guessing k-regular sequences #21204

dkrenn · 2016-08-10T13:12:40Z

Code for guessing the parameters of k-regular sequences from the first few entries of a list/sequence/etc.

See also Meta ticket #21202.

Depends on #21319

CC: @rwst

Component: combinatorics

Author: Daniel Krenn

Branch/Commit: u/cheuberg/sequences/k-regular-guess @ ae06d17

Reviewer: Clemens Heuberger

Issue created by migration from https://trac.sagemath.org/ticket/21204

The text was updated successfully, but these errors were encountered:

dkrenn · 2016-08-10T13:16:57Z

Branch: u/dkrenn/sequences/k-regular-guess

mantepse · 2016-08-10T14:36:51Z

Commit: 11c7ec4

mantepse · 2016-08-10T14:36:51Z

comment:2

Is it correct that (the generating function f of) every k-regular sequence satisfies a Mahler-type functional equation? That is, a polynomial p such that p(f(x), f(x<sup>2),...,f(x</sup>k)) = 0?

If so, then you could reuse the guessing functionality in FriCAS. Of course, FriCAS is not a standard package, although I think it should be, for many reasons. In any case, the guessing package within is fast and well tested, and covers a lot of ground - essentially everything which is available for 1-dimensional guessing. For example:

sage: l = [0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0]
sage: fricas.guessFE(l)
      n         3        2        2                             3      4
  [[[x ]f(x): (x  - x)f(x ) + (- x  + 1)f(x) - x= 0,f(x)= x + 2x  + O(x )]]

dkrenn · 2016-08-10T15:25:00Z

comment:3

Replying to @mantepse:

Is it correct that (the generating function f of) every k-regular sequence satisfies a Mahler-type functional equation? That is, a polynomial p such that p(f(x), f(x<sup>2),...,f(x</sup>k)) = 0?

I've found in the book of Allouche and Shallit:

"It is known that k-regular power series satisfy Mahler functional equations; see Rand ́e [1992], Dumas [1993a], and Becker [1994]."

so I believe the answer to your question is affirmative.

If so, then you could reuse the guessing functionality in FriCAS. Of course, FriCAS is not a standard package, although I think it should be, for many reasons. In any case, the guessing package within is fast and well tested, and covers a lot of ground - essentially everything which is available for 1-dimensional guessing.

I didn't know about this feature of FriCAS, but looks good :)

Are there any plans to make FriCAS standard (I tried with sage -i fricas, but installation fails :(

mantepse · 2016-08-10T15:34:16Z

comment:4

Replying to @dkrenn:

Are there any plans to make FriCAS standard (I tried with sage -i fricas, but installation fails :(

How does it fail? That's very bad indeed!

dkrenn · 2016-08-10T15:44:45Z

comment:5

Replying to @mantepse:

Are there any plans to make FriCAS standard (I tried with sage -i fricas, but installation fails :(

How does it fail? That's very bad indeed!

Posted on https://groups.google.com/forum/#!topic/sage-devel/6_rBQvKdGPo

dimpase · 2016-08-10T16:54:27Z

comment:6

this is to update FriCAS: #21209

probably the current version of FriCAS (1.2.7) should compile with current ECL just fine.

mantepse · 2016-08-21T15:37:09Z

comment:8

The new fricas interface is #21231. It would be very easy to add a dedicated command for guessing k-regular sequences to FriCAS. If this is interesting, I could probably further improve the interface to avoid pexpect and communicate with FriCAS more directly, Waldek Hebisch has already done the work on the FriCAS-side.

