--------------------------------------------------------------------------------- -- | -- 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 $ forAll ["x"] $ \x -> x `shiftL` 2 .== (x :: SWord8) -- Falsifiable. Counter-example: -- x = 51 :: SWord8 -- -- 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 signed and unsigned -- bit-vectors. Functions for checking satisfiability ('sat' and 'allSat') are also -- provided. -- -- In particular, the sbv library introduces the types: -- -- * 'SBool': Symbolic Booleans (bits). -- -- * 'SWord8', 'SWord16', 'SWord32', 'SWord64': Symbolic Words (unsigned). -- -- * 'SInt8', 'SInt16', 'SInt32', 'SInt64': Symbolic Ints (signed). -- -- * '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. -- -- 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, lazily. -- -- The sbv library uses third-party SMT solvers via the standard SMT-Lib interface: -- <http://goedel.cs.uiowa.edu/smtlib/>. -- -- The SBV library is designed to work with any SMT-Lib compliant SMT-solver. -- Currently, we support the Z3 SMT solver from Microsoft: <http://research.microsoft.com/en-us/um/redmond/projects/z3/> -- and the Yices SMT solver from SRI: <http://yices.csl.sri.com/>, out-of-the-box. Support for other solvers -- can be added with relative ease. --------------------------------------------------------------------------------- module Data.SBV ( -- * Programming with symbolic values -- $progIntro -- ** Symbolic types -- *** Symbolic bit SBool -- *** Unsigned symbolic bit-vectors , SWord8, SWord16, SWord32, SWord64 -- *** Signed symbolic bit-vectors , SInt8, SInt16, SInt32, SInt64 -- *** Signed unbounded integers -- $unboundedLimitations , SInteger -- *** Signed algebraic reals -- $algReals , SReal, AlgReal -- ** Creating a symbolic variable -- $createSym , sBool, sWord8, sWord16, sWord32, sWord64, sInt8, sInt16, sInt32, sInt64, sInteger -- ** Creating a list of symbolic variables -- $createSyms , sBools, sWord8s, sWord16s, sWord32s, sWord64s, sInt8s, sInt16s, sInt32s, sInt64s, sIntegers, sReal, sReals, toSReal -- *** Abstract SBV type , SBV -- *** Arrays of symbolic values , SymArray(..), SArray, SFunArray, mkSFunArray -- *** Full binary trees , STree, readSTree, writeSTree, mkSTree -- ** Operations on symbolic words -- *** Word level , sbvTestBit, sbvPopCount, sbvShiftLeft, sbvShiftRight, sbvSignedShiftArithRight, setBitTo, oneIf, lsb, msb -- *** List level , allEqual, allDifferent -- *** Blasting/Unblasting , blastBE, blastLE, FromBits(..) -- *** Splitting, joining, and extending , Splittable(..) -- *** Sign-casting , SignCast(..) -- ** Polynomial arithmetic and CRCs , Polynomial(..), crcBV, crc -- ** Conditionals: Mergeable values , Mergeable(..) -- ** Symbolic equality , EqSymbolic(..) -- ** Symbolic ordering , OrdSymbolic(..) -- ** Symbolic integral numbers , SIntegral -- ** Division , SDivisible(..) -- ** The Boolean class , Boolean(..) -- *** Generalizations of boolean operations , bAnd, bOr, bAny, bAll -- ** Pretty-printing and reading numbers in Hex & Binary , PrettyNum(..), readBin -- * Uninterpreted sorts, constants, and functions -- $uninterpreted , Uninterpreted(..), addAxiom -- * Properties, proofs, and satisfiability -- $proveIntro -- ** Predicates , Predicate, Provable(..), Equality(..) -- ** Proving properties , prove, proveWith, isTheorem, isTheoremWithin -- ** Checking satisfiability , sat, satWith, isSatisfiable, isSatisfiableWithin -- ** Finding all satisfying assignments , allSat, allSatWith, numberOfModels -- ** Satisfying a sequence of boolean conditions , solve -- ** Adding constraints -- $constrainIntro , constrain, pConstrain -- ** Checking constraint vacuity , isVacuous, isVacuousWith -- * Optimization -- $optimizeIntro , minimize, maximize, optimize , minimizeWith, maximizeWith, optimizeWith -- * Computing expected values , expectedValue, expectedValueWith -- * Model extraction -- $modelExtraction -- ** Inspecting proof results -- $resultTypes , ThmResult(..), SatResult(..), AllSatResult(..), SMTResult(..) -- ** Programmable model extraction -- $programmableExtraction , SatModel(..), Modelable(..), displayModels, extractModels -- * SMT Interface: Configurations and solvers , SMTConfig(..), OptimizeOpts(..), SMTSolver(..), yices, z3, defaultSMTCfg, sbvCheckSolverInstallation -- * Symbolic computations , Symbolic, output, SymWord(..) -- * Getting SMT-Lib output (for offline analysis) , compileToSMTLib, generateSMTBenchmarks -- * Test case generation , genTest, getTestValues, TestVectors, TestStyle(..), renderTest, CW(..), HasKind(..), Kind(..), cwToBool -- * Code generation from symbolic programs -- $cCodeGeneration , SBVCodeGen -- ** Setting code-generation options , cgPerformRTCs, cgSetDriverValues, cgGenerateDriver, cgGenerateMakefile -- ** Designating inputs , cgInput, cgInputArr -- ** Designating outputs , cgOutput, cgOutputArr -- ** Designating return values , cgReturn, cgReturnArr -- ** Code generation with uninterpreted functions , cgAddPrototype, cgAddDecl, cgAddLDFlags -- ** Code generation with 'SInteger' and 'SReal' types -- $unboundedCGen , cgIntegerSize, cgSRealType, CgSRealType(..) -- ** Compilation to C , compileToC, compileToCLib -- * Module exports -- $moduleExportIntro , module Data.Bits , module Data.Word , module Data.Int , module Data.Ratio ) where import Data.SBV.BitVectors.AlgReals import Data.SBV.BitVectors.Data import Data.SBV.BitVectors.Model import Data.SBV.BitVectors.PrettyNum import Data.SBV.BitVectors.SignCast import Data.SBV.BitVectors.Splittable import Data.SBV.BitVectors.STree import Data.SBV.Compilers.C import Data.SBV.Compilers.CodeGen import Data.SBV.Provers.Prover import Data.SBV.Tools.GenTest import Data.SBV.Tools.ExpectedValue import Data.SBV.Tools.Optimize import Data.SBV.Tools.Polynomial import Data.SBV.Utils.Boolean import Data.Bits import Data.Int import Data.Ratio import Data.Word -- 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. -} {- $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://goedel.cs.uiowa.edu/smtlib/>) to communicate with arbitrary SMT solvers. Unfortunately, the SMT-Lib version 1.X does not standardize how models are communicated back from solvers, so there is some work in parsing individual SMT solver output. The 2.X version of the SMT-Lib standard (not yet implemented by SMT solvers widely, unfortunately) will bring new standard features for getting models; at which time the SBV framework can be modified into a truly plug-and-play system where arbitrary SMT solvers can be used. -} {- $optimizeIntro Symbolic optimization. A call of the form: @minimize Quantified cost n valid@ returns @Just xs@, such that: * @xs@ has precisely @n@ elements * @valid xs@ holds * @cost xs@ is minimal. That is, for all sequences @ys@ that satisfy the first two criteria above, @cost xs .<= cost ys@ holds. If there is no such sequence, then 'minimize' will return 'Nothing'. The function 'maximize' is similar, except the comparator is '.>='. So the value returned has the largest cost (or value, in that case). The function 'optimize' allows the user to give a custom comparison function. The 'OptimizeOpts' argument controls how the optimization is done. If 'Quantified' is used, then the SBV optimization engine satisfies the following predicate: @exists xs. forall ys. valid xs && (valid ys ``implies`` (cost xs ``cmp`` cost ys))@ Note that this may cause efficiency problems as it involves alternating quantifiers. If 'OptimizeOpts' is set to 'Iterative' 'True', then SBV will programmatically search for an optimal solution, by repeatedly calling the solver appropriately. (The boolean argument controls whether progress reports are given. Use 'False' for quiet operation.) Note that the quantified and iterative versions are two different optimization approaches and may not necessarily yield the same results. In particular, the quantified version can find solutions where there is no global optimum value, while the iterative version would simply loop forever in such cases. On the other hand, the iterative version might be more suitable if the quantified version of the problem is too hard to deal with by the SMT