### unification-fd 0.7.0

The unification-fd package offers generic functions for single-sorted first-order structural unification (think of programming in Prolog, or of the metavariables in type inference)[1][2]. The library *is* sufficient for implementing higher-rank type systems à la Peyton Jones, Vytiniotis, Weirich, Shields, but bear in mind that unification variables are the metavariables of type inference— not the type-variables.

An effort has been made to make the package as portable as possible. However, because it uses the `ST`

monad and the mtl-2 package it can't be H98 nor H2010. However, it only uses the following common extensions which should be well supported[3]:

Rank2Types
MultiParamTypeClasses
FunctionalDependencies -- Alas, necessary for type inference
FlexibleContexts -- Necessary for practical use of MPTCs
FlexibleInstances -- Necessary for practical use of MPTCs
UndecidableInstances -- For Show instances due to two-level types

### Changes (since 0.6.0)

This release is another major API breaking release. Apologies, but things are a lot cleaner now and hopefully the API won't break again for a while. The biggest change is that the definition of terms has changed from the previous:

data MutTerm v t
= MutVar !v
| MutTerm !(t (MutTerm v t))

To the much nicer:

data UTerm t v
= UVar !v
| UTerm !(t (UTerm t v))

The old mnemonic of "mutable terms" was inherited from the code's previous life implementing a logic programming language; but when I was playing around with implementing a type checker I realized that the names don't really make sense outside of that original context. So the new mnemonic is "unification terms". In addition to being a bit shorter, it should help clarify the separation of concerns (e.g., between unification variables vs lambda-term variables, type variables, etc.).

The swapping of the type parameters is so that `UTerm`

can have instances for `Functor`

, `Monad`

, etc. This change should've been made along with the re-kinding of variable types back in version 0.6.0, since the `UTerm`

type is the free monad generated by `t`

. I've provided all the category theoretic instances I could imagine some plausible reason for wanting. Since it's free, there are a bunch more I haven't implemented since they don't really make sense for structural terms (e.g., `MonadTrans`

, `MonadWriter`

, `MonadReader`

, `MonadState`

, `MonadError`

, `MonadCont`

). If you can come up with some compelling reason to want those instances, I can add them in the future.

Since the order of type parameters to `BindingMonad`

, `UnificationFailure`

, `Rank`

, and `RankedBindingMonad`

was based on analogy to the order for terms, I've also swapped the order in all of them for consistency.

I've removed the `eqVar`

method of the `Variable`

class, and instead added an `Eq`

superclass constraint. Again, this should've happened with the re-kinding of variables back in version 0.6.0. A major benefit of this change is that now you can use all those library functions which require `Eq`

(e.g., many of the set-theoretic operations on lists, like `(\\)`

and `elem`

).

I've added new functions: `getFreeVarsAll`

, `applyBindingsAll`

, `freshenAll`

; which are like the versions without "All", except they're lifted to operate over `Foldable`

/`Traversable`

collections of terms. This is crucial for `freshenAll`

because it allows you to retain sharing of variables among the collection of terms. Whereas it's merely an optimization for the others (saves time for `getFreeVarsAll`

, saves space for `applyBindingsAll`

).

The type of the `seenAs`

function has also changed, to ensure that variables can only be seen as structure rather than as any `UTerm`

.

Thanks to Roman Cheplyaka for suggesting many of these changes.

### Description

The unification API is generic in the type of the structures being unified and in the implementation of unification variables, following the two-level types pearl of Sheard (2001). This style mixes well with Swierstra (2008), though an implementation of the latter is not included in this package.

That is, all you have to do is define the functor whose fixed-point is the recursive type you're interested in:

-- The non-recursive structure of terms
data S a = ...
-- The recursive term type
type PureTerm = Fix S

And then provide an instance for `Unifiable`

, where `zipMatch`

performs one level of equality testing for terms and returns the one-level spine filled with pairs of subterms to be recursively checked (or `Nothing`

if this level doesn't match).

class (Traversable t) => Unifiable t where
zipMatch :: t a -> t b -> Maybe (t (a,b))

The choice of which variable implementation to use is defined by similarly simple classes `Variable`

and `BindingMonad`

. We store the variable bindings in a monad, for obvious reasons. In case it's not obvious, see Dijkstra et al. (2008) for benchmarks demonstrating the cost of naively applying bindings eagerly.

