Re: numeration and order in a table

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Sujet : Re: numeration and order in a table
De : ross.a.finlayson (at) *nospam* gmail.com (Ross Finlayson)
Groupes : sci.math
Date : 12. Nov 2024, 05:47:00
Autres entêtes
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On 02/25/2019 05:17 PM, Ross A. Finlayson wrote:
On Saturday, February 16, 2019 at 8:29:24 AM UTC-8, frpatte wrote:
In an old text about versification, you can find algorithms to write all
the possibilities to alternate long (G) and short (L) syllables in a
verse. Here is the result for a 3 syllable verse:
>
GGG
LGG
GLG
LLG
GGL
LGL
GLL
LLL
>
As you can see, this table is ordered in reverse lexicographical order
from bottom to top.
>
The text gives a mean to find out the number of a line in the table when
you know its content: take the last L and write 2 under it. Then,
proceeding from right to left, double this number if the next syllable
is L, double it and subtract 1 if the syllable is G.
>
Exemple: LLG  write 2 under the last L, as the preceeding syllable is L,
double and you get 4 which is the number of the line.
>
LGL: 2 under the last L, under the G you get 2x2-1= 3, under the first L
you get 2x3 = 6.
>
This system works for every number of syllables, for instance for a 5
syllable verse (the table has 2^5 lines) the line LGGLL is the 26th one:
>
2 under the last L, double under the previous L: 4, double this 4 and
subtract 1 under the preceeding G: 8-1=7, double 7 and subtract 1 under
the preceeding G: 14-1= 13, double this 13 under the first L: 26.
>
Now the algorithm to construct these tables: take two copies of the
table for n syllables and put them one over the other and for the first
copy (on top) add a column of G on the right, and add a column of L on
the right of the 2nd copy (down) and you get the table for n+1 syllables.
>
My question:
>
Is there a relation between the algorithm used to construct the tables
and the way they use to calculate the rank of a line? Or between the
order of the lines (reverse lexicographical order from bottom to top)
and the calculation of this rank?
>
Thank you.
>
>
--
François Patte
Université Paris Descartes
>
>
It seems to be for splitting the table, into two sorts
that otherwise each together would move the sorts
from the "best case", in terms of maintaining the indices,
then to write the sorts and indices, as that arithmetically
the products return in bounds then arrangements, for
not moving the table nor sorting in-place.
>
Exhausting the inter-record distances in all the sorts,
is just making an expensive compression routine, here
for it falling out in arithmetic and entropy tables,
for the forward and reverse sorting, and the simple
algorithm to compress a table to smaller ordered
definitions of tables, then to reconstitute those to
the original tables with the offsets, entropy coding
(or making an array) the table types, by their values,
as that then the types maintain an arithmetic relation,
in offsets for whatever tuple maintains the identity of
related rows regardless their content, the alphabet.
Array also are the offsets, for computing arithmetically
the bounds of those and reconstituting tables from
ranges in sorts.
>
Addressing into the space of the entire language of
combinations of letters from the alphabet, this as
a different problem that where the words no way
fill the space of the transliteration of the space of
words to the space of letters of words in lengths,
the space of words is etymological segmentation,
thus about this words in the space of the letters of
words as about either forward, backward, or left, right,
what results as for example the eventual space of
"all the 0's and 1's", that is same in all directions,
contra here where the list of words in letters is
exponentially longer than the length of the words.
>
Here instead it seems the goal is deconstructing
an arithmetic representation of the words, as it
is the arithmetic representation of the addressing
as so happens to be an increment or particular distance.
>
>
"Is there a relation between the algorithm used to construct the tables
and the way they use to calculate the rank of a line? Or between the
order of the lines (reverse lexicographical order from bottom to top)
and the calculation of this rank? "
>
This combinatorial enumeration of the algorithm,
it's a two letter alphabet so it doesn't require memory
just alternation or toggle as the words are built right-to-left,
either maintaining the memory or recomputing what would
be the carry arithmetic or where to start the carry arithmetic
from the previous, the next, of the combinatorial enumeration.
>
So, there might be more than one way, to construct and deconstruct
the arithmetic.  (For the combinatorial enumeration as for example
a usual enumeration of integers to 2^n, or b^p, each of those an
arithmetization of the word from the language with whatever mapping
of digits for example positionally to letters.)
>
"Now the algorithm to construct these tables: take two copies of the
table for n syllables and put them one over the other and for the first
copy (on top) add a column of G on the right, and add a column of L on
the right of the 2nd copy (down) and you get the table for n+1 syllables. "
>
So, it seems you were building with having to compute the rank,
of the value, from the left as most significant, but positionally
you have changed the most significant bit, instead to the right.
>
That is your responsibility for maintenance of clock arithmetic
but it's no longer a clock arithmetic, that always it is left-to-right
most-significant-to-least-significant (or "big-endian" in usual
machine representation of binary data).
>
The original table as reverse ordered then, as that it was "constructed"
in reverse order a usual default integer assignment, each as computing
2^n then writing the value for 2^n-1 first instead of 0 first, it seems
that you constructed the table, then reversed it.  (It is most-significant
from the right again.)
>
Then, copying that into two consequential tables, and putting the
next most-significant half to the lesser, and half to the greater,
this is simpler to maintain for each line, in only having to compute
the length once, then remembering only 2^n many or for half the
length of the doubled table, the next value, with uniform random
access to the value at the position of the item of the table.
>
It seems you're interested in adjusting construction semantics (and
arithmetic semantics) as introduce access semantics (arithmetic).
>
If the table is left-to-right or right-to-left, differences from the less
significant side are in terms of powers of the size of the alphabet
as indiced from the other side.
>
Here it seems a feature of deconstruction of carry arithmetic (bounded).
>

Date Sujet#  Auteur
12 Nov 24 o Re: numeration and order in a table1Ross Finlayson

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