Interval exchange transformations

[vdelecroix]

This module implement Interval exchange transformations (iet) (and linear involutions (li)) from a combinatorial point of vue. It also makes the link with strata of Abelian differentials on Riemann surfaces. The three main objects defined in this module are:

• Special kinds of permutations
• Rauzy diagrams (oriented graph)
• Strata of differentials

There are different class of permutations associated to iet, but all are constructed within a class factory:

sage: p = iet.Permutation('a b c d','d c b a')
sage: p
a b c d
d c b a
sage: p.stratum()
H(2)

The object which links those permutations to the dynamic (Teichmüller flow) of strata of differentials is the Rauzy diagram:

sage: p = iet.Permutation('a b c','c b a')
sage: d = p.rauzy_diagram()
Rauzy diagram with 3 permutations

[slabbe]

First I must say that I am not expert at all in that field. I just know some of Interval exchange transformations. So, Vincent, I read through your patch today and I will post some small corrections in a patch soon. I have also some comments (sometimes suggestions open to you) below :

I would gather those related functionalities under a same bag where the user could found them (and all of them) more easily. For example.

sage: IET.[TAB]

or maybe

sage: from sage.combinat.iet.all import IET
sage: IET.[TAB]

For constructor.py, I suggest to

• Create a class constructor containing the functions PermutationsIET, PermutationIET, PermutationLI, GeneralizedPermutation, RauzyDiagram, IntervalExchangeTransformation.
• Create the object IET = constructor()
• Rename PermutationsIET to PermutationsIET_iterator
• Add a function named IntervalExchangeTransformation to the class constructor that wrapped the constructor of IET in iet.py.
• Do we want AbelianStratum, QuadraticStratum and AbelianStrata to be wrapped in constructor as well? I don't know, but if it is realated to everything in iet folder, then it would clearly help the user to know about it.

For iet.py

• Rename IntervalExchangeTransformation.__mul__ to something like multiply_lengths.
• Keep IntervalExchangeTransformation.__mul__ for multiplication of two IET.
• Change IntervalExchangeTransformation._repr_ to look more like an IET and less as two tuples. For example it could say Interval exchange transformation from [0, 1[ to [0, 1[ of permutation ?.
• Rename IntervalExchangeTransformation.copy as __copy__ or __deepcopy__ if it corresponds to what you want. This may applies to many other classes in the files.

I would like the following to work :

sage: p = Permutation([3,2,1])
sage: t = IntervalExchangeTransformation(p, [0.6, 0.1, 0.3])
Traceback (most recent call last):
...
TypeError: cannot concatenate 'str' and 'int' objects

where labels of the intervals are facultative:

sage: t = IntervalExchangeTransformation(p, [0.6, 0.1, 0.3], labels='abc')

Why do the following doesn't work:

sage: p = Permutation([3,2,1])
sage: PermutationIET(p)
1 2 3
3 2 1
sage: PermutationIET([3,2,1])
Traceback (most recent call last):
...
ValueError: your argument can not be split in two parts

• I don't know if the numerous _P_IET = PermutationIET really help the readability. I would not rename them and rather keep them as they are. Or, if so, I would use from this import longname as shortname instead because if shortname is new to me, than I don't have to go in the file this to understand that shortname simply means longname.

For template.py :

I am getting the following :

sage -t  "devel/sage-combinat/sage/combinat/iet/template.py"
**********************************************************************
File "/home/slabbe/sage-4.2/devel/sage-combinat/sage/combinat/iet/template.py", line 1898:
    sage: g == loads(dumps(g))
Exception raised:
    Traceback (most recent call last):
      File "/home/slabbe/sage-4.2/local/bin/ncadoctest.py", line 1231, in run_one_test
        self.run_one_example(test, example, filename, compileflags)
      File "/home/slabbe/sage-4.2/local/bin/sagedoctest.py", line 38, in run_one_example
        OrigDocTestRunner.run_one_example(self, test, example, filename, compileflags)
      File "/home/slabbe/sage-4.2/local/bin/ncadoctest.py", line 1172, in run_one_example
        compileflags, 1) in test.globs
      File "<doctest __main__.example_50[5]>", line 1, in <module>
        g == loads(dumps(g))###line 1898:
    sage: g == loads(dumps(g))
      File "sage_object.pyx", line 735, in sage.structure.sage_object.dumps (sage/structure/sage_object.c:8008)
      File "sage_object.pyx", line 165, in sage.structure.sage_object.SageObject.dumps (sage/structure/sage_object.c:2158)
    PicklingError: Can't pickle <class 'sage.combinat.iet.labeled.Path'>: attribute lookup sage.combinat.iet.labeled.Path failed

[slabbe]

I submitted before preview. So I repeat below the lists that I messed up above.

