OpenEuphoria

2. Re: Dimension of sequences

• Posted by Jason Gade <jaygade at ??hoo.com> Sep 14, 2007
• 669 views

Fernando Bauer wrote:
> 
> Hello All,
> 
> A basic question about sequences.
> 
> Suppose that a "retangular sequence" is a sequence generated by using
> iteratively
> the function repeat() beginning with an atom. Then, I think we can say that
> the dimension of a retangular sequence is the number of calls to repeat()
> function.
> (atom=dimension 0, vector=dimension 1, matrix=dimension 2, ...).
> 
> Now, let's say we have a sequence like { 1, {1,1} } which is not retangular.
> Then, a question arise:
> 
> What is the dimension of a non-retangular sequence ?
> 
> a) the maximum depth of the sequence.
> b) an integer number.
> c) a fractal number.
> d) a sequence which depends on the structure.
> e) the dimension concept does not apply.
> f) I don't know.
> g) other.

Interesting question. I think that (d) is probably the most correct answer.

The sequence is a one-dimensional sequence with a maximum depth of two. So it
depends on the structure.

I would say that "dimension" would refer to the minimum depth of the sequence's
elements. Someone who is more mathematically inclined will probably tell me how
wrong I am though.

> Trying to answer that question, others more basics and related to that arise
> to me (sorry if they are stupid!):
> 
> What is the dimension of the circumference ?
> a) 1 , because the area of the circumference is zero. It is a curved 1D
> object.
> b) 2 , because the circumference exists in a bidimensional space.
> c) Both. It has 2 types of dimensions!
> 
> Same question for a line not closed as, for example, the form of letter "U".

(c) as I answered above. A circumference in one dimension would consist of just
it's length, and would therefore be just a line. But the coordinates around the
circumference definitely have two dimensions.

> 
> Thanks for your reply,
> 
> Regards,
> Fernando Bauer


--
A complex system that works is invariably found to have evolved from a simple
system that works.
--John Gall's 15th law of Systemantics.

"Premature optimization is the root of all evil in programming."
--C.A.R. Hoare

j.

4. Re: Dimension of sequences

• Posted by Fernando Bauer <fmbauer at ?o?mail.com> Sep 15, 2007
• 633 views

CChris wrote:

Hi CChris. Thanks for your reply.
> 
> Fernando Bauer wrote:
> > 
> > Hello All,
> > 
> > A basic question about sequences.
> > 
> > Suppose that a "retangular sequence" is a sequence generated by using
> > iteratively
> > the function repeat() beginning with an atom. Then, I think we can say that
> > the dimension of a retangular sequence is the number of calls to repeat()
> > function.
> > (atom=dimension 0, vector=dimension 1, matrix=dimension 2, ...).
> > 
> 
> They call it the dimension of a tensor in physics. Well there are two
> dimensions
> because of covariant and contravariant indexes, but Euphoria ignores this.
Thanks for the information, even if I don't understand what are these indices..

> 
> > Now, let's say we have a sequence like { 1, {1,1} } which is not retangular.
> > Then, a question arise:
> > 
> > What is the dimension of a non-retangular sequence ?
> > 
> > a) the maximum depth of the sequence.
> > b) an integer number.
> > c) a fractal number.
> > d) a sequence which depends on the structure.
> > e) the dimension concept does not apply.
> > f) I don't know.
> > g) other.
> > 
> 
> Depends on what you wish to do with the dimension concept.
Are you saying that there isn't an unique definition for dimension of sequences
?

> In keeping with how the dimension of a topological space is defined, I'd say
> "the maximum depth" (a/).

Then, suppose you have the following sequence:
{
 {1,1,1},
 {1},
 {1,1,1}
}

If I can view this sequence as a representation of the letter "U", using your
answer below, I can conclude that the dimension of this sequence is 1.
But in this case the maximum depth is 2 (using Ricardo Forno's MaxDepth()
definition in genfunc.e library). So, what is the dimension in this case ? 1 or
2?

> 
> > Trying to answer that question, others more basics and related to that arise
> > to me (sorry if they are stupid!):
> > 
> > What is the dimension of the circumference ?
> > a) 1 , because the area of the circumference is zero. It is a curved 1D
> > object.
> > b) 2 , because the circumference exists in a bidimensional space.
> > c) Both. It has 2 types of dimensions!
> > 
> 
> No, the dimension is clearly one, if only because of the scaling issue. If you
> multiply the radius of a circle by c, the length of any arc on this circle is
> multiplied by c, which is power(c,1), as well, so the dimension is 1. For a
> disc, areas are multiplied by c squared, so discs have dimension 2. That's how
> Haüsdorff's dimension is defined, because the relevant measure is the length
> on circles and the area on discs.
> 
> > Same question for a line not closed as, for example, the form of letter "U".
> 
> Same answer: 1. Because there is a length, but not an area.
See above its possible implication for sequences.

> 
> However, if you do the following iterative process:
> 1/start with a line segment
> 2/for each line segment in your objct, replace the middle third segment by the
> other two sides of an equiklateral triangle. Try making orientation
> consistent.
> Then the limit object has dimension... ln(4)/ln(3). It isn't either a curve
> any longer - even though any finite application of algorithm above results in
> a curve - nor a surface yet. Consider that you replace three segments ___ with
> four _/\_ .

Ok, it's the Koch curve. And if the process starts with an equilateral triangle,
it is denominated, as you know, Koch snowflake, which encloses a finite area with
an infinite perimeter. A beautiful image!

> > 
> > Thanks for your reply,
> > 
> > Regards,
> > Fernando Bauer
> 
> CChris

Regards,
  Fernando

6. Re: Dimension of sequences

• Posted by Fernando Bauer <fmbauer at ?otmail.?om> Sep 15, 2007
• 653 views

Jason Gade wrote:

Hi Jason! Thanks for your reply. Interesting concepts.
> 
> Fernando Bauer wrote:
> > 
> > Hello All,
> > 
> > A basic question about sequences.
> > 
> > Suppose that a "retangular sequence" is a sequence generated by using
> > iteratively
> > the function repeat() beginning with an atom. Then, I think we can say that
> > the dimension of a retangular sequence is the number of calls to repeat()
> > function.
> > (atom=dimension 0, vector=dimension 1, matrix=dimension 2, ...).
> > 
> > Now, let's say we have a sequence like { 1, {1,1} } which is not retangular.
> > Then, a question arise:
> > 
> > What is the dimension of a non-retangular sequence ?
> > 
> > a) the maximum depth of the sequence.
> > b) an integer number.
> > c) a fractal number.
> > d) a sequence which depends on the structure.
> > e) the dimension concept does not apply.
> > f) I don't know.
> > g) other.
> 
> Interesting question. I think that (d) is probably the most correct answer.
> 
> The sequence is a one-dimensional sequence with a maximum depth of two. So it
> depends on the structure.

Could you explain better with an example or show the algorithm to generate that
sequence?

> 
> I would say that "dimension" would refer to the minimum depth of the
> sequence's
> elements. 

Isn't the minimum depth of a sequence always equal to 1? Could you explain
better how you obtain this minimum depth?
I'm using the concept of depth used by Forno's genfunc.e library.

> Someone who is more mathematically inclined will probably tell me
> how wrong I am though.
> 
> > Trying to answer that question, others more basics and related to that arise
> > to me (sorry if they are stupid!):
> > 
> > What is the dimension of the circumference ?
> > a) 1 , because the area of the circumference is zero. It is a curved 1D
> > object.
> > b) 2 , because the circumference exists in a bidimensional space.
> > c) Both. It has 2 types of dimensions!
> > 
> > Same question for a line not closed as, for example, the form of letter "U".
> 
> (c) as I answered above. A circumference in one dimension would consist of
> just
> it's length, and would therefore be just a line. But the coordinates around
> the circumference definitely have two dimensions.
Do you know the names of these 2 types of dimensions?

> 
> > 
> > Thanks for your reply,
> > 
> > Regards,
> > Fernando Bauer
> 
> 
> --
> A complex system that works is invariably found to have evolved from a simple
> system that works.
> --John Gall's 15th law of Systemantics.
> 
> "Premature optimization is the root of all evil in programming."
> --C.A.R. Hoare
> 
> j.

Regards,
 Fernando Bauer
 Porto Alegre - Brazil

8. Re: Dimension of sequences

• Posted by Igor Kachan <kinz at pet?rl?nk.ru> Sep 15, 2007
• 653 views

Fernando Bauer wrote:
> 
> Derek Parnell wrote:
> 
> Hi Derek. Thanks for your reply.
> 
> > A sequence doesn't have a 'dimension' in geometrical sense. A sequence is
> > not
> > a geometric object.
> 
> A vector or a matrix aren't geometric objects in mathematics or in programming
> either, however they
> have a dimension, as you know.