There are currently two implementations of variables provided: one based on `STRef`

s, and another based on a state monad carrying an `IntMap`

. The former has the benefit of O(1) access time, but the latter is plenty fast and has the benefit of supporting backtracking. Backtracking itself is provided by the logict package and is described in Kiselyov et al. (2005).

In addition to this modularity, unification-fd implements a number of optimizations over the algorithm presented in Sheard (2001)— which is also the algorithm presented in Cardelli (1987).

- Their implementation uses path compression, which we retain. Though we modify the compression algorithm in order to make sharing observable.
- In addition, we perform aggressive opportunistic observable sharing, a potentially novel method of introducing even more sharing than is provided by the monadic bindings. Basically, we make it so that we can use the observable sharing provided by the modified path compression as much as possible (without introducing any new variables).
- And we remove the notoriously expensive occurs-check, replacing it with visited-sets (which detect cyclic terms more lazily and without the asymptotic overhead of the occurs-check). A variant of unification which retains the occurs-check is also provided, in case you really need to fail fast.
- Finally, a highly experimental branch of the API performs
*weighted* path compression, which is asymptotically optimal. Unfortunately, the current implementation is quite a bit uglier than the unweighted version, and I haven't had a chance to perform benchmarks to see how the constant factors compare. Hence moving it to an experimental branch.

These optimizations pass a test suite for detecting obvious errors. If you find any bugs, do be sure to let me know. Also, if you happen to have a test suite or benchmark suite for unification on hand, I'd love to get a copy.

### Notes and limitations

[1] At present the library does not appear amenable for implementing higher-rank unification itself; i.e., for higher-ranked metavariables, or higher-ranked logic programming. To be fully general we'd have to abstract over which structural positions are co/contravariant, whether the unification variables should be predicative or impredicative, as well as the isomorphisms of moving quantifiers around. It's on my todo list, but it's certainly non-trivial. If you have any suggestions, feel free to contact me. [back]

[2] At present it is only suitable for single-sorted (aka untyped) unification, à la Prolog. In the future I aim to support multi-sorted (aka typed) unification, however doing so is complicated by the fact that it can lead to the loss of MGUs; so it will likely be offered as an alternative to the single-sorted variant, similar to how the weighted path-compression is currently offered as an alternative. [back]

[3] With the exception of fundeps which are notoriously difficult to implement. However, they are supported by Hugs and GHC 6.6, so I don't feel bad about requiring them. Once the API stabilizes a bit more I plan to release a unification-tf package which uses type families instead, for those who feel type families are easier to implement or use. There have been a couple requests for unification-tf, so I've bumped it up on my todo list. [back]

### References

- Luca Cardelli (1987)
*Basic polymorphic typechecking*.
Science of Computer Programming, 8(2): 147–172.
- Atze Dijkstra, Arie Middelkoop, S. Doaitse Swierstra (2008)
*Efficient Functional Unification and Substitution*.
Technical Report UU-CS-2008-027, Utrecht University.
- Simon Peyton Jones, Dimitrios Vytiniotis, Stephanie Weirich, Mark
Shields (2007)
*Practical type inference for arbitrary-rank types*.
JFP 17(1). The online version has some minor corrections/clarifications.
- Oleg Kiselyov, Chung-chieh Shan, Daniel P. Friedman, and Amr Sabry (2005)
*Backtracking, Interleaving, and Terminating Monad Transformers*.
ICFP.
- Tim Sheard (2001)
*Generic Unification via Two-Level Types and Paramterized Modules*,
Functional Pearl. ICFP.
- Tim Sheard and Emir Pasalic (2004)
*Two-Level Types and Parameterized Modules*.
JFP 14(5): 547–587.
This is an expanded version of Sheard (2001) with new examples.
- Wouter Swierstra (2008)
*Data types à la carte*,
Functional Pearl. JFP 18: 423–436.

### Links