----------------------------------------------------------------------------- -- | -- Module : Data.SBV -- Copyright : (c) Levent Erkok -- License : BSD3 -- Maintainer: erkokl@gmail.com -- Stability : experimental -- -- (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 -- -- The function 'prove' has the following type: -- -- @ -- 'prove' :: 'Provable' a => a -> 'IO' 'ThmResult' -- @ -- -- The class 'Provable' comes 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). -- -- * 'SWord' @n@: Generalized symbolic words of arbitrary bit-size. -- -- * 'SInt' @n@: 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 -- -- * '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 sums -- -- * 'SMaybe': Symbolic optional values -- -- * 'SSet': Symbolic sets -- -- * 'SArray', 'SFunArray': Flat 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. -- -- * Model validation: SBV can validate models returned by solvers, which allows -- for protection against bugs in SMT solvers and SBV itself. (See the 'validateModel' -- parameter.) -- -- 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 'prove' function) -- -- * checked for satisfiability (the 'sat', 'allSat' functions) -- -- * used in synthesis (the `sat` function 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 from Stanford: <https://cvc4.github.io/> -- -- * Boolector from Johannes Kepler University: <http://fmv.jku.at/boolector/> -- -- * 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. ----------------------------------------------------------------------------- {-# OPTIONS_GHC -Wall -Werror #-} module Data.SBV ( -- $progIntro -- * Symbolic types -- ** Booleans SBool -- *** Boolean values and functions , sTrue, sFalse, sNot, (.&&), (.||), (.<+>), (.~&), (.~|), (.=>), (.<=>), fromBool, oneIf -- *** Logical aggregations , sAnd, sOr, sAny, sAll -- ** Bit-vectors -- *** Unsigned bit-vectors , SWord8, SWord16, SWord32, SWord64, SWord, WordN -- *** Signed bit-vectors , SInt8, SInt16, SInt32, SInt64, SInt, IntN -- *** Converting between fixed-size and arbitrary bitvectors , IsNonZero, FromSized, ToSized, fromSized, toSized -- ** Unbounded integers -- $unboundedLimitations , SInteger -- ** Floating point numbers -- $floatingPoints , SFloat, SDouble -- ** Algebraic reals -- $algReals , SReal, AlgReal(..), sRealToSInteger, algRealToRational, RealPoint(..), realPoint, RationalCV(..) -- ** Characters, Strings and Regular Expressions -- $strings , SChar, SString -- ** Symbolic lists -- $lists , SList -- ** Tuples -- $tuples , SymTuple, STuple, STuple2, STuple3, STuple4, STuple5, STuple6, STuple7, STuple8 -- ** Sum types , SMaybe, SEither -- ** Sets , RCSet(..), SSet -- * Arrays of symbolic values , SymArray(readArray, writeArray, mergeArrays, sListArray), newArray_, newArray, SArray, SFunArray -- * Creating symbolic values -- ** Single value -- $createSym , sBool, sBool_ , sWord8, sWord8_, sWord16, sWord16_, sWord32, sWord32_, sWord64, sWord64_, sWord, sWord_ , sInt8, sInt8_, sInt16, sInt16_, sInt32, sInt32_, sInt64, sInt64_, sInt, sInt_ , sInteger, sInteger_ , sReal, sReal_ , sFloat, sFloat_ , sDouble, sDouble_ , sChar, sChar_ , sString, sString_ , sList, sList_ , sTuple, sTuple_ , sEither, sEither_ , sMaybe, sMaybe_ , sSet, sSet_ -- ** List of values -- $createSyms , sBools , sWord8s, sWord16s, sWord32s, sWord64s, sWords , sInt8s, sInt16s, sInt32s, sInt64s, sInts , sIntegers , sReals , sFloats , sDoubles , sChars , sStrings , sLists , sTuples , sEithers , sMaybes , sSets -- * Symbolic Equality and Comparisons , EqSymbolic(..), OrdSymbolic(..), Equality(..) -- * Conditionals: Mergeable values , Mergeable(..), ite, iteLazy -- * Symbolic integral numbers , SIntegral -- * Division and Modulus , SDivisible(..) -- * Bit-vector operations -- ** Conversions , sFromIntegral -- ** Shifts and rotates -- $shiftRotate , sShiftLeft, sShiftRight, sRotateLeft, sBarrelRotateLeft, sRotateRight, sBarrelRotateRight, sSignedShiftArithRight -- ** Finite bit-vector operations , SFiniteBits(..) -- ** Splitting, joining, and extending bit-vectors , bvExtract, (#), zeroExtend, signExtend, bvDrop, bvTake, ByteConverter(..) -- ** Exponentiation , (.