Yes, but their 'dimension' is just a special term, which means
just number of members. And at all another thing, if, for example,
some vector describes x-y-z coordinates of some point
in some 3-dimensional space. 

> > Why do you need to know the 'dimension' of a sequence? I mean if you knew
> > that
> > a given sequence has a dimension of 'x', what would you use 'x' for? 
> It is a theoretical question. Maybe there are rules or functions (e.g.
> compression
> factor?) that depends on this parameter. Why do the mathematicians and
> physicists
> calculate dimensions of fractal objects?

There is very simple theory of Euphoria sequences in Euphoria
documentation, firsthand about EU sequences.
These sequences have 'length', and nothing more.
But any member of any sequence can be a sequence itself
with its own length.
This 'length' is just number of members.
See, 'dimension' of vector is equal to 'length' of sequence.
So, your question is not 'theoretical' one, but is just some
question about sense of some terms in Euphoria.
This is a question about multimeaning of the word 'dimension',
I think.
Another thing that EU sequences can describe any
multidimensional real and unreal objects and things,
thanks to very flexible nature of these sequences.
 
> > Are you asking the right question?
> I think yes, since I understand your response as option e) (the concept does
> not apply to). I don't know the answer.

I think, the answer may be simple enough - dimension of
Euphoria sequence is just its length.

But we must not confuse sequence itself and thing,
which is described with that sequence.

Regards,
Igor Kachan
kinz at peterlink.ru

10. Re: Dimension of sequences

• Posted by Igor Kachan <kinz at peter?ink?ru> Sep 15, 2007
• 652 views

Derek Parnell wrote:
> 
> Igor Kachan wrote:
>  
> > But we must not confuse sequence itself and thing,
> > which is described with that sequence.
> 
> It is a very rare thing so I must make mention of it ... I agree with Igor.
>

Good Luck, Derek!  
But maybe Fernando wants to create some new brilliant
complete mathematical theory of the Euphoria sequences, who knows.
Good Luck, Fernando!  

Regards,
Igor Kachan
kinz at peterlink.ru

12. Re: Dimension of sequences

• Posted by Igor Kachan <kinz at pete?lin?.ru> Sep 16, 2007
• 651 views

Hello Fernando!

Fernando Bauer wrote:
> 
> Igor Kachan wrote:
> 
> Hello Igor! Thanks for the reply.
> > 
> > Fernando Bauer wrote:
> > > 
> > > Derek Parnell wrote:
> > > 
> > > Hi Derek. Thanks for your reply.
> > > 
> > > > A sequence doesn't have a 'dimension' in geometrical sense. A sequence
> > > > is not
> > > > a geometric object.
> > > 
> > > A vector or a matrix aren't geometric objects in mathematics or in
> > > programming
> > > either, however they
> > > have a dimension, as you know.
> > 
> > Yes, but their 'dimension' is just a special term, which means
> > just number of members. And at all another thing, if, for example,
> > some vector describes x-y-z coordinates of some point
> > in some 3-dimensional space. 
> 
> I think you misunderstood what I've tried to ask. Probably you didn't read the
> initial post.
> Let's use Euphoria to describe better:
> 
> }}}
<eucode>
> include genfunc.e
> atom a
> sequence seq1D, seq2D, seq3D
> -- an atom
> a = 1
> -- a 1D sequence
> seq1D = repeat(a, 2)
> -- a 2D sequence
> seq2D = repeat(seq1D, 2)
> -- a 3D sequence
> seq3D = repeat(seq2D, 2)
> 
> -- for rectangular sequences MaxDepth() compute the dimension
> procedure ShowDimension(object o)
> puts(1,"Dimension of ")
> print(1, o)
> printf(1," = %d\n",MaxDepth(o))
> end procedure
> 
> ShowDimension(a)
> ShowDimension(seq1D)
> ShowDimension(seq2D)
> ShowDimension(seq3D)
> </eucode>
{{{

> 
> I think my concept of dimension is like Rank concept in the J language.
> What I'm trying to ask is: What is the dimension for non-rectangular
> sequences?

Ok, good. I do understand your concept, be sure.
But it is something new. There is no any theory under
"dimension of Euphoria sequence".

Euphoria sequence as it is is invented by Robert Craig
and has just its length.

So you are free to define some *new* & *useful* parameters
of some *specific kind* of sequences and to name these
parameters as you wish - 'dimension' or something else,
say, 'length-depth'.

I do not think, for now at least, that we have to
care if the sequence is 'rectangular' or 'non-rectangular'.
But I think that sequences can describe any rectangular
and non-rectangular objects, and dimension of that
described object, for example, may be some dimension
of this describing sequence too.

I do not know what to do with this new parameter
of a sequence, with this new term, 'dimension'.

Ok, say, dimension of some sequence is 2.
So what?

But if I have some 2 components vector, I do know
what to do with that vector.

> > > > Why do you need to know the 'dimension' of a sequence? I mean if you
> > > > knew that
> > > > a given sequence has a dimension of 'x', what would you use 'x' for? 
> > > It is a theoretical question. Maybe there are rules or functions (e.g.
> > > compression
> > > factor?) that depends on this parameter. Why do the mathematicians and
> > > physicists
> > > calculate dimensions of fractal objects?
> > 
> > There is very simple theory of Euphoria sequences in Euphoria
> > documentation, firsthand about EU sequences.
> > These sequences have 'length', and nothing more.
> > But any member of any sequence can be a sequence itself
> > with its own length.
> > This 'length' is just number of members.
> > See, 'dimension' of vector is equal to 'length' of sequence.
> I think my "dimension" is your "number of dimensions". See the code above.
> 
> > So, your question is not 'theoretical' one, but is just some
> > question about sense of some terms in Euphoria.
> > This is a question about multimeaning of the word 'dimension',
> > I think.
> > Another thing that EU sequences can describe any
> > multidimensional real and unreal objects and things,
> > thanks to very flexible nature of these sequences.
> >  
> > > > Are you asking the right question?
> > > I think yes, since I understand your response as option e) (the concept
> > > does
> > > not apply to). I don't know the answer.
> > 
> > I think, the answer may be simple enough - dimension of
> > Euphoria sequence is just its length.
> See above other concept or meaning of "dimension", which is what I'm trying
> to ask about.

Ok, I see, I do understand this other concept,
but this concept itself is just some definition
of some new term for some specific kind of sequences,
so named rectangular sequences.
But 'rectangular' sequence itself needs some
definition now, and so on.

Ok, suppose, we now have all these new definitions -
'rectangular sequence' (RS), 'non-rectangular sequence' (NRS),
'dimension of RS', 'dimension of NRS'.

So what?  What will we do with all these new terms, with
all these new notions, with all these new concepts?

Euphoria itself is very simple language without complicated
abstractions. But can describe any complicated objects
using simple concept of sequence.

> > But we must not confuse sequence itself and thing,
> > which is described with that sequence.

Well, back to above your question, I do not think that someone
already knows how to define 'dimension of NRS' and so on and
for what purpose.

Ok, try to define it yourself, why not?
But remember please, Euphoria is simple thing, first of all.

Good Luck!

Regards,
Igor Kachan
kinz at peterlink.ru

14. Re: Dimension of sequences

• Posted by CChris <christian.cuvier at agric?lture.g?uv.fr> Sep 16, 2007
• 645 views

Derek Parnell wrote:
> 
> 
> I must disagree with Chris here.
> 
> A sequence doesn't have a 'dimension' in geometrical sense. A sequence is not
> a geometric object.
> 

Of course, but it is a data structure (a tree), and has a dimension as such.
There are dimensions in many other aspects of maths than geometry.

CChris

> Why do you need to know the 'dimension' of a sequence? I mean if you knew that
> a given sequence has a dimension of 'x', what would you use 'x' for? 
> 
> Are you asking the right question?
> 
>  
> Fernando Bauer wrote:
> > Then, suppose you have the following sequence:
> > {
> >  {1,1,1},
> >  {1},
> >  {1,1,1}
> > }
> > 
> > If I can view this sequence as a representation of the letter "U", 
> > using your answer below, I can conclude that the dimension of this
> > sequence is 1.
> 
> This sequence has a length of 3 and a depth of 1, so in that sense it has two
> dimensions I suppose - Length and Depth. It also has a content of 7 (the
> number
> of atoms in total) so maybe a sequence has three dimensions? It also has a
> sub-sequence
> of 3, so maybe a sequence has four dimensions - Length, Depth, Content,
> Sub-sequence.
> Each of these is independant of the other so they sort of qualify as
> dimensions,
> no? But then 'Content' is kinda like 'area' in concept so maybe it doesn't
> count
> after all.
> 
> 
> > > > Trying to answer that question, others more basics and related to that
> > > > arise
> > > > to me (sorry if they are stupid!):
> > > > 
> > > > What is the dimension of the circumference ?
> 
> Exactly my point. The dimension of a sequence is a meaningless concept. What
> are you really trying to know about the sequence?
> 
> -- 
> Derek Parnell
> Melbourne, Australia
> Skype name: derek.j.parnell

16. Re: Dimension of sequences

• Posted by Fernando Bauer <fmbauer at ho?mail.?om> Sep 18, 2007
• 657 views

Al Getz wrote:
> 
> Hello again,
> 

Hello Al! Thanks for your comments.

