Neighborhood of a mathematical expression using Haskell - haskell

Question

I'm trying to implement with Haskell an algorithm to manipulate mathematical expressions.
I have this data type :
data Exp = Var String | IVal Int | Add Exp Exp
This will be enough for my question.
Given a set of expression transformations, for example :
(Add a b) => (Add b a)
(Add (Add a b) c) => (Add a (Add b c))
And an expression, for example : x = (Add (Add x y) (Add z t)), I want to find all expressions in the neighborhood of x. Given that neighborhood of x is defined as: y in Neighborhood(x) if y can be reached from x within a single transformation.
I am new to Haskell. I am not even sure Haskell is the right tool for this job.
The final goal is to get a function : equivalent x which returns a set of all expressions that are equivalent to x. In other words, the set of all expressions that are in the closure of the neighborhood of x (given a set of transformations).
Right now, I have the following :
import Data.List(nub)
import Data.Set
data Exp = IVal Int
| Scalar String
| Add Exp Exp
deriving (Show, Eq, Ord)
commu (Add a b) = (Add b a)
commu x = x
assoc (Add (Add a b) c) = (Add a (Add b c))
assoc (Add a (Add b c)) = (Add (Add a b) c)
assoc x = x
neighbors x = [commu x, assoc x]
equiv :: [Exp] -> [Exp]
equiv closure
| closure == closureUntilNow = closure
| otherwise = equiv closureUntilNow
where closureUntilNow = nub $ closure ++ concat [neighbors x|x<-closure]
But It's probably slower than needed (nub is O(n^2)) and some terms are missing.
For example, if you have f = (x+y)+z, then, you will not get (x+z)+y, and some others.

Answer 1

Imports, etc. below. I'll be using the multiset package.
import Control.Monad
import Data.MultiSet as M
data Exp = Var String | IVal Int | Add Exp Exp deriving (Eq, Ord, Show, Read)
A bit of paper-and-pencil work shows the following fact: expressions e1 and e2 are in the congruence closure of your relation iff the multiset of leaves are equal. By leaves, I mean the Var and IVal values, e.g. the output of the following function:
leaves :: Exp -> MultiSet Exp
leaves (Add a b) = leaves a `union` leaves b
leaves e = singleton e
So this suggests a nice clean way to generate all the elements in a particular value's neighborhood (without attempting to generate any duplicates in the first place). First, generate the multiset of leaves; then nondeterministically choose a partition of the multiset and recurse. The code to generate partitions might look like this:
partitions :: Ord k => MultiSet k -> [(MultiSet k, MultiSet k)]
partitions = go . toOccurList where
go [] = [(empty, empty)]
go ((k, n):bag) = do
n' <- [0..n]
(left, right) <- go bag
return (insertMany k n' left, insertMany k (n-n') right)
Actually, we only want partitions where both the left and right part are non-empty. But we'll check that after we've generated them all; it's cheap, as there's only two that aren't like that per invocation of partitions. So now we can generate the whole neighborhood in one fell swoop:
neighborhood :: Exp -> [Exp]
neighborhood = go . leaves where
full = guard . not . M.null
go m
| size m == 1 = toList m
| otherwise = do
(leftBag, rightBag) <- partitions m
full leftBag
full rightBag
left <- go leftBag
right <- go rightBag
return (Add left right)
By the way, the reason you're not getting all the terms is because you're generating the reflexive, transitive closure but not the congruence closure: you need to apply your rewrite rules deep in the term, not just at the top level.