(Currently, passing long lists to FriCAS is very very slow. But I would like to have at least some confirmation of interest before I spend more time on the interface)

sagetrac-git · 2016-08-21T16:51:44Z

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

`4a1db0e`	`reorder imports`
`62ebcda`	`complete documentation`
`4331957`	`Merge branch 'u/dkrenn/sequences/recognizable' into u/dkrenn/sequences/k-regular`
`dab4db8`	`complete docstrings and 100% coverage`
`9afc4ac`	`cross-references`
`fd4f541`	`move "transpose" in docstring to correct place`
`4eb1544`	`Merge branch 'u/dkrenn/sequences/recognizable' into u/dkrenn/sequences/k-regular`
`cf1b8cc`	`correct one link`
`73ad7bd`	`Merge branch 'u/dkrenn/sequences/k-regular' into u/dkrenn/sequences/k-regular-guess`
`c349170`	`fix code and doctests to adapt with upstream recognizable series`

sagetrac-git · 2016-08-21T16:51:44Z

Changed commit from 11c7ec4 to c349170

dkrenn · 2016-08-21T17:12:42Z

comment:10

Replying to @mantepse:

The new fricas interface is #21231. It would be very easy to add a dedicated command for guessing k-regular sequences to FriCAS.

You mean guessing the Mahler functional equation, right?

If this is interesting, I could probably further improve the interface to avoid pexpect and communicate with FriCAS more directly, Waldek Hebisch has already done the work on the FriCAS-side.

I think having this would be interesting and good. I am still not sure how exactly I would get the k-regular sequence out of it; I have to dig into this topic a bit further, I guess. (At the moment, I do not see directly, where the parameter k comes into play, as the FriCAS guessing works without this parameter.)
This is not my top priority at the moment (as the simple implementation of guessing of this ticket satisfies my needs and gives exactly the kind of representation I need.

As FriCAS is an optional package, I would guess, when having the interface you speak about, would be used as an alternative choice which guessing-algorithm was used. (Maybe even if FriCAS was a standard package...not sure at the moment...)

(Currently, passing long lists to FriCAS is very very slow. But I would like to have at least some confirmation of interest before I spend more time on the interface)

From what you can suggest by reading the above, I am not sure what the implications for this ticket are. However, having such a functionality of guessing the functional equation would definitely something nice to have in SageMath.

mantepse · 2016-08-21T18:22:35Z

comment:11

Replying to @dkrenn:

Replying to @mantepse:

The new fricas interface is #21231. It would be very easy to add a dedicated command for guessing k-regular sequences to FriCAS.

You mean guessing the Mahler functional equation, right?

Hm. I actually meant to add a command that guesses the equation for the k-regular sequence. But I just found out that I was slightly mistaken about the nature of k-regular sequences: apparently there is no single equation determining it, but rather a bunch of them, right? It should still be possible to reuse the machinery.

sagetrac-git · 2016-08-22T13:17:50Z

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

`863d57a`	`remove None-sense to simplify code`
`aec95ce`	`extend element constructor`
`9e12383`	`add experimental warning`
`69289fe`	`Merge branch 'u/dkrenn/sequences/recognizable' into u/dkrenn/sequences/k-regular`
`71aef7f`	`fix doctests`
`15e4008`	`Merge branch 'u/dkrenn/sequences/k-regular' into u/dkrenn/sequences/k-regular-guess`
`1b34c92`	`put conversion code n --> index/word into separat method`
`eaf8df2`	`Merge branch 'u/dkrenn/sequences/k-regular' into u/dkrenn/sequences/k-regular-guess`
`355d24d`	`simplify code by using features of recognizable series better (and adapting to removing None-sense)`

