Chart: Implement pickling via __getstate__/__setstate__

[mkoeppe]

In this ticket, we implement the pickling protocol using __getstate__ and __setstate__

Depends on #31894

CC: @egourgoulhon @tscrim @mjungmath

Component: manifolds

Work Issues: redo on top of #31894

Branch/Commit: u/mkoeppe/chart__no_longer_use_uniquerepresentation__implement___getstate_____setstate__ @ 30a5d0b

Issue created by migration from https://trac.sagemath.org/ticket/31901

[egourgoulhon]

Replying to @mkoeppe:

In this ticket,

• we make it possible to pass the codomain restrictions already when initializing the chart

This would be nice! In the early stages of the manifold project, I did not manage to do it simply, i.e. without changing Sage's preparser, since

sage: D = Manifold(2, 'D')                                                                          
sage: X.<x,y> = D.chart(restrictions=x^2 + y^2 < 1)                                                 

would generate NameError: name 'y' is not defined. Of course, one could replace x^2 + y^2 < 1 by the string "x^2 + y^2 < 1", but this is not very user-friendly. Do you already have some solution in mind?

[egourgoulhon]

comment:3

Some thought: if UniqueRepresentation is abandoned for charts, it may be desirable to maintain WithEqualityById (one of the two mother classes of UniqueRepresentation) to have a fast equality operator. Indeed charts are heavily used as dictionary keys: for coordinates of manifold points, for coordinate expressions of scalar fields (the latter being used, among other things, to store the components of tensor fields), for coordinate expressions of continuous maps between manifolds (the keys being pairs of charts in that case), etc. Certainly WithEqualityById is the fastest equality operator and thus provides a fast access to the above dictionaries.

[mkoeppe]

comment:4

Replying to @egourgoulhon:

Replying to @mkoeppe:

sage: D = Manifold(2, 'D')                                                                          
sage: X.<x,y> = D.chart(restrictions=x^2 + y^2 < 1)                                                 

would generate NameError: name 'y' is not defined. Of course, one could replace x^2 + y^2 < 1 by the string "x^2 + y^2 < 1", but this is not very user-friendly. Do you already have some solution in mind?

Ah! That's of course a problem, I didn't realize this

[mkoeppe]

comment:6

Maybe a chart should become immutable "upon first use"?

[mkoeppe]

Branch: u/mkoeppe/chart__no_longer_use_uniquerepresentation__implement___getstate_____setstate__

[mkoeppe]

New commits:

5215e93

Chart: WIP WithEqualityById instead of UniqueRepresentation; disambiguate _restrictions -> _coordinate_restrictions

[mkoeppe]

Commit: 5215e93

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

facad97	ContinuousMap.preimage: Handle identity_map specially
c2ecf3e	ScalarField.preimage: Handle the case of the zero scalar field
3b1c428	ManifoldSubsetPullback: Make codomain_subset required 2nd init arg; fix pycodestyle/pyflakes warnings
d321b93	ContinuousMap: Return domain if the map's codomain is contained in the given subset
07aba9e	ManifoldSubsetPullback.is_closed: Preimage of finite sets is closed
653c651	src/sage/manifolds/subsets/pullback.py: Fix docstring markup
0f2e018	Merge #31688
e3e38ca	Chart.__init__: Move code that attaches self to the domain to TopologicalManifold.chart
3581c75	Chart.function_ring: Cache via @cached_method, not UniqueRepresentation
869599d	Chart: WIP equality/immutability/pickling

[sagetrac-git]

Changed commit from 5215e93 to 869599d

[mkoeppe]

Dependencies: #31688

[sagetrac-git]

Changed commit from 869599d to d071d56

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

d071d56

TopologicalSubmanifold: Adjust to renamed chart attribute _coord_restrictions

[mkoeppe]

comment:12

The last obstacle is this code which tries to hash a chart when it is still mutable.

        # The null and one functions of the coordinates:
        # Expression in self of the zero and one scalar fields of open sets
        # containing the domain of self:
        for dom in self.open_supersets():
            dom._zero_scalar_field._express[chart] = chart.function_ring().zero()
            dom._one_scalar_field._express[chart] = chart.function_ring().one()

[sagetrac-git]

Changed commit from d071d56 to 7b822b4

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

7b822b4

Fixup

[mkoeppe]

comment:14

Replying to @mkoeppe:

Maybe a chart should become immutable "upon first use"?

