalDiffTime, SatResult)]
- satConcurrentWithAny :: Satisfiable a => SMTConfig -> [Query b] -> a -> IO (Solver, NominalDiffTime, SatResult)
- satConcurrentWithAll :: Satisfiable a => SMTConfig -> [Query b] -> a -> IO [(Solver, NominalDiffTime, SatResult)]
- generateSMTBenchmarkSat :: SatisfiableM m a => a -> m String
- generateSMTBenchmarkProof :: ProvableM m a => a -> m String
- solve :: [SBool] -> Symbolic SBool
- constrain :: (SolverContext m, QuantifiedBool a) => a -> m ()
- softConstrain :: (SolverContext m, QuantifiedBool a) => a -> m ()
- class QuantifiedBool a
- quantifiedBool :: QuantifiedBool a => a -> SBool
- newtype Forall (nm :: Symbol) a = Forall (SBV a)
- newtype Exists (nm :: Symbol) a = Exists (SBV a)
- newtype ExistsUnique (nm :: Symbol) a = ExistsUnique (SBV a)
- newtype ForallN (n :: Nat) (nm :: Symbol) a = ForallN [SBV a]
- newtype ExistsN (n :: Nat) (nm :: Symbol) a = ExistsN [SBV a]
- class QNot a where
- class Skolemize a where
- skolemize :: (Constraint Symbolic (SkolemsTo a), Skolemize a) => a -> SkolemsTo a
- taggedSkolemize :: (Constraint Symbolic (SkolemsTo a), Skolemize a) => String -> a -> SkolemsTo a
- namedConstraint :: (SolverContext m, QuantifiedBool a) => String -> a -> m ()
- constrainWithAttribute :: (SolverContext m, QuantifiedBool a) => [(String, String)] -> a -> m ()
- pbAtMost :: [SBool] -> Int -> SBool
- pbAtLeast :: [SBool] -> Int -> SBool
- pbExactly :: [SBool] -> Int -> SBool
- pbLe :: [(Int, SBool)] -> Int -> SBool
- pbGe :: [(Int, SBool)] -> Int -> SBool
- pbEq :: [(Int, SBool)] -> Int -> SBool
- pbMutexed :: [SBool] -> SBool
- pbStronglyMutexed :: [SBool] -> SBool
- sAssert :: HasKind a => Maybe CallStack -> String -> SBool -> SBV a -> SBV a
- isSafe :: SafeResult -> Bool
- class ExtractIO m => SExecutable m a
- sName :: SExecutable IO a => a -> Symbolic ()
- safe :: SExecutable IO a => a -> IO [SafeResult]
- safeWith :: SExecutable IO a => SMTConfig -> a -> IO [SafeResult]
- sbvQuickCheck :: Symbolic SBool -> IO Bool
- data OptimizeStyle
- = Lexicographic
- | Independent
- | Pareto (Maybe Int)
- data Objective a
- class Metric a where
- type MetricSpace a :: Type
- toMetricSpace :: SBV a -> SBV (MetricSpace a)
- fromMetricSpace :: SBV (MetricSpace a) -> SBV a
- msMinimize :: (MonadSymbolic m, SolverContext m) => String -> SBV a -> m ()
- msMaximize :: (MonadSymbolic m, SolverContext m) => String -> SBV a -> m ()
- minimize :: Metric a => String -> SBV a -> Symbolic ()
- maximize :: Metric a => String -> SBV a -> Symbolic ()
- assertWithPenalty :: String -> SBool -> Penalty -> Symbolic ()
- data Penalty
- data ExtCV
- data GeneralizedCV
- newtype ThmResult = ThmResult SMTResult
- newtype SatResult = SatResult SMTResult
- data AllSatResult = AllSatResult {}
- newtype SafeResult = SafeResult (Maybe String, String, SMTResult)
- data OptimizeResult
- data SMTResult
- data SMTReasonUnknown
- observe :: SymVal a => String -> SBV a -> SBV a
- observeIf :: SymVal a => (a -> Bool) -> String -> SBV a -> SBV a
- sObserve :: SymVal a => String -> SBV a -> Symbolic ()
- class SatModel a where
- class Modelable a where
- modelExists :: a -> Bool
- getModelAssignment :: SatModel b => a -> Either String (Bool, b)
- getModelDictionary :: a -> Map String CV
- getModelValue :: SymVal b => String -> a -> Maybe b
- getModelUninterpretedValue :: String -> a -> Maybe String
- extractModel :: SatModel b => a -> Maybe b
- getModelObjectives :: a -> Map String GeneralizedCV
- getModelObjectiveValue :: String -> a -> Maybe GeneralizedCV
- getModelUIFuns :: a -> Map String (SBVType, Either String ([([CV], CV)], CV))
- getModelUIFunValue :: String -> a -> Maybe (SBVType, Either String ([([CV], CV)], CV))
- displayModels :: SatModel a => ([(Bool, a)] -> [(Bool, a)]) -> (Int -> (Bool, a) -> IO ()) -> AllSatResult -> IO Int
- extractModels :: SatModel a => AllSatResult -> [a]
- getModelDictionaries :: AllSatResult -> [Map String CV]
- getModelValues :: SymVal b => String -> AllSatResult -> [Maybe b]
- getModelUninterpretedValues :: String -> AllSatResult -> [Maybe String]
- data SMTConfig = SMTConfig {
- verbose :: Bool
- timing :: Timing
- printBase :: Int
- printRealPrec :: Int
- crackNum :: Bool
- satCmd :: String
- allSatMaxModelCount :: Maybe Int
- allSatPrintAlong :: Bool
- allSatTrackUFs :: Bool
- isNonModelVar :: String -> Bool
- validateModel :: Bool
- optimizeValidateConstraints :: Bool
- transcript :: Maybe FilePath
- smtLibVersion :: SMTLibVersion
- dsatPrecision :: Maybe Double
- solver :: SMTSolver
- extraArgs :: [String]
- roundingMode :: RoundingMode
- solverSetOptions :: [SMTOption]
- ignoreExitCode :: Bool
- redirectVerbose :: Maybe FilePath
- data Timing
- data SMTLibVersion = SMTLib2
- data Solver
- data SMTSolver = SMTSolver {
- name :: Solver
- executable :: String
- preprocess :: String -> String
- options :: SMTConfig -> [String]
- engine :: SMTEngine
- capabilities :: SolverCapabilities
- boolector :: SMTConfig
- bitwuzla :: SMTConfig
- cvc4 :: SMTConfig
- cvc5 :: SMTConfig
- yices :: SMTConfig
- dReal :: SMTConfig
- z3 :: SMTConfig
- mathSAT :: SMTConfig
- abc :: SMTConfig
- defaultSolverConfig :: Solver -> SMTConfig
- defaultSMTCfg :: SMTConfig
- defaultDeltaSMTCfg :: SMTConfig
- sbvCheckSolverInstallation :: SMTConfig -> IO Bool
- getAvailableSolvers :: IO [SMTConfig]
- setLogic :: SolverContext m => Logic -> m ()
- data Logic
- setOption :: SolverContext m => SMTOption -> m ()
- setInfo :: SolverContext m => String -> [String] -> m ()
- setTimeOut :: SolverContext m => Integer -> m ()
- data SBVException = SBVException {
- sbvExceptionDescription :: String
- sbvExceptionSent :: Maybe String
- sbvExceptionExpected :: Maybe String
- sbvExceptionReceived :: Maybe String
- sbvExceptionStdOut :: Maybe String
- sbvExceptionStdErr :: Maybe String
- sbvExceptionExitCode :: Maybe ExitCode
- sbvExceptionConfig :: SMTConfig
- sbvExceptionReason :: Maybe [String]
- sbvExceptionHint :: Maybe [String]
- data SBV a
- 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
- isRational :: a -> Bool
- isFP :: a -> Bool
- isUnbounded :: a -> Bool
- isUserSort :: a -> Bool
- isChar :: a -> Bool
- isString :: a -> Bool
- isList :: a -> Bool
- isSet :: a -> Bool
- isTuple :: a -> Bool
- isMaybe :: a -> Bool
- isEither :: a -> Bool
- showType :: a -> String
- data Kind
- class (HasKind a, Typeable a) => SymVal a
- free :: SymVal a => String -> Symbolic (SBV a)
- free_ :: SymVal a => Symbolic (SBV a)
- mkFreeVars :: SymVal a => Int -> Symbolic [SBV a]
- symbolic :: SymVal a => String -> Symbolic (SBV a)
- symbolics :: SymVal a => [String] -> Symbolic [SBV a]
- literal :: SymVal a => a -> SBV a
- unliteral :: SymVal a => SBV a -> Maybe a
- fromCV :: SymVal a => CV -> a
- isConcrete :: SymVal a => SBV a -> Bool
- isSymbolic :: SymVal a => SBV a -> Bool
- isConcretely :: SymVal a => SBV a -> (a -> Bool) -> Bool
- mkSymVal :: SymVal a => VarContext -> Maybe String -> Symbolic (SBV a)
- class MonadIO m => MonadSymbolic m where
- symbolicEnv :: m State
- type Symbolic = SymbolicT IO
- data SymbolicT m a
- label :: SymVal a => String -> SBV a -> SBV a
- output :: Outputtable a => a -> Symbolic a
- runSMT :: Symbolic a -> IO a
- runSMTWith :: SMTConfig -> Symbolic a -> IO a
- module Data.Bits
- module Data.Word
- module Data.Int
- module Data.Ratio
Documentation
The SBV library is really two things:
- A framework for writing symbolic programs in Haskell, i.e., programs operating on symbolic values along with the usual concrete counterparts.
- A framework for proving properties of such programs using SMT solvers.
The programming goal of SBV is to provide a seamless experience, i.e., let people program
in the usual Haskell style without distractions of symbolic coding. While Haskell helps
in some aspects (the Num and Bits classes simplify coding), it makes life harder
in others. For instance, if-then-else only takes Bool as a test in Haskell, and
comparisons (> etc.) only return Bools. Clearly we would like these values to be
symbolic (i.e., SBool), thus stopping us from using some native Haskell constructs.
When symbolic versions of operators are needed, they are typically obtained by prepending a dot,
for instance == becomes .==. Care has been taken to make the transition painless. In
particular, any Haskell program you build out of symbolic components is fully concretely
executable within Haskell, without the need for any custom interpreters. (They are truly
Haskell programs, not AST's built out of pieces of syntax.) This provides for an integrated
feel of the system, one of the original design goals for SBV.
Incremental query mode: SBV provides a wide variety of ways to utilize SMT-solvers, without requiring the user to deal with the solvers themselves. While this mode is convenient, advanced users might need access to the underlying solver at a lower level. For such use cases, SBV allows users to have an interactive session: The user can issue commands to the solver, inspect the values/results, and formulate new constraints. This advanced feature is available through the Data.SBV.Control module, where most SMTLib features are made available via a typed-API.
Symbolic types
Booleans
Boolean values and functions
oneIf :: (Ord a, Num a, SymVal a) => SBool -> SBV a Source #
Returns 1 if the boolean is sTrue, otherwise 0.
Logical aggregations
Bit-vectors
Unsigned bit-vectors
data WordN (n :: Nat) Source #
An unsigned bit-vector carrying its size info
Instances
Signed bit-vectors
A signed bit-vector carrying its size info
Instances
Converting between fixed-size and arbitrary bitvectors
type family BVIsNonZero (arg :: Nat) :: Constraint where ... Source #
Type family to create the appropriate non-zero constraint
Equations
| BVIsNonZero 0 = TypeError BVZeroWidth | |
| BVIsNonZero _ = () |
type family FromSized (t :: Type) :: Type where ... Source #
Capture the correspondence between sized and fixed-sized BVs
Equations
| FromSized (WordN 8) = Word8 | |
| FromSized (WordN 16) = Word16 | |
| FromSized (WordN 32) = Word32 | |
| FromSized (WordN 64) = Word64 | |
| FromSized (IntN 8) = Int8 | |
| FromSized (IntN 16) = Int16 | |
| FromSized (IntN 32) = Int32 | |
| FromSized (IntN 64) = Int64 | |
| FromSized (SWord 8) = SWord8 | |
| FromSized (SWord 16) = SWord16 | |
| FromSized (SWord 32) = SWord32 | |
| FromSized (SWord 64) = SWord64 | |
| FromSized (SInt 8) = SInt8 | |
| FromSized (SInt 16) = SInt16 | |
| FromSized (SInt 32) = SInt32 | |
| FromSized (SInt 64) = SInt64 |
type family ToSized (t :: Type) :: Type where ... Source #
Capture the correspondence between fixed-sized and sized BVs
Equations
| ToSized Word8 = WordN 8 | |
| ToSized Word16 = WordN 16 | |
| ToSized Word32 = WordN 32 | |
| ToSized Word64 = WordN 64 | |
| ToSized Int8 = IntN 8 | |
| ToSized Int16 = IntN 16 | |
| ToSized Int32 = IntN 32 | |
| ToSized Int64 = IntN 64 | |
| ToSized SWord8 = SWord 8 | |
| ToSized SWord16 = SWord 16 | |
| ToSized SWord32 = SWord 32 | |
| ToSized SWord64 = SWord 64 | |
| ToSized SInt8 = SInt 8 | |
| ToSized SInt16 = SInt 16 | |
| ToSized SInt32 = SInt 32 | |
| ToSized SInt64 = SInt 64 |
Unbounded integers
The SBV library supports unbounded signed integers with the type SInteger, which are not subject to
overflow/underflow as it is the case with the bounded types, such as SWord8, SInt16, etc. However,
some bit-vector based operations are not supported for the SInteger type while in the verification mode. That
is, you can use these operations on SInteger values during normal programming/simulation.
but the SMT translation will not support these operations since there corresponding operations are not supported in SMT-Lib.
Note that this should rarely be a problem in practice, as these operations are mostly meaningful on fixed-size
bit-vectors. The operations that are restricted to bounded word/int sizes are:
- Rotations and shifts:
rotateL,rotateR,shiftL,shiftR - Bitwise logical ops:
.&.,.|.,xor,complement - Extraction and concatenation:
bvExtract,#,zeroExtend,signExtend,bvDrop, andbvTake
Usual arithmetic (+, -, *, sQuotRem, sQuot, sRem, sDivMod, sDiv, sMod) and logical operations (.<, .<=, .>, .>=, .==, ./=) operations are
supported for SInteger fully, both in programming and verification modes.
Floating point numbers
Floating point numbers are defined by the IEEE-754 standard; and correspond to Haskell's
Float and Double types. For SMT support with floating-point numbers, see the paper
by Rummer and Wahl: http://www.philipp.ruemmer.org/publications/smt-fpa.pdf.
type family ValidFloat (eb :: Nat) (sb :: Nat) :: Constraint where ... Source #
A valid float has restrictions on eb/sb values. NB. In the below encoding, I found that CPP is very finicky about substitution of the machine-dependent macros. If you try to put the conditionals in the same line, it fails to substitute for some reason. Hence the awkward spacing. Filed this as a bug report for CPPHS at https://github.com/malcolmwallace/cpphs/issues/25.
Equations
| ValidFloat (eb :: Nat) (sb :: Nat) = (KnownNat eb, KnownNat sb, If ((((eb `CmpNat` 2) == 'EQ) || ((eb `CmpNat` 2) == 'GT)) && ((((eb `CmpNat` 61) == 'EQ) || ((eb `CmpNat` 61) == 'LT)) && ((((sb `CmpNat` 2) == 'EQ) || ((sb `CmpNat` 2) == 'GT)) && (((sb `CmpNat` 4611686018427387902) == 'EQ) || ((sb `CmpNat` 4611686018427387902) == 'LT))))) (() :: Constraint) (TypeError (InvalidFloat eb sb))) |
type SFloatingPoint (eb :: Nat) (sb :: Nat) = SBV (FloatingPoint eb sb) Source #
A symbolic arbitrary precision floating point value
data FloatingPoint (eb :: Nat) (sb :: Nat) Source #
A floating point value, indexed by its exponent and significand sizes.
An IEEE SP is FloatingPoint 8 24
DP is FloatingPoint 11 53
etc.
Instances
type FPHalf = FloatingPoint 5 11 Source #
Abbreviation for IEEE half precision float, bit width 16 = 5 + 11.
type FPBFloat = FloatingPoint 8 8 Source #
Abbreviation for brain-float precision float, bit width 16 = 8 + 8.
type FPSingle = FloatingPoint 8 24 Source #
Abbreviation for IEEE single precision float, bit width 32 = 8 + 24.
type FPDouble = FloatingPoint 11 53 Source #
Abbreviation for IEEE double precision float, bit width 64 = 11 + 53.
type FPQuad = FloatingPoint 15 113 Source #
Abbreviation for IEEE quadruble precision float, bit width 128 = 15 + 113.
Rationals
Algebraic reals
Algebraic reals are roots of single-variable polynomials with rational coefficients. (See
http://en.wikipedia.org/wiki/Algebraic_number.) Note that algebraic reals are infinite
precision numbers, but they do not cover all real numbers. (In particular, they cannot
represent transcendentals.) Some irrational numbers are algebraic (such as sqrt 2), while
others are not (such as pi and e).
SBV can deal with real numbers just fine, since the theory of reals is decidable. (See http://smtlib.cs.uiowa.edu/theories-Reals.shtml.) In addition, by leveraging backend solver capabilities, SBV can also represent and solve non-linear equations involving real-variables. (For instance, the Z3 SMT solver, supports polynomial constraints on reals starting with v4.0.)
