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Python stdlib has around 139 extension modules. Not all of them *need* to use internal C API. I'm interested to try to convert a bunch of them to the limited C API, as part of my work on the PEP 620. IMHO "Eating your own dog food" is a good practice to ensure that the limited C API is usable.
Currently, the stdlib has exactly one extension module using the limited C API: the xxlimited module built with Py_LIMITED_API macro defined as 0x03050000 in setup.py. By the way, maybe Py_LIMITED_API should be defined in xxlimited.c, rather than in setup.py.
xxlimited.c is not built if Python is built in debug mode. I'm not sure why.
The main limitation to use the limited C API for stdlib is Argument Clinic which attempts to always emit the most efficient code, like using the METH_FASTCALL calling convention and use private functions like _PyArg_CheckPositional() or "static _PyArg_Parser _parser".
Argument Clinic could be modified to have an option to only use C API of the limited C API. Cython is working on a similar option (restraint emitted code to the limited C API).
I already tried to convert stdlib extensions to the limited C API in bpo-39573. I found other issues:
* PyTypeObject is opaque and so it's not possible to implement a deallocator function (tp_dealloc) which calls tp_free like: Py_TYPE(self)->tp_free((PyObject*)self);
* _Py_IDENTIFIER() is not part of the limited C API
https://bugs.python.org/issue39573#msg361514 

See also bpo-27499 "PY_SSIZE_T_CLEAN conflicts with Py_LIMITED_API".

Other missing features of the limited C API:
* bpo-2897: "PyMemberDef missing in limited API / Deprecate structmember.h"
* Stefan Behnel wrote: "Some code cannot even be migrated at all,
e.g. because the entire buffer protocol is missing from it. Some bugs were
only fixed in Py3.9, time will tell if anything else is missing."
 https://mail.python.org/archives/list/python-dev@python.org/message/SENQBEJCJ7NYC72ZZ7BGIEDDBTUOXLI4/ 

PyType_FromSpec() doesn't support 11 slots:
* tp_dict
* tp_mro
* tp_cache
* tp_subclasses
* tp_weaklist
* tp_vectorcall
* tp_weaklistoffset (see PyMemberDef)
* tp_dictoffset (see PyMemberDef)
* tp_vectorcall_offset (see PyMemberDef)
* bf_getbuffer
* bf_releasebuffer
https://docs.python.org/dev/c-api/type.html#c.PyType_Slot
But it is possible to define tp_weaklistoffset, tp_dictoffset and tp_vectorcall_offset via Py_tp_members slot:
https://docs.python.org/dev/c-api/structures.html#pymemberdef-offsets
* "__dictoffset__" => tp_dictoffset
* "__weaklistoffset__" => tp_weaklistoffset
* "__vectorcalloffset__" => tp_vectorcall_offset
Maybe we can do add new members for the 8 missing slots, especially bf_getbuffer and bf_releasebuffer?
commit f7c4e236429606e1c982cacf24e10fc86ef4462f of bpo-40724 added Py_bf_getbuffer and Py_bf_releasebuffer slots to the C API, but these slots are still not available in the limited C API: see bpo-10181.

See also bpo-39123: "PyThread_xxx() not available when using limited API".

See also bpo-23903: "Generate PC/python3.def by scraping headers". Steve Dower suggests to convert macros to functions. I'm not sure which functions are still implemented as macro in the limited C API.

See also bpo-16731: "xxlimited/xxmodule docstrings ambiguous".

See also bpo-41073: "[C API] PyType_GetSlot() should accept static types".

See also ./Tools/scripts/abitype.py script and bpo-10943: "abitype: Need better support to port C extension modules to the stable C API".

See also bpo-40077: "Convert static types to PyType_FromSpec()".

See also bpo-29086: "Document C API that is not part of the limited API".

While this might make for an interesting validation experiment, I don't think the results should be checked in.
Code churn is to be avoided when there is no visible user benefit. It risks destabilizing code, introducing errors, and making the code less recognizable to the other developers who created and maintained it.

Some related changes in modules: posix, _hashopenssl, _ast (Python-ast.c), zlib, _struct, tkinter, _curses_panel.
commit b3966639d28313809774ca3859a347b9007be8d2
Author: Eddie Elizondo <eelizondo@fb.com>
Date: Tue Nov 5 07:16:14 2019 -0800
 bpo-35381 Remove all static state from posixmodule (GH-15892)
 
 
 
 After #9665, this moves the remaining types in posixmodule to be heap-allocated to make it compatible with PEP384 as well as modifying all the type accessors to fully make the type opaque.
 
 The original PR that got messed up a rebase: https://github.com/python/cpython/pull/10854. All the issues in that commit have now been addressed since https://github.com/python/cpython/pull/11661 got committed.
 