sagetrac-git · 2016-08-22T13:17:50Z

Changed commit from c349170 to 355d24d

sagetrac-git · 2016-08-22T15:23:47Z

Changed commit from 355d24d to 348fcff

sagetrac-git · 2016-08-22T15:23:47Z

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

`348fcff`	`solve transpositioning problem`

sagetrac-git · 2016-08-22T15:26:39Z

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

`2e99c5c`	`another example`

sagetrac-git · 2016-08-22T15:26:39Z

Changed commit from 348fcff to 2e99c5c

sagetrac-git · 2016-08-25T11:33:41Z

Changed commit from 2e99c5c to a68e20d

sagetrac-git · 2016-08-25T11:33:41Z

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

`b060652`	`minor rewrite in minimized`
`31e2fcf`	`Merge branch 'u/dkrenn/sequences/recognizable' into u/dkrenn/sequences/k-regular`
`893bd5c`	`experimental warning in k_regular_sequence`
`189ce50`	`pass keywords to coefficient_of_word`
`5919120`	`doctesting all parameters in coefficient_of_word`
`c7243a6`	`Merge tag '7.4.beta1' into t/21295/sequences/recognizable`
`4012778`	`Merge branch 't/21295/sequences/recognizable' into t/21203/sequences/k-regular`
`5dbedab`	`fix doctests (experimental warning)`
`143ea73`	`fix building of docs (no idea why it failed)`
`a68e20d`	`Merge branch 't/21203/sequences/k-regular' into t/21204/sequences/k-regular-guess`

sagetrac-git · 2016-08-25T11:53:00Z

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

`3023233`	`rewrite to make subsequence more efficient (dynamically add kernel)`
`51e7d72`	`minor rewrite`
`9311636`	`allow b to be a dict`
`624e6c0`	`forward/backward difference`
`45d9e67`	`minor rewrites`
`c430158`	`partial sums`
`7eb93e1`	`minor rewrites to use existing stuff better`
`6247a7f`	`Merge branch 't/21203/sequences/k-regular' into t/21318/sequences/rec-basic-arith`
`0f89159`	`Merge branch 't/21318/sequences/rec-basic-arith' into t/21325/sequences/k-regular-subseq`
`816fcfd`	`Merge branch 't/21325/sequences/k-regular-subseq' into t/21204/sequences/k-regular-guess`

sagetrac-git · 2016-08-25T11:53:00Z

Changed commit from a68e20d to 816fcfd

sagetrac-git · 2016-08-25T18:20:07Z

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

`a8e6334`	`MatrixSpace.some_elements`
`520aaf1`	`FreeModule_generic.some_elements`
`5afda8e`	`Merge branch 'u/dkrenn/matrices-some-elements' into t/21319/sequences/rec-hash`
`1e5754b`	`an_element and some_elements`
`becce48`	`pickling support`
`a4ccbec`	`TestSuite`
`c2f9c62`	`Merge branch 't/21325/sequences/k-regular-subseq' into t/21319/sequences/rec-hash`
`9818d94`	`Merge branch 't/21319/sequences/rec-hash' into t/21204/sequences/k-regular-guess`
`90e4ed5`	`doctests`
`899e56e`	`finish .guess (docstrings etc)`

sagetrac-git · 2016-08-25T18:20:07Z

Changed commit from 816fcfd to 899e56e

dkrenn · 2016-08-25T18:24:48Z

Changed dependencies from #21203 to #21203, #21325, #21319

dkrenn · 2016-08-25T18:24:48Z

comment:18

Merged in k-regular sequences: subsequences, forward/backward difference, partial sums #21325 to avoid a trivial merge conflict.
Merged in recognizable series: hash, comparison, TestSuite #21319 to have equality testing in the doctests, but not needed for the actual code.
Thus, this ticket essentially depends only on basic element and parent for k-regular sequences #21203.

cheuberg · 2022-10-02T08:27:19Z

Changed commit from 3c634df to 5cab567

cheuberg · 2022-10-02T08:30:00Z

comment:38

Sorry, that was a wrong push. I'll fix it soon.