Algebraic reals. Note that the representation is left abstract. We represent rational results explicitly, while the roots-of-polynomials are represented implicitly by their defining equation
Constructors
| AlgRational Bool Rational | bool says it's exact (i.e., SMT-solver did not return it with ? at the end.) |
| AlgPolyRoot (Integer, AlgRealPoly) (Maybe String) | which root of this polynomial and an approximate decimal representation with given precision, if available |
| AlgInterval (RealPoint Rational) (RealPoint Rational) | interval, with low and high bounds |
Instances
sRealToSInteger :: SReal -> SInteger Source #
Convert an SReal to an SInteger. That is, it computes the
largest integer n that satisfies sIntegerToSReal n <= r
essentially giving us the floor.
For instance, 1.3 will be 1, but -1.3 will be -2.
Is the endpoint included in the interval?
Constructors
| OpenPoint a | open: i.e., doesn't include the point |
| ClosedPoint a | closed: i.e., includes the point |
Instances
| Show a => Show (RealPoint a) Source # | |
| Eq a => Eq (RealPoint a) Source # | |
| Ord a => Ord (RealPoint a) Source # | |
Defined in Data.SBV.Core.AlgReals | |
data RationalCV Source #
Conversion from internal rationals to Haskell values
Constructors
| RatIrreducible AlgReal | Root of a polynomial, cannot be reduced |
| RatExact Rational | An exact rational |
| RatApprox Rational | An approximated value |
| RatInterval (RealPoint Rational) (RealPoint Rational) | Interval. Can be open/closed on both ends. |
Instances
| Show RationalCV Source # | |
Defined in Data.SBV.Core.AlgReals Methods showsPrec :: Int -> RationalCV -> ShowS # show :: RationalCV -> String # showList :: [RationalCV] -> ShowS # | |
Characters, Strings and Regular Expressions
Support for characters, strings, and regular expressions (initial version contributed by Joel Burget) adds support for QF_S logic, described here: http://smtlib.cs.uiowa.edu/theories-UnicodeStrings.shtml
See Data.SBV.Char, Data.SBV.String, Data.SBV.RegExp for related functions.
type SChar = SBV Char Source #
A symbolic character. Note that this is the full unicode character set. see: http://smtlib.cs.uiowa.edu/theories-UnicodeStrings.shtml for details.
type SString = SBV String Source #
A symbolic string. Note that a symbolic string is not a list of symbolic characters,
that is, it is not the case that SString = [SChar], unlike what one might expect following
Haskell strings. An SString is a symbolic value of its own, of possibly arbitrary but finite length,
and internally processed as one unit as opposed to a fixed-length list of characters.
Symbolic lists
Support for symbolic lists (initial version contributed by Joel Burget) adds support for sequence support.
See Data.SBV.List for related functions.
type SList a = SBV [a] Source #
A symbolic list of items. Note that a symbolic list is not a list of symbolic items,
that is, it is not the case that SList a = [a], unlike what one might expect following
haskell lists/sequences. An SList is a symbolic value of its own, of possibly arbitrary but finite
length, and internally processed as one unit as opposed to a fixed-length list of items.
Note that lists can be nested, i.e., we do allow lists of lists of ... items.
Tuples
Tuples can be used as symbolic values. This is useful in combination with lists, for example SBV [(Integer, String)] is a valid type. These types can be arbitrarily nested, eg SBV [(Integer, [(Char, (Integer, String))])]. Instances of upto 8-tuples are provided.
Identify tuple like things. Note that there are no methods, just instances to control type inference
Instances
| SymTuple () Source # | |
Defined in Data.SBV.Core.Model | |
| SymTuple (a, b) Source # | |
Defined in Data.SBV.Core.Model | |
| SymTuple (a, b, c) Source # | |
Defined in Data.SBV.Core.Model | |
| SymTuple (a, b, c, d) Source # | |
Defined in Data.SBV.Core.Model | |
| SymTuple (a, b, c, d, e) Source # | |
Defined in Data.SBV.Core.Model | |
| SymTuple (a, b, c, d, e, f) Source # | |
Defined in Data.SBV.Core.Model | |
| SymTuple (a, b, c, d, e, f, g) Source # | |
Defined in Data.SBV.Core.Model | |
| SymTuple (a, b, c, d, e, f, g, h) Source # | |
Defined in Data.SBV.Core.Model | |
Sum types
Sets
A RCSet is either a regular set or a set given by its complement from the corresponding universal set.
Constructors
| RegularSet (Set a) | |
| ComplementSet (Set a) |
Instances
Arrays of symbolic values
class SymArray array where Source #
Arrays of symbolic values
An array a b is an array indexed by the type SBV aSBV b
If a default value is supplied, then all the array elements will be initialized to this value. Otherwise, they will be left unspecified, i.e., a read from an unwritten location will produce an uninterpreted constant.
The reason for this class is rather historic. In the past, SBV provided two different kinds of
arrays: an SArray abstraction that mapped directly to SMTLib arrays (which is still available
today), and a functional notion of arrays that used internal caching, called SFunArray. The latter
has been removed as the code turned out to be rather tricky and hard to maintain; so we only
have one instance of this class. But end users can add their own instances, if needed.
NB. sListArray insists on a concrete initializer, because not having one would break
referential transparency. See https://github.com/LeventErkok/sbv/issues/553 for details.
Methods
sListArray :: (HasKind a, SymVal b) => b -> [(SBV a, SBV b)] -> array a b Source #
Create a literal array
readArray :: array a b -> SBV a -> SBV b Source #
Read the array element at a
writeArray :: SymVal b => array a b -> SBV a -> SBV b -> array a b Source #
Update the element at a to be b
mergeArrays :: SymVal b => SBV Bool -> array a b -> array a b -> array a b Source #
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
Instances
| SymArray SArray Source # | |
Defined in Data.SBV.Core.Data Methods newArray_ :: (MonadSymbolic m, HasKind a, HasKind b) => Maybe (SBV b) -> m (SArray a b) Source # newArray :: (MonadSymbolic m, HasKind a, HasKind b) => String -> Maybe (SBV b) -> m (SArray a b) Source # sListArray :: (HasKind a, SymVal b) => b -> [(SBV a, SBV b)] -> SArray a b Source # readArray :: SArray a b -> SBV a -> SBV b Source # writeArray :: SymVal b => SArray a b -> SBV a -> SBV b -> SArray a b Source # mergeArrays :: SymVal b => SBV Bool -> SArray a b -> SArray a b -> SArray a b Source # newArrayInState :: (HasKind a, HasKind b) => Maybe String -> Either (Maybe (SBV b)) String -> State -> IO (SArray a b) Source # | |
newArray_ :: (SymArray array, HasKind a, HasKind b) => Maybe (SBV b) -> Symbolic (array a b) Source #
Create a new anonymous array, possibly with a default initial value.
NB. For a version which generalizes over the underlying monad, see newArray_
newArray :: (SymArray array, HasKind a, HasKind b) => String -> Maybe (SBV b) -> Symbolic (array a b) Source #
Create a named new array, possibly with a default initial value.
NB. For a version which generalizes over the underlying monad, see newArray
Arrays implemented in terms of SMT-arrays: http://smtlib.cs.uiowa.edu/theories-ArraysEx.shtml
- Maps directly to SMT-lib arrays
- Reading from an uninitialized value is OK. If the default value is given in
newArray, it will be the result. Otherwise, the read yields an uninterpreted constant. - Can check for equality of these arrays
- Cannot be used in code-generation (i.e., compilation to C)
- Cannot quick-check theorems using
SArrayvalues
Instances
lambdaAsArray :: forall a b. (SymVal a, HasKind b) => (SBV a -> SBV b) -> SArray a b Source #
Using a lambda as an array
Creating symbolic values
Single value
These functions simplify declaring symbolic variables of various types. Strictly speaking, they are just synonyms
for free (specialized at the given type), but they might be easier to use. We provide both the named and anonymous
versions, latter with the underscore suffix.
sRational_ :: Symbolic SRational Source #
Declare an unnamed SRational.
NB. For a version which generalizes over the underlying monad, see sRational_
sFloatingPoint :: ValidFloat eb sb => String -> Symbolic (SFloatingPoint eb sb) Source #
Declare a named 'SFloatingPoint eb sb'
NB. For a version which generalizes over the underlying monad, see sFloatingPoint
sFloatingPoint_ :: ValidFloat eb sb => Symbolic (SFloatingPoint eb sb) Source #
Declare an unnamed SFloatingPoint eb sb
NB. For a version which generalizes over the underlying monad, see sFloatingPoint_
sFPSingle_ :: Symbolic SFPSingle Source #
Declare an unnamed SFPSingle
NB. For a version which generalizes over the underlying monad, see sFPSingle_
sFPDouble_ :: Symbolic SFPDouble Source #
Declare an unnamed SFPDouble
NB. For a version which generalizes over the underlying monad, see sFPDouble_
sTuple :: (SymTuple tup, SymVal tup) => String -> Symbolic (SBV tup) Source #
Declare a named tuple.
NB. For a version which generalizes over the underlying monad, see sTuple
sTuple_ :: (SymTuple tup, SymVal tup) => Symbolic (SBV tup) Source #
Declare an unnamed tuple.
NB. For a version which generalizes over the underlying monad, see sTuple_
List of values
These functions simplify declaring a sequence symbolic variables of various types. Strictly speaking, they are just synonyms
for mapM free (specialized at the given type), but they might be easier to use.
sRationals :: [String] -> Symbolic [SRational] Source #
Declare a list of SRational values.
NB. For a version which generalizes over the underlying monad, see sRationals
sFloatingPoints :: ValidFloat eb sb => [String] -> Symbolic [SFloatingPoint eb sb] Source #
Declare a list of SFloatingPoint eb sb's
NB. For a version which generalizes over the underlying monad, see sFloatingPoints
sFPBFloats :: [String] -> Symbolic [SFPBFloat] Source #
Declare a list of SFPQuads
NB. For a version which generalizes over the underlying monad, see sFPBFloats
sFPSingles :: [String] -> Symbolic [SFPSingle] Source #
Declare a list of SFPSingles
NB. For a version which generalizes over the underlying monad, see sFPSingles
sFPDoubles :: [String] -> Symbolic [SFPDouble] Source #
Declare a list of SFPDoubles
NB. For a version which generalizes over the underlying monad, see sFPDoubles
sTuples :: (SymTuple tup, SymVal tup) => [String] -> Symbolic [SBV tup] Source #
Declare a list of tuples.
NB. For a version which generalizes over the underlying monad, see sTuples
Symbolic Equality and Comparisons
Symbolic equality provides the notion of what it means to be equal, similar to Haskell's Eq class, except allowing comparison of
symbolic values. The methods are .== and ./=, returning SBool results. We also provide a notion of strong equality (.=== and ./==),
which is useful for floating-point value comparisons as it deals more uniformly with NaN and positive/negative zeros. Additionally, we
provide distinct that can be used to assert all elements of a list are different from each other, and distinctExcept which is similar
to distinct but allows for certain values to be considered different. These latter two functions are useful in modeling a variety of
puzzles and cardinality constraints:
>>>prove $ \a -> distinctExcept [a, a] [0::SInteger] .<=> a .== 0Q.E.D.>>>prove $ \a b -> distinctExcept [a, b] [0::SWord8] .<=> (a .== b .=> a .== 0)Q.E.D.>>>prove $ \a b c d -> distinctExcept [a, b, c, d] [] .== distinct [a, b, c, (d::SInteger)]Q.E.D.
class EqSymbolic a where Source #
Symbolic Equality. Note that we can't use Haskell's Eq class since Haskell insists on returning Bool
Comparing symbolic values will necessarily return a symbolic value.
Minimal complete definition: None, if the type is instance of Generic. Otherwise (.==).
Minimal complete definition
Nothing
Methods
(.==) :: a -> a -> SBool infix 4 Source #
Symbolic equality.
(./=) :: a -> a -> SBool infix 4 Source #
Symbolic inequality.
(.===) :: a -> a -> SBool infix 4 Source #
Strong equality. On floats (SFloat/SDouble), strong equality is object equality; that
is NaN == NaN holds, but +0 == -0 doesn't. On other types, (.===) is simply (.==).
Note that (.==) is the right notion of equality for floats per IEEE754 specs, since by
definition +0 == -0 and NaN equals no other value including itself. But occasionally
we want to be stronger and state NaN equals NaN and +0 and -0 are different from
each other. In a context where your type is concrete, simply use fpIsEqualObject. But in
a polymorphic context, use the strong equality instead.
NB. If you do not care about or work with floats, simply use (.==) and (./=).
(./==) :: a -> a -> SBool infix 4 Source #
Negation of strong equality. Equaivalent to negation of (.===) on all types.
distinct :: [a] -> SBool Source #
Returns (symbolic) sTrue if all the elements of the given list are different.
distinctExcept :: [a] -> [a] -> SBool Source #
Returns (symbolic) sTrue if all the elements of the given list are different. The second
list contains exceptions, i.e., if an element belongs to that set, it will be considered
distinct regardless of repetition.
allEqual :: [a] -> SBool Source #
Returns (symbolic) sTrue if all the elements of the given list are the same.
sElem :: a -> [a] -> SBool Source #
Symbolic membership test.
sNotElem :: a -> [a] -> SBool Source #
Symbolic negated membership test.
Instances
| EqSymbolic RegExp Source # | Regular expressions can be compared for equality. Note that we diverge here from the equality in the concrete sense; i.e., the Eq instance does not match the symbolic case. This is a bit unfortunate, but unavoidable with the current design of how we "distinguish" operators. Hopefully shouldn't be a big deal, though one should be careful. |
Defined in Data.SBV.Core.Model Methods (.==) :: RegExp -> RegExp -> SBool Source # (./=) :: RegExp -> RegExp -> SBool Source # (.===) :: RegExp -> RegExp -> SBool Source # (./==) :: RegExp -> RegExp -> SBool Source # distinct :: [RegExp] -> SBool Source # distinctExcept :: [RegExp] -> [RegExp] -> SBool Source # allEqual :: [RegExp] -> SBool Source # | |
| EqSymbolic Bool Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: Bool -> Bool -> SBool Source # (./=) :: Bool -> Bool -> SBool Source # (.===) :: Bool -> Bool -> SBool Source # (./==) :: Bool -> Bool -> SBool Source # distinct :: [Bool] -> SBool Source # distinctExcept :: [Bool] -> [Bool] -> SBool Source # allEqual :: [Bool] -> SBool Source # | |
| EqSymbolic (SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: SBV a -> SBV a -> SBool Source # (./=) :: SBV a -> SBV a -> SBool Source # (.===) :: SBV a -> SBV a -> SBool Source # (./==) :: SBV a -> SBV a -> SBool Source # distinct :: [SBV a] -> SBool Source # distinctExcept :: [SBV a] -> [SBV a] -> SBool Source # allEqual :: [SBV a] -> SBool Source # | |
| EqSymbolic a => EqSymbolic (S a) Source # | Symbolic equality for |
Defined in Documentation.SBV.Examples.ProofTools.BMC Methods (.==) :: S a -> S a -> SBool Source # (./=) :: S a -> S a -> SBool Source # (.===) :: S a -> S a -> SBool Source # (./==) :: S a -> S a -> SBool Source # distinct :: [S a] -> SBool Source # distinctExcept :: [S a] -> [S a] -> SBool Source # allEqual :: [S a] -> SBool Source # | |
| EqSymbolic a => EqSymbolic (Maybe a) Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: Maybe a -> Maybe a -> SBool Source # (./=) :: Maybe a -> Maybe a -> SBool Source # (.===) :: Maybe a -> Maybe a -> SBool Source # (./==) :: Maybe a -> Maybe a -> SBool Source # distinct :: [Maybe a] -> SBool Source # distinctExcept :: [Maybe a] -> [Maybe a] -> SBool Source # allEqual :: [Maybe a] -> SBool Source # | |
| EqSymbolic a => EqSymbolic [a] Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: [a] -> [a] -> SBool Source # (./=) :: [a] -> [a] -> SBool Source # (.===) :: [a] -> [a] -> SBool Source # (./==) :: [a] -> [a] -> SBool Source # distinct :: [[a]] -> SBool Source # distinctExcept :: [[a]] -> [[a]] -> SBool Source # allEqual :: [[a]] -> SBool Source # | |
| (EqSymbolic a, EqSymbolic b) => EqSymbolic (Either a b) Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: Either a b -> Either a b -> SBool Source # (./=) :: Either a b -> Either a b -> SBool Source # (.===) :: Either a b -> Either a b -> SBool Source # (./==) :: Either a b -> Either a b -> SBool Source # distinct :: [Either a b] -> SBool Source # distinctExcept :: [Either a b] -> [Either a b] -> SBool Source # allEqual :: [Either a b] -> SBool Source # | |
| EqSymbolic (SArray a b) Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: SArray a b -> SArray a b -> SBool Source # (./=) :: SArray a b -> SArray a b -> SBool Source # (.===) :: SArray a b -> SArray a b -> SBool Source # (./==) :: SArray a b -> SArray a b -> SBool Source # distinct :: [SArray a b] -> SBool Source # distinctExcept :: [SArray a b] -> [SArray a b] -> SBool Source # allEqual :: [SArray a b] -> SBool Source # | |
| (EqSymbolic a, EqSymbolic b) => EqSymbolic (a, b) Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: (a, b) -> (a, b) -> SBool Source # (./=) :: (a, b) -> (a, b) -> SBool Source # (.===) :: (a, b) -> (a, b) -> SBool Source # (./==) :: (a, b) -> (a, b) -> SBool Source # distinct :: [(a, b)] -> SBool Source # distinctExcept :: [(a, b)] -> [(a, b)] -> SBool Source # allEqual :: [(a, b)] -> SBool Source # | |
| (EqSymbolic a, EqSymbolic b, EqSymbolic c) => EqSymbolic (a, b, c) Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: (a, b, c) -> (a, b, c) -> SBool Source # (./=) :: (a, b, c) -> (a, b, c) -> SBool Source # (.===) :: (a, b, c) -> (a, b, c) -> SBool Source # (./==) :: (a, b, c) -> (a, b, c) -> SBool Source # distinct :: [(a, b, c)] -> SBool Source # distinctExcept :: [(a, b, c)] -> [(a, b, c)] -> SBool Source # allEqual :: [(a, b, c)] -> SBool Source # | |
| (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d) => EqSymbolic (a, b, c, d) Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source # (./=) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source # (.===) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source # (./==) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source # distinct :: [(a, b, c, d)] -> SBool Source # distinctExcept :: [(a, b, c, d)] -> [(a, b, c, d)] -> SBool Source # allEqual :: [(a, b, c, d)] -> SBool Source # sElem :: (a, b, c, d) -> [(a, b, c, d)] -> SBool Source # sNotElem :: (a, b, c, d) -> [(a, b, c, d)] -> SBool Source # | |
| (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d, EqSymbolic e) => EqSymbolic (a, b, c, d, e) Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source # (./=) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source # (.===) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source # (./==) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source # distinct :: [(a, b, c, d, e)] -> SBool Source # distinctExcept :: [(a, b, c, d, e)] -> [(a, b, c, d, e)] -> SBool Source # allEqual :: [(a, b, c, d, e)] -> SBool Source # sElem :: (a, b, c, d, e) -> [(a, b, c, d, e)] -> SBool Source # sNotElem :: (a, b, c, d, e) -> [(a, b, c, d, e)] -> SBool Source # | |
| (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d, EqSymbolic e, EqSymbolic f) => EqSymbolic (a, b, c, d, e, f) Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source # (./=) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source # (.===) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source # (./==) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source # distinct :: [(a, b, c, d, e, f)] -> SBool Source # distinctExcept :: [(a, b, c, d, e, f)] -> [(a, b, c, d, e, f)] -> SBool Source # allEqual :: [(a, b, c, d, e, f)] -> SBool Source # sElem :: (a, b, c, d, e, f) -> [(a, b, c, d, e, f)] -> SBool Source # sNotElem :: (a, b, c, d, e, f) -> [(a, b, c, d, e, f)] -> SBool Source # | |
| (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d, EqSymbolic e, EqSymbolic f, EqSymbolic g) => EqSymbolic (a, b, c, d, e, f, g) Source # | |
Defined in Data.SBV.Core.Model Methods (.==) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source # (./=) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source # (.===) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source # (./==) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source # distinct :: [(a, b, c, d, e, f, g)] -> SBool Source # distinctExcept :: [(a, b, c, d, e, f, g)] -> [(a, b, c, d, e, f, g)] -> SBool Source # allEqual :: [(a, b, c, d, e, f, g)] -> SBool Source # sElem :: (a, b, c, d, e, f, g) -> [(a, b, c, d, e, f, g)] -> SBool Source # sNotElem :: (a, b, c, d, e, f, g) -> [(a, b, c, d, e, f, g)] -> SBool Source # | |
class (Mergeable a, EqSymbolic a) => OrdSymbolic a where Source #
Symbolic Comparisons. Similar to Eq, we cannot implement Haskell's Ord class
since there is no way to return an Ordering value from a symbolic comparison.
