finite_state_machine: Add an example on the Gray code into the documentation

[cheuberg]

As an example on how to use cartesian products of transducers as introduced in #16061, a transducer converting standard binary expansion into Gray code is constructed by emulating the standard algorithm by xoring the the standard binary expansion with its shifted version.

The class collecting common transducers introduced in #16141 now also has a (hard-coded) version of the same transducer and it is made sure that both transducers agree.

Depends on #16132
Depends on #16141
Depends on #16142

CC: @sagetrac-skropf

Component: combinatorics

Author: Clemens Heuberger, Daniel Krenn, Sara Kropf

Branch/Commit: 301b5cb

Reviewer: Daniel Krenn, Sara Kropf

Issue created by migration from https://trac.sagemath.org/ticket/16143

[dkrenn]

Dependencies: #16141

[dkrenn]

Changed dependencies from #16141 to #16141, #16142

[dkrenn]

Reviewer: Daniel Krenn

[dkrenn]

Changed branch from u/cheuberg/fsm/example_gray_code to u/dkrenn/fsm/example_gray_code

[dkrenn]

comment:4

During the review I realized that the Gray-code transducer (in generators) didn't have any final states. I've corrected this. I added a doctest to print the Gray code of the first 10 non-negative integers.
Please review my changes.

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Last 10 new commits:

9029a07	Fixed doctest for operator
6fed278	Merge branch 'fsm/operator_transducers' into fsm/example_gray_code
f7b3a35	Gray code example modified and prepackaged transducer
7a907ac	minor corrections
4a53b09	Merge branch '#16132' into fsm/count_subblock_occurrences
5d65767	Edit doctests due to #16132 change.
e1bd1df	Merge branch 'fsm/count_subblock_occurrences' into fsm/operator_transducers
3110f9d	Edit doctests due to #16132 change.
6145b1f	Merge branch 'fsm/operator_transducers' into fsm/example_gray_code
1efa597	added final_states to Gray-code transducer, improved docstring

[dkrenn]

Changed author from Clemens Heuberger to Clemens Heuberger, Daniel Krenn

[dkrenn]

Changed commit from 6145b1f to 1efa597

[cheuberg]

Changed branch from u/dkrenn/fsm/example_gray_code to u/cheuberg/fsm/example_gray_code

[cheuberg]

Changed commit from 1efa597 to 5394265

[cheuberg]

comment:6

Modified example in finite_state_machine.py to also have correct final states and compare it with the final states given by transducers.GrayCode.

Corrected link from transducers.GrayCode to this example.

[dkrenn]

Last 10 new commits:

4a53b09	Merge branch '#16132' into fsm/count_subblock_occurrences
5d65767	Edit doctests due to #16132 change.
e1bd1df	Merge branch 'fsm/count_subblock_occurrences' into fsm/operator_transducers
3110f9d	Edit doctests due to #16132 change.
6145b1f	Merge branch 'fsm/operator_transducers' into fsm/example_gray_code
1efa597	added final_states to Gray-code transducer, improved docstring
1d3cb1e	changes in Gray-code example
16c62dd	Constructed Gray code now also has correct final states.
5394265	Corrected link from transducers.GrayCode to example in finite_state_machine module
82ceeba	Merge remote-tracking branch 'origin/u/cheuberg/fsm/example_gray_code' into u/cheuberg/fsm/example_gray_code

[dkrenn]

Changed branch from u/cheuberg/fsm/example_gray_code to u/dkrenn/fsm/example_gray_code

[dkrenn]

Changed commit from 5394265 to 82ceeba

[dkrenn]

comment:8

Now bases on beta8 + #16142.

---

Last 10 new commits:

6fed278	Merge branch 'fsm/operator_transducers' into fsm/example_gray_code
f7b3a35	Gray code example modified and prepackaged transducer
6145b1f	Merge branch 'fsm/operator_transducers' into fsm/example_gray_code
1efa597	added final_states to Gray-code transducer, improved docstring
1d3cb1e	changes in Gray-code example
16c62dd	Constructed Gray code now also has correct final states.
5394265	Corrected link from transducers.GrayCode to example in finite_state_machine module
82ceeba	Merge remote-tracking branch 'origin/u/cheuberg/fsm/example_gray_code' into u/cheuberg/fsm/example_gray_code
826b764	Merge remote-tracking branch 'origin/u/dkrenn/fsm/example_gray_code' into fsm/operator_transducers-on-beta8
2a6bafe	corrected one doctest and added another (left-shift transducer)

[dkrenn]

Changed commit from 82ceeba to 2a6bafe

[dkrenn]

Changed branch from u/dkrenn/fsm/example_gray_code to u/dkrenn/fsm/example_gray_code-on-beta8

[cheuberg]

comment:9

Cross-reviewed all changes by dkrenn, they are fine for me.

Corrected left-shift transducer (it is a right-shift transducer, as highlighted by the above doctest) and explained this.

[cheuberg]

Changed branch from u/dkrenn/fsm/example_gray_code-on-beta8 to u/cheuberg/fsm/example_gray_code

[cheuberg]

Changed commit from 2a6bafe to 8c1cbb6

[cheuberg]

Changed dependencies from #16141, #16142 to #16132, #16141, #16142

[dkrenn]

Changed branch from u/cheuberg/fsm/example_gray_code to u/dkrenn/fsm/example_gray_code

[dkrenn]

comment:12

Reviewed your changes...ok.

Rewrote/Extended the documentation of the 'shift'-part of the example. Please review.

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New commits:

24b6d69

explained the shifting in the Gray code-construction differently

[dkrenn]

Changed commit from 8c1cbb6 to 24b6d69

[cheuberg]

comment:13

Replying to @dkrenn:

Rewrote/Extended the documentation of the 'shift'-part of the example. Please review.

fine for me.

[sagetrac-skropf]

comment:14

I rewrote the documentation to clarify the position of the least significant digit and the direction of the shift. Please review.

Everthing else is fine for me.

[sagetrac-skropf]

Changed branch from u/dkrenn/fsm/example_gray_code to u/skropf/fsm/example_gray_code

[sagetrac-skropf]

Changed commit from 24b6d69 to 301b5cb

[cheuberg]

comment:15

fine for me.

[sagetrac-skropf]

Changed author from Clemens Heuberger, Daniel Krenn to Clemens Heuberger, Daniel Krenn, Sara Kropf

[sagetrac-skropf]

Changed reviewer from Daniel Krenn to Daniel Krenn, Sara Kropf

[dkrenn]

comment:17

Fome also for me. Since all participants in this ticket agree, I give it a positive review.

[vbraun]

Changed branch from u/skropf/fsm/example_gray_code to 301b5cb