For constructor.py, I suggest to

• Create a class constructor containing the functions PermutationsIET, PermutationIET, PermutationLI, GeneralizedPermutation, RauzyDiagram, IntervalExchangeTransformation.
• Create the object IET = constructor()
• Rename PermutationsIET to PermutationsIET_iterator
• Add a function named IntervalExchangeTransformation to the class constructor that wrapped the constructor of IET in iet.py.
• Do we want AbelianStratum, QuadraticStratum and AbelianStrata to be wrapped in constructor as well? I don't know, but if it is realated to everything in iet folder, then it would clearly help the user to know about it.

For iet.py

• Rename IntervalExchangeTransformation.__mul__ to something like multiply_lengths.
• Keep IntervalExchangeTransformation.__mul__ for multiplication of two IET.
• Change IntervalExchangeTransformation._repr_ to look more like an IET and less as two tuples. For example it could say Interval exchange transformation from [0, 1[ to [0, 1[ of permutation ?.
• Rename IntervalExchangeTransformation.copy as __copy__ or __deepcopy__ if it corresponds to what you want. This may applies to many other classes in the files.

[slabbe]

Applies over the precedent patch.

[slabbe]

Vincent, I tried your code again with Thierry Monteil who knows better this field and he liked all the functions you implemented.

He suggested that you change Permutation* to LabelledPermutation or PermutationwithLabel... up to you.

Also, I notice the following. I think it should be a method and not an attribute :

sage: p = Permutation([4,3,2,1])
sage: pp = PermutationIET(p)
sage: pp.alphabet
Ordered Alphabet [1, 2, 3, 4]

[vdelecroix]

Replying to slabbe:

Vincent, I tried your code again with Thierry Monteil who knows better this field and he liked all the functions you implemented.

He suggested that you change Permutation* to LabelledPermutation or PermutationwithLabel... up to you.

It won't be possible. I consider two kinds of permutations

• ReducedPermutation?
• LabelledPermutation?

A ReducedPermutation? can be identified with a permutation of an ordered alphabet. A Labeledpermutation is a couple of bijection (p_t,p_b) : alphabet -> [1,n]

Also, I notice the following. I think it should be a method and not an attribute :

sage: p = Permutation([4,3,2,1])
sage: pp = PermutationIET(p)
sage: pp.alphabet
Ordered Alphabet [1, 2, 3, 4]

The resason is because it's possible to change the alphabet sage: p = PermutationIET('a b c','c b a') sage: p a b c c b a sage: p.alphabet = [1,2,3] sage: p 1 2 3 3 2 1 sage: p.alphabet = 'cd' ... ValueError?: Your alphabet has not enough letters

[vdelecroix]

• Create a class constructor containing the functions PermutationsIET, PermutationIET, PermutationLI, GeneralizedPermutation, RauzyDiagram, IntervalExchangeTransformation.
• Create the object IET = constructor()

done. Everything is in constructor.py which is imported as iet. Then the iet.<tab> gives the (approximately) the following list

• Permutation
• RauzyDiagram?
• IntervalExchangeTransformation?

• Rename PermutationsIET to PermutationsIET_iterator

done

• Add a function named IntervalExchangeTransformation to the class constructor that wrapped the constructor of IET in iet.py.

done

• Do we want AbelianStratum, QuadraticStratum and AbelianStrata to be wrapped in constructor as well? I don't know, but if it is realated to everything in iet folder, then it would clearly help the user to know about it.

I don't know... the strata here means strata of Abelian differentials on Riemann surfaces. The fact is we need them to understand precisely the structure of Rauzy diagram. It is hence related to Rauzy diagram but somewhat independent of the theory of interval exchange transformations.