> 
> The game of chess if often referred to as having move sequences that
> exist on the edge of a multi-dimensional object, so i suppose
> you could say the same thing about a sequence.
> 

Sorry, I didn't understand this.

> My question would be say you defined the 'dimension' of a sequence
> as the product of the max of all the inner sequences written out
> such as 1x2x3 as is sometimes customary with arrays.  What would
> you use this information for in a program?
> 

Examples:
a)A 2D sequence (a matrix) can represent a system of simultaneous linear
equations. Then, you can use your description (a x b) to find out if the system
has possibly none, one or many solutions.
b)To know if a sequence represent a square matrix. For example, to know if you
can calculate its determinant.

> When i did my jpeg decoder program in Euphoria i used sequences
> like arrays where every level had the same length:
>   s=repeat(0,10)
>   s=repeat(s,4)
>   s=repeat(s,9)
>   s=repeat(s,5)
> and this builds up a structure similar to a C style array where
> every level has the same length.

That is what I'm calling rectangular sequence. Perhaps we could call it "array
sequence".

>  The above code creates 5 
> three dimensional 'arrays' that can be accessed by index.
> I would have no problem calling this a 10x4x9x5 or a 5x9x4x10
> 'sequence', 

Ok, you can call this way, but this isn't the dimension that I was talking
about. This is the number of atoms (after you compute the product).

>but the actual common useage seems to be to use
> every element, but then again i could easily see an app that
> although includes a given element, does not in fact use that
> element in the program, ever, and of course the implication of
> this is that an element missing altogether (only possible with 
> a sequence) does not necessarily decrease the length because
> the other sequences at the same level contain more elements:
>   s={
>       {1,2,3,4},
>       {5,6,7}
>     }
> is still a 2x4 sequence.
> 

If this is true, why *can't* I sum, subtract, multiply, etc, sequences with the
same description ?
The information of the structure of the sequence is lost in that representation.
I think if you eliminate an atom of a retangular sequence (array sequence), it
can't have the same description any longer.

> If you want to get more detailed, you might have to start enumerating
> every element and returning a sequence that corresponds to that
> sequence.  As with the above, this would be:
>   m={4,3}
> I guess this would simply be a sequence with the lengths of all 
> first level sequences arranged in the same structure as the
> original sequence, which would still have to be perused.
> Again i would have to wonder about how useful this would be unless
> the sequence isnt that big.  For example, "When do i need to know
> the actual structure of the sequence in full?"
> 

To serialize sequences, which is necessary in several situations: saving and
restoring sequences to/from files, transmission of sequences via serial networks.

> Some applications would be interesting to hear about at this point.
>

Any application that uses EDS, save and restore the full structures of sequences
to/from the database.
 
> 
> Al
> 
> E boa sorte com sua programacao Euphoria!

Obrigado. Para voce tambem!

> 
> 
> My bumper sticker: "I brake for LED's"
> 

Regards,
   Fernando

18. Re: Dimension of sequences

• Posted by Igor Kachan <kinz at pete?link.r?> Sep 18, 2007
• 649 views

Hello Fernando!

Fernando Bauer wrote:
> 
> Igor Kachan wrote:
> >

[snipped]

> > Ok, suppose, we now have all these new definitions -
> > 'rectangular sequence' (RS), 'non-rectangular sequence' (NRS),
> > 'dimension of RS', 'dimension of NRS'.
> 
> 'non-rectangular sequence' (NRS) = all sequences that aren't RS.
> 'dimension of RS' = maximum depth of the sequence.
> 'dimension of NRS' = that is the question!

Let's see the refman.doc file now.

Robert Craig wrotes:

 "Sequences can be nested to any depth, i.e. you can have sequences within
  sequences within sequences and so on to any depth (until you run out of
  memory)." 

I see Rob cares just about *maximum* depth here, so his definition
of depht, documented in refman.doc, is strongly equivalent to your
definition of 'dimension of RS'.
Same thing, you just introduce some new term, synonym.

No?

Then, I do not think now that the 'rectangular' word is very
good for your purpose, maybe, 'regular', as some short
for 'regularly nested', is better.

> > So what?  What will we do with all these new terms, with
> > all these new notions, with all these new concepts?
> 
> First of all, they facilitate our communication, since we don't have to say
> that whole definition phrase.
> Second, the terms RS and NRS can represent types in Euphoria, like vector or
> matrix, and so can be checked.
> Some algorithms can function with one and not with other type, etc..

Ok, but for now I do not see some real place for the 'dimension' word
here. Length & Depth. These old good words are very clear in 
the specific EU context. 

> > Euphoria itself is very simple language without complicated
> > abstractions. But can describe any complicated objects
> > using simple concept of sequence.
> > 
> > > > But we must not confuse sequence itself and thing,
> > > > which is described with that sequence.
> > 
> > Well, back to above your question, I do not think that someone
> > already knows how to define 'dimension of NRS' and so on and
> > for what purpose.
> 
> Maybe this answer and others can be found in Tree theory which was
> more studied.

Maybe, but that theory is not a theory of EU sequences,
so you'll need to redifine some their terms into native
EU terms first of all, if there are needed to you
concepts in that theory.  

> 
> > 
> > Ok, try to define it yourself, why not?
> > But remember please, Euphoria is simple thing, first of all.
> > 
> > Good Luck!
> 
> Thank you!

Good Luck again!

Regards,
Igor Kachan
kinz at peterlink.ru

20. Re: Dimension of sequences

• Posted by Fernando Bauer <fmbauer at hotma?l.com> Sep 19, 2007
• 650 views

Igor Kachan wrote:
> 
> Hello Fernando!

Hello Igor!

> 
> Fernando Bauer wrote:
> > 
> > Igor Kachan wrote:
> > >
> 
> [snipped]
> 
> > > Ok, suppose, we now have all these new definitions -
> > > 'rectangular sequence' (RS), 'non-rectangular sequence' (NRS),
> > > 'dimension of RS', 'dimension of NRS'.
> > 
> > 'non-rectangular sequence' (NRS) = all sequences that aren't RS.
> > 'dimension of RS' = maximum depth of the sequence.
> > 'dimension of NRS' = that is the question!
> 
> Let's see the refman.doc file now.
> 
> Robert Craig wrotes:
> 
>  "Sequences can be nested to any depth, i.e. you can have sequences within
>   sequences within sequences and so on to any depth (until you run out of
>   memory)." 
> 
> I see Rob cares just about *maximum* depth here, so his definition
> of depht, documented in refman.doc, is strongly equivalent to your
> definition of 'dimension of RS'.
> Same thing, you just introduce some new term, synonym.
> 
> No?

Ok. We can use "depth" as synonym of "dimension" as you prefer. Then my question
is:
What is the depth of a NRS? How can I obtain it? For now, I only know the depth
of a RS.

> 
> Then, I do not think now that the 'rectangular' word is very
> good for your purpose, maybe, 'regular', as some short
> for 'regularly nested', is better.
> 
Ok. But I think "rectangular" is sligthly more precise than "regular", since I
could think that the following sequence is regular:
repeat({1,{1,1}},n) where n is any integer number. And that sequence is NRS.
Well, this is only a definition problem.

> > > So what?  What will we do with all these new terms, with
> > > all these new notions, with all these new concepts?
> > 
> > First of all, they facilitate our communication, since we don't have to say
> > that whole definition phrase.
> > Second, the terms RS and NRS can represent types in Euphoria, like vector or
> > matrix, and so can be checked.
> > Some algorithms can function with one and not with other type, etc..
> 
> Ok, but for now I do not see some real place for the 'dimension' word
> here. Length & Depth. These old good words are very clear in 
> the specific EU context. 

Ok. But the Depth concept is not so clear as it seems, since nobody defined what
it is for NRS yet.

