Sunday, 19 August 2007

What is a Ring?

The concept of a ring is fundamental to advanced algebra, with applications throughout pure maths. Basically, a ring is the mathematical concept that the Haskell Num type class is trying to capture (but not quite succeeding).

A ring is a set R with two binary operations, addition and multiplication, satisfying certain rules. Specifically:

R is a commutative group with respect to addition
  • Addition is associative: a+(b+c) = (a+b)+c
  • There is an additive identity, 0, such that a+0 = 0+a = a
  • There is an additive inverse, -a, such that a+(-a) = (-a)+a = 0
  • Addition is commutative: a+b = b+a
R is a semigroup with respect to multiplication
  • Multiplication is associative: a*(b*c) = (a*b)*c
  • Often, there will be a multiplicative identity, 1, such that 1*a = a*1 = a
  • Quite often, multiplication will be commutative
Multiplication distributes over addition
  • a*(b+c) = a*b + a*c
  • (a+b)*c = a*c + b*c
Now of course, there are plenty of examples of rings - in fact they're the sort of thing that we run into all the time. Here are some of the most obvious:
  • The integers. These are really the prototype for the concept. However the integers are actually quite a special ring, having many properties that other rings can fail to have. Part of understanding rings is understanding which properties of the integers don't hold for all rings. Mathematicians use the abbreviation Z for the integers (from the German Zahlen, meaning numbers)
  • The rational numbers (Q), real numbers (R), and complex numbers (C).
  • For fixed n, the set of n*n matrices. For example, the set of 2*2 matrices.
  • The set of polynomials in x
And so on. It's also worth pointing out a couple of non-examples:
  • The natural numbers N (the positive integers). They're not a ring because we don't have additive inverses (the negative integers).
  • Vectors with the cross product as multiplication. The problem is that the cross product isn't associative.
Time for some Haskell. Let's start with 2*2 matrices:

{-# OPTIONS_GHC -fglasgow-exts #-}

module Rings where

newtype Matrix r = M [[r]] deriving (Eq,Show)

instance Num r => Num (Matrix r) where
M [[a,b],[c,d]] + M [[a',b'],[c',d']] = M [[a+a',b+b'],[c+c',d+d']]
negate (M [[a,b],[c,d]]) = M [[-a,-b],[-c,-d]]
M [[a,b],[c,d]] * M [[e,f],[g,h]] = M [[a*e+b*g, a*f+b*h] ,[c*e+d*g, c*f+d*h]]
fromInteger n = M [[fromInteger n, 0],[0, fromInteger n]]

It wouldn't have been much harder to handle the n*n case rather than just the 2*2 case, but I thought the code was clearer this way.

Let's quickly test it:

> M [[1,2],[3,4]] - M [[1,0],[0,1]]
M [[0,2],[3,3]]
> M [[2,0],[0,3]] * M [[1,2],[3,4]]
M [[2,4],[9,12]]

Notice that in Haskell, the Num type class is what you use when you want to define a ring. It provides the (+), (-), and (*) operators that you need. We also need a way to say what the 0 and 1 are in our ring. This is what the fromInteger function is for. If we wanted, we could have just defined fromInteger 0 and fromInteger 1, and left the other cases unmatched, like so:

fromInteger 0 = M [[0,0],[0,0]]
fromInteger 1 = M [[1,0],[0,1]]

As far as defining a ring goes, this would have been fine. But once someone knows what fromInteger 1 is, they can use the ring laws to work out fromInteger n as follows:

fromInteger n | n >= 0 = sum $ replicate n $ fromInteger 1
| otherwise = negate $ fromInteger $ negate n

So we might as well save them the trouble and just do it for them.

