Hasmtlib - Haskell SMTLib MTL Library

Hasmtlib is a library with high-level-abstraction for generating SMTLib2-problems using a monad. It takes care of encoding your problem, marshaling the data to an external solver and parsing and interpreting the result into Haskell types. It is highly inspired by ekmett/ersatz which does the same for QSAT. Communication with external solvers is handled by tweag/smtlib-backends.

Building expressions with type-level representations of the SMTLib2-Sorts guarantees type-safety when communicating with external solvers.

While formula construction is entirely pure, Hasmtlib - just like ersatz - makes use of observable sharing for expressions.

This allows you to use the much richer subset of Haskell than a purely monadic meta-language would, which ultimately results in extremely compact code.

For instance, to define the addition of two V3 containing Real-SMT-Expressions:

v3Add :: V3 (Expr RealSort) -> V3 (Expr RealSort) -> V3 (Expr RealSort)
v3Add = liftA2 (+)

Even better, the Expr-GADT allows a polymorph definition:

v3Add :: Num (Expr t) => V3 (Expr t) -> V3 (Expr t) -> V3 (Expr t)
v3Add = liftA2 (+)

This looks a lot like the definition of Num for V3 a:

instance Num a => Num (V3 a) where
  (+) :: V3 a -> V3 a -> V3 a
  (+) = liftA2 (+)

Hence, no extra definition is needed at all. We can use the existing instances:

{-# LANGUAGE DataKinds #-}

import Language.Hasmtlib
import Linear

-- instances with default impl
instance Codec a => Codec (V3 a)
instance Variable a => Variable (V3 a)

main :: IO ()
main = do
  res <- solveWith @SMT (solver cvc5) $ do
    setLogic "QF_NRA"

    u :: V3 (Expr RealSort) <- variable
    v <- variable

    assert $ dot u v === 5

    return (u,v)

  print res

May print: (Sat,Just (V3 (-2.0) (-1.0) 0.0,V3 (-2.0) (-1.0) 0.0))

Features

• SMTLib2-Sorts in the Haskell-Type to guarantee well-typed expressions

    data SMTSort =
        BoolSort
      | IntSort
      | RealSort
      | BvSort BvEnc Nat
      | ArraySort SMTSort SMTSort
      | StringSort
    data Expr (t :: SMTSort) where ...
  
    ite :: Expr BoolSort -> Expr t -> Expr t -> Expr t

• Full SMTLib 2.6 standard support for Sorts Bool, Int, Real, BitVec, Array & String
• Type-level length-indexed Bitvectors with type-level encoding (Signed/Unsigned) for BitVec
• Pure API with plenty common instances: Num, Floating, Bounded, Bits, Ixed and many more
• Add your own solvers via the Solver type
• Solvers via external processes: CVC5, Z3, Yices2-SMT, MathSAT, OptiMathSAT, OpenSMT & Bitwuzla
• Support for incremental solving
• Pure quantifiers for_all and exists

    solveWith @SMT (solver z3) $ do
      setLogic "LIA"
      z <- var @IntSort
      assert $ z === 0
      assert $
        for_all $ \x ->
            exists $ \y ->
              x + y === z
      return z

• Optimization Modulo Theories (OMT) / MaxSMT

    res <- solveWith @OMT (solver z3) $ do
      setLogic "QF_LIA"
      x <- var @IntSort
  
      assert $ x >? -2
      assertSoftWeighted (x >? -1) 5.0
      minimize x
  
      return x

Related work

• ekmett/ersatz: Huge inspiration for this library (some code stolen). We do for SMT what they do for SAT.
• hgoes/smtlib2: Their eDSL is highly expressive and focuses on well-typed SMT-expressions. But their approach is a little verbose and makes usage feel quite heavy. Their eDSL is also entirely monadic and therefore formula construction is painful.
• yav/simple-smt: They are lightweight but their types are weak and their API is barely embedded into Haskell.
• LevantErkok/sbv: While they "express properties about Haskell programs and automatically prove them using SMT", we instead use Haskell to simplify the encoding of SMT-Problems. They can do a whole lot (C-Code-Gen, Crypto-Stuff,...), which is cool, but adds weight.

If you want highly specific implementations for different solvers, all their individual configurations and swallow the awkward typing, then use hgoes/smtlib2.

If you want to express properties about Haskell programs and automatically prove them using SMT, then use LevantErkok/sbv.

If you want to encode SMT-problems as lightweight and close to Haskell as possible, then use this library. I personally use it for scheduling/resource-allocation-problems.

Examples

There are some examples in here.

Contact information

Contributions, critics and bug reports are welcome!

Please feel free to contact me through GitHub.