For iet.py

• Rename IntervalExchangeTransformation.__mul__ to something like multiply_lengths.

done. In fact, it's implemented in .normalize()

• Keep IntervalExchangeTransformation.__mul__ for multiplication of two IET.

done. But a little todo remains. I choose to create canonic labels from the labels of the two iets and this force a conversion of labels to strings. It's not so beautiful if we do not need the labels.

• Change IntervalExchangeTransformation._repr_ to look more like an IET and less as two tuples. For example it could say Interval exchange transformation from [0, 1[ to [0, 1[ of permutation ?.

done.

• Rename IntervalExchangeTransformation.copy as __copy__ or __deepcopy__ if it corresponds to what you want. This may applies to many other classes in the files.

done.

[slabbe]

Dear Vincent,

Thanks for doing all those changes.

I am now looking at your recent patch and I have a few more comments. I am sorry I was not able to tell them before. I will try now to make a complete review with all my (hopefully) final remarks. After those FIVE remarks answered, I think we will be very near of a positive review!

Sébastien

ONE.

The resason is because it's possible to change the alphabet sage: p = PermutationIET('a b c','c b a') sage: p a b c c b a sage: p.alphabet = [1,2,3] sage: p 1 2 3 3 2 1 sage: p.alphabet = 'cd' ... ValueError??: Your alphabet has not enough letters

I am not sure this is a good reason for using an attribute instead of a method. See the following example :

sage: g = Graph()
sage: p = g.plot()
sage: p.xmin()
-1.0
sage: p.xmin(-2)
sage: p.xmin()
-2.0

TWO.

The documentation and doctest coverage is very good (96%) but not perfect :

slabbe@pol:~/sage-4.2/devel/sage-combinat/sage/combinat/iet$ sage -coverage .
constructors.py: 75% (6 of 8)
iet.py: 100% (23 of 23)
labelled.py: 95% (57 of 60)
reduced.py: 100% (55 of 55)
strata.py: 100% (39 of 39)
template.py: 96% (89 of 92)

Overall weighted coverage score:  96.9%
Total number of functions:  277
We need    5 more function to get to 99% coverage.

Moreover, sage -coverage * tells many function name doesn't occur in doctests. Maybe you could add #indirect doctest in doctests that are indirect.

While you are at it, you may consider to add some INPUT and OUTPUT where there are missing. See the discussion  sage-devel: input and output in docstrings

THREE.

I am getting 2 doctests failures :

sage -t  "devel/sage-combinat/sage/combinat/iet/constructors.py"
**********************************************************************
File "/home/slabbe/sage-4.2/devel/sage-combinat/sage/combinat/iet/constructors.py", line 531:
    sage: for p in P: print p, "* *\n"
Expected:
    a b
    b a
    * *
Got:
    a b
    b a * *
    <BLANKLINE>
    b a
    a b * *
    <BLANKLINE>
**********************************************************************
File "/home/slabbe/sage-4.2/devel/sage-combinat/sage/combinat/iet/constructors.py", line 536:
    sage: for p in P: print p, "\* * *\n"
Expected:
    a b c
    c b a
    * * *
Got:
    a b c
    b c a \* * *
    <BLANKLINE>
    a b c
    c a b \* * *
    <BLANKLINE>
    a b c
    c b a \* * *
    <BLANKLINE>
    a c b
    b a c \* * *
    <BLANKLINE>
    a c b
    b c a \* * *
    <BLANKLINE>
    a c b
    c b a \* * *
    <BLANKLINE>
    b a c
    a c b \* * *
    <BLANKLINE>
    b a c
    c a b \* * *
    <BLANKLINE>
    b a c
    c b a \* * *
    <BLANKLINE>
    b c a
    a b c \* * *
    <BLANKLINE>
    b c a
    a c b \* * *
    <BLANKLINE>
    b c a
    c a b \* * *
    <BLANKLINE>
    c a b
    a b c \* * *
    <BLANKLINE>
    c a b
    b a c \* * *
    <BLANKLINE>
    c a b
    b c a \* * *
    <BLANKLINE>
    c b a
    a b c \* * *
    <BLANKLINE>
    c b a
    a c b \* * *
    <BLANKLINE>
    c b a
    b a c \* * *
    <BLANKLINE>
**********************************************************************
1 items had failures:
   2 of   6 in __main__.example_4
***Test Failed*** 2 failures.
For whitespace errors, see the file /home/slabbe/.sage//tmp/.doctest_constructors.py
	 [3.0 s]
exit code: 1024

FOUR.