Furthermore, OrdSymbolic requires Mergeable to implement if-then-else, for the
benefit of implementing symbolic versions of max and min functions.
Minimal complete definition
Methods
(.<) :: a -> a -> SBool infix 4 Source #
Symbolic less than.
(.<=) :: a -> a -> SBool infix 4 Source #
Symbolic less than or equal to.
(.>) :: a -> a -> SBool infix 4 Source #
Symbolic greater than.
(.>=) :: a -> a -> SBool infix 4 Source #
Symbolic greater than or equal to.
Symbolic minimum.
Symbolic maximum.
inRange :: a -> (a, a) -> SBool Source #
Is the value within the allowed inclusive range?
Instances
| (Ord a, SymVal a) => OrdSymbolic (SBV a) Source # | If comparison is over something SMTLib can handle, just translate it. Otherwise desugar. |
| OrdSymbolic a => OrdSymbolic (Maybe a) Source # | |
Defined in Data.SBV.Core.Model | |
| OrdSymbolic a => OrdSymbolic [a] Source # | |
| (OrdSymbolic a, OrdSymbolic b) => OrdSymbolic (Either a b) Source # | |
Defined in Data.SBV.Core.Model Methods (.<) :: Either a b -> Either a b -> SBool Source # (.<=) :: Either a b -> Either a b -> SBool Source # (.>) :: Either a b -> Either a b -> SBool Source # (.>=) :: Either a b -> Either a b -> SBool Source # smin :: Either a b -> Either a b -> Either a b Source # smax :: Either a b -> Either a b -> Either a b Source # inRange :: Either a b -> (Either a b, Either a b) -> SBool Source # | |
| (OrdSymbolic a, OrdSymbolic b) => OrdSymbolic (a, b) Source # | |
| (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c) => OrdSymbolic (a, b, c) Source # | |
Defined in Data.SBV.Core.Model Methods (.<) :: (a, b, c) -> (a, b, c) -> SBool Source # (.<=) :: (a, b, c) -> (a, b, c) -> SBool Source # (.>) :: (a, b, c) -> (a, b, c) -> SBool Source # (.>=) :: (a, b, c) -> (a, b, c) -> SBool Source # smin :: (a, b, c) -> (a, b, c) -> (a, b, c) Source # smax :: (a, b, c) -> (a, b, c) -> (a, b, c) Source # inRange :: (a, b, c) -> ((a, b, c), (a, b, c)) -> SBool Source # | |
| (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d) => OrdSymbolic (a, b, c, d) Source # | |
Defined in Data.SBV.Core.Model Methods (.<) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source # (.<=) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source # (.>) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source # (.>=) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source # smin :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) Source # smax :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) Source # inRange :: (a, b, c, d) -> ((a, b, c, d), (a, b, c, d)) -> SBool Source # | |
| (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d, OrdSymbolic e) => OrdSymbolic (a, b, c, d, e) Source # | |
Defined in Data.SBV.Core.Model Methods (.<) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source # (.<=) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source # (.>) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source # (.>=) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source # smin :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) Source # smax :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) Source # inRange :: (a, b, c, d, e) -> ((a, b, c, d, e), (a, b, c, d, e)) -> SBool Source # | |
| (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d, OrdSymbolic e, OrdSymbolic f) => OrdSymbolic (a, b, c, d, e, f) Source # | |
Defined in Data.SBV.Core.Model Methods (.<) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source # (.<=) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source # (.>) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source # (.>=) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source # smin :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) Source # smax :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) Source # inRange :: (a, b, c, d, e, f) -> ((a, b, c, d, e, f), (a, b, c, d, e, f)) -> SBool Source # | |
| (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d, OrdSymbolic e, OrdSymbolic f, OrdSymbolic g) => OrdSymbolic (a, b, c, d, e, f, g) Source # | |
Defined in Data.SBV.Core.Model Methods (.<) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source # (.<=) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source # (.>) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source # (.>=) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source # smin :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) Source # smax :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) Source # inRange :: (a, b, c, d, e, f, g) -> ((a, b, c, d, e, f, g), (a, b, c, d, e, f, g)) -> SBool Source # | |
class Equality a where Source #
Equality as a proof method. Allows for very concise construction of equivalence proofs, which is very typical in bit-precise proofs.
Instances
Conditionals: Mergeable values
class Mergeable a where Source #
Symbolic conditionals are modeled by the Mergeable class, describing
how to merge the results of an if-then-else call with a symbolic test. SBV
provides all basic types as instances of this class, so users only need
to declare instances for custom data-types of their programs as needed.
A Mergeable instance may be automatically derived for a custom data-type
with a single constructor where the type of each field is an instance of
Mergeable, such as a record of symbolic values. Users only need to add
Generic and Mergeable to the deriving clause for the data-type. See
Status for an example and an
illustration of what the instance would look like if written by hand.
The function select is a total-indexing function out of a list of choices
with a default value, simulating array/list indexing. It's an n-way generalization
of the ite function.
Minimal complete definition: None, if the type is instance of Generic. Otherwise
symbolicMerge. Note that most types subject to merging are likely to be
trivial instances of Generic.
Minimal complete definition
Nothing
Methods
symbolicMerge :: Bool -> SBool -> a -> a -> a Source #
Merge two values based on the condition. The first argument states whether we force the then-and-else branches before the merging, at the word level. This is an efficiency concern; one that we'd rather not make but unfortunately necessary for getting symbolic simulation working efficiently.
select :: (Ord b, SymVal b, Num b) => [a] -> a -> SBV b -> a Source #
Total indexing operation. select xs default index is intuitively
the same as xs !! index, except it evaluates to default if index
underflows/overflows.
Instances
ite :: Mergeable a => SBool -> a -> a -> a Source #
If-then-else. This is by definition symbolicMerge with both
branches forced. This is typically the desired behavior, but also
see iteLazy should you need more laziness.
iteLazy :: Mergeable a => SBool -> a -> a -> a Source #
A Lazy version of ite, which does not force its arguments. This might cause issues for symbolic simulation with large thunks around, so use with care.
Symbolic integral numbers
class (SymVal a, Num a, Bits a, Integral a) => SIntegral a Source #
Symbolic Numbers. This is a simple class that simply incorporates all number like
base types together, simplifying writing polymorphic type-signatures that work for all
symbolic numbers, such as SWord8, SInt8 etc. For instance, we can write a generic
list-minimum function as follows:
mm :: SIntegral a => [SBV a] -> SBV a mm = foldr1 (a b -> ite (a .<= b) a b)
It is similar to the standard Integral class, except ranging over symbolic instances.
Instances
| SIntegral Int16 Source # | |
Defined in Data.SBV.Core.Model | |
| SIntegral Int32 Source # | |
Defined in Data.SBV.Core.Model | |
| SIntegral Int64 Source # | |
Defined in Data.SBV.Core.Model | |
| SIntegral Int8 Source # | |
Defined in Data.SBV.Core.Model | |
| SIntegral Word16 Source # | |
Defined in Data.SBV.Core.Model | |
| SIntegral Word32 Source # | |
Defined in Data.SBV.Core.Model | |
| SIntegral Word64 Source # | |
Defined in Data.SBV.Core.Model | |
| SIntegral Word8 Source # | |
Defined in Data.SBV.Core.Model | |
| SIntegral Integer Source # | |
Defined in Data.SBV.Core.Model | |
| (KnownNat n, BVIsNonZero n) => SIntegral (IntN n) Source # | |
Defined in Data.SBV.Core.Sized | |
| (KnownNat n, BVIsNonZero n) => SIntegral (WordN n) Source # | |
Defined in Data.SBV.Core.Sized | |
Division and Modulus
class SDivisible a where Source #
The SDivisible class captures the essence of division.
Unfortunately we cannot use Haskell's Integral class since the Real
and Enum superclasses are not implementable for symbolic bit-vectors.
However, quotRem and divMod both make perfect sense, and the SDivisible class captures
this operation. One issue is how division by 0 behaves. The verification
technology requires total functions, and there are several design choices
here. We follow Isabelle/HOL approach of assigning the value 0 for division
by 0. Therefore, we impose the following pair of laws:
xsQuotRem0 = (0, x) xsDivMod0 = (0, x)
Note that our instances implement this law even when x is 0 itself.
NB. quot truncates toward zero, while div truncates toward negative infinity.
C code generation of division operations
In the case of division or modulo of a minimal signed value (e.g. -128 for
SInt8) by -1, SMTLIB and Haskell agree on what the result should be.
Unfortunately the result in C code depends on CPU architecture and compiler
settings, as this is undefined behaviour in C. **SBV does not guarantee**
what will happen in generated C code in this corner case.
Instances
Euclidian division and modulus
Euclidian division and modulus for integers differ from regular division modulus when
the divisor is negative. It satisfies the following desirable property: For any m, n, we have:
Givenm,n, s.t., n /= 0 Let (q, r) = msEDivModn Then: m = n * q + r and 0 <= r <= |n| - 1
That is, the modulus is always positive. There's no standard Haskell function that performs this operation. The main reason to prefer this function is that SMT solvers can deal with them better. Compare:
>>>sDivMod @SInteger 3 (-2)(-2 :: SInteger,-1 :: SInteger)>>>sEDivMod 3 (-2)(-1 :: SInteger,1 :: SInteger)>>>prove $ \x y -> y .> 0 .=> x `sDivMod` y .== x `sEDivMod` yQ.E.D.
Bit-vector operations
Conversions
sFromIntegral :: forall a b. (Integral a, HasKind a, Num a, SymVal a, HasKind b, Num b, SymVal b) => SBV a -> SBV b Source #
Conversion between integral-symbolic values, akin to Haskell's fromIntegral
Shifts and rotates
Symbolic words (both signed and unsigned) are an instance of Haskell's Bits class, so regular
bitwise operations are automatically available for them. Shifts and rotates, however, require
specialized type-signatures since Haskell insists on an Int second argument for them.
sShiftRight :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a Source #
Generalization of shiftR, when the shift-amount is symbolic. Since Haskell's
shiftR only takes an Int as the shift amount, it cannot be used when we have
a symbolic amount to shift with.
NB. If the shiftee is signed, then this is an arithmetic shift; otherwise it's logical,
following the usual Haskell convention. See sSignedShiftArithRight for a variant
that explicitly uses the msb as the sign bit, even for unsigned underlying types.
sBarrelRotateLeft :: (SFiniteBits a, SFiniteBits b) => SBV a -> SBV b -> SBV a Source #
An implementation of rotate-left, using a barrel shifter like design. Only works when both
arguments are finite bitvectors, and furthermore when the second argument is unsigned.
The first condition is enforced by the type, but the second is dynamically checked.
We provide this implementation as an alternative to sRotateLeft since SMTLib logic
does not support variable argument rotates (as opposed to shifts), and thus this
implementation can produce better code for verification compared to sRotateLeft.
>>>prove $ \x y -> (x `sBarrelRotateLeft` y) `sBarrelRotateRight` (y :: SWord32) .== (x :: SWord64)Q.E.D.
sBarrelRotateRight :: (SFiniteBits a, SFiniteBits b) => SBV a -> SBV b -> SBV a Source #
An implementation of rotate-right, using a barrel shifter like design. See comments
for sBarrelRotateLeft for details.
>>>prove $ \x y -> (x `sBarrelRotateRight` y) `sBarrelRotateLeft` (y :: SWord32) .== (x :: SWord64)Q.E.D.
sSignedShiftArithRight :: (SFiniteBits a, SIntegral b) => SBV a -> SBV b -> SBV a Source #
Arithmetic shift-right with a symbolic unsigned shift amount. This is equivalent
to sShiftRight when the argument is signed. However, if the argument is unsigned,
then it explicitly treats its msb as a sign-bit, and uses it as the bit that
gets shifted in. Useful when using the underlying unsigned bit representation to implement
custom signed operations. Note that there is no direct Haskell analogue of this function.
Finite bit-vector operations
class (Ord a, SymVal a, Num a, Bits a) => SFiniteBits a where Source #
Finite bit-length symbolic values. Essentially the same as SIntegral, but further leaves out Integer. Loosely
based on Haskell's FiniteBits class, but with more methods defined and structured differently to fit into the
symbolic world view. Minimal complete definition: sFiniteBitSize.
Minimal complete definition
Methods
sFiniteBitSize :: SBV a -> Int Source #
Bit size.
lsb :: SBV a -> SBool Source #
Least significant bit of a word, always stored at index 0.
msb :: SBV a -> SBool Source #
Most significant bit of a word, always stored at the last position.
blastBE :: SBV a -> [SBool] Source #
Big-endian blasting of a word into its bits.
blastLE :: SBV a -> [SBool] Source #
Little-endian blasting of a word into its bits.
fromBitsBE :: [SBool] -> SBV a Source #
Reconstruct from given bits, given in little-endian.
fromBitsLE :: [SBool] -> SBV a Source #
Reconstruct from given bits, given in little-endian.
sTestBit :: SBV a -> Int -> SBool Source #
sExtractBits :: SBV a -> [Int] -> [SBool] Source #
Variant of sTestBit, where we want to extract multiple bit positions.
sPopCount :: SBV a -> SWord8 Source #
Variant of popCount, returning a symbolic value.
setBitTo :: SBV a -> Int -> SBool -> SBV a Source #
fullAdder :: SBV a -> SBV a -> (SBool, SBV a) Source #
Full adder, returns carry-out from the addition. Only for unsigned quantities.
fullMultiplier :: SBV a -> SBV a -> (SBV a, SBV a) Source #
Full multiplier, returns both high and low-order bits. Only for unsigned quantities.
sCountLeadingZeros :: SBV a -> SWord8 Source #
Count leading zeros in a word, big-endian interpretation.
sCountTrailingZeros :: SBV a -> SWord8 Source #
Count trailing zeros in a word, big-endian interpretation.
Instances
Splitting, joining, and extending bit-vectors
Arguments
| :: forall i j n bv proxy. (KnownNat n, BVIsNonZero n, SymVal (bv n), KnownNat i, KnownNat j, (i + 1) <= n, j <= i, BVIsNonZero ((i - j) + 1)) | |
| => proxy i |
|
| -> proxy j |
|
| -> SBV (bv n) | Input bit vector of size |
| -> SBV (bv ((i - j) + 1)) | Output is of size |
Extract a portion of bits to form a smaller bit-vector.
>>>prove $ \x -> bvExtract (Proxy @7) (Proxy @3) (x :: SWord 12) .== bvDrop (Proxy @4) (bvTake (Proxy @9) x)Q.E.D.
