OpenEuphoria

Re: Dimension of sequences

• Posted by Igor Kachan <kinz at peterlin?.r?> Sep 20, 2007
• 645 views

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
> 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.

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

Regards,
Igor Kachan
kinz at peterlink.ru