sagetrac-git · 2022-10-02T08:33:28Z

Changed commit from 5cab567 to 8ee2328

sagetrac-git · 2022-10-02T08:33:28Z

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

`bde8e67`	`Merge branch 't/21319/sequences/rec-hash' into t/21204/sequences/k-regular-guess`
`622c3f6`	`Merge branch 'u/dkrenn/sequences/rec-hash' into u/dkrenn/sequences/k-regular-guess`
`4e7f615`	`Merge branch 'u/dkrenn/sequences/rec-hash' into u/dkrenn/sequences/k-regular-guess`
`e726278`	`Merge branch 't/21319/sequences/rec-hash' into t/21204/sequences/k-regular-guess`
`dcc7cd1`	`Trac #21204: cherry-pick to avoid merge conflict`
`c0519f0`	`Trac #21204: fixup code and tests`
`5ce5276`	`Merge branch 't/21319/sequences/rec-hash' into t/21204/sequences/k-regular-guess`
`00047ce`	`Trac #21204: fix punctuation`
`3c634df`	`Merge branch 't/21319/sequences/rec-hash' into t/21204/sequences/k-regular-guess`
`8ee2328`	`Merge tag '9.8.beta1' into u/cheuberg/sequences/k-regular-guess`

sagetrac-git · 2022-10-02T18:48:22Z

Changed commit from 8ee2328 to 4337bd3

sagetrac-git · 2022-10-02T18:48:22Z

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

`4337bd3`	`Fix pycodestyle issue E306`

sagetrac-git · 2022-10-11T17:00:38Z

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

`ae06d17`	`Trac #21204: Fix typo`

sagetrac-git · 2022-10-11T17:00:38Z

Changed commit from 4337bd3 to ae06d17

cheuberg · 2022-10-11T17:09:43Z

Reviewer: Clemens Heuberger

cheuberg · 2022-10-11T17:09:43Z

comment:42

I started reading this and have a lot of questions. I might then have to reread the code because I am not yet confident that I understand everything, so I apologise in advance if some further questions pop up at a later iteration.

While reading the code, I tried to make some notes on what the various routines are doing; I pushed those notes to branch u/cheuberg/sequences/k-regular-guess-documentation; after polishing those, it might be worthwhile to include these.
Due to my question 16 below, these notes might still require substantial rewrite (I am not sure about left/right).

k_regular_sequence.value in the docstring: the \operator does not yield the desired output in the HTML documentation.
k_regular_sequence.split_interlace: from the description, I do not understand what "split a digit into p parts" shall mean. Instead of the many examples (which might be moved to TESTS) I'd prefer to see fewer examples with an explanation what the output means. Perhaps "split into p parts" could be made explicit into saying that it will be expanded as a k/p-ary expansion? And also say that "value of the expansion" means expansion in base k/p?
kRegularSequenceSpace.guess: One-sentence description: perhaps "Guess a k-regular sequence whose first terms coincide with ..."?
kRegularSequenceSpace.guess: it is unclear to me what "bootstrapping this guessing" means.
kRegularSequenceSpace.guess: please use the "new" method linear_representation for all examples
kRegularSequenceSpace.guess: In the first example, it may be helpful to explain what "variant 1" and "variant 2" are (variant 1: without a bootstrapping sequence; variant 2: with the constant sequence as a bootstrapping sequence).
kRegularSequenceSpace.guess: I suggest to compare the outputs of Variant 1 and Variant 2 (that they yield the same sequence).
kRegularSequenceSpace.guess: it would be helpful to indicate the vector valued sequences used in the various examples (Sum of digits, variant 1: (s(n), s(2n+1)); variant 2: (1, s(n)), all natural numbers: (n, 2n+1), indicator function of even integers: ([n even], 1), indicator function of odd integers: )
kRegularSequenceSpace.guess: I suggest to rename all functions containing linear_dependence in their name to linear_combination (we are actually interested in the coefficients, not the existence of coefficients)
kRegularSequenceSpace.guess: I suggest to rename guess_linear_dependence to linear_combination_candidate: we are not guessing here, but we are deriving the only candidate and are checking it in the next step.
kRegularSequenceSpace.guess: I am surprised that verify_linear_dependence has (n_max - r_L) // k**t_L + 1 as an upper bound for the loop. I think that the code does not guarantee that it will never evaluate f for some higher values than n_max, that might happen in some_inverse_U_matrix. On the other hand, something might be considered to be linearly dependent on the available functions while in truth it is not; this should not cause real problems.