[snipped]

> Regards,
> Igor Kachan
> kinz at peterlink.ru

Regards,
  Fernando

22. Re: Dimension of sequences

• Posted by CChris <christian.cuvier at ??riculture.gouv.fr> Sep 19, 2007
• 639 views

Igor Kachan wrote:
> 
> Fernando Bauer wrote:
> > 
> > Igor Kachan wrote:
> > > 
> > > > 'non-rectangular sequence' (NRS) = all sequences that aren't RS.
> > > > 'dimension of RS' = maximum depth of the sequence.
> > > > 'dimension of NRS' = that is the question!
> > > 
> > > Let's see the refman.doc file now.
> > > 
> > > Robert Craig writes:
> > > 
> > >  "Sequences can be nested to any depth, i.e. you can have sequences within
> > >   sequences within sequences and so on to any depth (until you run out of
> > >   memory)." 
> > > 
> > > I see Rob cares just about *maximum* depth here, so his definition
> > > of depht, documented in refman.doc, is strongly equivalent to your
> > > definition of 'dimension of RS'.
> > > Same thing, you just introduce some new term, synonym.
> > > 
> > > No?
> > 
> > Ok. We can use "depth" as synonym of "dimension" as you prefer. Then my
> > question
> > is:
> > What is the depth of a NRS? How can I obtain it? For now, I only know the
> > depth
> > of a RS.
> 
> Hello, Fernando!
> 
> OK, let's use just that MaxDepth() function by Ricardo,
> it seems to work properly not only on 'simple' sequences,
> but on any ones, it is recursive, and gives us that
> pure maximum maximorum depth of any sequence.
> 
> See please:
> }}}
<eucode>
> global function MaxDepth(object o)
>     integer n, x
>     if atom(o) then
>        return 0
>     else
>        n = 0
> 	for i = 1 to length(o) do
> 	    x = MaxDepth(o[i])
> 	    if x > n then
> 		n = x
> 	    end if
> 	end for
> 	return n + 1
>     end if
> end function -- it is function by Ricardo Forno, genfunc.e lib 
> 
> ? MaxDepth({{{{}}},{},{{},{{{{{{{{{{{{}}}}}}}}}}}}},{{{{{{{{}}}}}}}}})
> ? MaxDepth({{{{1}}},{1},{{1},{{{{{{{{{{{{1}}}}}}}}}}}}},{{{{{{{{1}}}}}}}}})
> </eucode>
{{{
 
> The maximum depth is 14, all right.
> Count please the '{' signs, opening the deepest sequence,
> to be sure.
> It seems to be a good parameter for description of a sequence.
> Now a sequence has its own second dimension, and may be
> considered as some 2-dimensional object, which can contain
> description of any-dimentional real and unreal objects -
> vectors, matrixes, tensors, lists, arrays, trees, books,
> images ... etc, etc.
> Sequence itself really has two own dimensions - length and depth,
> I think now. So attempts to give it some *single* dimension
> may be not very productive.
> 
> > > Then, I do not think now that the 'rectangular' word is very
> > > good for your purpose, maybe, 'regular', as some short
> > > for 'regularly nested', is better.
> > > 
> > Ok. But I think "rectangular" is sligthly more precise than "regular", since
> > I could think that the following sequence is regular:
> > repeat({1,{1,1}},n) where n is any integer number. And that sequence is NRS.
> > Well, this is only a definition problem.
> 
> Ricardo names these sequences as 'simple sequences'.
> 
> > > > > So what?  What will we do with all these new terms, with
> > > > > all these new notions, with all these new concepts?
> > > > 
> > > > First of all, they facilitate our communication, since we don't have to
> > > > say
> > > > that whole definition phrase.
> > > > Second, the terms RS and NRS can represent types in Euphoria, like
> > > > vector or
> > > > matrix, and so can be checked.
> > > > Some algorithms can function with one and not with other type, etc..
> > > 
> > > Ok, but for now I do not see some real place for the 'dimension' word
> > > here. Length & Depth. These old good words are very clear in 
> > > the specific EU context. 
> > 
> > Ok. But the Depth concept is not so clear as it seems, since nobody defined
> > what it is for NRS yet.
> 
> OK, now we can calculate that Depth (good, let's name the maximum depth
> as Depth) with Ricardo's function.
> 
> Regards,
> Igor Kachan
> kinz at peterlink.ru

Eu sequences model trees with atomic labels allowed, and mandatory, only on leaf
nodes.
For instance, {1,{2,3},4} is the same as
root -> node1, node2, node3
node1: 1
node2 -> node4, node5
node3: 4
node4: 2
node5: 3

with hopefully obvious meaning of such a description of a tree.

Now google for some litterature on trees. You'lll see that most specific form of
trees define their own dimension, depth or height.
Concepts that can be defined always include:
* the depth of a tree, which is the height of its root;
* the depth of a node in a tree, which is its distance to the root (ie the
number of edges to go from the node to the root);
* the height of a node, which is the length of the longest path from the node to
a leaf node in its subtree.

The length of a sequence is simply the number of children of the root. This has
not much importance in tree theory, but is essential for _some_ sorts of them,
namely strings of arbitrary things, among which trees. It has been chosen to
build this special notion in the language, and to leave out the more general
ones. Other trees will have their own notion of depth or somsuch, which depends
on the speifics of the layout an uses of the given family of trees.
So yes, sequences don't have a single dimension, although some sorts of
sequences do have a more intinsic kind of dimension. Strings are an example, for
which the length is the "natural" dimension.

CChris

24. Re: Dimension of sequences

• Posted by CChris <christian.cuvier at agri?u?ture.gouv.fr> Sep 19, 2007
• 640 views

Igor Kachan wrote:
> 
> CChris wrote:
> > 
> > Igor Kachan wrote:
> > > 
> > > Fernando Bauer wrote:
> > > > 
> > > > Igor Kachan wrote:
> > > > > 
> > > > > > 'non-rectangular sequence' (NRS) = all sequences that aren't RS.
> > > > > > 'dimension of RS' = maximum depth of the sequence.
> > > > > > 'dimension of NRS' = that is the question!
> > > > > 
> > > > > Let's see the refman.doc file now.
> > > > > 
> > > > > Robert Craig writes:
> > > > > 
> > > > >  "Sequences can be nested to any depth, i.e. you can have sequences
> > > > >  within
> > > > >   sequences within sequences and so on to any depth (until you run out
> > > > >   of
> > > > >   memory)." 
> > > > > 
> > > > > I see Rob cares just about *maximum* depth here, so his definition
> > > > > of depht, documented in refman.doc, is strongly equivalent to your
> > > > > definition of 'dimension of RS'.
> > > > > Same thing, you just introduce some new term, synonym.
> > > > > 
> > > > > No?
> > > > 
> > > > Ok. We can use "depth" as synonym of "dimension" as you prefer. Then my
> > > > question
> > > > is:
> > > > What is the depth of a NRS? How can I obtain it? For now, I only know
> > > > the depth
> > > > of a RS.
> > > 
> > > Hello, Fernando!
> > > 
> > > OK, let's use just that MaxDepth() function by Ricardo,
> > > it seems to work properly not only on 'simple' sequences,
> > > but on any ones, it is recursive, and gives us that
> > > pure maximum maximorum depth of any sequence.
> > > 
> > > See please:
> > > }}}
<eucode>
> > > global function MaxDepth(object o)
> > >     integer n, x
> > >     if atom(o) then
> > >        return 0
> > >     else
> > >        n = 0
> > > 	for i = 1 to length(o) do
> > > 	    x = MaxDepth(o[i])
> > > 	    if x > n then
> > > 		n = x
> > > 	    end if
> > > 	end for
> > > 	return n + 1
> > >     end if
> > > end function -- it is function by Ricardo Forno, genfunc.e lib 
> > > 
> > > ? MaxDepth({{{{}}},{},{{},{{{{{{{{{{{{}}}}}}}}}}}}},{{{{{{{{}}}}}}}}})
> > > ?
> > > MaxDepth({{{{1}}},{1},{{1},{{{{{{{{{{{{1}}}}}}}}}}}}},{{{{{{{{1}}}}}}}}})
> > > </eucode>
{{{
 
