Tomorrow we will think about why the last two definitions work, and then continue from there.
Representing Multidimensional Trees v0.3
Posted by kernco at July 14, 2004 06:26 PMThis is looking very nice. A few more ideas:
The last sentence before Def 2 (the bit about n-branching --> binary branching)
might be better placed as a footnote (\footnote{...}) following the first
sentence after Def 1.
First line of section 1.3. There is some cognitive dissonance in use of "term"
in its algebraic sense in the previous paragraph (which I like very much) and
"term" in its linguistic sense here. Maybe "terminology" here instead.
Def 3. Should this be "Local Yield" to distinguish it from the yields we are
about to define? (e.g. 1-d yield of 3-d structure, etc.) This introduces
another word which will have to appear frequently
(x query-replaceyieldlocal yield ), but it avoids having to
change terminology later. Also, this should note that the n-dimensional yield
is always a (n-1)-dimensional tree, which is to say a singleton forest---there
must be exactly one minimum point wrt the (n-1)-dimensional ordering. (This
will give us the base case of the lemma to follow.) Finally, since the yield
in the nth-dimension is an (n-1)-dimensional tree, I think we should be
referring to it as the (n-1)-dimensional yield, as you do in the very next
definition. (Where the yield of an n-dim local tree is an (n-1)-dim yield.)
I don't know if the child structure should be n-dim or (n-1)-dim. Both seem
better in some contexts than the other. In keeping with the (i,d)-forest
notation, I think it should probably be that the child structure rooted in the
yield of an i-dim local tree (which is to say an (i-1)-dimensional local yield)
should be an (i-1)-dimensional child structure.
I suspect that Def 3 and Def 4 should be reversed. A local tree is a tree of
height (i,d)
But I'm not sure we can actually maintain it and still be able to plug in
non-trivial linearly ordered forests in the t_1 position.
Lemma 1, and its corollaries can be moved earlier? Corollary 2 is a clause of
Def. 7, is it not?
Section 2:
This still doesn't have the structure I'm looking for. The idea is:
o we have defined class of mathematical objects
o To actually work with these we need to realize them in concrete form
o For processing, this is ADT realized as C++ class.
o What is data the ADT must store/member vars of class
o What access methods/member functions do we need
o Give a (basic) decl of the class
o For file I/O useful to have linearized form
o Interpret \Sigma as function symbols
o X(t1,...) r/t T(X,...)
o Then linear form is just term algebra generated by the functions symbols
\Sigma from the single ground term ~.
Gotta go,
Jim