This is the approach that I have been trying out in this branch.
But it does not work in examples like this one:

            sage: M = Manifold(2, 'M', structure='topological')
            sage: X.<x,y> = M.chart()
            sage: U = M.open_subset('U', coord_def={X: [x>1, y<pi]})

because X in coord_def is unhashable, leading to an error.

So perhaps an API change is needed, after all...

[egourgoulhon]

comment:15

In the current branch, Chart inherits from WithEqualityById but redefines __eq__ (in a rather complicated and possibly CPU time consuming way). Doesn't this defeat the purpose? In particular in view of comment:3.

In the code of __hash__, isn't

if not self.is_immutable():
    raise TypeError('Charts are unhashable until set_immutable() has been called')

an unnecessary limitation?
More generally, why do you want to make Chart mutable? It is logically immutable: a chart is entirely defined by its domain and its coordinates (i.e. a tuple of symbolic variables). Coordinate restrictions, which can be added only after __init__ via add_restrictions in the current implementation, do not (mathematically) modify the object: there cannot be two charts with the same domain and the same coordinates, but with different coordinate restrictions.

[egourgoulhon]

comment:16

A possible option to get rid of the 2-stages construction of a chart (i.e. to get rid of the requirement of invoking add_restriction after __init__) could be to add the keyword argument restrictions to Chart.__init__, which can be either

• a string (case with no predefined coordinate variables, i.e. chart created from scratch)
• a nested list/tuple of symbolic expressions (case of a chart created from another one, as in Chart.restrict)
  After all, when creating a chart from scratch, we are already passing a string if we want to specify the coordinate ranges and LaTeX symbols; so maybe we could admit having a second string...
  For instance, we could have something like

sage: H = Manifold(2, 'H')  # half-disk
sage: X.<x,y> = H.chart(r"x y:(0,+oo)", restrictions="x^2 + y^2 < 1")

or equivalently

sage: X.<x,y> = H.chart(restrictions="[x^2 + y^2 < 1, y>0]")

Then we could get rid of the method add_restriction and consider that the chart is a fully immutable object (of course, technically speaking, the sheafy attributes can be muted).

[mkoeppe]

comment:17

Replying to @egourgoulhon:

In the current branch, Chart inherits from WithEqualityById but redefines __eq__ (in a rather complicated and possibly CPU time consuming way). Doesn't this defeat the purpose?

Yes, this part is not finished. I started with WithEqualityById as per your suggestion, but I wanted to implement actual pickling (not just passing the pickling-related testsuite) as a step to actual pickling of manifolds. Then the identity and unique-representation tricks are not sufficient any more, and one needs to implement full comparison.

One can still have a fast path for equality using id, and fast lookup in sets etc. via a fast hash.

[mkoeppe]

comment:18

Replying to @egourgoulhon:

More generally, why do you want to make Chart mutable?

I don't really, it was just an attempt to keep the current API (initialization, followed by add_restrictions) but implementing pickling.

[mkoeppe]

comment:19

When discussing a possible redesign of chart initialization, I think a relevant issue is the current use of the symbolic assumptions facility for the coordinates.

Are two charts with a different tuple of variables allowed to share some variables? If yes, does the shared variable have to have the same global assumptions and periodicity?

Also I note that creating a RealChart puts global assumptions on the variables of the chart. So creating a subchart will affect simplification of expressions that relate to computation on the original chart. This should probably be revised.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

4b930ff	Merge tag '9.4.beta4' into t/31688/pullbacks_of_manifold_subsets_under_continuous_maps
96e1fbf	Merge branch 't/31688/pullbacks_of_manifold_subsets_under_continuous_maps' into t/31901/chart__no_longer_use_uniquerepresentation__implement___getstate_____setstate__

[mkoeppe]

comment:23

Replying to @egourgoulhon:

Replying to @mkoeppe:

Also I note that creating a RealChart puts global assumptions on the variables of the chart. [...]

Yes, absolutely. This is something I have in mind for a long time but did not find the time to work on it. The assumptions should not be global but chart-wise. The way Sage's symbolic calculus is implemented, I guess this means that they should be switched on (via assume) before each calculus at the chart function level and switched off once the calculus is done (via forget).

There is already a context manager assuming for that. We could create it at initialization and invoke it using with whenever computations are done with this chart.

