--------------------------------------------------------------------------------- -- | -- Module : Data.SBV -- Copyright : (c) Levent Erkok -- License : BSD3 -- Maintainer : erkokl@gmail.com -- Stability : experimental -- Portability : portable -- -- (The sbv library is hosted at <http://github.com/LeventErkok/sbv>. -- Comments, bug reports, and patches are always welcome.) -- -- SBV: Symbolic Bit Vectors in Haskell -- -- Express properties about bit-precise 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 = 128 :: 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 -- -- 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 -- bit-precise 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 Yices SMT solver from SRI: <http://yices.csl.sri.com/>, -- and the Z3 SMT solver from Microsoft: <http://research.microsoft.com/en-us/um/redmond/projects/z3/>. -- -- You /should/ download and install Yices on your machine, and make sure the -- @yices@ executable is in your path before using the sbv library, as it is the -- current default solver. Alternatively, you can specify the location of yices -- executable in the environment variable @SBV_YICES@ and the options to yices -- in @SBV_YICES_OPTIONS@. -- -- Use of quantified variables require an installation of z3. Again, -- z3 must be in your path. Or, you can use the @SBV_Z3@ and @SBV_Z3_OPTIONS@ -- environment variables to set the executable and the options. --------------------------------------------------------------------------------- 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 -- *** 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 , bitValue, 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(..) -- ** Division , BVDivisible(..) -- ** The Boolean class , Boolean(..) -- *** Generalizations of boolean operations , bAnd, bOr, bAny, bAll -- ** Pretty-printing and reading numbers in Hex & Binary , PrettyNum(..), readBin -- * Uninterpreted constants and functions , Uninterpreted(..) -- ** Accessing the handle , SBVUF, sbvUFName -- ** Adding axioms , addAxiom -- * Proving properties -- $proveIntro -- ** Predicates , Predicate, Provable(..), Equality(..) -- ** Proving properties , prove, proveWith, isTheorem, isTheoremWithin -- ** Checking satisfiability , sat, satWith, isSatisfiable, isSatisfiableWithin -- ** Finding all satisfying assignments , allSat, allSatWith, numberOfModels -- * Optimization -- $optimizeIntro , minimize, maximize, optimize -- * Model extraction -- $modelExtraction -- ** Inspecting proof results -- $resultTypes , ThmResult(..), SatResult(..), AllSatResult(..), SMTResult(..) -- ** Programmable model extraction -- $programmableExtraction , SatModel(..), getModel, displayModels -- * SMT Interface: Configurations and solvers , SMTConfig(..), SMTSolver(..), yices, z3 -- * Symbolic computations , Symbolic, output, SymWord(..) -- * Getting SMT-Lib output (for offline analysis) , compileToSMTLib -- * Code generation from symbolic programs -- $cCodeGeneration , SBVCodeGen -- ** Setting code-generation options , cgPerformRTCs, cgSetDriverValues, cgGenerateDriver -- ** Designating inputs , cgInput, cgInputArr -- ** Designating outputs , cgOutput, cgOutputArr -- ** Designating return values , cgReturn, cgReturnArr -- ** Code generation with uninterpreted functions , cgAddPrototype, cgAddDecl, cgAddLDFlags, cgIntegerSize -- ** Compilation to C , compileToC, compileToCLib -- * Module exports -- $moduleExportIntro , module Data.Bits , module Data.Word , module Data.Int ) where import Data.SBV.BitVectors.Data import Data.SBV.BitVectors.Model import Data.SBV.BitVectors.Optimize import Data.SBV.BitVectors.PrettyNum import Data.SBV.BitVectors.Polynomial 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.Utils.Boolean import Data.Bits import Data.Word import Data.Int -- Haddock section documentation {- $progIntro The SBV library is really two things: * A framework for writing bit-precise programs in Haskell * A framework for proving properties of such programs using SMT solvers In this first section we will look at the constructs that will let us construct such programs in Haskell. The goal is to have a "seamless" experience, i.e., 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 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. Logically, the SBV optimization engine satisfies the following predicate: @exists xs. forall ys. valid xs && (valid ys ``implies`` (cost xs ``cmp`` cost ys))@ -} {- $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. -} {- $moduleExportIntro The SBV library exports the following modules wholesale, as user programs will have to import these three modules to make any sensible use of the SBV functionality. -} {- $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 ('+', '-', '*', 'bvQuotRem') and logical operations ('.<', '.<=', '.>', '.>=', '.==', './=') operations are supported for 'SInteger' fully, both in programming and verification modes. -}