^) -- * IEEE-floating point numbers , IEEEFloating(..), RoundingMode(..), SRoundingMode, nan, infinity, sNaN, sInfinity -- ** Rounding modes , sRoundNearestTiesToEven, sRoundNearestTiesToAway, sRoundTowardPositive, sRoundTowardNegative, sRoundTowardZero, sRNE, sRNA, sRTP, sRTN, sRTZ -- ** Conversion to/from floats , IEEEFloatConvertible(..) -- ** Bit-pattern conversions , sFloatAsSWord32, sWord32AsSFloat, sDoubleAsSWord64, sWord64AsSDouble, blastSFloat, blastSDouble -- * Enumerations -- $enumerations , mkSymbolicEnumeration -- * Uninterpreted sorts, axioms, constants, and functions -- $uninterpreted , mkUninterpretedSort, Uninterpreted(..), addAxiom -- * Properties, proofs, and satisfiability -- $proveIntro -- $noteOnNestedQuantifiers -- $multiIntro , Predicate, Goal , Provable, forAll_, forAll, forSome_, forSome , prove, proveWith , dprove, dproveWith , sat, satWith , dsat, dsatWith , allSat, allSatWith , optimize, optimizeWith, isVacuous , isVacuousWith, isTheorem, isTheoremWith, isSatisfiable, isSatisfiableWith , proveWithAll, proveWithAny, satWithAll , proveConcurrentWithAny, proveConcurrentWithAll, satConcurrentWithAny, satConcurrentWithAll , satWithAny, generateSMTBenchmark , solve -- * Constraints -- $constrainIntro -- ** General constraints -- $generalConstraints , constrain, softConstrain -- ** Constraint Vacuity -- $constraintVacuity -- ** Named constraints and attributes -- $namedConstraints , namedConstraint, constrainWithAttribute -- ** Unsat cores -- $unsatCores -- ** Cardinality constraints -- $cardIntro , pbAtMost, pbAtLeast, pbExactly, pbLe, pbGe, pbEq, pbMutexed, pbStronglyMutexed -- * Checking safety -- $safeIntro , sAssert, isSafe, SExecutable, sName_, sName, safe, safeWith -- * Quick-checking , sbvQuickCheck -- * Optimization -- $optiIntro -- ** Multiple optimization goals -- $multiOpt , OptimizeStyle(..) -- ** Objectives and Metrics , Objective(..) , Metric(..), minimize, maximize -- ** Soft assertions -- $softAssertions , assertWithPenalty , Penalty(..) -- ** Field extensions -- | If an optimization results in an infinity/epsilon value, the returned `CV` value will be in the corresponding extension field. , ExtCV(..), GeneralizedCV(..) -- * Model extraction -- $modelExtraction -- ** Inspecting proof results -- $resultTypes , ThmResult(..), SatResult(..), AllSatResult(..), SafeResult(..), OptimizeResult(..), SMTResult(..), SMTReasonUnknown(..) -- ** Observing expressions -- $observeInternal , observe, observeIf, sObserve -- ** Programmable model extraction -- $programmableExtraction , SatModel(..), Modelable(..), displayModels, extractModels , getModelDictionaries, getModelValues, getModelUninterpretedValues -- * SMT Interface , SMTConfig(..), Timing(..), SMTLibVersion(..), Solver(..), SMTSolver(..) -- ** Controlling verbosity -- $verbosity -- ** Solvers , boolector, cvc4, yices, dReal, z3, mathSAT, abc -- ** Configurations , defaultSolverConfig, defaultSMTCfg, defaultDeltaSMTCfg, sbvCheckSolverInstallation, getAvailableSolvers , setLogic, Logic(..), setOption, setInfo, setTimeOut -- ** SBV exceptions , SBVException(..) -- * Abstract SBV type , SBV, HasKind(..), Kind(..) , SymVal, forall, forall_, mkForallVars, exists, exists_, mkExistVars, free , free_, mkFreeVars, symbolic, symbolics, literal, unliteral, fromCV , isConcrete, isSymbolic, isConcretely, mkSymVal , MonadSymbolic(..), Symbolic, SymbolicT, label, output, runSMT, runSMTWith -- * Module exports -- $moduleExportIntro , module Data.Bits , module Data.Word , module Data.Int , module Data.Ratio ) where import Data.SBV.Core.AlgReals import Data.SBV.Core.Data hiding (forall, forall_, mkForallVars, exists, exists_, mkExistVars, free, free_, mkFreeVars, output, symbolic, symbolics, mkSymVal, newArray, newArray_) import Data.SBV.Core.Model hiding (assertWithPenalty, minimize, maximize, forall, forall_, exists, exists_, solve, sBool, sBool_, sBools, sChar, sChar_, sChars, sDouble, sDouble_, sDoubles, sFloat, sFloat_, sFloats, sInt8, sInt8_, sInt8s, sInt16, sInt16_, sInt16s, sInt32, sInt32_, sInt32s, sInt64, sInt64_, sInt64s, sInteger, sInteger_, sIntegers, sList, sList_, sLists, sTuple, sTuple_, sTuples, sReal, sReal_, sReals, sString, sString_, sStrings, sWord8, sWord8_, sWord8s, sWord16, sWord16_, sWord16s, sWord32, sWord32_, sWord32s, sWord64, sWord64_, sWord64s, sMaybe, sMaybe_, sMaybes, sEither, sEither_, sEithers, sSet, sSet_, sSets) import