spe alternatives and similar packages
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hgeometry
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eigen
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Access the most powerful time series database as a service
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They vary from L1 to L5 with "L5" being the highest.
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README
The spe
package — species lite
A simple library for combinatorial species with no dependencies but base. For a quick taste, look at the examples below. For further information, see the reference documentation on Hackage, and the full (but short!) source code: Math/Spe.hs.
Examples
Octopodes
[octopus](img/oct.png)
An octopus is a cycle of nonempty lists:
oct = cyc `o` nonEmpty list
Connected lists
A connected list is a nonempty list that begins with its smallest
element. E.g, [3,5,9,7]
is connected but [2,4,1]
is not. Using the
ordinal product we can define the Lspecies of connected lists by
clist = x <*. list
in which x
is the singleton species. Let us print a few of these
connected lists to get a feeling for how this works:
> mapM_ print $ clist [2,5,6,9]
(2,[9,6,5])
(2,[9,5,6])
(2,[6,9,5])
(2,[6,5,9])
(2,[5,9,6])
(2,[5,6,9])
If we prefer to see these simply as lists rather than pairs, then we would define
clist = map (uncurry (:)) . (x <*. list)
Indeed, with this definition we have
> mapM_ print $ clist [2,5,6,9]
[2,9,6,5]
[2,9,5,6]
[2,6,9,5]
[2,6,5,9]
[2,5,9,6]
[2,5,6,9]
You might wonder why these lists are called "connected". It has to do
with the species list
and set `o` listc
being isomorphic. In
general, if f
and g
are species and f = set `o` g
then g
can
be seen as (connected) components of f
, and those components may be
called connected f
structures.
Binary trees
Here's an example of a recursively defined species.
data BTree a = Empty  BNode a (BTree a) (BTree a) deriving (Show, Eq)
bTree :: Spe a (BTree a)
bTree [] = [ Empty ]
bTree xs = [ BNode v l r
 (v,(l,r)) < x .*. (bTree .*. bTree) $ xs
]
We could similary use the ordinal product to define increasing binary trees, also called minheaps:
heap :: Spe a (BTree a)
heap [] = [ Empty ]
heap xs = [ BNode v l r
 (v,(l,r)) < x <*. (heap .*. heap) $ xs
]
Derivatives
The following expression evaluates to true and illustrates that the derivative of the species of cycles is isomorphic to the species of lists (linear orders):
(map catMaybes $ dx cyc [1..5]) == list [1..5]
Pointing
A pointed Fstructure is an Fstructure together with a distinguished element (the element pointed at):
> mapM_ print $ pointed set [1..3]
([1,2,3],1)
([1,2,3],2)
([1,2,3],3)
One may note that pointed f
is isomorphic x .*. dx f
. For example,
> mapM_ print $ (x .*. dx set) [1..3]
(3,[Nothing,Just 1,Just 2])
(2,[Nothing,Just 1,Just 3])
(1,[Nothing,Just 2,Just 3])
Cartesian product
The Cartesian product (><)
of two species F and G on a set U is
obtained by superimposing both an Fstructure and a Gstructure on
U. For instance, pointing can be expressed using the Cartesian (and
the ordinary) species product:
pointed f = f >< (x .*. set)
Functorial composition
The Spe
package doesn't provide an operator for the functorial
composition of species. This is because, for our definition of species,
it coincides with the usual composition of functions. For instance, the
species of simple undirected graphs can be defined like this:
graph = subset . kSubset 2
Ballot matrices
A ballot (or ordered set partition) is a list of blocks, where a block is simply a nonempty set. We may give ballots this type:
type Bal a = [[a]]
The ballot species can be defined by list `o` nonEmpty set
. The type
of this expression, Spe a ([[a]],[[a]])
, is, however, a bit more
complicated than we intended. Looking at the definition of partitional
composition in
Math/Spe.hs we
realize that mapM (nonEmpty set) bs == bs
for any set partition
bs
. Thus the second component is redundant, and a better definition of
the species of ballots would be
bal :: Spe a (Bal a)
bal = map fst . (list `o` nonEmpty set)
The species of ballots is already defined in Math/Spe.hs. The definition given there is a bit different for reasons of efficiency.
In a recent preprint, Stuart Hannah and I study uppertriangular matrices of ballots without empty rows. Those can be defined as follows:
type Row a = [Bal a]
type BalMat a = [Row a]
rowOfLength :: Int > Spe a (Row a)
rowOfLength i = bal .^ i
balMatOfDim :: Int > Spe a (BalMat a)
balMatOfDim k = prod [ nonEmpty (rowOfLength i)  i < [1..k] ]
balMat :: Spe a (BalMat a)
balMat xs = assemble [ balMatOfDim k  k < [0..length xs] ] xs
Further, we define the sign of a ballot matrix by
sign :: BalMat a > Int
sign m = (1)^(dim m + blk m)
where
dim = length  Matrix dimension
blk = length . concat . concat  Total number of blocks
Let us count ballot matrices with respect to this sign:
> [ sum . map sign $ balMat [1..n]  n<[0..6] ]
[1,1,3,19,207,3451,81663]
Looking up this sequence in OEIS, here using the
command line utility sloane
, we find:
$ sloane 1,1,3,19,207,3451,81663
S A079144 1,1,3,19,207,3451,81663,2602699,107477247,5581680571,356046745023,
N A079144 Number of labeled interval orders on n elements: (2+2)free posets.
In the aforementioned preprint we prove this surprising result using a sign reversing involution.
Notes
Transport of structure
Assuming that h . h' = h' . h = id
, we can define the transport of
structure as follows.
transport :: Functor f => (a > b) > (b > a) > Spe a (f a) > Spe b (f b)
transport h h' spe = map (fmap h) . spe . map h'
For this to work consistently we would, however, have to define some
additional newtypes and functor instances. For instance, in
bal :: Spe a [[a]]
we would like fmap
to map two levels deep instead
of one level deep in [[a]]
.
The species
package by Brent Yorgey
This species module was created for use in my own research. It is
sufficient for my needs. If you want something more substantial, then
you will most likely be happier with the excellent
species
package
by Brent Yorgey.