solver. -} {- $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. -} {- $cCodeGeneration The SBV library can generate straight-line executable code in C. (While other target languages are certainly possible, currently only C is supported.) The generated code will perform no run-time memory-allocations, (no calls to @malloc@), so its memory usage can be predicted ahead of time. Also, the functions will execute precisely the same instructions in all calls, so they have predictable timing properties as well. The generated code has no loops or jumps, and is typically quite fast. While the generated code can be large due to complete unrolling, these characteristics make them suitable for use in hard real-time systems, as well as in traditional computing. -} {- $unboundedCGen The types 'SInteger' and 'SReal' are unbounded quantities that have no direct counterparts in the C language. Therefore, it is not possible to generate standard C code for SBV programs using these types, unless custom libraries are available. To overcome this, SBV allows the user to explicitly set what the corresponding types should be for these two cases, using the functions below. Note that while these mappings will produce valid C code, the resulting code will be subject to overflow/underflows for 'SInteger', and rounding for 'SReal', so there is an implicit loss of precision. If the user does /not/ specify these mappings, then SBV will refuse to compile programs that involve these types. -} {- $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. -} {- $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: 'split', '#', and 'extend' (see the 'Splittable' class) 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://goedel.cs.uiowa.edu/smtlib/theories/Reals.smt2>.) 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.) -} {- $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 '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. 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 ('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' 'false'@.) A probabilistic constraint (see 'pConstrain') attaches a probability threshold for the constraint to be considered. For instance: @ 'pConstrain' 0.8 c @ will make sure that the condition @c@ is satisfied 80% of the time (and correspondingly, falsified 20% of the time), in expectation. This variant is useful for 'genTest' and 'quickCheck' functions, where we want to filter the test cases according to some probability distribution, to make sure that the test-vectors are drawn from interesting subsets of the input space. For instance, if we were to generate 100 test cases with the above constraint, we'd expect about 80 of them to satisfy the condition @c@, while about 20 of them will fail it. The following properties hold: @ 'constrain' = 'pConstrain' 1 'pConstrain' t c = 'pConstrain' (1-t) (not c) @ Note that while 'constrain' can be used freely, 'pConstrain' is only allowed in the contexts of 'genTest' or 'quickCheck'. Calls to 'pConstrain' in a prove/sat call will be rejected as SBV does not deal with probabilistic constraints when it comes to satisfiability and proofs. Also, both 'constrain' and 'pConstrain' calls during code-generation will also be rejected, for similar reasons. -} {- $uninterpreted Users can introduce new uninterpreted sorts simply by defining a data-type in Haskell and registering it as such. The following example demonstrates: @ data B = B deriving (Eq, Ord, Data, Typeable) instance SymWord B instance HasKind B @ (Note that you'll also need to use the language pragma @DeriveDataTypeable@, and import @Data.Generics@ for the above to work.) Once GHC implements derivable user classes (<http://hackage.haskell.org/trac/ghc/ticket/5462>), we will be able to simplify this to: @ data B = B deriving (Eq, Ord, Data, Typeable, SymWord, HasKind) @ 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. Uninterpreted functions over both uninterpreted and regular sorts can be declared using the facilities introduced by the 'Uninterpreted' class. -} {-# ANN module "HLint: ignore Use import/export shortcut" #-}