This term, I’m listening to a lecture course on Computational Group Theory. As a good exercise, I plan to implement everything we talk about as Haskell functions as well.
The first lecture was mainly an introduction to the area, ending with a very naďve algorithm to generate a permutation group from a set of generators. Next week will bring less naďve algorithms with not quite as horrible complexity.
Before the algorithm can be brought, we’d want some undergrowth: we’d want to be able to work with permutations at all. So, we’ll start with the basic group theory and permutation implementations. A lot of this is stolen or rewritten from this permutation group code.
Our code will make use of two libraries, so if you collate code snippets while reading this, you’ll want to use
If you don’t want to bother with that, the code is available here.
One of the things I love the most with Haskell is the way that you just write down what you think it means, and end up with good code. So, what is a group? It is this type class:
identity :: Int -> a
isIdentity :: a -> Bool
(*>) :: a -> a -> a – right multiplication
inverse :: a -> a
with some conditions on the functions given. This is geared towards implementing permutation groups in the first line — so the idea is that you give your group as a subgroup of Sn, and you can retrieve the identity in Sn by using the identity function.
We tend to want to be able to conjugate elements and multiply a whole list of elements, so let’s add a few functions:
prod gs = foldl1 (*>) gs
And now we can start looking at permutations. We will store a permutation as the list of images, in order. So for instance, the permutation given by (1 2)(1 3) will be stored as the list [2,3,1]. (note that in the good tradition of computational group theory, we act from the right – thus motivating this result and not [2,1,3])
The interesting bit about permutations, though, is that they form a group; so we instantiate our type class as well.
x .^ (PL g) = g !! (x-1)
instance Group Permutation where
identity n = PL [1..n]
isIdentity (PL xs) = all (\(i,j) -> i==j) (zip [1..] xs)
(PL g) *> h = PL (map (.^ h) g)
inverse (PL g) = PL (map snd (sort (zip g [1..])))
We are used to being able to write permutations in cycle notation, so we’d want to be able to convert to and from that, too.
pointActions = concat (map fromCycle cs) fromCycle is = zip is (rotateL is) fromCycles’ n cs = PL (elems (array (1,n) [(i,i) | i <- [1..n]] // (concat (map fromCycle cs))))
findOrbit (PL permutation) elt =
let buildOrbit os ps e =
if (elem e os)
else let ei = elemIndex e ps in
case ei of
Nothing -> os
Just e’ -> buildOrbit ([e] ++ os) ps (e’+1)
in buildOrbit  permutation elt toCycles (PL permutation) =
getOrbits out  per = out
getOrbits out els per =
orbit = findOrbit per (head els)
els’ = els \\ orbit
in getOrbits (out ++ [orbit]) els’ per
in getOrbits  permutation (PL permutation)
rotateL (x:xs) = xs ++ [x]
saturateCycles :: [[Int]] -> Int -> [[Int]] saturateCycles cs n = cs ++ ( group ( [1..n] \\ foldl1 (++) cs ) )
desaturateCycles cs = filter (\x -> length x > 1) cs
We can use saturateCycles and desaturateCycles in order to convert to and from the normal way of writing permutations. The toCycles and fromCycles functions both expect fixpoints to be listed as well. We need, however, to tell the saturateCycles which symmetric group we want to live in.
Now, the most naďve algorithm to list all elements in a group starts with the one element we know will live in the group: the identity. And for each iteration, all generators are multiplied onto the first known candidate in a list of group element candidates, and all results that are not yet known are added to this list.
In Haskell code:
let g0 = head generators
i = g0 *> inverse g0
addElts group  gens = group
addElts group cands gens =
let cand = take 1 cands
group’ = group ++ cand
newcands = map (*> (head cand)) gens
cands’ = (tail cands) ++ (newcands \\ (group’ ++ cands))
addElts group’ cands’ gens
in addElts  [i] generators
We keep on adding tested candidates to the first list, and anything not yet seen that we get from our multiplication with generators gets added to the middle list. Once the middle list is empty, we’re done.