Arguments
| :: (KnownNat n, BVIsNonZero n, SymVal (bv n), KnownNat m, BVIsNonZero m, SymVal (bv m)) | |
| => SBV (bv n) | First input, of size |
| -> SBV (bv m) | Second input, of size |
| -> SBV (bv (n + m)) | Concatenation, of size |
Join two bitvectors.
>>>prove $ \x y -> x .== bvExtract (Proxy @79) (Proxy @71) ((x :: SWord 9) # (y :: SWord 71))Q.E.D.
Arguments
| :: forall n m bv. (KnownNat n, BVIsNonZero n, SymVal (bv n), KnownNat m, BVIsNonZero m, SymVal (bv m), (n + 1) <= m, SIntegral (bv (m - n)), BVIsNonZero (m - n)) | |
| => SBV (bv n) | Input, of size |
| -> SBV (bv m) | Output, of size |
Zero extend a bit-vector.
>>>prove $ \x -> bvExtract (Proxy @20) (Proxy @12) (zeroExtend (x :: SInt 12) :: SInt 21) .== 0Q.E.D.
Arguments
| :: forall n m bv. (KnownNat n, BVIsNonZero n, SymVal (bv n), KnownNat m, BVIsNonZero m, SymVal (bv m), (n + 1) <= m, SFiniteBits (bv n), SIntegral (bv (m - n)), BVIsNonZero (m - n)) | |
| => SBV (bv n) | Input, of size |
| -> SBV (bv m) | Output, of size |
Sign extend a bit-vector.
>>>prove $ \x -> sNot (msb x) .=> bvExtract (Proxy @20) (Proxy @12) (signExtend (x :: SInt 12) :: SInt 21) .== 0Q.E.D.>>>prove $ \x -> msb x .=> bvExtract (Proxy @20) (Proxy @12) (signExtend (x :: SInt 12) :: SInt 21) .== complement 0Q.E.D.
Arguments
| :: forall i n m bv proxy. (KnownNat n, BVIsNonZero n, KnownNat i, (i + 1) <= n, ((i + m) - n) <= 0, BVIsNonZero (n - i)) | |
| => proxy i |
|
| -> SBV (bv n) | Input, of size |
| -> SBV (bv m) | Output, of size |
Drop bits from the top of a bit-vector.
>>>prove $ \x -> bvDrop (Proxy @0) (x :: SWord 43) .== xQ.E.D.>>>prove $ \x -> bvDrop (Proxy @20) (x :: SWord 21) .== ite (lsb x) 1 (0 :: SWord 1)Q.E.D.
Arguments
| :: forall i n bv proxy. (KnownNat n, BVIsNonZero n, KnownNat i, BVIsNonZero i, i <= n) | |
| => proxy i |
|
| -> SBV (bv n) | Input, of size |
| -> SBV (bv i) | Output, of size |
Take bits from the top of a bit-vector.
>>>prove $ \x -> bvTake (Proxy @13) (x :: SWord 13) .== xQ.E.D.>>>prove $ \x -> bvTake (Proxy @1) (x :: SWord 13) .== ite (msb x) 1 0Q.E.D.>>>prove $ \x -> bvTake (Proxy @4) x # bvDrop (Proxy @4) x .== (x :: SWord 23)Q.E.D.
class ByteConverter a where Source #
A helper class to convert sized bit-vectors to/from bytes.
Methods
toBytes :: a -> [SWord 8] Source #
Convert to a sequence of bytes
>>>prove $ \a b c d -> toBytes ((fromBytes [a, b, c, d]) :: SWord 32) .== [a, b, c, d]Q.E.D.
fromBytes :: [SWord 8] -> a Source #
Convert from a sequence of bytes
>>>prove $ \r -> fromBytes (toBytes r) .== (r :: SWord 64)Q.E.D.
Instances
| ByteConverter (SWord 8) Source # |
|
| ByteConverter (SWord 16) Source # |
|
| ByteConverter (SWord 32) Source # |
|
| ByteConverter (SWord 64) Source # |
|
| ByteConverter (SWord 128) Source # |
|
| ByteConverter (SWord 256) Source # |
|
| ByteConverter (SWord 512) Source # |
|
| ByteConverter (SWord 1024) Source # |
|
Exponentiation
(.^) :: (Mergeable b, Num b, SIntegral e) => b -> SBV e -> b Source #
Symbolic exponentiation using bit blasting and repeated squaring.
N.B. The exponent must be unsigned/bounded if symbolic. Signed exponents will be rejected.
IEEE-floating point numbers
class (SymVal a, RealFloat a) => IEEEFloating a where Source #
A class of floating-point (IEEE754) operations, some of which behave differently based on rounding modes. Note that unless the rounding mode is concretely RoundNearestTiesToEven, we will not concretely evaluate these, but rather pass down to the SMT solver.
Minimal complete definition
Nothing
Methods
fpAbs :: SBV a -> SBV a Source #
Compute the floating point absolute value.
fpNeg :: SBV a -> SBV a Source #
Compute the unary negation. Note that 0 - x is not equivalent to -x for floating-point, since -0 and 0 are different.
fpAdd :: SRoundingMode -> SBV a -> SBV a -> SBV a Source #
Add two floating point values, using the given rounding mode
fpSub :: SRoundingMode -> SBV a -> SBV a -> SBV a Source #
Subtract two floating point values, using the given rounding mode
fpMul :: SRoundingMode -> SBV a -> SBV a -> SBV a Source #
Multiply two floating point values, using the given rounding mode
fpDiv :: SRoundingMode -> SBV a -> SBV a -> SBV a Source #
Divide two floating point values, using the given rounding mode
fpFMA :: SRoundingMode -> SBV a -> SBV a -> SBV a -> SBV a Source #
Fused-multiply-add three floating point values, using the given rounding mode. fpFMA x y z = x*y+z but with only
one rounding done for the whole operation; not two. Note that we will never concretely evaluate this function since
Haskell lacks an FMA implementation.
fpSqrt :: SRoundingMode -> SBV a -> SBV a Source #
Compute the square-root of a float, using the given rounding mode
fpRem :: SBV a -> SBV a -> SBV a Source #
Compute the remainder: x - y * n, where n is the truncated integer nearest to x/y. The rounding mode
is implicitly assumed to be RoundNearestTiesToEven.
fpRoundToIntegral :: SRoundingMode -> SBV a -> SBV a Source #
Round to the nearest integral value, using the given rounding mode.
fpMin :: SBV a -> SBV a -> SBV a Source #
Compute the minimum of two floats, respects infinity and NaN values
fpMax :: SBV a -> SBV a -> SBV a Source #
Compute the maximum of two floats, respects infinity and NaN values
fpIsEqualObject :: SBV a -> SBV a -> SBool Source #
Are the two given floats exactly the same. That is, NaN will compare equal to itself, +0 will not compare
equal to -0 etc. This is the object level equality, as opposed to the semantic equality. (For the latter, just use .==.)
fpIsNormal :: SBV a -> SBool Source #
Is the floating-point number a normal value. (i.e., not denormalized.)
fpIsSubnormal :: SBV a -> SBool Source #
Is the floating-point number a subnormal value. (Also known as denormal.)
fpIsZero :: SBV a -> SBool Source #
Is the floating-point number 0? (Note that both +0 and -0 will satisfy this predicate.)
fpIsInfinite :: SBV a -> SBool Source #
Is the floating-point number infinity? (Note that both +oo and -oo will satisfy this predicate.)
fpIsNaN :: SBV a -> SBool Source #
Is the floating-point number a NaN value?
fpIsNegative :: SBV a -> SBool Source #
Is the floating-point number negative? Note that -0 satisfies this predicate but +0 does not.
fpIsPositive :: SBV a -> SBool Source #
Is the floating-point number positive? Note that +0 satisfies this predicate but -0 does not.
fpIsNegativeZero :: SBV a -> SBool Source #
Is the floating point number -0?
fpIsPositiveZero :: SBV a -> SBool Source #
Is the floating point number +0?
fpIsPoint :: SBV a -> SBool Source #
Is the floating-point number a regular floating point, i.e., not NaN, nor +oo, nor -oo. Normals or denormals are allowed.
Instances
data RoundingMode Source #
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.
Constructors
| 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.) |
Instances
type SRoundingMode = SBV RoundingMode Source #
The symbolic variant of RoundingMode
sNaN :: (Floating a, SymVal a) => SBV a Source #
Symbolic variant of Not-A-Number. This value will inhabit
SFloat, SDouble and SFloatingPoint. types.
sInfinity :: (Floating a, SymVal a) => SBV a Source #
Symbolic variant of infinity. This value will inhabit both
SFloat, SDouble and SFloatingPoint. types.
Rounding modes
sRoundNearestTiesToEven :: SRoundingMode Source #
Symbolic variant of RoundNearestTiesToEven
sRoundNearestTiesToAway :: SRoundingMode Source #
Symbolic variant of RoundNearestTiesToAway
sRoundTowardPositive :: SRoundingMode Source #
Symbolic variant of RoundTowardPositive
sRoundTowardNegative :: SRoundingMode Source #
Symbolic variant of RoundTowardNegative
sRoundTowardZero :: SRoundingMode Source #
Symbolic variant of RoundTowardZero
sRNE :: SRoundingMode Source #
Alias for sRoundNearestTiesToEven
sRNA :: SRoundingMode Source #
Alias for sRoundNearestTiesToAway
sRTP :: SRoundingMode Source #
Alias for sRoundTowardPositive
sRTN :: SRoundingMode Source #
Alias for sRoundTowardNegative
sRTZ :: SRoundingMode Source #
Alias for sRoundTowardZero
Conversion to/from floats
Capture convertability from/to FloatingPoint representations.
Conversions to float: toSFloat and toSDouble simply return the
nearest representable float from the given type based on the rounding
mode provided. Similarly, toSFloatingPoint converts to a generalized
floating-point number with specified exponent and significand bith widths.
Conversions from float: fromSFloat, fromSDouble, fromSFloatingPoint functions do
the reverse conversion. However some care is needed when given values
that are not representable in the integral target domain. For instance,
converting an SFloat to an SInt8 is problematic. The rules are as follows:
If the input value is a finite point and when rounded in the given rounding mode to an integral value lies within the target bounds, then that result is returned. (This is the regular interpretation of rounding in IEEE754.)
Otherwise (i.e., if the integral value in the float or double domain) doesn't
fit into the target type, then the result is unspecified. Note that if the input
is +oo, -oo, or NaN, then the result is unspecified.
Due to the unspecified nature of conversions, SBV will never constant fold conversions from floats to integral values. That is, you will always get a symbolic value as output. (Conversions from floats to other floats will be constant folded. Conversions from integral values to floats will also be constant folded.)
Note that unspecified really means unspecified: In particular, SBV makes
no guarantees about matching the behavior between what you might get in
Haskell, via SMT-Lib, or the C-translation. If the input value is out-of-bounds
as defined above, or is NaN or oo or -oo, then all bets are off. In particular
C and SMTLib are decidedly undefine this case, though that doesn't mean they do the
same thing! Same goes for Haskell, which seems to convert via Int64, but we do
not model that behavior in SBV as it doesn't seem to be intentional nor well documented.
You can check for NaN, oo and -oo, using the predicates fpIsNaN, fpIsInfinite,
and fpIsPositive, fpIsNegative predicates, respectively; and do the proper conversion
based on your needs. (0 is a good choice, as are min/max bounds of the target type.)
Currently, SBV provides no predicates to check if a value would lie within range for a particular conversion task, as this depends on the rounding mode and the types involved and can be rather tricky to determine. (See http://github.com/LeventErkok/sbv/issues/456 for a discussion of the issues involved.) In a future release, we hope to be able to provide underflow and overflow predicates for these conversions as well.
Some examples to illustrate the behavior follows:
>>>:{roundTrip :: forall a. (Eq a, IEEEFloatConvertible a) => SRoundingMode -> SBV a -> SBool roundTrip m x = fromSFloat m (toSFloat m x) .== x :}
>>>prove $ roundTrip @Int8Q.E.D.>>>prove $ roundTrip @Word8Q.E.D.>>>prove $ roundTrip @Int16Q.E.D.>>>prove $ roundTrip @Word16Q.E.D.>>>prove $ roundTrip @Int32Falsifiable. Counter-example: s0 = RoundNearestTiesToAway :: RoundingMode s1 = -740395022 :: Int32
Note how we get a failure on Int32. The counter-example value is not representable exactly as a single precision float:
>>>toRational (-740395022 :: Float)(-740395008) % 1
Note how the numerator is different, it is off by 14. This is hardly surprising, since floats become sparser as the magnitude increases to be able to cover all the integer values representable.
>>>:{roundTrip :: forall a. (Eq a, IEEEFloatConvertible a) => SRoundingMode -> SBV a -> SBool roundTrip m x = fromSDouble m (toSDouble m x) .== x :}
>>>prove $ roundTrip @Int8Q.E.D.>>>prove $ roundTrip @Word8Q.E.D.>>>prove $ roundTrip @Int16Q.E.D.>>>prove $ roundTrip @Word16Q.E.D.>>>prove $ roundTrip @Int32Q.E.D.>>>prove $ roundTrip @Word32Q.E.D.>>>prove $ roundTrip @Int64Falsifiable. Counter-example: s0 = RoundTowardPositive :: RoundingMode s1 = 1152921504606846896 :: Int64
Just like in the SFloat case, once we reach 64-bits, we no longer can exactly represent the
integer value for all possible values:
>>>toRational (fromIntegral (1152921504606846896 :: Int64) :: Double)1152921504606846848 % 1
In this case the numerator is off by 48.
class SymVal a => IEEEFloatConvertible a where Source #
Conversion to and from floats
Minimal complete definition
Nothing
Methods
fromSFloat :: SRoundingMode -> SFloat -> SBV a Source #
Convert from an IEEE74 single precision float.
toSFloat :: SRoundingMode -> SBV a -> SFloat Source #
Convert to an IEEE-754 Single-precision float.
fromSDouble :: SRoundingMode -> SDouble -> SBV a Source #
Convert from an IEEE74 double precision float.
toSDouble :: SRoundingMode -> SBV a -> SDouble Source #
Convert to an IEEE-754 Double-precision float.
fromSFloatingPoint :: ValidFloat eb sb => SRoundingMode -> SFloatingPoint eb sb -> SBV a Source #
Convert from an arbitrary floating point.
toSFloatingPoint :: ValidFloat eb sb => SRoundingMode -> SBV a -> SFloatingPoint eb sb Source #
Convert to an arbitrary floating point.
default toSFloatingPoint :: (Integral a, ValidFloat eb sb) => SRoundingMode -> SBV a -> SFloatingPoint eb sb Source #
Instances
Bit-pattern conversions
sFloatAsSWord32 :: SFloat -> SWord32 Source #
Convert an SFloat to an SWord32, preserving the bit-correspondence. Note that since the
representation for NaNs are not unique, this function will return a symbolic value when given a
concrete NaN.
Implementation note: Since there's no corresponding function in SMTLib for conversion to bit-representation due to partiality, we use a translation trick by allocating a new word variable, converting it to float, and requiring it to be equivalent to the input. In code-generation mode, we simply map it to a simple conversion.
sWord32AsSFloat :: SWord32 -> SFloat Source #
Reinterpret the bits in a 32-bit word as a single-precision floating point number
sDoubleAsSWord64 :: SDouble -> SWord64 Source #
Convert an SDouble to an SWord64, preserving the bit-correspondence. Note that since the
representation for NaNs are not unique, this function will return a symbolic value when given a
concrete NaN.
See the implementation note for sFloatAsSWord32, as it applies here as well.
sWord64AsSDouble :: SWord64 -> SDouble Source #
Reinterpret the bits in a 32-bit word as a single-precision floating point number
sFloatingPointAsSWord :: forall eb sb. (ValidFloat eb sb, KnownNat (eb + sb), BVIsNonZero (eb + sb)) => SFloatingPoint eb sb -> SWord (eb + sb) Source #
Convert a float to the word containing the corresponding bit pattern
sWordAsSFloatingPoint :: forall eb sb. (KnownNat (eb + sb), BVIsNonZero (eb + sb), ValidFloat eb sb) => SWord (eb + sb) -> SFloatingPoint eb sb Source #
Convert a word to an arbitrary float, by reinterpreting the bits of the word as the corresponding bits of the float.
Extracting bit patterns from floats
blastSFloat :: SFloat -> (SBool, [SBool], [SBool]) Source #
Extract the sign/exponent/mantissa of a single-precision float. The output will have 8 bits in the second argument for exponent, and 23 in the third for the mantissa.
blastSDouble :: SDouble -> (SBool, [SBool], [SBool]) Source #
Extract the sign/exponent/mantissa of a single-precision float. The output will have 11 bits in the second argument for exponent, and 52 in the third for the mantissa.
blastSFloatingPoint :: forall eb sb. (ValidFloat eb sb, KnownNat (eb + sb), BVIsNonZero (eb + sb)) => SFloatingPoint eb sb -> (SBool, [SBool], [SBool]) Source #
Extract the sign/exponent/mantissa of an arbitrary precision float. The output will have
eb bits in the second argument for exponent, and sb-1 bits in the third for mantissa.
Showing values in detail
crack :: SBV a -> String Source #
Show a value in detailed (cracked) form, if possible. This makes most sense with numbers, and especially floating-point types.
Enumerations
If the uninterpreted sort definition takes the form of an enumeration (i.e., a simple data type with all nullary constructors), then
you can use the mkSymbolicEnumeration function to turn it into an enumeration in SMTLib.