> > > The maximum depth is 14, all right.
> > > Count please the '{' signs, opening the deepest sequence,
> > > to be sure.
> > > It seems to be a good parameter for description of a sequence.
> > > Now a sequence has its own second dimension, and may be
> > > considered as some 2-dimensional object, which can contain
> > > description of any-dimentional real and unreal objects -
> > > vectors, matrixes, tensors, lists, arrays, trees, books,
> > > images ... etc, etc.
> > > Sequence itself really has two own dimensions - length and depth,
> > > I think now. So attempts to give it some *single* dimension
> > > may be not very productive.
> > > 
> > > > > Then, I do not think now that the 'rectangular' word is very
> > > > > good for your purpose, maybe, 'regular', as some short
> > > > > for 'regularly nested', is better.
> > > > > 
> > > > Ok. But I think "rectangular" is sligthly more precise than "regular",
> > > > since
> > > > I could think that the following sequence is regular:
> > > > repeat({1,{1,1}},n) where n is any integer number. And that sequence is
> > > > NRS.
> > > > Well, this is only a definition problem.
> > > 
> > > Ricardo names these sequences as 'simple sequences'.
> > > 
> > > > > > > So what?  What will we do with all these new terms, with
> > > > > > > all these new notions, with all these new concepts?
> > > > > > 
> > > > > > First of all, they facilitate our communication, since we don't have
> > > > > > to
> say</font></i>
> > > > > > that whole definition phrase.
> > > > > > Second, the terms RS and NRS can represent types in Euphoria, like
> > > > > > vector
> > or</font></i>
> > > > > > matrix, and so can be checked.
> > > > > > Some algorithms can function with one and not with other type, etc..
> > > > > 
> > > > > Ok, but for now I do not see some real place for the 'dimension' word
> > > > > here. Length & Depth. These old good words are very clear in 
> > > > > the specific EU context. 
> > > > 
> > > > Ok. But the Depth concept is not so clear as it seems, since nobody
> > > > defined
> > > > what it is for NRS yet.
> > > 
> > > OK, now we can calculate that Depth (good, let's name the maximum depth
> > > as Depth) with Ricardo's function.
> > > 
> > > Regards,
> > > Igor Kachan
> > > kinz at peterlink.ru
> > 
> > Eu sequences model trees with atomic labels allowed, and mandatory, only on
> > leaf nodes.
> > For instance, {1,{2,3},4} is the same as
> > root -> node1, node2, node3
> > node1: 1
> > node2 -> node4, node5
> > node3: 4
> > node4: 2
> > node5: 3
> > 
> > with hopefully obvious meaning of such a description of a tree.
> > 
> > Now google for some litterature on trees. You'lll see that most specific
> > form
> > of trees define their own dimension, depth or height.
> > Concepts that can be defined always include:
> > * the depth of a tree, which is the height of its root;
> 
> OK, I like this theory very much as math at all, some good
> thing to get crazy, if you do not have or do not like vodka, sorry.  
> 
> Why do they need these topsyturvy trees, Cris?
> 

They are the most common objects in decision theory (think of a chess computer
program) or in searching/retrieval problems (remember the contest Derek had
hosted a couple years ago?). And many others, but hese may be the most obvious.

> I can understand depth of a root and height of a tree,
> but not on the contrary.
> 

This has to do with our habit to read a sheet of paper from top to bottom, which
is why the root is usually on top 
  
> > * the depth of a node in a tree, which is its distance to the root (ie the
> > number
> > of edges to go from the node to the root);
> > * the height of a node, which is the length of the longest path from the
> > node
> > to a leaf node in its subtree.
> 
> Same thing, very-very demonstrative concept.  
> 
> > The length of a sequence is simply the number of children of the root. This
> > has not much importance in tree theory, but is essential for _some_ sorts of
> > them, namely strings of arbitrary things, among which trees. It has been
> > chosen
> > to build this special notion in the language, and to leave out the more
> > general
> > ones. Other trees will have their own notion of depth or somsuch, which
> > depends
> > on the speifics of the layout an uses of the given family of trees.
> > So yes, sequences don't have a single dimension, although some sorts of
> > sequences
> > do have a more intinsic kind of dimension. Strings are an example, for which
> > the length is the "natural" dimension.
> 
> OK, but depth of a string exists and is 1. So sequence seems to be
> some 2-dimensional object anyway and always.
> 
2,3,27,... depending on the exact notion of dimension you need. A sequence 
whose elements may freely be atoms or not has dimension 1, and strings are
special cases of them. Otherwise, as I said, the concept is not uniquely defined,
and the value for the dimension is always 2 or more.

CChris


> Regards,
> Igor Kachan
> kinz at peterlink.ru

26. Re: Dimension of sequences

• Posted by CChris <christian.cuvier at agricultu?e.gouv.fr> Sep 20, 2007
• 638 views

Igor Kachan wrote:
> 
> CChris wrote:
> > 
> > Igor Kachan wrote:
> > > 
> > > CChris wrote:
> > > > 
> > > > Igor Kachan wrote:
> > > > > 
> > > > > Fernando Bauer wrote:
> > > > > > 
> > > > > > Igor Kachan wrote:
> > > > > > >
> 
> [snip]
>  
> > > > Eu sequences model trees with atomic labels allowed, and mandatory, only
> > > > on
> > > > leaf nodes.
> > > > For instance, {1,{2,3},4} is the same as
> > > > root -> node1, node2, node3
> > > > node1: 1
> > > > node2 -> node4, node5
> > > > node3: 4
> > > > node4: 2
> > > > node5: 3
> > > > 
> > > > with hopefully obvious meaning of such a description of a tree.
> > > > 
> > > > Now google for some litterature on trees. You'lll see that most specific
> > > > form
> > > > of trees define their own dimension, depth or height.
> > > > Concepts that can be defined always include:
> > > > * the depth of a tree, which is the height of its root;
> > > 
> > > OK, I like this theory very much as math at all, some good
> > > thing to get crazy, if you do not have or do not like vodka, sorry.  
> > > 
> > > Why do they need these topsyturvy trees, Cris?
> > > 
> > 
> > They are the most common objects in decision theory (think of a chess
> > computer
> > program) or in searching/retrieval problems (remember the contest Derek had
> > hosted a couple years ago?). And many others, but hese may be the most
> > obvious.
> > 
> > > I can understand depth of a root and height of a tree,
> > > but not on the contrary.
> > > 
> > 
> > This has to do with our habit to read a sheet of paper from top to bottom,
> which is why the root is usually on</font></i>
> > top 
> >   
> > > > * the depth of a node in a tree, which is its distance to the root (ie
> > > > the
> number</font></i>
> > > > of edges to go from the node to the root);
> > > > * the height of a node, which is the length of the longest path from the
> > > > node
> > > > to a leaf node in its subtree.
> > > 
> > > Same thing, very-very demonstrative concept.  
> > > 
> > > > The length of a sequence is simply the number of children of the root.
> > > > This
> > > > has not much importance in tree theory, but is essential for _some_
> > > > sorts of
> > > > them, namely strings of arbitrary things, among which trees. It has been
> > > > chosen
> > > > to build this special notion in the language, and to leave out the more
> > > > general
> > > > ones. Other trees will have their own notion of depth or somsuch, which
> > > > depends
> > > > on the speifics of the layout an uses of the given family of trees.
> > > > So yes, sequences don't have a single dimension, although some sorts of
> > > > sequences
> > > > do have a more intinsic kind of dimension. Strings are an example, for
> > > > which
> > > > the length is the "natural" dimension.
> > > 
> > > OK, but depth of a string exists and is 1. So sequence seems to be
> > > some 2-dimensional object anyway and always.
> > > 
> > 2,3,27,... depending on the exact notion of dimension you need. A sequence 
> > whose elements may freely be atoms or not has dimension 1, and strings are
> > special
> > cases of them. Otherwise, as I said, the concept is not uniquely defined,
> > and
> > the value for the dimension is always 2 or more.
> 
> Here, in Russia, we use 2 words for these things:
> "izmerenie" -- it is just single "dimension"
> and
> "razmernost'" --  it is common number of dimensions, "dimensionality".
> So I'd prefer to say that any sequence has dimensionality 2,
> and its dimensions are named as "length" and "depth".

This looks ok to me, except that the length is really a very special case.

> But this 2-dimensional sequence can describe and contain
> multi-dimensional objects.
> Anyway, in machine memory, it is just 1-dimensional *range* of
> addresses.
> 

Exactly. So the "length" of a general sequence would be the total number of
nodes it holds. The current definition of length() is very adequate for linear
trees - one root with length(s) children, the nature of which is not taken into
account -. For general trees, however, it barely makes sense.

CChris

> Regards,
> Igor Kachan
> kinz at peterlink.ru

28. Re: Dimension of sequences

• Posted by Fernando Bauer <fmbauer at h?t?ail.com> Sep 20, 2007
• 640 views

Hello Igor!

