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unification-sop

This is a haskell package that implements typed terms with logical variables in a prolog-style, using generics-sop.

Consider the datatype:

data Foo = FooI Int | FooS String Foo

exFoo1, exFoo2 :: Foo
exFoo1 = FooI 3
exFoo2 = FooS "hey" (FooI 4)

With this library you can express terms like:

exTermFoo1, exTermFoo2, exTermFoo3 :: Term Foo
exTermFoo1 = fooI (Var 1)                    -- prolog: fooI(X)
exTermFoo2 = fooR (Var 1) (fooI (Var 1))     -- prolog: fooR(X, fooI(Y))
exTermFoo3 = fooR (Con "hey") (fooI (Var 1)) -- prolog: fooR ("hey", fooI(X))

while maintaining all the well-typedness guarantees of haskell. In the example above, exTermFoo1 would be written in prolog as fooI(X), and exTermFoo2 as fooR(X, fooI(Y)), because, since the variables are typed, Var 1 has type String in the first case and Int in the second.

As an important side effect, you can also talk about unification for types that are not structurally recursive: I see this as an important expressive improvement on the classic unification-fd approach. This also opens the way for optimizations, because if in our problems we care only about non-recursive data types, we can omit the occurs check while maintaining soundness.

You can find more information in the tutorial, or see the haddock documentation.

A quick survey of the organization of the code:

  • Generic.Unification.Tutorial

    Here I wrote a more complete version of the ideas expressed in the first section of the readme.

  • Generic.Unification.Term

    Here I define the Term datatype, which is the structure that lets us express the term with logical variable idea.

  • Generic.Unification.Substitution

    Here we define a concept of substitution, which is essentially a map from variables to terms; since in our case variables are typed, a substitution is denotationally a map (a :: Type, var :: a) -> Term a (using a notation borrowed from dependent types).

  • Generic.Unification.Unification

    Here we implement the quite fast unification algorithm described in Efficient functional Unification and Substitution by A.Dijkstra (and used for example in his EHC haskell compiler).

  • Generic.Unification.Hinze

    Here I wrote a complete prolog (with cut) using the machinery defined in the other modules, and following the paper Prolog control constructs in a functional setting by Hinze, augmented with explicit logical variables. This is an experimental module meant to demonstrate a possible use of the package, and will probably be removed in the future.

FAQ

  • Why do you want to use logical variables when you can instead use a purely monadic representation indicating logic computation, like the one in Hinze's paper?

    My tentative answer is that I sometime use logical variables to structure data (think graphs) so that I can use unification to quickly prototype algorithms.

  • How is the Term datatype constructed since you use a non recursive generic implementation (generics-sop instead of, say, generics-mrsop or multirec)?

    I was surprised too, and the approach works by defining the base case of the recursion as special overlapping instances. I hope to detail the approach in a blog post sometimes.

References

Efficient functional Unification and Substitution by A. Dijkstra

Prolog control constructs in a functional settings by R.Hinze

Contributing

I wrote this package as a proof of concept for another one I needed (which is closed source for now) which uses the same Term idea but delegates unification to the z3 solver. As I'm using the new package I don't plan on improving this one much further, but if you have suggestions, want me to develop some feature, or want to send me a patch, please open an issue here on github: all issues are very welcome :)