unification-fd-0.9.0: Simple generic unification algorithms.

CopyrightCopyright (c) 2007--2014 wren gayle romano
LicenseBSD
Maintainerwren@community.haskell.org
Stabilityexperimental
Portabilitysemi-portable (MPTCs, FlexibleContexts)
Safe HaskellNone
LanguageHaskell98

Control.Unification

Contents

Description

This module provides first-order structural unification over general structure types. It also provides the standard suite of functions accompanying unification (applying bindings, getting free variables, etc.).

The implementation makes use of numerous optimization techniques. First, we use path compression everywhere (for weighted path compression see Control.Unification.Ranked). Second, we replace the occurs-check with visited-sets. Third, we use a technique for aggressive opportunistic observable sharing; that is, we track as much sharing as possible in the bindings (without introducing new variables), so that we can compare bound variables directly and therefore eliminate redundant unifications.

Synopsis

Data types, classes, etc

Unification terms

data UTerm t v Source

The type of terms generated by structures t over variables v. The structure type should implement Unifiable and the variable type should implement Variable.

The Show instance doesn't show the constructors, in order to improve legibility for large terms.

All the category theoretic instances (Functor, Foldable, Traversable,...) are provided because they are often useful; however, beware that since the implementations must be pure, they cannot read variables bound in the current context and therefore can create incoherent results. Therefore, you should apply the current bindings before using any of the functions provided by those classes.

Constructors

UVar !v

A unification variable.

UTerm !(t (UTerm t v))

Some structure containing subterms.

Instances

Alternative t => Alternative (UTerm t) 
Functor t => Monad (UTerm t) 
Functor t => Functor (UTerm t) 
(Functor t, MonadPlus t) => MonadPlus (UTerm t) 
Functor t => Applicative (UTerm t) 
Foldable t => Foldable (UTerm t) 
Traversable t => Traversable (UTerm t) 
(Show v, Show (t (UTerm t v))) => Show (UTerm t v) 

freeze :: Traversable t => UTerm t v -> Maybe (Fix t) Source

O(n). Extract a pure term from a mutable term, or return Nothing if the mutable term actually contains variables. N.B., this function is pure, so you should manually apply bindings before calling it.

unfreeze :: Functor t => Fix t -> UTerm t v Source

O(n). Embed a pure term as a mutable term.

Errors

data UnificationFailure t v Source

The possible failure modes that could be encountered in unification and related functions. While many of the functions could be given more accurate types if we used ad-hoc combinations of these constructors (i.e., because they can only throw one of the errors), the extra complexity is not considered worth it.

Updated: 0.8.1 added Functor, Foldable, and Traversable instances.

Constructors

OccursIn v (UTerm t v)

A cyclic term was encountered (i.e., the variable occurs free in a term it would have to be bound to in order to succeed). Infinite terms like this are not generally acceptable, so we do not support them. In logic programming this should simply be treated as unification failure; in type checking this should result in a "could not construct infinite type a = Foo a" error.

Note that since, by default, the library uses visited-sets instead of the occurs-check these errors will be thrown at the point where the cycle is dereferenced/unrolled (e.g., when applying bindings), instead of at the time when the cycle is created. However, the arguments to this constructor should express the same context as if we had performed the occurs-check, in order for error messages to be intelligable.

TermMismatch (t (UTerm t v)) (t (UTerm t v))

The top-most level of the terms do not match (according to zipMatch). In logic programming this should simply be treated as unification failure; in type checking this should result in a "could not match expected type Foo with inferred type Bar" error.

UnknownError String

Required for the Error instance, which in turn is required to appease ErrorT in the MTL. We do not use this anywhere.

Basic type classes

class Traversable t => Unifiable t where Source

An implementation of syntactically unifiable structure. The Traversable constraint is there because we also require terms to be functors and require the distributivity of sequence or mapM.

Methods

zipMatch :: t a -> t a -> Maybe (t (Either a (a, a))) Source

Perform one level of equality testing for terms. If the term constructors are unequal then return Nothing; if they are equal, then return the one-level spine filled with resolved subterms and/or pairs of subterms to be recursively checked.

class Eq v => Variable v where Source

An implementation of unification variables. The Eq requirement is to determine whether two variables are equal as variables, without considering what they are bound to. We use Eq rather than having our own eqVar method so that clients can make use of library functions which commonly assume Eq.