Note that because Haskell will do fromInteger calls implicitly for us, we can write things like this:

> 2 * M [[1,2],[3,4]]
M [[2,4],[6,8]]

A couple of things to point out about the ring of 2*2 matrices:
  • Multiplication is not commutative
  • There are zero divisors - we can find matrices a/=0, b/=0 such that a*b==0

Okay, next, polynomials in x:

newtype UPoly a = UP [a] deriving (Eq)
-- the list [a_0, a_1, ..., a_n] represents the polynomial a_0 + a_1 x + ... + a_n x^n

x = UP [0,1] :: UPoly Integer

instance (Show a, Num a) => Show (UPoly a) where
show (UP []) = "0"
show (UP as) = let powers = reverse $ filter ( (/=0) . fst ) $ zip as [0..]
c:cs = concatMap showTerm powers
in if c == '+' then cs else c:cs
where showTerm (a,i) = showCoeff a ++ showPower a i
showCoeff a | a == 1 = "+"
| a == -1 = "-"
| otherwise = let cs = show a
in if head cs == '-' then cs else '+':cs
showPower a i | i == 0 = if a `elem` [1,-1] then "1" else ""
| i == 1 = "x"
| i > 1 = "x^" ++ show i

instance Num a => Num (UPoly a) where
UP as + UP bs = toUPoly $ as <+> bs
negate (UP as) = UP $ map negate as
UP as * UP bs = toUPoly $ as <*> bs
fromInteger 0 = UP []
fromInteger a = UP [fromInteger a]

toUPoly as = UP (reverse (dropWhile (== 0) (reverse as)))

(a:as) <+> (b:bs) = (a+b) : (as <+> bs)
as <+> [] = as
[] <+> bs = bs

[] <*> _ = []
_ <*> [] = []
(a:as) <*> (b:bs) = [a*b] <+> (0 : map (a*) bs) <+> (0 : map (*b) as) <+> (0 : 0 : as <*> bs)

Quick test:

> (x+1)^3

Unlike the matrices, this ring of polynomials over the integers (which mathematicians write as Z[x]) is quite a nicely behaved ring. Multiplication is commutative, and there are no zero divisors.

There are many other important rings. Let's look at one more: the integers modulo n (for fixed n) - also known as "clock arithmetic", and denoted Zn by mathematicians. Here I'm going to use some phantom type trickery, so the following code may not compile in all environments:

class IntegerAsType a where
value :: a -> Integer

data T12
instance IntegerAsType T12 where value _ = 12

data T10
instance IntegerAsType T10 where value _ = 10

newtype Zn n = Zn Integer deriving (Eq)

instance Show (Zn n) where
show (Zn x) = show x

instance IntegerAsType n => Num (Zn n) where
Zn x + Zn y = Zn $ (x+y) `mod` value (undefined :: n)
negate (Zn 0) = 0
negate (Zn x) = Zn $ value (undefined :: n) - x
Zn x * Zn y = Zn $ (x*y) `mod` value (undefined :: n)
fromInteger n = Zn $ n `mod` value (undefined :: n)
When using this code, I need to make sure that the interpreter knows which type I am working in:

> 2*9 :: Zn T12
> 2*9 :: Zn T10
Zn is a commutative ring. However, when n is composite (not a prime), then it has zero divisors.

So there you have it, a few examples of the concept of ring.

At the beginning, I said that the Num type class didn't quite succeed at capturing the concept of ring. That's because it has a couple of other functions that we've been ignoring up to now - abs and signum. This is the little blot in Haskell's copybook. abs and signum don't have anything to do with rings. They don't make sense for most of the rings we've discussed. In fact they're only really relevant to rings which can be embedded into the complex numbers. I'm hoping that in the next version of Haskell, this little wrinkle will be ironed out.

Next time: fields


Thiagarajan said...

I liked your haskell math code very much...
i got a question there anyway to enforce commutativity of operations in haskell type classes

DavidA said...

You mean, can you get the language to enforce commutativity (eg by signalling an error at compile time)? I don't think so. The usual approach when defining type classes is to state the invariants that you want to hold, and just trust people defining instances to respect the invariants.

RedGlow said...

Pity you stopped writing articles like this, it was very well done : /

mladen said...

Agree :/

Elin said...

Good for people to know.

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