Igor Kachan wrote:
> 
> Fernando Bauer wrote:
> > 
> > Igor Kachan wrote:
> > > 
> > > > 'non-rectangular sequence' (NRS) = all sequences that aren't RS.
> > > > 'dimension of RS' = maximum depth of the sequence.
> > > > 'dimension of NRS' = that is the question!
> > > 
> > > Let's see the refman.doc file now.
> > > 
> > > Robert Craig writes:
> > > 
> > >  "Sequences can be nested to any depth, i.e. you can have sequences within
> > >   sequences within sequences and so on to any depth (until you run out of
> > >   memory)." 
> > > 
> > > I see Rob cares just about *maximum* depth here, so his definition
> > > of depht, documented in refman.doc, is strongly equivalent to your
> > > definition of 'dimension of RS'.
> > > Same thing, you just introduce some new term, synonym.
> > > 
> > > No?
> > 
> > Ok. We can use "depth" as synonym of "dimension" as you prefer. Then my
> > question
> > is:
> > What is the depth of a NRS? How can I obtain it? For now, I only know the
> > depth
> > of a RS.
> 
> Hello, Fernando!
> 
> OK, let's use just that MaxDepth() function by Ricardo,
> it seems to work properly not only on 'simple' sequences,
> but on any ones, it is recursive, and gives us that
> pure maximum maximorum depth of any sequence.
Ok. This is a possibility, and was the first reply posted by CChris to define
the dimension of NRS.
But, for now, I'm not sure.
> 
> See please:
> }}}
<eucode>
> global function MaxDepth(object o)
>     integer n, x
>     if atom(o) then
>        return 0
>     else
>        n = 0
> 	for i = 1 to length(o) do
> 	    x = MaxDepth(o[i])
> 	    if x > n then
> 		n = x
> 	    end if
> 	end for
> 	return n + 1
>     end if
> end function -- it is function by Ricardo Forno, genfunc.e lib 
> 
> ? MaxDepth({{{{}}},{},{{},{{{{{{{{{{{{}}}}}}}}}}}}},{{{{{{{{}}}}}}}}})
> ? MaxDepth({{{{1}}},{1},{{1},{{{{{{{{{{{{1}}}}}}}}}}}}},{{{{{{{{1}}}}}}}}})
> </eucode>
{{{
 
> The maximum depth is 14, all right.
> Count please the '{' signs, opening the deepest sequence,
> to be sure.
> It seems to be a good parameter for description of a sequence.
> Now a sequence has its own second dimension, and may be
> considered as some 2-dimensional object, which can contain
> description of any-dimentional real and unreal objects -
> vectors, matrixes, tensors, lists, arrays, trees, books,
> images ... etc, etc.
> Sequence itself really has two own dimensions - length and depth,
> I think now. So attempts to give it some *single* dimension
> may be not very productive.

Just to make clear, this concept of 2-dimensional object, or dimensionality 2 as
you define, is a higher abstract concept, and is not my original definition of
"dimension" which is your Depth.

> 
> > > Then, I do not think now that the 'rectangular' word is very
> > > good for your purpose, maybe, 'regular', as some short
> > > for 'regularly nested', is better.
> > > 
> > Ok. But I think "rectangular" is sligthly more precise than "regular", since
> > I could think that the following sequence is regular:
> > repeat({1,{1,1}},n) where n is any integer number. And that sequence is NRS.
> > Well, this is only a definition problem.
> 
> Ricardo names these sequences as 'simple sequences'.

No. Ricardo defines "simple sequences" as "sequences formed only by atoms"
(depth=1). The sequence repeat({1,{1,1}},n) has sub-sequences therefore it's not
a simple sequence.
A simple sequence is always a RS, but not always a RS is a simple sequence.

> 
> > > > > So what?  What will we do with all these new terms, with
> > > > > all these new notions, with all these new concepts?
> > > > 
> > > > First of all, they facilitate our communication, since we don't have to
> > > > say
> > > > that whole definition phrase.
> > > > Second, the terms RS and NRS can represent types in Euphoria, like
> > > > vector or
> > > > matrix, and so can be checked.
> > > > Some algorithms can function with one and not with other type, etc..
> > > 
> > > Ok, but for now I do not see some real place for the 'dimension' word
> > > here. Length & Depth. These old good words are very clear in 
> > > the specific EU context. 
> > 
> > Ok. But the Depth concept is not so clear as it seems, since nobody defined
> > what it is for NRS yet.
> 
> OK, now we can calculate that Depth (good, let's name the maximum depth
> as Depth) with Ricardo's function.
But then think about this (bad) analogy:
If someone ask you about the depth of the Pacific Ocean, then you would say the
maximum depth of it, since you have defined this way. However we know that its
depth depends on the place.
(notice that this depth is not our discussed concept of Depth).
> 
> Regards,
> Igor Kachan
> kinz at peterlink.ru

Regards,
   Fernando

30. Re: Dimension of sequences

• Posted by Fernando Bauer <fmbauer at hotmail??om> Sep 20, 2007
• 646 views

Hello Al,

Al Getz wrote:
> 
> Fernando Bauer wrote:
> > 
> > Al Getz wrote:
> > > 
> > > Hello again,
> > > 
> > 
> > Hello Al! Thanks for your comments.
> > 
> > > 
> > > The game of chess if often referred to as having move sequences that
> > > exist on the edge of a multi-dimensional object, so i suppose
> > > you could say the same thing about a sequence.
> > > 
> > 
> > Sorry, I didn't understand this.
> > 
> > > My question would be say you defined the 'dimension' of a sequence
> > > as the product of the max of all the inner sequences written out
> > > such as 1x2x3 as is sometimes customary with arrays.  What would
> > > you use this information for in a program?
> > > 
> > 
> > Examples:
> > a)A 2D sequence (a matrix) can represent a system of simultaneous linear
> > equations.
> > Then, you can use your description (a x b) to find out if the system has
> > possibly
> > none, one or many solutions.
> > b)To know if a sequence represent a square matrix. For example, to know if
> > you
> > can calculate its determinant.
> > 
> > > When i did my jpeg decoder program in Euphoria i used sequences
> > > like arrays where every level had the same length:
> > >   s=repeat(0,10)
> > >   s=repeat(s,4)
> > >   s=repeat(s,9)
> > >   s=repeat(s,5)
> > > and this builds up a structure similar to a C style array where
> > > every level has the same length.
> > 
> > That is what I'm calling rectangular sequence. Perhaps we could call it
> > "array
> > sequence".
> > 
> > >  The above code creates 5 
> > > three dimensional 'arrays' that can be accessed by index.
> > > I would have no problem calling this a 10x4x9x5 or a 5x9x4x10
> > > 'sequence', 
> > 
> > Ok, you can call this way, but this isn't the dimension that I was talking
> > about.
> > This is the number of atoms (after you compute the product).
> > 
> > >but the actual common useage seems to be to use
> > > every element, but then again i could easily see an app that
> > > although includes a given element, does not in fact use that
> > > element in the program, ever, and of course the implication of
> > > this is that an element missing altogether (only possible with 
> > > a sequence) does not necessarily decrease the length because
> > > the other sequences at the same level contain more elements:
> > >   s={
> > >       {1,2,3,4},
> > >       {5,6,7}
> > >     }
> > > is still a 2x4 sequence.
> > > 
> > 
> > If this is true, why *can't* I sum, subtract, multiply, etc, sequences with
> > the same description ?
> > The information of the structure of the sequence is lost in that
> > representation.
> > I think if you eliminate an atom of a retangular sequence (array sequence),
> > it can't have the same description any longer.
> > 
> > > If you want to get more detailed, you might have to start enumerating
> > > every element and returning a sequence that corresponds to that
> > > sequence.  As with the above, this would be:
> > >   m={4,3}
> > > I guess this would simply be a sequence with the lengths of all 
> > > first level sequences arranged in the same structure as the
> > > original sequence, which would still have to be perused.
> > > Again i would have to wonder about how useful this would be unless
> > > the sequence isnt that big.  For example, "When do i need to know
> > > the actual structure of the sequence in full?"
> > > 
> > 
> > To serialize sequences, which is necessary in several situations: saving and
> > restoring sequences to/from files, transmission of sequences via serial
> > networks.
> > 
> > > Some applications would be interesting to hear about at this point.
> > >
> > 
> > Any application that uses EDS, save and restore the full structures of
> > sequences
> > to/from the database.
> >  
> > > 
> > > Al
> > > 
> > > E boa sorte com sua programacao Euphoria!
> > 
> > Obrigado. Para voce tambem!
> > 
> > > 
> > > My bumper sticker: "I brake for LED's"
> > > 
> > 
> > Regards,
> >    Fernando
> 
> Hi Fernando,
> 
> Ok, since you seem to have shown that there would be some use lets
> look a little farther...
Ok. But just to be clear: you are talking about structure of sequences, and my
original question is about "dimension" of sequences (or "Depth" as Igor prefers).