I use sage-4.2 and I have problem to docbuild a branch of sage. It docbuilds the sage main branch, so I am not able to test if there are no Sphinx warnings and if everythings looks good on the html doc. I am waiting sage-4.3 to come out to test your patch on it and see the doc.

FIVE.

I changed a little bit the _repr_ of Interval exchange transformation. See the patch attached. Hope you agree.

[slabbe]

Applies over the precedent 3 patches.

[vdelecroix]

Replying to slabbe: Dear Sebastien,

Thanks for doing all those changes.

Thanks for this nice review ! It will be a *very* good patch.

ONE.

The resason is because it's possible to change the alphabet sage: p = PermutationIET('a b c','c b a') sage: p a b c c b a sage: p.alphabet = [1,2,3] sage: p 1 2 3 3 2 1 sage: p.alphabet = 'cd' ... ValueError??: Your alphabet has not enough letters

I am not sure this is a good reason for using an attribute instead of a method. See the following example :

sage: g = Graph()
sage: p = g.plot()
sage: p.xmin()
-1.0
sage: p.xmin(-2)
sage: p.xmin()
-2.0

It works now as

sage: p = iet.Permutation('a b c','c b a')
sage: print p
a b c
c b a
sage: p.alphabet([1,2,3])
sage: print p
1 2 3
3 2 1

TWO.

The documentation and doctest coverage is very good (96%) but not perfect :

up to 100% now

While you are at it, you may consider to add some INPUT and OUTPUT where there are missing. See the discussion  sage-devel: input and output in docstrings

lot of INPUT/OUTPUT added

THREE.

I am getting 2 doctests failures :

corrected

FIVE.

I changed a little bit the _repr_ of Interval exchange transformation. See the patch attached. Hope you agree.

I realy agree. In France, we use the coma instead of points to distinguish the integer part and the decimal that why we use dot comman for intervals... I keep it for the french translation ;-)

[vdelecroix]

Applies over the 4 precedings

[slabbe]

Great. Point ONE above was addressed. The documentation coverage is now 100% perfect. All tests passed.

I builded the documentation and there was several sphinx issues. I corrected them in a patch that I will upload shortly.

[slabbe]

Applies over all the precedent patches.

[slabbe]

Vincent, can you review my last patch and make sure everything is OK with the sphinx output? My patch solved some sphinx syntax, but I feel like more improvements could be done.

[vdelecroix]

Hi,

Vincent, can you review my last patch and make sure everything is OK with the sphinx output? My patch solved some sphinx syntax, but I feel like more improvements could be done.

I'm not sure about what you did with the OUTPUT part of the documentation string. It does not correspond to what is written in the "developer's guide"

OUTPUT:

    integer -- the ...

What do I do for this ?

I will make a complete review of the documentation after the answer.

[slabbe]

I'm not sure about what you did with the OUTPUT part of the documentation string. It does not correspond to what is written in the "developer's guide"

My fault. In fact, I should update the way I write the OUTPUT part. I will take a closer look to the developer's guide. Feel free to change those.

[vdelecroix]

I made a complete review of the doc. There is no warning when building the documentation and there is INPUT and OUTPUT fields where there are needed.

I replace your patch trac_7145-review3-sl.patch with my trac-7145-documentation-review-vd.patch (because I find it more natural to erase the OUTPUT field modification than modify it back).

I also made little modifications on default argument on some method (.flips() and .list() of FlippedPermutation?) and a change of name (.rauzy_1n() becomes .cylindric()). The latter is due to a forthcoming paper.

[vdelecroix]

applies over (trac_7145-iet-vd.patch, trac_7145-review-sl.patch, trac_7145-corrections-vd.patch, trac_7145-review2-sl ,trac_7145-corrections2-vd.patch)

[vdelecroix]

applies over (trac_7145-iet-vd.patch, trac_7145-review-sl.patch, trac_7145-corrections-vd.patch, trac_7145-review2-sl ,trac_7145-corrections2-vd.patch)

[slabbe]

Apply only this one.

[slabbe]

I folded the relevant patches together : trac_7145-iet-final.patch .

Positive review.

[slabbe]

Positive review.

I just want to add that I tested it on a fresh Sage-4.3.