A simple example is:
data X = A | B | C
mkSymbolicEnumeration ''X
Note the magic incantation mkSymbolicEnumeration ''X. For this to work, you need to have the following
options turned on:
LANGUAGE TemplateHaskell LANGUAGE StandaloneDeriving LANGUAGE DeriveDataTypeable LANGUAGE DeriveAnyClass
The declaration will automatically introduce the type:
type SX = SBV X
along with symbolic values of each of the enumerated values sA, sB, and sC. This way,
you can refer to the symbolic version as SX, treating it as a regular symbolic type ranging over the values A, B, and C. Such values can be compared for equality, and with the usual
other comparison operators, such as .==, ./=, .>, .>=, <, and <=. For each enumerated value X, the symbolic versions sX is defined to be equal to literal X.
A simple query would look like:
allSat $ x -> x .== (x :: SX)
which would list all three elements of this domain as satisfying solutions.
Solution #1:
s0 = A :: X
Solution #2:
s0 = B :: X
Solution #3:
s0 = C :: X
Found 3 different solutions.
Note that the result is properly typed as X elements; these are not mere strings.
See Documentation.SBV.Examples.Misc.Enumerate for an extended example on how to use symbolic enumerations.
Uninterpreted sorts, constants, and functions
Users can introduce new uninterpreted sorts simply by defining an empty data-type in Haskell and registering it as such. The following example demonstrates:
data B
mkUninterpretedSort ''B
(Note that you'll also need to use pragmas TemplateHaskell, StandAloneDeriving, DeriveDataTypeable, and DeriveAnyClass for this to work, follow GHC's error messages!)
This is all it takes to introduce B as an uninterpreted sort in SBV, which makes the type SBV B automagically become available as the type
of symbolic values that ranges over B values. Note that this will also introduce the type SB into your environment, which is a synonym
for SBV B.
Uninterpreted functions over both uninterpreted and regular sorts can be declared using the facilities introduced by
the SMTDefinable class.
Stopping unrolling: Defined functions
class SMTDefinable a where Source #
SMT definable constants and functions, which can also be uninterpeted. This class captures functions that we can generate standalone-code for in the SMT solver. Note that we also allow uninterpreted constants and functions too. An uninterpreted constant is a value that is indexed by its name. The only property the prover assumes -- about these values are that they are equivalent to themselves; i.e., (for functions) they return the same results when applied to same arguments. We support uninterpreted-functions as a general means of black-box'ing operations that are irrelevant for the purposes of the proof; i.e., when the proofs can be performed without any knowledge about the function itself.
Minimal complete definition: sbvDefineValue. However, most instances in
practice are already provided by SBV, so end-users should not need to define their
own instances.
Minimal complete definition
Methods
smtFunction :: Lambda Symbolic a => String -> a -> a Source #
Generate the code for this value as an SMTLib function, instead of the usual unrolling semantics. This is useful for generating sub-functions in generated SMTLib problem, or handling recursive (and mutually-recursive) definitions that wouldn't terminate in an unrolling symbolic simulation context.
IMPORTANT NOTE The string argument names this function. Note that SBV will identify
this function with that name, i.e., if you use this function twice (or use it recursively),
it will simply assume this name uniquely identifies the function being defined. Hence,
the user has to assure that this string is unique amongst all the functions you use.
Furthermore, if the call to smtFunction happens in the scope of a parameter, you
must make sure the string is chosen to keep it unique per parameter value. For instance,
if you have:
bar :: SInteger -> SInteger -> SInteger bar k = smtFunction "bar" (x -> x+k) -- Note the capture of k!
and you call bar 2 and bar 3, you *will* get the same SMTLib function. Obviously
this is unsound. The reason is that the parameter value isn't captured by the name. In general,
you should simply not do this, but if you must, have a concrete argument to make sure you can
create a unique name. Something like:
bar :: String -> SInteger -> SInteger -> SInteger
bar tag k = smtFunction ("bar_" ++ tag) (x -> x+k) -- Tag should make the name unique!
Then, make sure you use bar "two" 2 and bar "three" 3 etc. to preserve the invariant.
Note that this is a design choice, to keep function creation as easy to use as possible. SBV
could've made smtFunction a monadic call and generated the name itself to avoid all these issues.
But the ergonomics of that is worse, and doesn't fit with the general design philosophy. If you
can think of a solution (perhaps using some nifty GHC tricks?) to avoid this issue without making
smtFunction return a monadic result, please get in touch!
uninterpret :: String -> a Source #
Uninterpret a value, i.e., add this value as a completely undefined value/function that the solver is free to instantiate to satisfy other constraints.
cgUninterpret :: String -> [String] -> a -> a Source #
Uninterpret a value, only for the purposes of code-generation. For execution and verification the value is used as is. For code-generation, the alternate definition is used. This is useful when we want to take advantage of native libraries on the target languages.
sbvDefineValue :: String -> UIKind a -> a Source #
Most generalized form of uninterpretation, this function should not be needed by end-user-code, but is rather useful for the library development.
A synonym for uninterpret. Allows us to create variables without
having to call free explicitly, i.e., without being in the symbolic monad.
Instances
| HasKind a => SMTDefinable (SBV a) Source # | |
| (SymVal b, HasKind a) => SMTDefinable (SBV b -> SBV a) Source # | |
Defined in Data.SBV.Core.Model | |
| (SymVal c, SymVal b, HasKind a) => SMTDefinable (SBV c -> SBV b -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> (SBV c -> SBV b -> SBV a) -> SBV c -> SBV b -> SBV a Source # uninterpret :: String -> SBV c -> SBV b -> SBV a Source # cgUninterpret :: String -> [String] -> (SBV c -> SBV b -> SBV a) -> SBV c -> SBV b -> SBV a Source # sbvDefineValue :: String -> UIKind (SBV c -> SBV b -> SBV a) -> SBV c -> SBV b -> SBV a Source # | |
| (SymVal d, SymVal c, SymVal b, HasKind a) => SMTDefinable (SBV d -> SBV c -> SBV b -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> (SBV d -> SBV c -> SBV b -> SBV a) -> SBV d -> SBV c -> SBV b -> SBV a Source # uninterpret :: String -> SBV d -> SBV c -> SBV b -> SBV a Source # cgUninterpret :: String -> [String] -> (SBV d -> SBV c -> SBV b -> SBV a) -> SBV d -> SBV c -> SBV b -> SBV a Source # sbvDefineValue :: String -> UIKind (SBV d -> SBV c -> SBV b -> SBV a) -> SBV d -> SBV c -> SBV b -> SBV a Source # | |
| (SymVal e, SymVal d, SymVal c, SymVal b, HasKind a) => SMTDefinable (SBV e -> SBV d -> SBV c -> SBV b -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> (SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # uninterpret :: String -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # cgUninterpret :: String -> [String] -> (SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # sbvDefineValue :: String -> UIKind (SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # sym :: String -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # | |
| (SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, HasKind a) => SMTDefinable (SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> (SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # uninterpret :: String -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # cgUninterpret :: String -> [String] -> (SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # sbvDefineValue :: String -> UIKind (SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # sym :: String -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # | |
| (SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, HasKind a) => SMTDefinable (SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> (SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # uninterpret :: String -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # cgUninterpret :: String -> [String] -> (SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # sbvDefineValue :: String -> UIKind (SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # sym :: String -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # | |
| (SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, HasKind a) => SMTDefinable (SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> (SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # uninterpret :: String -> SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # cgUninterpret :: String -> [String] -> (SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # sbvDefineValue :: String -> UIKind (SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) -> SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # sym :: String -> SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a Source # | |
| (SymVal c, SymVal b, HasKind a) => SMTDefinable ((SBV c, SBV b) -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> ((SBV c, SBV b) -> SBV a) -> (SBV c, SBV b) -> SBV a Source # uninterpret :: String -> (SBV c, SBV b) -> SBV a Source # cgUninterpret :: String -> [String] -> ((SBV c, SBV b) -> SBV a) -> (SBV c, SBV b) -> SBV a Source # sbvDefineValue :: String -> UIKind ((SBV c, SBV b) -> SBV a) -> (SBV c, SBV b) -> SBV a Source # | |
| (SymVal d, SymVal c, SymVal b, HasKind a) => SMTDefinable ((SBV d, SBV c, SBV b) -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> ((SBV d, SBV c, SBV b) -> SBV a) -> (SBV d, SBV c, SBV b) -> SBV a Source # uninterpret :: String -> (SBV d, SBV c, SBV b) -> SBV a Source # cgUninterpret :: String -> [String] -> ((SBV d, SBV c, SBV b) -> SBV a) -> (SBV d, SBV c, SBV b) -> SBV a Source # sbvDefineValue :: String -> UIKind ((SBV d, SBV c, SBV b) -> SBV a) -> (SBV d, SBV c, SBV b) -> SBV a Source # | |
| (SymVal e, SymVal d, SymVal c, SymVal b, HasKind a) => SMTDefinable ((SBV e, SBV d, SBV c, SBV b) -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> ((SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV e, SBV d, SBV c, SBV b) -> SBV a Source # uninterpret :: String -> (SBV e, SBV d, SBV c, SBV b) -> SBV a Source # cgUninterpret :: String -> [String] -> ((SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV e, SBV d, SBV c, SBV b) -> SBV a Source # sbvDefineValue :: String -> UIKind ((SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV e, SBV d, SBV c, SBV b) -> SBV a Source # sym :: String -> (SBV e, SBV d, SBV c, SBV b) -> SBV a Source # | |
| (SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, HasKind a) => SMTDefinable ((SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> ((SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # uninterpret :: String -> (SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # cgUninterpret :: String -> [String] -> ((SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # sbvDefineValue :: String -> UIKind ((SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # sym :: String -> (SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # | |
| (SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, HasKind a) => SMTDefinable ((SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> ((SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # uninterpret :: String -> (SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # cgUninterpret :: String -> [String] -> ((SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # sbvDefineValue :: String -> UIKind ((SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # sym :: String -> (SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # | |
| (SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, HasKind a) => SMTDefinable ((SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) Source # | |
Defined in Data.SBV.Core.Model Methods smtFunction :: String -> ((SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # uninterpret :: String -> (SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # cgUninterpret :: String -> [String] -> ((SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # sbvDefineValue :: String -> UIKind ((SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) -> (SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # sym :: String -> (SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a Source # | |
Special relations
A special relation is a binary relation that has additional properties. SBV allows for the checking of various kinds of special relations respecting various axioms, and allows for creating transitive closures. See Documentation.SBV.Examples.Misc.FirstOrderLogic for several examples.
isPartialOrder :: SymVal a => String -> Relation a -> SBool Source #
Check if a relation is a partial order. The string argument must uniquely identify this order.
isLinearOrder :: SymVal a => String -> Relation a -> SBool Source #
Check if a relation is a linear order. The string argument must uniquely identify this order.
isTreeOrder :: SymVal a => String -> Relation a -> SBool Source #
Check if a relation is a tree order. The string argument must uniquely identify this order.
isPiecewiseLinearOrder :: SymVal a => String -> Relation a -> SBool Source #
Check if a relation is a piece-wise linear order. The string argument must uniquely identify this order.
mkTransitiveClosure :: forall a. SymVal a => String -> Relation a -> Relation a Source #
Create the transitive closure of a given relation. The string argument must uniquely identify the newly created relation.
Properties, proofs, and satisfiability
The SBV library provides a "push-button" verification system via automated SMT solving. The design goal is to let SMT solvers be used without any knowledge of how SMT solvers work or how different logics operate. The details are hidden behind the SBV framework, providing Haskell programmers with a clean API that is unencumbered by the details of individual solvers. To that end, we use the SMT-Lib standard (http://smtlib.cs.uiowa.edu/) to communicate with arbitrary SMT solvers.
Using multiple solvers
On a multi-core machine, it might be desirable to try a given property using multiple SMT solvers, using parallel threads. Even with machines with single-cores, threading can be helpful if you want to try out multiple-solvers but do not know which one would work the best for the problem at hand ahead of time.
SBV allows proving/satisfiability-checking with multiple backends at the same time. Each function comes in two variants, one that returns the results from all solvers, the other that returns the fastest one.
The All variants, (i.e., proveWithAll, satWithAll) run all solvers and
return all the results. SBV internally makes sure that the result is lazily generated; so,
the order of solvers given does not matter. In other words, the order of results will follow
the order of the solvers as they finish, not as given by the user. These variants are useful when you
want to make sure multiple-solvers agree (or disagree!) on a given problem.
The Any variants, (i.e., proveWithAny, satWithAny) will run all the solvers
in parallel, and return the results of the first one finishing. The other threads will then be killed. These variants
are useful when you do not care if the solvers produce the same result, but rather want to get the
solution as quickly as possible, taking advantage of modern many-core machines.
Note that the function getAvailableSolvers will return all the installed solvers, which can be
used as the first argument to all these functions, if you simply want to try all available solvers on a machine.
type Predicate = Symbolic SBool Source #
A predicate is a symbolic program that returns a (symbolic) boolean value. For all intents and
purposes, it can be treated as an n-ary function from symbolic-values to a boolean. The Symbolic
monad captures the underlying representation, and can/should be ignored by the users of the library,
unless you are building further utilities on top of SBV itself. Instead, simply use the Predicate
type when necessary.
type ConstraintSet = Symbolic () Source #
A constraint set is a symbolic program that returns no values. The idea is that the constraints/min-max goals will serve as the collection of constraints that will be used for sat/optimize calls.
type Satisfiable = SatisfiableM IO Source #
Satisfiable is specialization of SatisfiableM to the IO monad. Unless you are using
transformers explicitly, this is the type you should prefer.
prove :: Provable a => a -> IO ThmResult Source #
Prove a predicate, using the default solver.
NB. For a version which generalizes over the underlying monad, see prove
proveWith :: Provable a => SMTConfig -> a -> IO ThmResult Source #
Prove the predicate using the given SMT-solver.
NB. For a version which generalizes over the underlying monad, see proveWith
dprove :: Provable a => a -> IO ThmResult Source #
Prove a predicate with delta-satisfiability, using the default solver.
NB. For a version which generalizes over the underlying monad, see prove
dproveWith :: Provable a => SMTConfig -> a -> IO ThmResult Source #
Prove the predicate with delta-satisfiability using the given SMT-solver.
NB. For a version which generalizes over the underlying monad, see proveWith
sat :: Satisfiable a => a -> IO SatResult Source #
Find a satisfying assignment for a predicate, using the default solver.
NB. For a version which generalizes over the underlying monad, see sat
satWith :: Satisfiable a => SMTConfig -> a -> IO SatResult Source #
Find a satisfying assignment using the given SMT-solver.
NB. For a version which generalizes over the underlying monad, see satWith
dsat :: Satisfiable a => a -> IO SatResult Source #
Find a delta-satisfying assignment for a predicate, using the default solver for delta-satisfiability.
NB. For a version which generalizes over the underlying monad, see dsat
dsatWith :: Satisfiable a => SMTConfig -> a -> IO SatResult Source #
Find a satisfying assignment using the given SMT-solver.
NB. For a version which generalizes over the underlying monad, see satWith
allSat :: Satisfiable a => a -> IO AllSatResult Source #
Find all satisfying assignments, using the default solver.
Equivalent to allSatWith defaultSMTCfgallSatWith for details.
NB. For a version which generalizes over the underlying monad, see allSat
allSatWith :: Satisfiable a => SMTConfig -> a -> IO AllSatResult Source #
Return all satisfying assignments for a predicate.
Note that this call will block until all satisfying assignments are found. If you have a problem
with infinitely many satisfying models (consider SInteger) or a very large number of them, you
might have to wait for a long time. To avoid such cases, use the allSatMaxModelCount
parameter in the configuration.
NB. Uninterpreted constant/function values and counter-examples for array values are ignored for
the purposes of allSat. That is, only the satisfying assignments modulo uninterpreted functions and
array inputs will be returned. This is due to the limitation of not having a robust means of getting a
function counter-example back from the SMT solver.
Find all satisfying assignments using the given SMT-solver
NB. For a version which generalizes over the underlying monad, see allSatWith
optimize :: Satisfiable a => OptimizeStyle -> a -> IO OptimizeResult Source #
optimizeWith :: Satisfiable a => SMTConfig -> OptimizeStyle -> a -> IO OptimizeResult Source #
Optimizes the objectives using the given SMT-solver.
NB. For a version which generalizes over the underlying monad, see optimizeWith
isVacuousProof :: Provable a => a -> IO Bool Source #
Check if the constraints given are consistent in a prove call using the default solver.
NB. For a version which generalizes over the underlying monad, see isVacuousProof
isVacuousProofWith :: Provable a => SMTConfig -> a -> IO Bool Source #
Determine if the constraints are vacuous in a SAT call using the given SMT-solver.
NB. For a version which generalizes over the underlying monad, see isVacuousProofWith
isTheorem :: Provable a => a -> IO Bool Source #
Checks theoremhood using the default solver.
NB. For a version which generalizes over the underlying monad, see isTheorem
isTheoremWith :: Provable a => SMTConfig -> a -> IO Bool Source #
Check whether a given property is a theorem.
NB. For a version which generalizes over the underlying monad, see isTheoremWith
isSatisfiable :: Satisfiable a => a -> IO Bool Source #
Checks satisfiability using the default solver.
NB. For a version which generalizes over the underlying monad, see isSatisfiable
isSatisfiableWith :: Satisfiable a => SMTConfig -> a -> IO Bool Source #
Check whether a given property is satisfiable.