> 
> First off, i made a mistake when i showed the possible detailed
> dimensions for the sequence
>   s={
>       {1,2,3,4},
>       {5,6,7}
>     }
> 
> which should probably be more like this:
>   m={2,{4,3}}
> 
> but does this even really work for all sequences?
> Also notice this will get a little hard to read too, almost
> as bad as trying to find the individual lengths of the original
> sub sequences and if there even are any.
> 
> I did a math program a while back that used matrixes too, and what
> i did to see if one matrix was compatible with another was to test
> for the two dimensions (or only one if needed).  For
>   s={{1,2},{3,4},{5,6}}
> the two tests would be
>   v=length(s)
> and
>   h=length(s[1])
> With this info i was able to tell if a multiplication could
> be done with either a vector or another matrix, or if the
> maxtrix was square, or even if the matrix was really a vector.
> 
> Going to one more dimension, you probably have this being application
> specific where you know in advance that all the sub sequences are
> matixes and you have to test them as any other one, except first
> you have to extract the individual matrix:
>   s={{1,2},{3,4},{5,6}}
>   MyMatrixes={s,s,s}
>   for k=1 to length(MyMatrixes) do
>     Dims=Test(MyMatrixes[k])
>   end for
> 
> Compare two more complex matrixes:
>   s={1,{2,3},4,5}
>   t={1,2,{3,4},5}
> 
> How do we indicate the dims without repeating almost the
> whole sequence?
>  ms={1,2,1,1}
>  mt={1,1,2,1}
> perhaps?
Using your concept of dim, I think it's more correct:
ms = {a,2,a,a}
mt = {a,a,2,a}

where a = dim of an atom. (a!= 0) because the dim of {} is zero.

> 
> Then the matrix might be even more complex:
>   s={1,{2,{3,6}},4,5}
> and dims:
>   m={1,{1,{2}},1,1}
> and reading or parsing the dims sequence might be almost as hard
> as parsing the original sequence to query its dimensions.
> 
> On the other hand, if we do define the dims as the set of
> max dims we might end up with something like this:
>   m={4,{2{1}}}
> which is at least a bit simpler than the original sequence.
> 
> BTW, by 1x2x3 that doesnt mean multiply all the numbers to get 6,
> but rather shows the dimensions in the same way that the dimensions
> of C arrays.  The numbers dont get multiplied but stay separate:
>   (1,2,3)
Yes. I said that because in Euphoria 1x2x3 is an expression which results in an
atom.
Then it would be more correct to say, for example, {1,2,3} in this case.

> 
> Lastly, you'd have to think about how this is going to be 
> implemented in the language.  During the build of any sequence
> the dimension array would have to be maintained and this would
> slow down the sequence a bit more too.  You'd have to see just
> how much effect this would have on the overall speed.
> 
> 
> Al
> 
> E boa sorte com sua programacao Euphoria!
> 
> 
> My bumper sticker: "I brake for LED's"
> 

Regards,
   Fernando

32. Re: Dimension of sequences

• Posted by Dan Moyer <danielmoyer at ??odigy.net> Sep 21, 2007
• 626 views

c.k.,

I think that of the two possibilities, too much or too little trimming,
too MUCH trimming is the worst:  a reader can always skip *past* the
"too much text", but they can NOT read what's not there.

Dan Moyer

c.k.lester wrote:
> 
> Fernando Bauer wrote:
> > Al Getz wrote:
> 
> You guys need to trim your posts better! That's a lot of quoted text just
> for a few lines of response. :)

34. Re: Dimension of sequences

• Posted by Dan Moyer <danielmoyer at prod??y.net> Sep 21, 2007
• 622 views

Derek Parnell wrote:
> 
> response
> full
> for
> below
> see
> 
> > c.k.lester wrote:
> > > 
> > > Fernando Bauer wrote:
> > > > Al Getz wrote:
> > > 
> > > You guys need to trim your posts better! ...
> 
> Dan Moyer wrote:
> > I think that of the two possibilities, too much or too little trimming, ...
> 
> However, there is actually a third possibility, that of appropriate trimming.
> Getting rid of stuff that is no longer in context is a good thing. 

Agreed.

> 
> Also, top posting is such an uncomfortable style of responding, no?

Yep, you're right, sorry, I forgot 

Dan

> -- 
> Derek Parnell
> Melbourne, Australia
> Skype name: derek.j.parnell

36. Re: Dimension of sequences

• Posted by Igor Kachan <kinz at peter?ink?ru> Sep 24, 2007
• 659 views

Hello Fernando!

Fernando Bauer wrote:
> 
> Hello Igor!
> 
> Igor Kachan wrote:
> >

[snip]
 
> > 
> > See please:
> > }}}
<eucode>
> > global function MaxDepth(object o)
> >     integer n, x
> >     if atom(o) then
> >        return 0
> >     else
> >        n = 0
> > 	for i = 1 to length(o) do
> > 	    x = MaxDepth(o[i])
> > 	    if x > n then
> > 		n = x
> > 	    end if
> > 	end for
> > 	return n + 1
> >     end if
> > end function -- it is function by Ricardo Forno, genfunc.e lib 
> > 
> > ? MaxDepth({{{{}}},{},{{},{{{{{{{{{{{{}}}}}}}}}}}}},{{{{{{{{}}}}}}}}})
> > ? MaxDepth({{{{1}}},{1},{{1},{{{{{{{{{{{{1}}}}}}}}}}}}},{{{{{{{{1}}}}}}}}})
> > </eucode>
{{{
 
> > The maximum depth is 14, all right.
> > Count please the '{' signs, opening the deepest sequence,
> > to be sure.
> > It seems to be a good parameter for description of a sequence.
> > Now a sequence has its own second dimension, and may be
> > considered as some 2-dimensional object, which can contain
> > description of any-dimentional real and unreal objects -
> > vectors, matrixes, tensors, lists, arrays, trees, books,
> > images ... etc, etc.
> > Sequence itself really has two own dimensions - length and depth,
> > I think now. So attempts to give it some *single* dimension
> > may be not very productive.
> 
> Just to make clear, this concept of 2-dimensional object, or dimensionality
> 2 as you define, is a higher abstract concept, and is not my original
> definition
> of "dimension" which is your Depth.

I do not think that it is more abstract, it is very clear, if
you'll just draw a sequence as the pretti_print() draws or as
it is drawn below? for example:

?
MaxDepth({{{{1}}},{1},{{1},{{{{{{{{{{{1,{1},1}}}}}}}}}}}},1,{{{{{{{{1}}}}}}}}})
-- --{                                                                   }
--    {     },{1},{                                 },1,{               }
--     {   }       {1},{                           }     {             }
--      {1}             {                         }       {           }
--                       {                       }         {         }
--                        {                     }           {       }
--                         {                   }             {     }
--                          {                 }               {   }
--                           {               }                 {1}
--                            {             }
--                             {           }
--                              {         }
--                               {1,   ,1}
--                                  {1}

or:

-- --{                         }
--   { },{1},{           },1,{ }
--   { }     {1},{       }   { }
--   {1}         {       }   { }
--               {       }   { }
--               {       }   { }
--               {       }   { }
--               {       }   { }
--               {       }   {1}
--               {       }
--               {       }
--               {       }
--               {1,   ,1}
--                  {1}

It is something like to just icicles or stalactites.
Did you see the icicles in Brasilia?   

And these dimensions are very demonstative, not abstract at all.
They are just dimensions of some sheet of paper, needed to draw
this graphical representation of some sequence.

But you'll need not only the *maximum* depth, but also the
*maximum* length. The maximum length can be on any depth,
not only on depth 1. The function for calculating of
maximum length may be very simple.

[snip]
 
> > > > Then, I do not think now that the 'rectangular' word is very
> > > > good for your purpose, maybe, 'regular', as some short
> > > > for 'regularly nested', is better.
> > > > 
> > > Ok. But I think "rectangular" is sligthly more precise than "regular",
> > > since
> > > I could think that the following sequence is regular:
> > > repeat({1,{1,1}},n) where n is any integer number. And that sequence is
> > > NRS.
> > > Well, this is only a definition problem.
> > 
> > Ricardo names these sequences as 'simple sequences'.
> 
> No. Ricardo defines "simple sequences" as "sequences formed only by atoms"
> (depth=1).
> The sequence repeat({1,{1,1}},n) has sub-sequences therefore it's not a simple
> sequence.
> A simple sequence is always a RS, but not always a RS is a simple sequence.

Ok, no matter, the Ricardo's function works properly with any
sequence, so if it is "simple" or it is "complicated" really
is not a question.

> > OK, now we can calculate that Depth (good, let's name the maximum depth
> > as Depth) with Ricardo's function.
> But then think about this (bad) analogy:
> If someone ask you about the depth of the Pacific Ocean, then you would say
> the maximum depth of it, since you have defined this way. However we know that
> its depth depends on the place.
> (notice that this depth is not our discussed concept of Depth).

Good analogy, not bad, I always can say that the depth of the Pacific Ocean
is *down to about* 11000 meters.

But if I'll know the Depth of the Pacific Ocean, I do know the
dimensions of a sheet of paper to draw the depth profiles of
that ocean.

Regards,
Igor Kachan
kinz at peterlink.ru

38. Re: Dimension of sequences

• Posted by Fernando Bauer <fmbauer at hot?ail.c?m> Sep 25, 2007
• 626 views

Hello Igor!