 This change also removes any state from the data segment and onto the module state itself.
 
 
 https://bugs.python.org/issue35381
 
 
 
 Automerge-Triggered-By: @encukou
commit df69e75edcc08475bc9a57a5a76df8a45bfc3c34
Author: Christian Heimes <christian@python.org>
Date: Wed Sep 25 23:03:30 2019 +0200
 bpo-38142: Updated _hashopenssl.c to be PEP 384 compliant (#16071)
 
 * Updated _hashopenssl.c to be PEP 384 compliant
 * Remove refleak test from test_hashlib. The updated type no longer accepts random arguments to __init__.
commit ac46eb4ad6662cf6d771b20d8963658b2186c48c
Author: Dino Viehland <dinoviehland@fb.com>
Date: Wed Sep 11 10:16:34 2019 -0700
 bpo-38113: Update the Python-ast.c generator to PEP384 (gh-15957)
 
 Summary: This mostly migrates Python-ast.c to PEP384 and removes all statics from the whole file. This modifies the generator itself that generates the Python-ast.c. It leaves in the usage of _PyObject_LookupAttr even though it's not fully PEP384 compatible (this could always be shimmed in by anyone who needs it).
commit a1ffad07195b8b976f8c371a92028240946d4e76
Author: Dino Viehland <dinoviehland@gmail.com>
Date: Tue Sep 10 11:27:03 2019 +0100
 bpo-38074: Make zlib extension module PEP-384 compatible (GH-15792)
 
 Updated zlibmodule.c to be PEP 384 compliant.
commit 4f384af067d05b16a554bfd976934fca9f87a1cf
Author: Dino Viehland <dinoviehland@gmail.com>
Date: Tue Sep 10 11:18:37 2019 +0100
 bpo-38076: Make struct module PEP-384 compatible (#15805)
 
 * PEP-384 _struct
 
 * More PEP-384 fixes for _struct
 
 Summary: Add a couple of more fixes for `_struct` that were previously missed such as removing `tp_*` accessors and using `PyBytesWriter` instead of calling `PyBytes_FromStringAndSize` with `NULL`. Also added a test to confirm that `iter_unpack` type is still uninstantiable.
 
 * 📜🤖 Added by blurb_it.
commit d2217a83d4e2ee9aec1a0bf590820aa77b7ed5e7
Author: Andrew Svetlov <andrew.svetlov@gmail.com>
Date: Tue Oct 30 22:49:16 2012 +0200
 Issue #15721: apply PEP 384 Refactoring to tkinter module.
commit bc07cb883e5c03b8c108c2c9d86bc0a158d62c27
Author: Martin v. Löwis <martin@v.loewis.de>
Date: Thu Jun 14 16:01:23 2012 +0200
 Issue #14936: curses_panel was converted to PEP 3121 and PEP 384 API.
 Patch by Robin Schreiber.

> See also bpo-40077: "Convert static types to PyType_FromSpec()".
In bpo-1635741, many extension modules have been ported to the multi-phase initilization API (PEP 489), and many static types have been converted to heap types.

> Maybe we can do add new members for the 8 missing slots
See: https://bugs.python.org/issue41618?#msg380414 

> By the way, maybe Py_LIMITED_API should be defined in xxlimited.c, rather than in setup.py.
+1. Defining Py_LIMITED_API in xxlimited.c is more direct than in setup.py. so I created the PR-25115.

See also bpo-43688: [C API] Support the limited C API in debug mode (Py_INCREF and Py_DECREF).

New changeset 240bcf82a11fe7433a61da70605e924c53b88096 by Victor Stinner in branch 'master':
bpo-41111: xxlimited.c defines Py_LIMITED_API (GH-25151)
https://github.com/python/cpython/commit/240bcf82a11fe7433a61da70605e924c53b88096

The latest commit seems to break the build if configured --with-trace-refs.

I should revise that comment. The xxlimited and xxlimited_35 modules fail to build. That seems suboptimal, but perhaps is to be expected. Perhaps it would be better that compiling them not be attempted with configuring --with-trace-refs?

> Perhaps it would be better that compiling them not be attempted with configuring --with-trace-refs?
+1. The limited C API can't be build under Py_TRACE_REFS now.

New changeset 5787ba4a45492e232f5470c7d2e93763198e4b22 by Hai Shi in branch 'master':
bpo-41111: Don't build xxlimited with Py_TRACE_REFS macro (GH-25180)
https://github.com/python/cpython/commit/5787ba4a45492e232f5470c7d2e93763198e4b22

Skip Montanaro: "The latest commit seems to break the build if configured --with-trace-refs."
Oops, I forgot about this special build mode. Thanks for the reminder Skip, and thanks for the fix Hai ;-)
Skip: By the way, I'm curious, why do you use --with-trace-refs?

>
> Skip: By the way, I'm curious, why do you use --with-trace-refs?
>
I'm still horsing around with register opcodes and got in the habit of
building with pydebug and trace refs enabled.

> The main limitation to use the limited C API for stdlib is Argument Clinic which attempts to always emit the most efficient code, like using the METH_FASTCALL calling convention and use private functions like _PyArg_CheckPositional() or "static _PyArg_Parser _parser".
PR-26080 adds the feature that Argument Clinic to support to use the Limited C API.
METH_FASTCALL is the part of the stable ABI in PR-23009.
Do we need convert METH_VARARGS and METH_KEYWORDS as the part of the stable ABI too(something like METH_FASTCALL)?

METH_VARARGS and METH_KEYWORDS are part of the stable ABI.

> METH_VARARGS and METH_KEYWORDS are part of the stable ABI.
 Copy that, thanks, petr.