NB. For a version which generalizes over the underlying monad, see isSatisfiableWith
proveWithAny :: Provable a => [SMTConfig] -> a -> IO (Solver, NominalDiffTime, ThmResult) Source #
Prove a property with multiple solvers, running them in separate threads. Only the result of the first one to finish will be returned, remaining threads will be killed. Note that we send an exception to the losing processes, but we do *not* actually wait for them to finish. In rare cases this can lead to zombie processes. In previous experiments, we found that some processes take their time to terminate. So, this solution favors quick turnaround.
proveWithAll :: Provable a => [SMTConfig] -> a -> IO [(Solver, NominalDiffTime, ThmResult)] Source #
Prove a property with multiple solvers, running them in separate threads. The results will be returned in the order produced.
proveConcurrentWithAny :: Provable a => SMTConfig -> [Query b] -> a -> IO (Solver, NominalDiffTime, ThmResult) Source #
Prove a property by running many queries each isolated to their own thread concurrently and return the first that finishes, killing the others
proveConcurrentWithAll :: Provable a => SMTConfig -> [Query b] -> a -> IO [(Solver, NominalDiffTime, ThmResult)] Source #
Prove a property by running many queries each isolated to their own thread concurrently and wait for each to finish returning all results
satWithAny :: Satisfiable a => [SMTConfig] -> a -> IO (Solver, NominalDiffTime, SatResult) Source #
Find a satisfying assignment to a property with multiple solvers, running them in separate threads. Only the result of the first one to finish will be returned, remaining threads will be killed. Note that we send an exception to the losing processes, but we do *not* actually wait for them to finish. In rare cases this can lead to zombie processes. In previous experiments, we found that some processes take their time to terminate. So, this solution favors quick turnaround.
satWithAll :: Satisfiable a => [SMTConfig] -> a -> IO [(Solver, NominalDiffTime, SatResult)] Source #
Find a satisfying assignment to a property with multiple solvers, running them in separate threads. The results will be returned in the order produced.
satConcurrentWithAny :: Satisfiable a => SMTConfig -> [Query b] -> a -> IO (Solver, NominalDiffTime, SatResult) Source #
Find a satisfying assignment to a property using a single solver, but providing several query problems of interest, with each query running in a separate thread and return the first one that returns. This can be useful to use symbolic mode to drive to a location in the search space of the solver and then refine the problem in query mode. If the computation is very hard to solve for the solver than running in concurrent mode may provide a large performance benefit.
satConcurrentWithAll :: Satisfiable a => SMTConfig -> [Query b] -> a -> IO [(Solver, NominalDiffTime, SatResult)] Source #
Find a satisfying assignment to a property using a single solver, but run
each query problem in a separate isolated thread and wait for each thread to
finish. See satConcurrentWithAny for more details.
generateSMTBenchmarkSat :: SatisfiableM m a => a -> m String Source #
Create an SMT-Lib2 benchmark, for a SAT query.
generateSMTBenchmarkProof :: ProvableM m a => a -> m String Source #
Create an SMT-Lib2 benchmark, for a Proof query.
solve :: [SBool] -> Symbolic SBool Source #
Form the symbolic conjunction of a given list of boolean conditions. Useful in expressing problems with constraints, like the following:
sat $ do [x, y, z] <- sIntegers ["x", "y", "z"]
solve [x .> 5, y + z .< x]
NB. For a version which generalizes over the underlying monad, see solve
Constraints and Quantifiers
A constraint is a means for restricting the input domain of a formula. Here's a simple example:
do x <-free"x" y <-free"y"constrain$ x .> yconstrain$ x + y .>= 12constrain$ y .>= 3 ...
The first constraint requires x to be larger than y. The scond one says that
sum of x and y must be at least 12, and the final one says that y to be at least 3.
Constraints provide an easy way to assert additional properties on the input domain, right at the point of
the introduction of variables.
Note that the proper reading of a constraint depends on the context:
- In a
sat(orallSat) call: The constraint added is asserted conjunctively. That is, the resulting satisfying model (if any) will always satisfy all the constraints given. - In a
provecall: In this case, the constraint acts as an implication. The property is proved under the assumption that the constraint holds. In other words, the constraint says that we only care about the input space that satisfies the constraint. - In a
quickCheckcall: The constraint acts as a filter forquickCheck; if the constraint does not hold, then the input value is considered to be irrelevant and is skipped. Note that this is similar toprove, but is stronger: We do not accept a test case to be valid just because the constraints fail on them, although semantically the implication does hold. We simply skip that test case as a bad test vector. - In a
genTestcall: Similar toquickCheckandprove: If a constraint does not hold, the input value is ignored and is not included in the test set.
A good use case (in fact the motivating use case) for constrain is attaching a
constraint to a variable at the time of its creation.
Also, the conjunctive semantics for sat and the implicative
semantics for prove simplify programming by choosing the correct interpretation
automatically. However, one should be aware of the semantic difference. For instance, in
the presence of constraints, formulas that are provable are not necessarily
satisfiable. To wit, consider:
do x <-free"x"constrain$ x .< x return $ x .< (x ::SWord8)
This predicate is unsatisfiable since no element of SWord8 is less than itself. But
it's (vacuously) true, since it excludes the entire domain of values, thus making the proof
trivial. Hence, this predicate is provable, but is not satisfiable. To make sure the given
constraints are not vacuous, the functions isVacuousProof (and isVacuousProofWith) can be used.
Also note that this semantics imply that test case generation (genTest) and
quick-check can take arbitrarily long in the presence of constraints, if the random input values generated
rarely satisfy the constraints. (As an extreme case, consider constrain sFalse
constrain :: (SolverContext m, QuantifiedBool a) => a -> m () Source #
Add a constraint, any satisfying instance must satisfy this condition.
softConstrain :: (SolverContext m, QuantifiedBool a) => a -> m () Source #
Add a soft constraint. The solver will try to satisfy this condition if possible, but won't if it cannot.
Quantified constraints, quantifier elimination, and skolemization
You can write quantified formulas, and reason with them as in first-order logic. Here is a simple example:
constrain $ \(Forall x) (Exists y) -> y .> (x :: SInteger)
You can nest quantifiers as you wish, and the quantified parameters can be of arbitrary symbolic type.
Additionally, you can convert such a quantified formula to a regular boolean, via a call to quantifiedBool
function, essentially performing quantifier elimination:
other_condition .&& quantifiedBool (\(Forall x) (Exists y) -> y .> (x :: SInteger))
Or you can prove/sat quantified formulas directly:
prove $ \(Forall x) (Exists y) -> y .> (x :: SInteger)
This facility makes quantifiers part of the regular SBV language, allowing them to be mixed/matched with all your other symbolic computations. See the following files demonstrating reasoning with quantifiers:
- Documentation.SBV.Examples.Puzzles.Birthday
- Documentation.SBV.Examples.Puzzles.KnightsAndKnaves
- Documentation.SBV.Examples.Puzzles.Rabbits
- Documentation.SBV.Examples.Misc.FirstOrderLogic
SBV also supports the constructors ExistsUnique to create unique existentials, in addition to
ForallN and ExistsN for creating multiple variables at the same time.
In general, SBV will not display the values of quantified variables for a satisfying instance. For a satisfiability problem, you can apply skolemization manually to have these values computed by the backend solver. Note that skolemization will produce functions for existentials under universals, and SBV generally cannot translate function values back to Haskell, except in certain simple cases. However, for prefix existentials, the manual transformation can pay off. As an example, compare:
>>>sat $ \(Exists x) (Forall y) -> x .<= (y :: SWord8)Satisfiable
to:
>>>sat $ do { x <- free "x"; pure (quantifiedBool (\(Forall y) -> x .<= (y :: SWord8))) }Satisfiable. Model: x = 0 :: Word8
where we have skolemized the top-level existential out, and received a witness value for it.
class QuantifiedBool a Source #
A value that can be used as a quantified boolean
Minimal complete definition
Instances
| QuantifiedBool SBool Source # | Base case of quantification, simple booleans |
Defined in Data.SBV.Core.Data Methods quantifiedBool :: SBool -> SBool Source # | |
| Constraint Symbolic a => QuantifiedBool a Source # | A constraint can be turned into a boolean |
Defined in Data.SBV.Lambda Methods quantifiedBool :: a -> SBool Source # | |
quantifiedBool :: QuantifiedBool a => a -> SBool Source #
Turn a quantified boolean into a regular boolean. That is, this function turns an exists/forall quantified formula to a simple boolean that can be used as a regular boolean value. An example is:
quantifiedBool $ \(Forall x) (Exists y) -> y .> (x :: SInteger)
is equivalent to sTrue. You can think of this function as performing quantifier-elimination: It takes
a quantified formula, and reduces it to a simple boolean that is equivalent to it, but has no quantifiers.
newtype Forall (nm :: Symbol) a Source #
A universal symbolic variable, used in building quantified constraints. The name attached via the symbol is used during skolemization. It names the corresponding argument to the skolem-functions within the scope of this quantifier.
Instances
newtype Exists (nm :: Symbol) a Source #
An existential symbolic variable, used in building quantified constraints. The name attached via the symbol is used during skolemization to create a skolem-function name when this variable is eliminated.
Instances
newtype ExistsUnique (nm :: Symbol) a Source #
An existential unique symbolic variable, used in building quantified constraints. The name
attached via the symbol is used during skolemization. It's split into two extra names, suffixed
_eu1 and _eu2, to name the universals in the equivalent formula:
\(\exists! x\,P(x)\Leftrightarrow \exists x\,P(x) \land \forall x_{eu1} \forall x_{eu2} (P(x_{eu1}) \land P(x_{eu2}) \Rightarrow x_{eu1} = x_{eu2}) \)
Constructors
| ExistsUnique (SBV a) |
Instances
newtype ForallN (n :: Nat) (nm :: Symbol) a Source #
Exactly n universal symbolic variables, used in in building quantified constraints. The name attached
will be prefixed in front of _1, _2, ..., _n to form the names of the variables.
Instances
newtype ExistsN (n :: Nat) (nm :: Symbol) a Source #
Exactly n existential symbolic variables, used in building quantified constraints. The name attached
will be prefixed in front of _1, _2, ..., _n to form the names of the variables.
Instances
Class of things that we can logically negate
Methods
qNot :: a -> NegatesTo a Source #
Negation of a quantified formula. This operation essentially lifts sNot to quantified formulae.
Note that you can achieve the same using sNot . quantifiedBool
Instances
| QNot SBool Source # | Base case; pure symbolic boolean |
| QNot r => QNot (Exists nm a -> r) Source # | Negate over an existential quantifier. Switches to universal. |
| QNot r => QNot (ExistsN nm n a -> r) Source # | Negate over a number of existential quantifiers |
| (QNot r, QuantifiedBool r, EqSymbolic (SBV a)) => QNot (ExistsUnique nm a -> r) Source # | Negate over an unique existential quantifier |
Defined in Data.SBV.Core.Data Associated Types type NegatesTo (ExistsUnique nm a -> r) Source # Methods qNot :: (ExistsUnique nm a -> r) -> NegatesTo (ExistsUnique nm a -> r) Source # | |
| QNot r => QNot (Forall nm a -> r) Source # | Negate over a universal quantifier. Switches to existential. |
| QNot r => QNot (ForallN nm n a -> r) Source # | Negate over a number of universal quantifiers |
class Skolemize a where Source #
A class of values that can be skolemized. Note that we don't export this class. Use
the skolemize function instead.
Minimal complete definition
skolem
Methods
skolemize :: (Constraint Symbolic (SkolemsTo a), Skolemize a) => a -> SkolemsTo a Source #
Skolemization. For any formula, skolemization gives back an equisatisfiable formula that has no existential quantifiers in it. You have to provide enough names for all the existentials in the argument. (Extras OK, so you can pass an infinite list if you like.) The names should be distinct, and also different from any other uninterpreted name you might have elsewhere.
taggedSkolemize :: (Constraint Symbolic (SkolemsTo a), Skolemize a) => String -> a -> SkolemsTo a Source #
If you use the same names for skolemized arguments in different functions, they will
collide; which is undesirable. Unfortunately there's no easy way for SBV to detect this.
In such cases, use taggedSkolemize to add a scope to the skolem-function names generated.
Instances
| Skolemize (SBV a) Source # | Base case; pure symbolic values |
| (HasKind a, KnownSymbol nm, Skolemize r) => Skolemize (Exists nm a -> r) Source # | Skolemize over an existential quantifier |
Defined in Data.SBV.Core.Data Associated Types type SkolemsTo (Exists nm a -> r) | |
| (HasKind a, KnownNat n, KnownSymbol nm, Skolemize r) => Skolemize (ExistsN n nm a -> r) Source # | Skolemize over a number of existential quantifiers |
Defined in Data.SBV.Core.Data Associated Types type SkolemsTo (ExistsN n nm a -> r) | |
| (HasKind a, EqSymbolic (SBV a), KnownSymbol nm, QuantifiedBool r, Skolemize (Forall (AppendSymbol nm "_eu1") a -> Forall (AppendSymbol nm "_eu2") a -> SBool)) => Skolemize (ExistsUnique nm a -> r) Source # | Skolemize over a unique existential quantifier |
Defined in Data.SBV.Core.Data Associated Types type SkolemsTo (ExistsUnique nm a -> r) Methods skolem :: String -> [(SVal, String)] -> (ExistsUnique nm a -> r) -> SkolemsTo (ExistsUnique nm a -> r) skolemize :: (ExistsUnique nm a -> r) -> SkolemsTo (ExistsUnique nm a -> r) Source # taggedSkolemize :: String -> (ExistsUnique nm a -> r) -> SkolemsTo (ExistsUnique nm a -> r) Source # | |
| (KnownSymbol nm, Skolemize r) => Skolemize (Forall nm a -> r) Source # | Skolemize over a universal quantifier |
Defined in Data.SBV.Core.Data Associated Types type SkolemsTo (Forall nm a -> r) | |
| (KnownSymbol nm, Skolemize r) => Skolemize (ForallN n nm a -> r) Source # | Skolemize over a number of universal quantifiers |
Defined in Data.SBV.Core.Data Associated Types type SkolemsTo (ForallN n nm a -> r) | |
Constraint Vacuity
When adding constraints, one has to be careful about
making sure they are not inconsistent. The function isVacuousProof can be used for this purpose.
Here is an example. Consider the following predicate:
>>>let pred = do { x <- free "x"; constrain $ x .< x; return $ x .>= (5 :: SWord8) }
This predicate asserts that all 8-bit values are larger than 5, subject to the constraint that the
values considered satisfy x .< x, i.e., they are less than themselves. Since there are no values that
satisfy this constraint, the proof will pass vacuously:
>>>prove predQ.E.D.
We can use isVacuousProof to make sure to see that the pass was vacuous:
>>>isVacuousProof predTrue
While the above example is trivial, things can get complicated if there are multiple constraints with non-straightforward relations; so if constraints are used one should make sure to check the predicate is not vacuously true. Here's an example that is not vacuous:
>>>let pred' = do { x <- free "x"; constrain $ x .> 6; return $ x .>= (5 :: SWord8) }
This time the proof passes as expected:
>>>prove pred'Q.E.D.
And the proof is not vacuous:
>>>isVacuousProof pred'False
Named constraints and attributes
Constraints can be given names:
namedConstraint "a is at least 5" $ a .>= 5Similarly, arbitrary term attributes can also be associated:
constrainWithAttribute [(":solver-specific-attribute", "value")] $ a .>= 5Note that a namedConstraint is equivalent to a constrainWithAttribute call, setting the `":named"' attribute.
namedConstraint :: (SolverContext m, QuantifiedBool a) => String -> a -> m () Source #
Add a named constraint. The name is used in unsat-core extraction.
constrainWithAttribute :: (SolverContext m, QuantifiedBool a) => [(String, String)] -> a -> m () Source #
Add a constraint, with arbitrary attributes.
Unsat cores
Named constraints are useful when used in conjunction with getUnsatCore function
where the backend solver can be queried to obtain an unsat core in case the constraints are unsatisfiable.
See getUnsatCore for details and Documentation.SBV.Examples.Queries.UnsatCore for an example use case.
Cardinality constraints
A pseudo-boolean function (http://en.wikipedia.org/wiki/Pseudo-Boolean_function) is a
function from booleans to reals, basically treating True as 1 and False as 0. They
are typically expressed in polynomial form. Such functions can be used to express cardinality
constraints, where we want to count how many things satisfy a certain condition.
One can code such constraints using regular SBV programming: Simply walk over the booleans and the corresponding coefficients, and assert the required relation. For instance:
[b0, b1, b2, b3] `pbAtMost` 2
is precisely equivalent to:
sum (map (\b -> ite b 1 0) [b0, b1, b2, b3]) .<= 2
and they both express that at most two of b0, b1, b2, and b3 can be sTrue.
However, the equivalent forms give rise to long formulas and the cardinality constraint
can get lost in the translation. The idea here is that if you use these functions instead, SBV will
produce better translations to SMTLib for more efficient solving of cardinality constraints, assuming
the backend solver supports them. Currently, only Z3 supports pseudo-booleans directly. For all other solvers,
SBV will translate these to equivalent terms that do not require special functions.
Checking safety
The sAssert function allows users to introduce invariants to make sure
certain properties hold at all times. This is another mechanism to provide further documentation/contract info
into SBV code. The functions safe and safeWith can be used to statically discharge these proof assumptions.
If a violation is found, SBV will print a model showing which inputs lead to the invariant being violated.
Here's a simple example. Let's assume we have a function that does subtraction, and requires its first argument to be larger than the second:
>>>let sub x y = sAssert Nothing "sub: x >= y must hold!" (x .>= y) (x - y)
Clearly, this function is not safe, as there's nothing that stops us from passing it a larger second argument.
We can use safe to statically see if such a violation is possible before we use this function elsewhere.
>>>safe (sub :: SInt8 -> SInt8 -> SInt8)[sub: x >= y must hold!: Violated. Model: s0 = 0 :: Int8 s1 = 1 :: Int8]
What happens if we make sure to arrange for this invariant? Consider this version:
>>>let safeSub x y = ite (x .>= y) (sub x y) 0
Clearly, safeSub must be safe. And indeed, SBV can prove that:
>>>safe (safeSub :: SInt8 -> SInt8 -> SInt8)[sub: x >= y must hold!: No violations detected]
Note how we used sub and safeSub polymorphically. We only need to monomorphise our types when a proof
attempt is done, as we did in the safe calls.