Methods

getVarID :: v -> Int Source

Return a unique identifier for this variable, in order to support the use of visited-sets instead of occurs-checks. This function must satisfy the following coherence law with respect to the Eq instance:

x == y if and only if getVarID x == getVarID y

Instances

class (Unifiable t, Variable v, Applicative m, Monad m) => BindingMonad t v m | m t -> v, v m -> t where Source

The basic class for generating, reading, and writing to bindings stored in a monad. These three functionalities could be split apart, but are combined in order to simplify contexts. Also, because most functions reading bindings will also perform path compression, there's no way to distinguish "true" mutation from mere path compression.

The superclass constraints are there to simplify contexts, since we make the same assumptions everywhere we use BindingMonad.

Minimal complete definition

lookupVar, freeVar, bindVar

Methods

lookupVar :: v -> m (Maybe (UTerm t v)) Source

Given a variable pointing to UTerm t v, return the term it's bound to, or Nothing if the variable is unbound.

freeVar :: m v Source

Generate a new free variable guaranteed to be fresh in m.

newVar :: UTerm t v -> m v Source

Generate a new variable (fresh in m) bound to the given term. The default implementation is:

newVar t = do { v <- freeVar ; bindVar v t ; return v }

bindVar :: v -> UTerm t v -> m () Source

Bind a variable to a term, overriding any previous binding.

Operations on one term

getFreeVars :: BindingMonad t v m => UTerm t v -> m [v] Source

Walk a term and determine which variables are still free. N.B., this function does not detect cyclic terms (i.e., throw errors), but it will return the correct answer for them in finite time.

applyBindings Source

Arguments

:: (BindingMonad t v m, MonadTrans e, Functor (e m), MonadError (UnificationFailure t v) (e m)) 
=> UTerm t v 
-> e m (UTerm t v) 

Apply the current bindings from the monad so that any remaining variables in the result must, therefore, be free. N.B., this expensively clones term structure and should only be performed when a pure term is needed, or when OccursIn exceptions must be forced. This function does preserve sharing, however that sharing is no longer observed by the monad.

If any cyclic bindings are detected, then an OccursIn exception will be thrown.

freshen Source

Arguments

:: (BindingMonad t v m, MonadTrans e, Functor (e m), MonadError (UnificationFailure t v) (e m)) 
=> UTerm t v 
-> e m (UTerm t v) 

Freshen all variables in a term, both bound and free. This ensures that the observability of sharing is maintained, while freshening the free variables. N.B., this expensively clones term structure and should only be performed when necessary.

If any cyclic bindings are detected, then an OccursIn exception will be thrown.

Operations on two terms

Symbolic names

(===) infix 4 Source

Arguments

:: BindingMonad t v m 
=> UTerm t v 
-> UTerm t v 
-> m Bool 

(=~=) infix 4 Source

Arguments

:: BindingMonad t v m 
=> UTerm t v 
-> UTerm t v 
-> m (Maybe (IntMap Int)) 

(=:=) infix 4 Source

Arguments

:: (BindingMonad t v m, MonadTrans e, Functor (e m), MonadError (UnificationFailure t v) (e m)) 
=> UTerm t v 
-> UTerm t v 
-> e m (UTerm t v) 

(<:=) infix 4 Source

Arguments

:: (BindingMonad t v m, MonadTrans e, Functor (e m), MonadError (UnificationFailure t v) (e m)) 
=> UTerm t v 
-> UTerm t v 
-> e m Bool 

Textual names

equals infix 4 Source

Arguments

:: BindingMonad t v m 
=> UTerm t v 
-> UTerm t v 
-> m Bool 

Determine if two terms are structurally equal. This is essentially equivalent to (==) except that it does not require applying bindings before comparing, so it is more efficient. N.B., this function does not consider alpha-variance, and thus variables with different names are considered unequal. Cf., equiv.

equiv infix 4 Source

Arguments

:: BindingMonad t v m 
=> UTerm t v 
-> UTerm t v 
-> m (Maybe (IntMap Int)) 

Determine if two terms are structurally equivalent; that is, structurally equal modulo renaming of free variables. Returns a mapping from variable IDs of the left term to variable IDs of the right term, indicating the renaming used.