Data.SBV.Core.Sized hiding (sWord, sWord_, sWords, sInt, sInt_, sInts) import Data.SBV.Core.Floating import Data.SBV.Core.Symbolic (MonadSymbolic(..), SymbolicT) import Data.SBV.Provers.Prover hiding (forAll_, forAll, forSome_, forSome, prove, proveWith, sat, satWith, allSat, dsat, dsatWith, dprove, dproveWith, allSatWith, optimize, optimizeWith, isVacuous, isVacuousWith, isTheorem, isTheoremWith, isSatisfiable, isSatisfiableWith, runSMT, runSMTWith, sName_, sName, safe, safeWith) import Data.SBV.Client import Data.SBV.Client.BaseIO import Data.SBV.Utils.TDiff (Timing(..)) import Data.Bits import Data.Int import Data.Ratio import Data.Word import Data.SBV.SMT.Utils (SBVException(..)) import Data.SBV.Control.Types (SMTReasonUnknown(..), Logic(..)) -- Haddock section documentation {- $progIntro 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 'Bool's. 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. -} {- $proveIntro 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. -} {- $multiIntro === 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. -} {- $safeIntro 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. -} {- $optiIntro SBV can optimize metric functions, i.e., those that generate both bounded @SIntN@, @SWordN@, and unbounded 'SInteger' types, along with those produce 'SReal's. 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: * 'minimize': Minimize a given arithmetic goal * 'maximize': Minimize a given arithmetic goal 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*y @ minimizes 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 -} {- $multiOpt 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.) -} {- $softAssertions 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. -} {- $modelExtraction 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. -} {- $resultTypes '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. -} {- $programmableExtraction 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. -} {- $moduleExportIntro 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. -} {- $createSym 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. -} {- $createSyms 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. -} {- $unboundedLimitations 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', and 'bvTake' Usual arithmetic ('+', '-', '*', 'sQuotRem', 'sQuot', 'sRem', 'sDivMod', 'sDiv', 'sMod') and logical operations ('.<', '.<=', '.>', '.>=', '.==', './=') operations are supported for 'SInteger' fully, both in programming and verification modes. -} {- $algReals 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.) -} {- $floatingPoints 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>. -} {- $strings Support for characters, strings, and regular expressions (intial version contributed by Joel Burget) adds support for QF_S logic, described here: <http://smtlib.cs.uiowa.edu/theories-UnicodeStrings.shtml> and here: <http://rise4fun.com/z3/tutorialcontent/sequences>. Note that this logic is still not part of official SMTLib (as of March 2018), so it should be considered experimental. See "Data.SBV.Char", "Data.SBV.String", "Data.SBV.RegExp" for related functions. -} {- $lists Support for symbolic lists (intial version contributed by Joel Burget) adds support for sequence support, described here: <http://rise4fun.com/z3/tutorialcontent/sequences>. Note that this logic is still not part of official SMTLib (as of March 2018), so it should be considered experimental. See "Data.SBV.List" for related functions. -} {- $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. -} {- $shiftRotate 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. -} {- $constrainIntro A constraint is a means for restricting the input domain of a formula. Here's a simple example: @ do x <- 'exists' \"x\" y <- 'exists' \"y\" 'constrain' $ x .> y 'constrain' $ x + y .