If required, the user can pass a CallStack through the first argument to sAssert, which will be used
by SBV to print a diagnostic info to pinpoint the failure.
Also see Documentation.SBV.Examples.Misc.NoDiv0 for the classic div-by-zero example.
sAssert :: HasKind a => Maybe CallStack -> String -> SBool -> SBV a -> SBV a Source #
Symbolic assert. Check that the given boolean condition is always sTrue in the given path. The
optional first argument can be used to provide call-stack info via GHC's location facilities.
isSafe :: SafeResult -> Bool Source #
Check if a safe-call was safe or not, turning a SafeResult to a Bool.
class ExtractIO m => SExecutable m a Source #
Symbolically executable program fragments. This class is mainly used for safe calls, and is sufficiently populated internally to cover most use
cases. Users can extend it as they wish to allow safe checks for SBV programs that return/take types that are user-defined.
Minimal complete definition
Instances
sName :: SExecutable IO a => a -> Symbolic () Source #
Create an argument for a name used in a safety-checking call.
NB. For a version which generalizes over the underlying monad, see sName_
safe :: SExecutable IO a => a -> IO [SafeResult] Source #
Check safety using the default solver.
NB. For a version which generalizes over the underlying monad, see safe
safeWith :: SExecutable IO a => SMTConfig -> a -> IO [SafeResult] Source #
Quick-checking
sbvQuickCheck :: Symbolic SBool -> IO Bool Source #
Quick check an SBV property. Note that a regular quickCheck call will work just as
well. Use this variant if you want to receive the boolean result.
Optimization
SBV can optimize metric functions, i.e., those that generate both bounded SIntN, SWordN, and unbounded SInteger
types, along with those produce SReals. That is, it can find models satisfying all the constraints while minimizing
or maximizing user given metrics. Currently, optimization requires the use of the z3 SMT solver as the backend,
and a good review of these features is given
in this paper: http://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/nbjorner-scss2014.pdf.
Goals can be lexicographically (default), independently, or pareto-front optimized. The relevant functions are:
Goals can be optimized at a regular or an extended value: An extended value is either positive or negative infinity (for unbounded integers and reals) or positive or negative epsilon differential from a real value (for reals).
For instance, a call of the form
minimize "name-of-goal" $ x + 2*yminimizes the arithmetic goal x+2*y, where x and y can be signed/unsigned bit-vectors, reals,
or integers.
A simple example
Here's an optimization example in action:
>>>optimize Lexicographic $ \x y -> minimize "goal" (x+2*(y::SInteger))Optimal in an extension field: goal = -oo :: Integer
We will describe the role of the constructor Lexicographic shortly.
Of course, this becomes more useful when the result is not in an extension field:
>>>:{optimize Lexicographic $ do x <- sInteger "x" y <- sInteger "y" constrain $ x .> 0 constrain $ x .< 6 constrain $ y .> 2 constrain $ y .< 12 minimize "goal" $ x + 2 * y :} Optimal model: x = 1 :: Integer y = 3 :: Integer goal = 7 :: Integer
As usual, the programmatic API can be used to extract the values of objectives and model-values (getModelObjectives,
getModelAssignment, etc.) to access these values and program with them further.
The following examples illustrate the use of basic optimization routines:
- Documentation.SBV.Examples.Optimization.LinearOpt: Simple linear-optimization example.
- Documentation.SBV.Examples.Optimization.Production: Scheduling machines in a shop
- Documentation.SBV.Examples.Optimization.VM: Scheduling virtual-machines in a data-center
Multiple optimization goals
Multiple goals can be specified, using the same syntax. In this case, the user gets to pick what style of
optimization to perform, by passing the relevant OptimizeStyle as the first argument to optimize.
- [
Lexicographic]. The solver will optimize the goals in the given order, optimizing the latter ones under the model that optimizes the previous ones. - [
Independent]. The solver will optimize the goals independently of each other. In this case the user will be presented a model for each goal given. - [
Pareto]. Finally, the user can query for pareto-fronts. A pareto front is an model such that no goal can be made "better" without making some other goal "worse."
Pareto fronts only make sense when the objectives are bounded. If there are unbounded objective values, then the
backend solver can loop infinitely. (This is what z3 does currently.) If you are not sure the objectives are
bounded, you should first use Independent mode to ensure the objectives are bounded, and then switch to
pareto-mode to extract them further.
The optional number argument to Pareto specifies the maximum number of pareto-fronts the user is asking
to get. If Nothing, SBV will query for all pareto-fronts. Note that pareto-fronts can be really large,
so if Nothing is used, there is a potential for waiting indefinitely for the SBV-solver interaction to finish. (If
you suspect this might be the case, run in verbose mode to see the interaction and put a limiting factor
appropriately.)
data OptimizeStyle Source #
Style of optimization. Note that in the pareto case the user is allowed
to specify a max number of fronts to query the solver for, since there might
potentially be an infinite number of them and there is no way to know exactly
how many ahead of time. If Nothing is given, SBV will possibly loop forever
if the number is really infinite.
Constructors
| Lexicographic | Objectives are optimized in the order given, earlier objectives have higher priority. |
| Independent | Each objective is optimized independently. |
| Pareto (Maybe Int) | Objectives are optimized according to pareto front: That is, no objective can be made better without making some other worse. |
Instances
| Show OptimizeStyle Source # | |
Defined in Data.SBV.Core.Symbolic Methods showsPrec :: Int -> OptimizeStyle -> ShowS # show :: OptimizeStyle -> String # showList :: [OptimizeStyle] -> ShowS # | |
| NFData OptimizeStyle Source # | |
Defined in Data.SBV.Core.Symbolic Methods rnf :: OptimizeStyle -> () # | |
| Eq OptimizeStyle Source # | |
Defined in Data.SBV.Core.Symbolic Methods (==) :: OptimizeStyle -> OptimizeStyle -> Bool # (/=) :: OptimizeStyle -> OptimizeStyle -> Bool # | |
Objectives and Metrics
Objective of optimization. We can minimize, maximize, or give a soft assertion with a penalty for not satisfying it.
Constructors
| Minimize String a | Minimize this metric |
| Maximize String a | Maximize this metric |
| AssertWithPenalty String a Penalty | A soft assertion, with an associated penalty |
Class of metrics we can optimize for. Currently, booleans,
bounded signed/unsigned bit-vectors, unbounded integers,
algebraic reals and floats can be optimized. You can add
your instances, but bewared that the MetricSpace should
map your type to something the backend solver understands, which
are limited to unsigned bit-vectors, reals, and unbounded integers
for z3.
A good reference on these features is given in the following paper: http://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/nbjorner-scss2014.pdf.
Minimal completion: None. However, if MetricSpace is not identical to the type, you want
to define toMetricSpace and possibly minimize/maximize to add extra constraints as necessary.
Minimal complete definition
Nothing
Associated Types
type MetricSpace a :: Type Source #
The metric space we optimize the goal over. Usually the same as the type itself, but not always!
For instance, signed bit-vectors are optimized over their unsigned counterparts, floats are
optimized over their Word32 comparable counterparts, etc.
type MetricSpace a = a
Methods
toMetricSpace :: SBV a -> SBV (MetricSpace a) Source #
Compute the metric value to optimize.
default toMetricSpace :: a ~ MetricSpace a => SBV a -> SBV (MetricSpace a) Source #
fromMetricSpace :: SBV (MetricSpace a) -> SBV a Source #
Compute the value itself from the metric corresponding to it.
default fromMetricSpace :: a ~ MetricSpace a => SBV (MetricSpace a) -> SBV a Source #
msMinimize :: (MonadSymbolic m, SolverContext m) => String -> SBV a -> m () Source #
Minimizing a metric space
msMaximize :: (MonadSymbolic m, SolverContext m) => String -> SBV a -> m () Source #
Maximizing a metric space
Instances
minimize :: Metric a => String -> SBV a -> Symbolic () Source #
Minimize a named metric
NB. For a version which generalizes over the underlying monad, see minimize
maximize :: Metric a => String -> SBV a -> Symbolic () Source #
Maximize a named metric
NB. For a version which generalizes over the underlying monad, see maximize
Soft assertions
Related to optimization, SBV implements soft-asserts via assertWithPenalty calls. A soft assertion
is a hint to the SMT solver that we would like a particular condition to hold if **possible*.
That is, if there is a solution satisfying it, then we would like it to hold, but it can be violated
if there is no way to satisfy it. Each soft-assertion can be associated with a numeric penalty for
not satisfying it, hence turning it into an optimization problem.
Note that assertWithPenalty works well with optimization goals (minimize/maximize etc.),
and are most useful when we are optimizing a metric and thus some of the constraints
can be relaxed with a penalty to obtain a good solution. Again
see http://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/nbjorner-scss2014.pdf
for a good overview of the features in Z3 that SBV is providing the bridge for.
A soft assertion can be specified in one of the following three main ways:
assertWithPenalty "bounded_x" (x .< 5) DefaultPenalty
assertWithPenalty "bounded_x" (x .< 5) (Penalty 2.3 Nothing)
assertWithPenalty "bounded_x" (x .< 5) (Penalty 4.7 (Just "group-1"))In the first form, we are saying that the constraint x .< 5 must be satisfied, if possible,
but if this constraint can not be satisfied to find a model, it can be violated with the default penalty of 1.
In the second case, we are associating a penalty value of 2.3.
Finally in the third case, we are also associating this constraint with a group. The group name is only needed if we have classes of soft-constraints that should be considered together.
assertWithPenalty :: String -> SBool -> Penalty -> Symbolic () Source #
Introduce a soft assertion, with an optional penalty
NB. For a version which generalizes over the underlying monad, see assertWithPenalty
Penalty for a soft-assertion. The default penalty is 1, with all soft-assertions belonging
to the same objective goal. A positive weight and an optional group can be provided by using
the Penalty constructor.
Constructors
| DefaultPenalty | Default: Penalty of |
| Penalty Rational (Maybe String) | Penalty with a weight and an optional group |
Field extensions
If an optimization results in an infinity/epsilon value, the returned CV value will be in the corresponding extension field.
A simple expression type over extended values, covering infinity, epsilon and intervals.
Constructors
| Infinite Kind | |
| Epsilon Kind | |
| Interval ExtCV ExtCV | |
| BoundedCV CV | |
| AddExtCV ExtCV ExtCV | |
| MulExtCV ExtCV ExtCV |
Instances
| Show ExtCV Source # | Show instance, shows with the kind |
| HasKind ExtCV Source # | Kind instance for Extended CV |
Defined in Data.SBV.Core.Concrete Methods kindOf :: ExtCV -> Kind Source # hasSign :: ExtCV -> Bool Source # intSizeOf :: ExtCV -> Int Source # isBoolean :: ExtCV -> Bool Source # isBounded :: ExtCV -> Bool Source # isReal :: ExtCV -> Bool Source # isFloat :: ExtCV -> Bool Source # isDouble :: ExtCV -> Bool Source # isRational :: ExtCV -> Bool Source # isFP :: ExtCV -> Bool Source # isUnbounded :: ExtCV -> Bool Source # isUserSort :: ExtCV -> Bool Source # isChar :: ExtCV -> Bool Source # isString :: ExtCV -> Bool Source # isList :: ExtCV -> Bool Source # isSet :: ExtCV -> Bool Source # isTuple :: ExtCV -> Bool Source # isMaybe :: ExtCV -> Bool Source # | |
data GeneralizedCV Source #
A generalized CV allows for expressions involving infinite and epsilon values/intervals Used in optimization problems.
Constructors
| ExtendedCV ExtCV | |
| RegularCV CV |
Instances
Model extraction
The default Show instances for prover calls provide all the counter-example information in a
human-readable form and should be sufficient for most casual uses of sbv. However, tools built
on top of sbv will inevitably need to look into the constructed models more deeply, programmatically
extracting their results and performing actions based on them. The API provided in this section
aims at simplifying this task.
Inspecting proof results
ThmResult, SatResult, and AllSatResult are simple newtype wrappers over SMTResult. Their
main purpose is so that we can provide custom Show instances to print results accordingly.
Instances
| Show ThmResult Source # | |
| NFData ThmResult Source # | |
Defined in Data.SBV.SMT.SMT | |
| Modelable ThmResult Source # |
|
Defined in Data.SBV.SMT.SMT Methods modelExists :: ThmResult -> Bool Source # getModelAssignment :: SatModel b => ThmResult -> Either String (Bool, b) Source # getModelDictionary :: ThmResult -> Map String CV Source # getModelValue :: SymVal b => String -> ThmResult -> Maybe b Source # getModelUninterpretedValue :: String -> ThmResult -> Maybe String Source # extractModel :: SatModel b => ThmResult -> Maybe b Source # getModelObjectives :: ThmResult -> Map String GeneralizedCV Source # getModelObjectiveValue :: String -> ThmResult -> Maybe GeneralizedCV Source # getModelUIFuns :: ThmResult -> Map String (SBVType, Either String ([([CV], CV)], CV)) Source # getModelUIFunValue :: String -> ThmResult -> Maybe (SBVType, Either String ([([CV], CV)], CV)) Source # | |
A sat call results in a SatResult
The reason for having a separate SatResult is to have a more meaningful Show instance.
Instances
| Show SatResult Source # | |
| NFData SatResult Source # | |
Defined in Data.SBV.SMT.SMT | |
| Modelable SatResult Source # |
|
Defined in Data.SBV.SMT.SMT Methods modelExists :: SatResult -> Bool Source # getModelAssignment :: SatModel b => SatResult -> Either String (Bool, b) Source # getModelDictionary :: SatResult -> Map String CV Source # getModelValue :: SymVal b => String -> SatResult -> Maybe b Source # getModelUninterpretedValue :: String -> SatResult -> Maybe String Source # extractModel :: SatModel b => SatResult -> Maybe b Source # getModelObjectives :: SatResult -> Map String GeneralizedCV Source # getModelObjectiveValue :: String -> SatResult -> Maybe GeneralizedCV Source # getModelUIFuns :: SatResult -> Map String (SBVType, Either String ([([CV], CV)], CV)) Source # getModelUIFunValue :: String -> SatResult -> Maybe (SBVType, Either String ([([CV], CV)], CV)) Source # | |
data AllSatResult Source #
An allSat call results in a AllSatResult
Constructors
| AllSatResult | |
Fields
| |
Instances
| Show AllSatResult Source # | |
Defined in Data.SBV.SMT.SMT Methods showsPrec :: Int -> AllSatResult -> ShowS # show :: AllSatResult -> String # showList :: [AllSatResult] -> ShowS # | |
newtype SafeResult Source #
A safe call results in a SafeResult
Constructors
| SafeResult (Maybe String, String, SMTResult) |
Instances
| Show SafeResult Source # | |
Defined in Data.SBV.SMT.SMT Methods showsPrec :: Int -> SafeResult -> ShowS # show :: SafeResult -> String # showList :: [SafeResult] -> ShowS # | |
data OptimizeResult Source #
An optimize call results in a OptimizeResult. In the ParetoResult case, the boolean is True
if we reached pareto-query limit and so there might be more unqueried results remaining. If False,
it means that we have all the pareto fronts returned. See the Pareto OptimizeStyle for details.
Constructors
| LexicographicResult SMTResult | |
| ParetoResult (Bool, [SMTResult]) | |
| IndependentResult [(String, SMTResult)] |
Instances
| Show OptimizeResult Source # | |
Defined in Data.SBV.SMT.SMT Methods showsPrec :: Int -> OptimizeResult -> ShowS # show :: OptimizeResult -> String # showList :: [OptimizeResult] -> ShowS # | |
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.)
Constructors
| Unsatisfiable SMTConfig (Maybe [String]) | Unsatisfiable. If unsat-cores are enabled, they will be returned in the second parameter. |
| Satisfiable SMTConfig SMTModel | Satisfiable with model |
| DeltaSat SMTConfig (Maybe String) SMTModel | Delta satisfiable with queried string if available and model |
| SatExtField SMTConfig SMTModel | Prover returned a model, but in an extension field containing Infinite/epsilon |
| Unknown SMTConfig SMTReasonUnknown | Prover returned unknown, with the given reason |
| ProofError SMTConfig [String] (Maybe SMTResult) | Prover errored out, with possibly a bogus result |
Instances
| NFData SMTResult Source # | |
Defined in Data.SBV.Core.Symbolic | |
| Modelable SMTResult Source # |
|
Defined in Data.SBV.SMT.SMT Methods modelExists :: SMTResult -> Bool Source # getModelAssignment :: SatModel b => SMTResult -> Either String (Bool, b) Source # getModelDictionary :: SMTResult -> Map String CV Source # getModelValue :: SymVal b => String -> SMTResult -> Maybe b Source # getModelUninterpretedValue :: String -> SMTResult -> Maybe String Source # extractModel :: SatModel b => SMTResult -> Maybe b Source # getModelObjectives :: SMTResult -> Map String GeneralizedCV Source # getModelObjectiveValue :: String -> SMTResult -> Maybe GeneralizedCV Source # getModelUIFuns :: SMTResult -> Map String (SBVType, Either String ([([CV], CV)], CV)) Source # getModelUIFunValue :: String -> SMTResult -> Maybe (SBVType, Either String ([([CV], CV)], CV)) Source # | |
data SMTReasonUnknown Source #
Reason for reporting unknown.
