Hi Daniel, It depends on what you want to use the normalized / canonical form for. If it's to reduce semantic equivalence testing to simple syntactic equality, e.g. (A == B) = (canonize(A) == canonize(B)), then you could just use the fully expanded form, which isn't really simplification. :) I'm doing something similar here (for polynomial expressions over an inner product space): https://github.com/mkovacs/ipoly/blob/master/Poly.hs Cheers, Máté On Sat, Jun 16, 2012 at 2:17 PM, Daniel Hlynskyi wrote:

Hello.

I am new to typefull programming, so I've got a question. I have a simple mathematical expression (addition, product and exponentiation only):

...

data Expr = I Int -- integer constant           | V -- symbolic variable           | Sum [Expr]           | Prod [Expr]           | Pow Expr Expr

What I want is normalize\simplify this expression. Eg `Prod [Pow V (I 0), Pow V (I 1)] ` must be simplified to just `V`. What techniques should I use to write `normalize` function? Simplification rules are quite simple:

...

normalize (Sum [a]) = normalize a normalize (Sum xs) | (I 0) `elem` xs = map nomalize . Sum $ filter (/= I 0) xs                          | otherwise = map normalize xs normalize (Prod xs) | (I 0) `elem` xs = I 0 normalize (Prod xs) | (I 1) `elem` xs = map nomalize . Prod $ filter (/= I 1) xs                          | otherwise = map normalize xs normalize (Pow a (I 0)) = I 1 normalize (Pow a (I 1)) = normalize a

and so on. But rules like theese cannot simplify some expressions, for example, `Prod [Pow V (I 0), Pow V (I 1)] `.

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On Sat, Jun 16, 2012 at 11:17 PM, Daniel Hlynskyi wrote:

Hello. Simplification rules are quite simple:

...

normalize (Sum [a]) = normalize a normalize (Sum xs) | (I 0) `elem` xs = map nomalize . Sum $ filter (/= I 0) xs                          | otherwise = map normalize xs normalize (Prod xs) | (I 0) `elem` xs = I 0 normalize (Prod xs) | (I 1) `elem` xs = map nomalize . Prod $ filter (/= I 1) xs                          | otherwise = map normalize xs normalize (Pow a (I 0)) = I 1 normalize (Pow a (I 1)) = normalize a

and so on. But rules like theese cannot simplify some expressions, for example, `Prod [Pow V (I 0), Pow V (I 1)] `.

It's because you're doing it in the wrong direction : in a tree always normalize leaves before you try to normalize branches. normalize (Sum xs) = case sort . filter (/= I 0) . map normalize $ xs of [] -> I 0 [a] -> a ys -> sumPrefix ys where sumPrefix (I n : I m : ys) = sumPrefix $ I (n+m) : ys sumPrefix ys = ys See how I normalize the elements of xs before anything else. Note that this isn't quite the right way to represent a poly if you want to get to a canonical representation. -- Jedaï