I think you’ve all heard me ramble on about the libraries I find interesting, so today it’s with great pleasure that I turn the spotlight over to Tikhon Jelvis. Enjoy!
Today we’re going to talk about sbv, a library for interacting with SMT solvers. SBV stands for “SMT-based verification”; SMT itself stands for “SAT modulo theories” and SAT is “Boolean satisfiability” — three layers of acronyms! Luckily, they’re reasonably easy to follow. Let’s start from the bottom.
Boolean satisfiability (SAT) is one of the canonical NP-complete problems. The problem is based on Boolean formulas: boolean variables combined with AND (∧), OR (∨) and NOT (¬). Here’s an example formula with variables a, b, c, d and e:
(a∨b∨c) ∧ (a∨¬d∨¬e) ∧ (¬y∨c∨e)
The goal is to figure out if a Boolean formula is “satisfiable” — that is, if there is some way to assign all the variables to true or false that results in the entire formula evaluating to true.
SAT is interesting because it can be easily used to express other (usually also NP-complete) problems. While SAT, being NP-complete, is hard in general, most real-world instances tend to be reasonably tractable. A fast algorithm to solve SAT could be used in quite a surprising variety of real-world contexts — things like verifying and optimizing circuit design or automated theorem proving.
SMT is a popular extension to SAT which adds “theories” to the basic logic SAT supports. These theories let us write formulas using more than just Boolean constraints: examples include bitvectors, arbitrary-sized integers and arrays, records or even algebraic numbers. These extensions make encoding various domain-specific problems much easier with SMT than with just SAT; moreover, SMT solvers can use high-level knowledge about the constraints to implement particular theories more efficiently.
SMT solvers are useful for most problems that can be recast as a set of constraints, like dependency resolution for package managers. They are also very useful for working with programming languages: most common language features are easy to express using basic SMT theories. SMT solvers can be used to verify code against some sort of logical specification (like Liquid Haskell) or even synthesize new algorithms1.
sbv is a library for conveniently interacting with SMT solvers. It lets you write Haskell programs that produce a formula that can be sent off to an SMT solver. These formulas are created by using “symbolic” versions of normal data types like Booleans and various Words, prefixed by an “S”, such as SBool, SWord and so on. “Symbolic” refers to the fact that the types do not represent actual booleans or numbers in Haskell; instead, they are variables in a formula to be passed to the solver. Functions2 written with these symbolic types can be compiled to logic formulas and passed to the solver.
To work with sbv, you need to install an external SMT solver. The library currently supports four different backends, with Microsoft’s Z3 having the most features, making it the default choice. Currently, Z3 is the fastest overall solver based on the SMT competition. It is available under an MIT license, suitable for any purpose including commercial use.
Once you’ve installed Z3 and have it in your path, you can start playing around in GHCi by importing Data.SBV.
The simplest example we can do is prove something like the fact that x + x always equals x * 2. This use is very similar to QuickCheck except that the solver is exhaustive — it proves the property for all possible values:
> prove . forAll ["x"] $ \ (x::SWord8) -> x * 2 .== x + x
Q.E.D.The string "x" in the forAll is what the corresponding variable will be named in the solver. This is the name used in the output of the solver and is really useful for debugging. It helps to call the corresponding Haskell variable (the argument in the lambda) the same name, but it isn’t obligatory.
If the property does not hold, the solver gives us a counter-example:
> prove . forAll ["x"] $ \ (x::SWord8) -> x * 3 .== x + x
Falsifiable. Counter-example:
x = 255 :: SWord8
We can also solve for values that satisfy our constraints. For example, we could find a solution to a system of equations:
> sat . forSome ["x", "y"] $ \ (x::SInteger) y ->
x^2 + y^2 .== 25 &&& 3 * x + 4 * y .== 0
Satisfiable. Model:
x = 4 :: SInteger
y = -3 :: SInteger
These constraints don’t even have to have a solution. If there is no solution, the solver can often tell you, even if you’re using unbounded integers!
> sat . forSome ["x", "y"] $ \ (x::SInteger) y ->
x^2 + y^2 .== 42 &&& 3 * x + 4 * y .== 0
UnsatisfiableIf there is more than one solution, the solver will just choose one.
> sat . forSome ["x", "y", "z"] $ \ (x::SInteger) y z ->
x + 2 * y .== 3 * y + z + x^2 &&& x ./= y
Satisfiable. Model:
x = 1 :: SInteger
y = 0 :: SInteger
z = 0 :: SInteger
You could instead get all the satisfying assignments by using allSat in place of sat.
While certainly interesting as an interactive tool, the solver really comes into its own when you can use its results in the rest of your program. We can do this with extractModel which returns a “model” for our formula, which is just a set of assignments to variables that satisfies all the constraints — a single solution to our formula.
> res <- sat . forSome ["x", "y"] $ \(x::SWord8) y ->
x^2 + y .== 10 &&& y^2 + x^3 .== 100
> extractModel res :: Maybe (Word8, Word8)
Just (251,241)extractModel is polymorphic over the sort of types it can output, and sbv knows how to construct different containers. For example, you could get a list instead of a tuple:
> extractModel res :: Maybe [Word8]
Just [251,241]In the case of prove, extractModel is reversed: if the formula is satisfiable, it returns Nothing and otherwise it returns a counterexample. For a function like allSat with multiple results, you can use the extractModels function to get all of the models.
Playing around with sbv at the REPl is a good way to get a handle on how the package works and how SMT solvers can be used. For actual programs, chances are you will want to work with significantly more complicated SMT formulas; sbv provides a monad for stringing together different assertions in a formula and generating variables programmatically which can be very useful.
The sbv documentation is a good reference for the capabilities and constructs sbv offers. The package also comes with a few deeper examples which are well worth a look.
This post is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
This article from the Communications of the ACM contains more examples, use cases and background information on SMT solvers.↩︎
The solver works on a bunch of different types so most of the operators are polymorphic. This means that you have to specify the type you’re working with somewhere. In these examples, I add a type annotation to a lambda:
\ (x::SInteger) y -> ...To do this, you have to turn on ScopedTypeVariables. In GHCi, you can do this with:
> :set -XScopedTypeVariablesChances are that you will have to do this more often in GHCi than in normal code because GHCi leaves less scope for types to be inferred.↩︎