Igor Kachan wrote:
> 
> Hello Fernando!
> 
> Fernando Bauer wrote:
> > 
> > Hello Igor!
> > 
> > Igor Kachan wrote:
> > >
> 
> [snip]
>  
> > > 
> > > See please:
> > > }}}
<eucode>
> > > global function MaxDepth(object o)
> > >     integer n, x
> > >     if atom(o) then
> > >        return 0
> > >     else
> > >        n = 0
> > > 	for i = 1 to length(o) do
> > > 	    x = MaxDepth(o[i])
> > > 	    if x > n then
> > > 		n = x
> > > 	    end if
> > > 	end for
> > > 	return n + 1
> > >     end if
> > > end function -- it is function by Ricardo Forno, genfunc.e lib 
> > > 
> > > ? MaxDepth({{{{}}},{},{{},{{{{{{{{{{{{}}}}}}}}}}}}},{{{{{{{{}}}}}}}}})
> > > ?
> > > MaxDepth({{{{1}}},{1},{{1},{{{{{{{{{{{{1}}}}}}}}}}}}},{{{{{{{{1}}}}}}}}})
> > > </eucode>
{{{
 
> > > The maximum depth is 14, all right.
> > > Count please the '{' signs, opening the deepest sequence,
> > > to be sure.
> > > It seems to be a good parameter for description of a sequence.
> > > Now a sequence has its own second dimension, and may be
> > > considered as some 2-dimensional object, which can contain
> > > description of any-dimentional real and unreal objects -
> > > vectors, matrixes, tensors, lists, arrays, trees, books,
> > > images ... etc, etc.
> > > Sequence itself really has two own dimensions - length and depth,
> > > I think now. So attempts to give it some *single* dimension
> > > may be not very productive.
Ok. Thanks to your fantastic picture below, now I understand your concept!

> > 
> > Just to make clear, this concept of 2-dimensional object, or dimensionality
> > 2 as you define, is a higher abstract concept, and is not my original
> > definition
> > of "dimension" which is your Depth.
> 
> I do not think that it is more abstract, it is very clear, if
> you'll just draw a sequence as the pretti_print() draws or as
> it is drawn below? for example:
> }}}
<eucode>
> ?
> MaxDepth({{{{1}}},{1},{{1},{{{{{{{{{{{1,{1},1}}}}}}}}}}}},1,{{{{{{{{1}}}}}}}}})
> -- --{                                                                   }
> --    {     },{1},{                                 },1,{               }
> --     {   }       {1},{                           }     {             }
> --      {1}             {                         }       {           }
> --                       {                       }         {         }
> --                        {                     }           {       }
> --                         {                   }             {     }
> --                          {                 }               {   }
> --                           {               }                 {1}
> --                            {             }
> --                             {           }
> --                              {         }
> --                               {1,   ,1}
> --                                  {1}
> 
> or:
> 
> -- --{                         }
> --   { },{1},{           },1,{ }
> --   { }     {1},{       }   { }
> --   {1}         {       }   { }
> --               {       }   { }
> --               {       }   { }
> --               {       }   { }
> --               {       }   { }
> --               {       }   {1}
> --               {       }
> --               {       }
> --               {       }
> --               {1,   ,1}
> --                  {1}
> </eucode>
{{{

> 
Very elucidative pictures! Now, I understand your concept of dimensionality 2.
Maybe we could call it Igor's plane? Kachan's plane?
And, as you said, it can represent all the sequences!

Now, following your reasoning:
1) a sequence can contain description of any-dimensional real and unreal objects
- vectors, matrixes, tensors, lists, arrays, trees, books,images ... etc, etc.
2) all sequences can be represented in the Kachan's plane.
3) Kachan's plane can be represented in a sheet of paper.
Then:
- Any-dimensional real and unreal objects can be represented in a sheet of
  paper!
or maybe:
  - Universe can be represented in a sheet of paper!!

Now, I can't even imagine something as abstract as this!! 
(abstract= "the concentrated essence of a larger whole")
This concept is so wide-ranging that I don't know if it has some use, since all
the sequences have the same dimension 2.

> It is something like to just icicles or stalactites.
> Did you see the icicles in Brasilia?   
So far, I didn't. :)

> 
> And these dimensions are very demonstative, not abstract at all.
> They are just dimensions of some sheet of paper, needed to draw
> this graphical representation of some sequence.
Ok. Your picture shows this very well!

[snipped]

>  
> Regards,
> Igor Kachan
> kinz at peterlink.ru

I appreciated very much your view about this.
I still have another approach about the "dimension of sequences", but before I
post it, I want to implement it in Euphoria.
The line of reasoning is: a NRS is a set of RS where each one has a known
integer dimension. Then, the dimension of a NRS is a sequence formed by that
dimensions.

Thanks,
  Fernando

40. Re: Dimension of sequences

• Posted by Ricardo Forno <ricardoforno at tut?p?a.com> Sep 25, 2007
• 659 views

<snip>
> >   - Universe can be represented in a sheet of paper!!
> 
> It is not that big news, all our old good books consist
> just of sheets of paper, and they represent not only
> Universe, but much more, sometimes.    
> Same about sequences.
<snip>

Universe is so small, compared against other things...
For example: How many different pictures may be represented on the screen you
have in front of your eyes?
This code tryied to show this number, but it failed. So I modified it:

atom x, z
-- x = power(256.0 * 256.0 * 256.0, 1024.0 * 768.0)
-- printf(1, "%g\n", x)
z = log(256.0 * 256.0 * 256.0) * 1024.0 * 768.0
printf(1, "%g\n", z)

Best regards.

42. Re: Dimension of sequences

• Posted by Igor Kachan <kinz at p?terl?nk.ru> Oct 14, 2007
• 691 views

Fernando Bauer wrote:

> 
> Hello Igor!
> 
> [snipped]
> > [snipped]

Hello Fernando!

> > OK, interesting, the specific structure of a sequence is involved.
> > Let's wait.
> > 
> 
> Here is my current implementation of "dimension of sequences" involving its
> structure:
> }}}
<eucode>
> function Dimension(object o)
> integer ini
> object base
> sequence s
> 
>    if atom(o) then return 0 end if
>    if length(o)=0 then return 1 end if
>    if length(o)=1 then return 1+Dimension(o[1]) end if   
>    s = {}
>    ini = 1
>    o = o = o
>    base = o[1]
>    for i = 2 to length(o) do
>       if not equal(base, o[i]) then
>          if i > ini+1 then 
>             s = append(s,1+Dimension(base))
>          else
>             s = append(s,Dimension(base))
>          end if
>          base = o[i]
>          ini = i
>       end if
>    end for
>    if ini = 1          then return 1+Dimension(o[1]) end if
>    if ini = length(o)  then return append(s,Dimension(o[ini])) end if
>    return append(s,1+Dimension(o[ini]))
> end function
> 
> procedure ShowDimension(object o)
> puts(1,"Dimension of ")
> print(1, o)
> puts(1," = ")
> print(1,Dimension(o))
> puts(1,"\n")
> end procedure
> 
> object obj
> -- Create an object formed by rectangular sequences with incremental dimension
> obj = {1}
> for i = 2 to 4 do
>    obj = append(obj, repeat(obj[i-1],1))
> end for
> 
> for i=1 to length(obj) do
>    ShowDimension(obj[i])
> end for
> 
> ShowDimension(1)
> ShowDimension({})
> ShowDimension({2})
> ShowDimension({3,4})
> ShowDimension({5,{6}})
> ShowDimension({7,{8},9})
> ShowDimension({10,{11},{12}})
> ShowDimension({13,{14,15}})
> ShowDimension({{16,17},{18,19}})
> ShowDimension({{20,21},{22}})
> ShowDimension({23,{24,{{25,26},{27,28}}}})
> ShowDimension({1,{1},{1,1},{1,1,1},{1,1},{1},1})
> ShowDimension({1,{1},{1},{1,1},{1,1},{{1,1},{1,1},{1,1}}})
> ShowDimension({obj[1],obj[2],obj[3],obj[4]})
> ShowDimension({obj[4],obj[3],obj[2],obj[1]})
> ShowDimension({obj[1],{obj[2],{obj[3],{obj[4]}}},obj[1]})
> 
> for i = 1 to length(obj) do
>    for j = 1 to length(obj) do
>       for k = 1 to length(obj) do
>          ShowDimension({obj[i], obj[j], obj[k]})
>       end for
>    end for
> end for
> </eucode>
{{{

> 
> Thanks to everyone who has contributed to this topic!

I'm sorry I was off line all those days, and will try your
new current implementation for "dimension" next week, if I
have some spare time for it.

But your initial question about dimension seems to be answered
already, you do have your own vision and solution of the problem
about "NRS dimension" now, so let's finish this thread, OK?

Regards,
Igor Kachan
kinz at peterlink.ru