unify infix 4 Source

Arguments

:: (BindingMonad t v m, MonadTrans e, Functor (e m), MonadError (UnificationFailure t v) (e m)) 
=> UTerm t v 
-> UTerm t v 
-> e m (UTerm t v) 

Unify two terms, or throw an error with an explanation of why unification failed. Since bindings are stored in the monad, the two input terms and the output term are all equivalent if unification succeeds. However, the returned value makes use of aggressive opportunistic observable sharing, so it will be more efficient to use it in future calculations than either argument.

unifyOccurs Source

Arguments

:: (BindingMonad t v m, MonadTrans e, Functor (e m), MonadError (UnificationFailure t v) (e m)) 
=> UTerm t v 
-> UTerm t v 
-> e m (UTerm t v) 

A variant of unify which uses occursIn instead of visited-sets. This should only be used when eager throwing of OccursIn errors is absolutely essential (or for testing the correctness of unify). Performing the occurs-check is expensive. Not only is it slow, it's asymptotically slow since it can cause the same subterm to be traversed multiple times.

subsumes infix 4 Source

Arguments

:: (BindingMonad t v m, MonadTrans e, Functor (e m), MonadError (UnificationFailure t v) (e m)) 
=> UTerm t v 
-> UTerm t v 
-> e m Bool 

Determine whether the left term subsumes the right term. That is, whereas (tl =:= tr) will compute the most general substitution s such that (s tl === s tr), (tl <:= tr) computes the most general substitution s such that (s tl === tr). This means that tl is less defined than and consistent with tr.

N.B., this function updates the monadic bindings just like unify does. However, while the use cases for unification often want to keep the bindings around, the use cases for subsumption usually do not. Thus, you'll probably want to use a binding monad which supports backtracking in order to undo the changes. Unfortunately, leaving the monadic bindings unaltered and returning the necessary substitution directly imposes a performance penalty or else requires specifying too much about the implementation of variables.

Operations on many terms

getFreeVarsAll :: (BindingMonad t v m, Foldable s) => s (UTerm t v) -> m [v] Source

Same as getFreeVars, but works on several terms simultaneously. This is more efficient than getting the free variables for each of the terms separately because we can make use of sharing across the whole collection. That is, each time we move to the next term, we still remember the bound variables we've already looked at (and therefore do not need to traverse, since we've already seen whatever free variables there are down there); whereas we would forget between each call to getFreeVars.

Since: 0.7.0

applyBindingsAll Source

Arguments

:: (BindingMonad t v m, MonadTrans e, Functor (e m), MonadError (UnificationFailure t v) (e m), Traversable s) 
=> s (UTerm t v) 
-> e m (s (UTerm t v)) 

Same as applyBindings, but works on several terms simultaneously. This function preserves sharing across the entire collection of terms, whereas applying the bindings to each term separately would only preserve sharing within each term.

Since: 0.7.0

freshenAll Source

Arguments

:: (BindingMonad t v m, MonadTrans e, Functor (e m), MonadError (UnificationFailure t v) (e m), Traversable s) 
=> s (UTerm t v) 
-> e m (s (UTerm t v)) 

Same as freshen, but works on several terms simultaneously. This is different from freshening each term separately, because freshenAll preserves the relationship between the terms. For instance, the result of

mapM freshen [UVar 1, UVar 1]

would be [UVar 2, UVar 3] or something alpha-equivalent, whereas the result of

freshenAll [UVar 1, UVar 1]

would be [UVar 2, UVar 2] or something alpha-equivalent.

Since: 0.7.0

Helper functions

Client code should not need to use these functions, but they are exposed just in case they are needed.

fullprune :: BindingMonad t v m => UTerm t v -> m (UTerm t v) Source

Canonicalize a chain of variables so they all point directly to the term at the end of the chain (or the free variable, if the chain is unbound), and return that end.

N.B., this is almost never the function you want. Cf., semiprune.

semiprune :: BindingMonad t v m => UTerm t v -> m (UTerm t v) Source

Canonicalize a chain of variables so they all point directly to the last variable in the chain, regardless of whether it is bound or not. This allows detecting many cases where multiple variables point to the same term, thereby allowing us to avoid re-unifying the term they point to.

occursIn :: BindingMonad t v m => v -> UTerm t v -> m Bool Source

Determine if a variable appears free somewhere inside a term. Since occurs checks only make sense when we're about to bind the variable to the term, we do not bother checking for the possibility of the variable occuring bound in the term.