kRegularSequenceSpace.guess: I do not understand the if seq(0): part: Why does it make a difference whether the zeroth element of the given sequence vanishes or not? I do not know whether it is related, but the following is not what I expect:

sage: def s(k):
sage:     return k
sage: Seq2 = kRegularSequenceSpace(2, ZZ)
sage: S1 = Seq2.guess(s)
sage: S2 = Seq2.guess(s, sequence=S1)
sage: S1
2-regular sequence 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
sage: S2
2-regular sequence 0, -2, -2, -6, -2, -6, -10, -14, -2, -6, ...

kRegularSequenceSpace.guess: I am surprised by if t_R >= max_dimension:: I thought that max_dimension will be the dimension of the matrices of the resulting regular sequence, but it seems to be the exponent? Shouldn't it be called like that in this case?
kRegularSequenceSpace.guess: I do not understand why the parameter s is actually stored into lines, I do not think it is ever used.
I do not understand why k_regular_sequence.split_interlace is defined in this ticket, it do not see a usage of it.
I find it puzzling that the values function contains seq which in turn has the left vector of the given sequence times the matrix product, and this at the very end ends up in right.

…-guess-documentation' into u/dkrenn/sequences/k-regular-guess * cheuberg/u/cheuberg/sequences/k-regular-guess-documentation: Add some further comments Add ALGORITHM section to docstring Trac sagemath#21204: Fix typo Fix pycodestyle issue E306

…ar-warning * t/21204/sequences/k-regular-guess: (11522 commits) Trac sagemath#21204: fixup code and tests Trac sagemath#21204: cherry-pick to avoid merge conflict Trac sagemath#21319: fixup due to changes in dependencies Trac sagemath#21318: fixup due to recent changes in dependencies Updated SageMath version to 9.3 build/pkgs/fplll/spkg-install.in: Configure --without-qd if we use gcc from spkg build/pkgs/fplll/spkg-configure.m4: Add depcheck on gcc build/pkgs/ppl/spkg-configure.m4: Add depcheck on gcc build/pkgs/brial/spkg-configure.m4: Add depcheck on gcc build/pkgs/{freetype,libgd}/spkg-configure.m4: Add depcheck for gcc build/pkgs/zeromq/spkg-configure.m4: Add depcheck for gcc build/pkgs/ntl/spkg-configure.m4: Add depcheck for gcc Trac sagemath#21295 review issue 29: notice minimize vs field Trac sagemath#21295 review issue 7: document accessing coefficients Trac sagemath#21295 review issue 33: rename to number_of_zeros (as it should be) Trac sagemath#21203 review issue 4: rename to coefficient ring Trac sagemath#21295: rename to coefficient_ring Trac sagemath#21203 review issue 3: example for __getitem__ and __iter__ Trac sagemath#21203 review issue 2: extend odds in Pascal's triangle Trac sagemath#21203 review issue 1: better binary sum of digits ...

…ar-warning * t/21204/sequences/k-regular-guess: Trac sagemath#21204: fix punctuation Trac sagemath#21319: fix punctuation Trac sagemath#21318: rmul/lmul preserve identity for multiplying by 1 Trac sagemath#21325: fixup test .subsequence being identity Trac sagemath#21318: rmul/lmul preserve identity for multiplying by 1 Trac sagemath#21325: use (new) decorator minimize_result Trac sagemath#21325: remove empty lines, fix punctuation Trac sagemath#21325: remove iteritems Trac sagemath#21318: fix rmul/lmul issues Trac sagemath#21318: use "correct" 1 Trac sagemath#21318: use tensor_product in method hadamard_product Trac sagemath#21318: use is_trivial_zero in doctest of _neg_ Trac sagemath#21318: use (new) .linear_representation in doctests Trac sagemath#21318: fix empty lines and punctuation Trac sagemath#21318: decorator minimize_result Trac sagemath#21203 review issue 10: use "raise ... from None" where approriate