Constructors
| UnknownMemOut | |
| UnknownIncomplete | |
| UnknownTimeOut | |
| UnknownOther String |
Instances
| Show SMTReasonUnknown Source # | Show instance for unknown |
Defined in Data.SBV.Control.Types Methods showsPrec :: Int -> SMTReasonUnknown -> ShowS # show :: SMTReasonUnknown -> String # showList :: [SMTReasonUnknown] -> ShowS # | |
| NFData SMTReasonUnknown Source # | Trivial rnf instance |
Defined in Data.SBV.Control.Types Methods rnf :: SMTReasonUnknown -> () # | |
Observing expressions
The observe command can be used to trace values of arbitrary expressions during a sat, prove, or perhaps more
importantly, in a quickCheck call with the sObserve variant.. This is useful for, for instance, recording expected vs. obtained expressions
as a symbolic program is executing.
>>>:{prove $ do a1 <- free "i1" a2 <- free "i2" let spec, res :: SWord8 spec = a1 + a2 res = ite (a1 .== 12 .&& a2 .== 22) -- insert a malicious bug! 1 (a1 + a2) return $ observe "Expected" spec .== observe "Result" res :} Falsifiable. Counter-example: Expected = 34 :: Word8 Result = 1 :: Word8 i1 = 12 :: Word8 i2 = 22 :: Word8
The observeIf variant allows the user to specify a boolean condition when the value is interesting to observe. Useful when
you have lots of "debugging" points, but not all are of interest. Use the sObserve variant when you are at the Symbolic
monad, which also supports quick-check applications.
observe :: SymVal a => String -> SBV a -> SBV a Source #
Observe the value of an expression, unconditionally. See observeIf for a generalized version.
observeIf :: SymVal a => (a -> Bool) -> String -> SBV a -> SBV a Source #
Observe the value of an expression, if the given condition holds. Such values are useful in model construction, as they are printed part of a satisfying model, or a
counter-example. The same works for quick-check as well. Useful when we want to see intermediate values, or expected/obtained
pairs in a particular run. Note that an observed expression is always symbolic, i.e., it won't be constant folded. Compare this to label
which is used for putting a label in the generated SMTLib-C code.
sObserve :: SymVal a => String -> SBV a -> Symbolic () Source #
A variant of observe that you can use at the top-level. This is useful with quick-check, for instance.
Programmable model extraction
While default Show instances are sufficient for most use cases, it is sometimes desirable (especially
for library construction) that the SMT-models are reinterpreted in terms of domain types. Programmable
extraction allows getting arbitrarily typed models out of SMT models.
class SatModel a where Source #
Instances of SatModel can be automatically extracted from models returned by the
solvers. The idea is that the sbv infrastructure provides a stream of CV's (constant values)
coming from the solver, and the type a is interpreted based on these constants. Many typical
instances are already provided, so new instances can be declared with relative ease.
Minimum complete definition: parseCVs
Minimal complete definition
Nothing
Methods
parseCVs :: [CV] -> Maybe (a, [CV]) Source #
Given a sequence of constant-words, extract one instance of the type a, returning
the remaining elements untouched. If the next element is not what's expected for this
type you should return Nothing
cvtModel :: (a -> Maybe b) -> Maybe (a, [CV]) -> Maybe (b, [CV]) Source #
Given a parsed model instance, transform it using f, and return the result.
The default definition for this method should be sufficient in most use cases.
Instances
class Modelable a where Source #
Various SMT results that we can extract models out of.
Minimal complete definition
modelExists, getModelAssignment, getModelDictionary, getModelObjectives, getModelUIFuns
Methods
modelExists :: a -> Bool Source #
Is there a model?
getModelAssignment :: SatModel b => a -> Either String (Bool, b) Source #
Extract assignments of a model, the result is a tuple where the first argument (if True) indicates whether the model was "probable". (i.e., if the solver returned unknown.)
getModelDictionary :: a -> Map String CV Source #
Extract a model dictionary. Extract a dictionary mapping the variables to
their respective values as returned by the SMT solver. Also see getModelDictionaries.
getModelValue :: SymVal b => String -> a -> Maybe b Source #
Extract a model value for a given element. Also see getModelValues.
getModelUninterpretedValue :: String -> a -> Maybe String Source #
Extract a representative name for the model value of an uninterpreted kind.
This is supposed to correspond to the value as computed internally by the
SMT solver; and is unportable from solver to solver. Also see getModelUninterpretedValues.
extractModel :: SatModel b => a -> Maybe b Source #
A simpler variant of getModelAssignment to get a model out without the fuss.
getModelObjectives :: a -> Map String GeneralizedCV Source #
Extract model objective values, for all optimization goals.
getModelObjectiveValue :: String -> a -> Maybe GeneralizedCV Source #
Extract the value of an objective
getModelUIFuns :: a -> Map String (SBVType, Either String ([([CV], CV)], CV)) Source #
Extract model uninterpreted-functions
getModelUIFunValue :: String -> a -> Maybe (SBVType, Either String ([([CV], CV)], CV)) Source #
Extract the value of an uninterpreted-function as an association list
Instances
displayModels :: SatModel a => ([(Bool, a)] -> [(Bool, a)]) -> (Int -> (Bool, a) -> IO ()) -> AllSatResult -> IO Int Source #
Given an allSat call, we typically want to iterate over it and print the results in sequence. The
displayModels function automates this task by calling disp on each result, consecutively. The first
Int argument to disp 'is the current model number. The second argument is a tuple, where the first
element indicates whether the model is alleged (i.e., if the solver is not sure, returning Unknown).
The arrange argument can sort the results in any way you like, if necessary.
extractModels :: SatModel a => AllSatResult -> [a] Source #
Return all the models from an allSat call, similar to extractModel but
is suitable for the case of multiple results.
getModelDictionaries :: AllSatResult -> [Map String CV] Source #
Get dictionaries from an all-sat call. Similar to getModelDictionary.
getModelValues :: SymVal b => String -> AllSatResult -> [Maybe b] Source #
Extract value of a variable from an all-sat call. Similar to getModelValue.
getModelUninterpretedValues :: String -> AllSatResult -> [Maybe String] Source #
Extract value of an uninterpreted variable from an all-sat call. Similar to getModelUninterpretedValue.
SMT Interface
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 end 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.
The printBase field can be used to print numbers in base 2, 10, or 16.
The crackNum field can be used to display numbers in detail, all its bits and how they are laid out in memory. Works with all bounded number types
(i.e., SWord and SInt), but also with floats. It is particularly useful with floating-point numbers, as it shows you how they are laid out in
memory following the IEEE754 rules.
Constructors
| SMTConfig | |
Fields
| |
Specify how to save timing information, if at all.
Constructors
| NoTiming | |
| PrintTiming | |
| SaveTiming (IORef NominalDiffTime) |
data SMTLibVersion Source #
Representation of SMTLib Program versions. As of June 2015, we're dropping support for SMTLib1, and supporting SMTLib2 only. We keep this data-type around in case SMTLib3 comes along and we want to support 2 and 3 simultaneously.
Constructors
| SMTLib2 |
Instances
| Bounded SMTLibVersion Source # | |
Defined in Data.SBV.Core.Symbolic | |
| Enum SMTLibVersion Source # | |
Defined in Data.SBV.Core.Symbolic Methods succ :: SMTLibVersion -> SMTLibVersion # pred :: SMTLibVersion -> SMTLibVersion # toEnum :: Int -> SMTLibVersion # fromEnum :: SMTLibVersion -> Int # enumFrom :: SMTLibVersion -> [SMTLibVersion] # enumFromThen :: SMTLibVersion -> SMTLibVersion -> [SMTLibVersion] # enumFromTo :: SMTLibVersion -> SMTLibVersion -> [SMTLibVersion] # enumFromThenTo :: SMTLibVersion -> SMTLibVersion -> SMTLibVersion -> [SMTLibVersion] # | |
| Show SMTLibVersion Source # | |
Defined in Data.SBV.Core.Symbolic Methods showsPrec :: Int -> SMTLibVersion -> ShowS # show :: SMTLibVersion -> String # showList :: [SMTLibVersion] -> ShowS # | |
| NFData SMTLibVersion Source # | |
Defined in Data.SBV.Core.Symbolic Methods rnf :: SMTLibVersion -> () # | |
| Eq SMTLibVersion Source # | |
Defined in Data.SBV.Core.Symbolic Methods (==) :: SMTLibVersion -> SMTLibVersion -> Bool # (/=) :: SMTLibVersion -> SMTLibVersion -> Bool # | |
Solvers that SBV is aware of
An SMT solver
Constructors
| SMTSolver | |
Fields
| |
Controlling verbosity
SBV provides various levels of verbosity to aid in debugging, by using the SMTConfig fields:
- [
verbose] Print on stdout a shortened account of what is sent/received. This is specifically trimmed to reduce noise and is good for quick debugging. The output is not supposed to be machine-readable. - [
redirectVerbose] Send the verbose output to a file. Note that you still have to set `verbose=True` for redirection to take effect. Otherwise, the output is the same as what you would see inverbose. - [
transcript] Produce a file that is valid SMTLib2 format, containing everything sent and received. In particular, one can directly feed this file to the SMT-solver outside of the SBV since it is machine-readable. This is good for offline analysis situations, where you want to have a full account of what happened. For instance, it will print time-stamps at every interaction point, so you can see how long each command took.
Solvers
Configurations
defaultSolverConfig :: Solver -> SMTConfig Source #
The default configs corresponding to supported SMT solvers
defaultSMTCfg :: SMTConfig Source #
The default solver used by SBV. This is currently set to z3.
defaultDeltaSMTCfg :: SMTConfig Source #
The default solver used by SBV for delta-satisfiability problems. This is currently set to dReal, which is also the only solver that supports delta-satisfiability.
sbvCheckSolverInstallation :: SMTConfig -> IO Bool Source #
Check whether the given solver is installed and is ready to go. This call does a simple call to the solver to ensure all is well.
getAvailableSolvers :: IO [SMTConfig] Source #
Return the known available solver configs, installed on your machine.
setLogic :: SolverContext m => Logic -> m () Source #
Set the logic.
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 a call to setLogic This is especially handy if one is experimenting with custom
logics that might be supported on new solvers. See http://smtlib.cs.uiowa.edu/logics.shtml for the official list.
Constructors
| 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. |
| 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_UFNIRA | Unquantified non-linear real integer arithmetic with uninterpreted sort and function symbols. |
| UFLRA | Linear real arithmetic with uninterpreted sort and function symbols. |
| UFNIA | Non-linear integer arithmetic with uninterpreted sort and function symbols. |
| QF_FPBV | Quantifier-free formulas over the theory of floating point numbers, arrays, and bit-vectors. |
| QF_FP | Quantifier-free formulas over the theory of floating point numbers. |
| QF_FD | Quantifier-free finite domains. |
| QF_S | Quantifier-free formulas over the theory of strings. |
| Logic_ALL | The catch-all value. |
| Logic_NONE | Use this value when you want SBV to simply not set the logic. |
| CustomLogic String | In case you need a really custom string! |
setOption :: SolverContext m => SMTOption -> m () Source #
Set an option.
setInfo :: SolverContext m => String -> [String] -> m () Source #
Set info. Example: setInfo ":status" ["unsat"].
setTimeOut :: SolverContext m => Integer -> m () Source #
Set a solver time-out value, in milli-seconds. This function
essentially translates to the SMTLib call (set-info :timeout val),
and your backend solver may or may not support it! The amount given
is in milliseconds. Also see the function timeOut for finer level
control of time-outs, directly from SBV.
SBV exceptions
data SBVException Source #
An exception thrown from SBV. If the solver ever responds with a non-success value for a command,
SBV will throw an SBVException, it so the user can process it as required. The provided Show instance
will render the failure nicely. Note that if you ever catch this exception, the solver is no longer alive:
You should either -- throw the exception up, or do other proper clean-up before continuing.
Constructors
Instances
| Exception SBVException Source # | SBVExceptions are throwable. A simple "show" will render this exception nicely though of course you can inspect the individual fields as necessary. |
Defined in Data.SBV.SMT.Utils Methods toException :: SBVException -> SomeException # fromException :: SomeException -> Maybe SBVException # displayException :: SBVException -> String # | |
| Show SBVException Source # | A fairly nice rendering of the exception, for display purposes. |
Defined in Data.SBV.SMT.Utils Methods showsPrec :: Int -> SBVException -> ShowS # show :: SBVException -> String # showList :: [SBVException] -> ShowS # | |
Abstract SBV type
The Symbolic value. The parameter a is phantom, but is
extremely important in keeping the user interface strongly typed.
Instances
class HasKind a where Source #
A class for capturing values that have a sign and a size (finite or infinite)
minimal complete definition: kindOf, unless you can take advantage of the default
signature: This class can be automatically derived for data-types that have
a Data instance; this is useful for creating uninterpreted sorts. So, in
reality, end users should almost never need to define any methods.
Minimal complete definition
Nothing
Methods
intSizeOf :: a -> Int Source #
isBoolean :: a -> Bool Source #
isBounded :: a -> Bool Source #
isDouble :: a -> Bool Source #
isRational :: a -> Bool Source #
isUnbounded :: a -> Bool Source #
isUserSort :: a -> Bool Source #
isString :: a -> Bool Source #
Instances
Kind of symbolic value
Constructors
| KBool | |
| KBounded !Bool !Int | |
| KUnbounded | |
| KReal | |
| KUserSort String (Maybe [String]) | |
| KFloat | |
| KDouble | |
| KFP !Int !Int | |
| KChar | |
| KString | |
| KList Kind | |
| KSet Kind | |
| KTuple [Kind] | |
| KMaybe Kind | |
| KRational | |
| KEither Kind Kind |
Instances
| Data Kind Source # | |
Defined in Data.SBV.Core.Kind Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Kind -> c Kind # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c Kind # dataTypeOf :: Kind -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c Kind) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Kind) # gmapT :: (forall b. Data b => b -> b) -> Kind -> Kind # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Kind -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Kind -> r # gmapQ :: (forall d. Data d => d -> u) -> Kind -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Kind -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Kind -> m Kind # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Kind -> m Kind # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Kind -> m Kind # | |
| Show Kind Source # | The interesting about the show instance is that it can tell apart two kinds nicely; since it conveniently
ignores the enumeration constructors. Also, when we construct a |
| NFData Kind Source # | |
Defined in Data.SBV.Core.Symbolic | |
| Eq Kind Source # | |
| Ord Kind Source # | |
| HasKind Kind Source # | |
Defined in Data.SBV.Core.Kind Methods kindOf :: Kind -> Kind Source # hasSign :: Kind -> Bool Source # intSizeOf :: Kind -> Int Source # isBoolean :: Kind -> Bool Source # isBounded :: Kind -> Bool Source # isReal :: Kind -> Bool Source # isFloat :: Kind -> Bool Source # isDouble :: Kind -> Bool Source # isRational :: Kind -> Bool Source # isUnbounded :: Kind -> Bool Source # isUserSort :: Kind -> Bool Source # isChar :: Kind -> Bool Source # isString :: Kind -> Bool Source # isList :: Kind -> Bool Source # isSet :: Kind -> Bool Source # isTuple :: Kind -> Bool Source # isMaybe :: Kind -> Bool Source # | |
class (HasKind a, Typeable a) => SymVal a Source #
A SymVal is a potential symbolic value that can be created instances of to be fed to a symbolic program.
Instances
free :: SymVal a => String -> Symbolic (SBV a) Source #
Create a free variable, universal in a proof, existential in sat
NB. For a version which generalizes over the underlying monad, see free
free_ :: SymVal a => Symbolic (SBV a) Source #
Create an unnamed free variable, universal in proof, existential in sat
NB. For a version which generalizes over the underlying monad, see free_
mkFreeVars :: SymVal a => Int -> Symbolic [SBV a] Source #
Create a bunch of free vars
NB. For a version which generalizes over the underlying monad, see mkFreeVars
symbolic :: SymVal a => String -> Symbolic (SBV a) Source #
Similar to free; Just a more convenient name
NB. For a version which generalizes over the underlying monad, see symbolic
symbolics :: SymVal a => [String] -> Symbolic [SBV a] Source #
Similar to mkFreeVars; but automatically gives names based on the strings
NB. For a version which generalizes over the underlying monad, see symbolics
isConcretely :: SymVal a => SBV a -> (a -> Bool) -> Bool Source #
Does it concretely satisfy the given predicate?
mkSymVal :: SymVal a => VarContext -> Maybe String -> Symbolic (SBV a) Source #
One stop allocator
NB. For a version which generalizes over the underlying monad, see mkSymVal
class MonadIO m => MonadSymbolic m where Source #
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.
Computations which support symbolic operations
Minimal complete definition
Nothing
Methods
symbolicEnv :: m State Source #
default symbolicEnv :: (MonadTrans t, MonadSymbolic m', m ~ t m') => m State Source #
Instances
A generalization of Symbolic.
Instances
label :: SymVal a => String -> SBV a -> SBV a Source #
label: Label the result of an expression. This is essentially a no-op, but useful as it generates a comment in the generated C/SMT-Lib code.
Note that if the argument is a constant, then the label is dropped completely, per the usual constant folding strategy. Compare this to observe
which is good for printing counter-examples.
output :: Outputtable a => a -> Symbolic a Source #
Mark an interim result as an output. Useful when constructing Symbolic programs that return multiple values, or when the result is programmatically computed.
NB. For a version which generalizes over the underlying monad, see output
runSMT :: Symbolic a -> IO a Source #
Run an arbitrary symbolic computation, equivalent to runSMTWith defaultSMTCfg
NB. For a version which generalizes over the underlying monad, see runSMT
runSMTWith :: SMTConfig -> Symbolic a -> IO a Source #
Runs an arbitrary symbolic computation, exposed to the user in SAT mode
NB. For a version which generalizes over the underlying monad, see runSMTWith
Module exports
The SBV library exports the following modules wholesale, as user programs will have to import these modules to make any sensible use of the SBV functionality.
module Data.Bits
module Data.Word
module Data.Int
module Data.Ratio