>= 12 'constrain' $ 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' (or 'allSat') call: The constraint added is asserted conjunctively. That is, the resulting satisfying model (if any) will always satisfy all the constraints given. * In a 'prove' call: 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 @quickCheck@ call: The constraint acts as a filter for @quickCheck@; if the constraint does not hold, then the input value is considered to be irrelevant and is skipped. Note that this is similar to 'prove', 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 'Data.SBV.Tools.GenTest.genTest' call: Similar to @quickCheck@ and 'prove': If a constraint does not hold, the input value is ignored and is not included in the test set. -} {- $generalConstraints A good use case (in fact the motivating use case) for 'constrain' is attaching a constraint to a 'forall' or 'exists' 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 <- 'exists' \"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 'isVacuous' (and 'isVacuousWith') can be used. Also note that this semantics imply that test case generation ('Data.SBV.Tools.GenTest.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'@.) -} {- $constraintVacuity When adding constraints, one has to be careful about making sure they are not inconsistent. The function 'isVacuous' can be use 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 pred Q.E.D. We can use 'isVacuous' to make sure to see that the pass was vacuous: >>> isVacuous pred True 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: >>> isVacuous pred' False -} {- $namedConstraints Constraints can be given names: @ 'namedConstraint' "a is at least 5" $ a .>= 5@ Similarly, arbitrary term attributes can also be associated: @ 'constrainWithAttribute' [(":solver-specific-attribute", "value")] $ a .>= 5@ Note that a 'namedConstraint' is equivalent to a 'constrainWithAttribute' call, setting the `":named"' attribute. -} {- $unsatCores Named constraints are useful when used in conjunction with 'Data.SBV.Control.getUnsatCore' function where the backend solver can be queried to obtain an unsat core in case the constraints are unsatisfiable. See 'Data.SBV.Control.getUnsatCore' for details and "Documentation.SBV.Examples.Queries.UnsatCore" for an example use case. -} {- $uninterpreted 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 'Data.SBV.Core.Model.Uninterpreted' class. -} {- $enumerations If the uninterpreted sort definition takes the form of an enumeration (i.e., a simple data type with all nullary constructors), then tou can use the 'mkSymbolicEnumeration' functio 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. -} {- $noteOnNestedQuantifiers === A note on reasoning in the presence of quantifiers Note that SBV allows reasoning with quantifiers: Inputs can be existentially or universally quantified. Predicates can be built with arbitrary nesting of such quantifiers as well. However, SBV always /assumes/ that the input is in prenex-normal form: <http://en.wikipedia.org/wiki/Prenex_normal_form>. That is, all the input declarations are treated as happening at the beginning of a predicate, followed by the actual formula. Unfortunately, the way predicates are written can be misleading at times, since symbolic inputs can be created at arbitrary points; interleaving them with other code. The rule is simple, however: All inputs are assumed at the top, in the order declared, regardless of their quantifiers. SBV will apply skolemization to get rid of existentials before sending predicates to backend solvers. However, if you do want nested quantification, you will manually have to first convert to prenex-normal form (which produces an equisatisfiable but not necessarily equivalent formula), and code that explicitly in SBV. See <http://github.com/LeventErkok/sbv/issues/256> for a detailed discussion of this issue. -} {- $cardIntro 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. -} {- $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 in `verbose`. * ['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. -} {- $observeInternal 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. -} {-# ANN module ("HLint: ignore Use import/export shortcut" :: String) #-}