gh-35896: deal with bad chosen representations of k-regular sequences ### 📚 Description It deals with the situation when mu[0]*right != right in k-regular sequences. Fixes #21343. This PR is created from the branch/code that has been on the corresponding trac ticket + merging in a current SageMath version. See also meta issue #21202. ### 📝 Checklist - [x] The title is concise, informative, and self-explanatory. - [x] The description explains in detail what this PR is about. - [x] I have linked a relevant issue or discussion. - [x] I have created tests covering the changes. - [x] I have updated the documentation accordingly. ### ⌛ Dependencies - #21204: guessing k-regular sequences from the first few values No formal dependencies, but there are trivial merge conflicts---in both branches, methods are inserted at the same position in the code--- between: - #21343: deal with mu[0]*right != right in k-regular sequences - #35894: Regular sequences: implement convolution / ring structure URL: #35896 Reported by: Daniel Krenn Reviewer(s): cheuberg, Daniel Krenn

* t/21343/k-regular-warning: (11523 commits) Trac sagemath#21343: adapt to removed transpose-property in dependency Trac sagemath#21204: fixup code and tests Trac sagemath#21204: cherry-pick to avoid merge conflict Trac sagemath#21319: fixup due to changes in dependencies Trac sagemath#21318: fixup due to recent changes in dependencies Updated SageMath version to 9.3 build/pkgs/fplll/spkg-install.in: Configure --without-qd if we use gcc from spkg build/pkgs/fplll/spkg-configure.m4: Add depcheck on gcc build/pkgs/ppl/spkg-configure.m4: Add depcheck on gcc build/pkgs/brial/spkg-configure.m4: Add depcheck on gcc build/pkgs/{freetype,libgd}/spkg-configure.m4: Add depcheck for gcc build/pkgs/zeromq/spkg-configure.m4: Add depcheck for gcc build/pkgs/ntl/spkg-configure.m4: Add depcheck for gcc Trac sagemath#21295 review issue 29: notice minimize vs field Trac sagemath#21295 review issue 7: document accessing coefficients Trac sagemath#21295 review issue 33: rename to number_of_zeros (as it should be) Trac sagemath#21203 review issue 4: rename to coefficient ring Trac sagemath#21295: rename to coefficient_ring Trac sagemath#21203 review issue 3: example for __getitem__ and __iter__ Trac sagemath#21203 review issue 2: extend odds in Pascal's triangle ...

dkrenn added c: combinatorics labels Aug 10, 2016

dkrenn added the s: needs review label Aug 25, 2016

cheuberg added s: needs info and removed s: needs work labels Oct 11, 2022

mkoeppe removed this from the sage-9.8 milestone Jan 29, 2023

This was referenced Dec 29, 2022

Meta ticket: k-regular sequences #21202

Open

deal with bad chosen representations of k-regular sequences #21343

Closed

dkrenn added a commit to dkrenn/sage that referenced this issue May 25, 2023

Trac sagemath#21204: cherry-pick to avoid merge conflict

dcc7cd1

dkrenn added a commit to dkrenn/sage that referenced this issue May 25, 2023

Trac sagemath#21204: fixup code and tests

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dkrenn added a commit to dkrenn/sage that referenced this issue May 25, 2023

Trac sagemath#21204: fix punctuation

00047ce

dkrenn mentioned this issue May 25, 2023

guessing k-regular sequences #35682

Merged

cheuberg added a commit to cheuberg/sage that referenced this issue May 25, 2023

Trac sagemath#21204: Fix typo

ae06d17

cheuberg added a commit to cheuberg/sage that referenced this issue May 25, 2023

Trac sagemath#21204: Fix typo

bb32fee

dkrenn mentioned this issue Jul 4, 2023

deal with bad chosen representations of k-regular sequences #35896

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5 tasks

vbraun closed this as completed in fba3ecf Jul 20, 2023

mkoeppe added this to the sage-10.1 milestone Jul 20, 2023

kwankyu removed the s: needs info label Jan 31, 2025

guessing k-regular sequences #21204

guessing k-regular sequences #21204

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