The tricky bit is when two charts are involved with overlapping variables. Then some renaming may have to be done first. A possible route is through #32089.

[mkoeppe]

comment:24

I have opened #32102 (Chart: Add init argument coord_restrictions, deprecate method add_restrictions)

[mkoeppe]

comment:25

Replying to @egourgoulhon:

A connected question is how to pass these chart-based assumptions to SymPy, in case the latter has been chosen as the default symbolic backend on the manifold (via M.set_calculus_method('sympy')) or on a given chart X (via X.calculus_method().set('sympy')). For the time being, calculus with SymPy is possible but the assumptions are not taken into account on the SymPy side...

I suppose we can go through a new method assuming of CalculusMethod that dispatches in the same way as the simplify method does.

[egourgoulhon]

comment:26

Replying to @mkoeppe:

OK. Then I think there is little point in trying to provide better syntax for creating the variables than asking the user to do var('x, y, z') before creating a chart with coordinate restrictions.

The issue here is that on real manifolds (by far the most used ones), the symbolic variables are created with the extra parameter domain='real' (cf. RealChart._init_coordinates), so the user must actually do var('x y z', domain='real'), which is not very intuitive. For some reason, it is not equivalent to assume(x, 'real'), which is performed in RealChart._init_coordinates as well. Both proved to be necessary for some simplifications.

Then the restrictions can just be passed as symbolic inequalities to the constructor. (I don't think going the string route would be an improvement.)

I don't like strings either, but I would prefer this to a 2-stage declaration involving var. Anyway, let us continue the discussion on #32102.

We could deprecate add_restrictions then.

Yes.

[egourgoulhon]

comment:27

Replying to @mkoeppe:

Replying to @egourgoulhon:

Replying to @mkoeppe:

Also I note that creating a RealChart puts global assumptions on the variables of the chart. [...]

Yes, absolutely. This is something I have in mind for a long time but did not find the time to work on it. The assumptions should not be global but chart-wise. The way Sage's symbolic calculus is implemented, I guess this means that they should be switched on (via assume) before each calculus at the chart function level and switched off once the calculus is done (via forget).

There is already a context manager assuming for that. We could create it at initialization and invoke it using with whenever computations are done with this chart.

The tricky bit is when two charts are involved with overlapping variables. Then some renaming may have to be done first.

I don't see why: two computations on two distinct charts can set different assumptions on the same variable. There is no ambiguity as long as assumptions are tied to charts.

[mkoeppe]

comment:28

Replying to @egourgoulhon:

Replying to @mkoeppe:

When discussing a possible redesign of chart initialization, I think a relevant issue is the current use of the symbolic assumptions facility for the coordinates.

Are two charts with a different tuple of variables allowed to share some variables?

Yes, definitely. This is a desirable feature.

A follow-up question on this:

sage: M = Manifold(2, 'M', structure='topological')
....: U = M.open_subset('U')
....: V = M.open_subset('V')
sage: XU = U.chart(names=("x", "y"))
sage: XV = V.chart(names=("x", "y"))
sage: M.atlas()
[Chart (U, (x, y)), Chart (V, (x, y))]
sage: M.top_charts()
[Chart (U, (x, y))]

Clearly something went wrong here.
Should there have been a declaration of something on M regarding the intention to declare charts with the coordinate pair (x,y) on subsets?

[mkoeppe]

comment:29

Replying to @egourgoulhon:

Replying to @mkoeppe:
I think there is little point in trying to provide better syntax for creating the variables than asking the user to do var('x, y, z') before creating a chart with coordinate restrictions.

The issue here is that on real manifolds (by far the most used ones), the symbolic variables are created with the extra parameter domain='real' (cf. RealChart._init_coordinates), so the user must actually do var('x y z', domain='real'), which is not very intuitive.

How about something like M.var('x y z') then?

[mkoeppe]

Changed dependencies from #31688 to #31688, #32102

[egourgoulhon]

comment:31

Replying to @mkoeppe:

A follow-up question on this:

sage: M = Manifold(2, 'M', structure='topological')
....: U = M.open_subset('U')
....: V = M.open_subset('V')
sage: XU = U.chart(names=("x", "y"))
sage: XV = V.chart(names=("x", "y"))
sage: M.atlas()
[Chart (U, (x, y)), Chart (V, (x, y))]
sage: M.top_charts()
[Chart (U, (x, y))]

Clearly something went wrong here.

Good catch!

Should there have been a declaration of something on M regarding the intention to declare charts with the coordinate pair (x,y) on subsets?

No, the above is actually a bug in Chart.__init__. I've opened #32112 for it and submitted a fix there.

[sagetrac-git]

Changed commit from 96e1fbf to d2e3ff7

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

4e316e9	Fix bug in Chart.__init__ regarding the determination of top charts (Trac #32112)
d2e3ff7	Merge #32112

[mkoeppe]

Changed dependencies from #31688, #32102 to #31688, #32112, #32102

[sagetrac-git]

Changed commit from d2e3ff7 to a328e91

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

a328e91

Merge #32112

[mkoeppe]

Work Issues: redo without mutability on top of #32102

[mkoeppe]

comment:36

Replying to @egourgoulhon:

creating a RealChart puts global assumptions on the variables of the chart. So creating a subchart will affect simplification of expressions that relate to computation on the original chart.

The assumptions should not be global but chart-wise.

I've opened #32120 (Chart-wise assumptions) for this.

[mkoeppe]

comment:37

Replying to @egourgoulhon:

Replying to @mkoeppe:

The tricky bit is when two charts are involved with overlapping variables. Then some renaming may have to be done first.

I don't see why: two computations on two distinct charts can set different assumptions on the same variable. There is no ambiguity as long as assumptions are tied to charts.

I was thinking of computations with coordinate changes here, which involve two charts simultaneously.

[mkoeppe]

comment:38

Replying to @egourgoulhon:

A connected question is how to pass these chart-based assumptions to SymPy, in case the latter has been chosen as the default symbolic backend on the manifold (via M.set_calculus_method('sympy')) or on a given chart X (via X.calculus_method().set('sympy')). For the time being, calculus with SymPy is possible but the assumptions are not taken into account on the SymPy side...

We already have a meta-ticket for this, #31958 (Use the SymPy assumptions facility). I was surprised that SymPy's assumptions are not really well connected to SymPy's sets (for which I have been building some glue code in #31926).

[mkoeppe]

comment:39

Given that #32102 is making mutability unnecessary, it seems we can keep UniqueRepresentation for charts for now (until we see another reason to drop it - such as unbounded memory) and pickle through that.

I'm setting the ticket to "sage-wishlist" for this reason.

[egourgoulhon]

comment:40

Replying to @mkoeppe:

Replying to @egourgoulhon:

Replying to @mkoeppe:

The tricky bit is when two charts are involved with overlapping variables. Then some renaming may have to be done first.

I don't see why: two computations on two distinct charts can set different assumptions on the same variable. There is no ambiguity as long as assumptions are tied to charts.

I was thinking of computations with coordinate changes here, which involve two charts simultaneously.

There should not by any issue with coordinate changes: a coordinate change X1 --> X2 involves only a set of chart functions based on X1 (stored as a MultiCoordFunction in the attribute CoordChange._transf). So only assumptions relative to X1 will be invoked in calculus involving the coordinate change X1 --> X2 (such as the Jacobian matrix). An exception is the
computation of the inverse in CoordChange.inverse. There only assumptions on X2 should matter. However, if X1 and X2 share the same variables, the inverse is trivial. To summarize, I don't think there is a case where both sets of assumptions are required simultaneously in coordinate changes.

[mkoeppe]

comment:41

Thanks for the explanation, that's good news.

[sagetrac-git]

Changed commit from a328e91 to 30a5d0b

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. Last 10 new commits:

bf62543	Merge branch 't/32089/conditionset__conditionset_callable_symbolic_expression' into t/32009/eliminate_direct_use_of_the_chart__domain_attribute
2f47485	Merge #32009
21297f3	Merge #32009
deace83	Chart: Replace _init_coordinates by _parse_coordinates
4db4995	Chart: Fix up `__classcall__` and _parse_coordinates by avoiding unhashable things
fc59c9d	Chart.__classcall__: Add doctest
1742fdc	Merge #32009
9ac1834	src/sage/manifolds/chart.py: Add raw string marker
d7f9d17	Merge #32116
30a5d0b	Chart: WIP getstate/setstate, no UniqueRepresentation

[mkoeppe]

Changed dependencies from #31688, #32112, #32102 to #31894

[mkoeppe]

Changed work issues from redo without mutability on top of #32102 to redo on top of #31894