On Sun, Jul 9, 2017 at 11:14 PM, Bob Coecke <[email protected]> wrote:
> Thanks for the response. Some later date, I would like to talk more. Yes,
> its the non-Cartesian-ness of it all that I think I now know how to
> handle. Very briefly: in the original papers on link-grammar (1991-1993),
> they explained it by drawing pictures of jigsaw puzzle-pieces. One of the
> pop-sci reports of your work has a diagram of ... jigsaw puzzle pieces. I
> slapped my forehead. The current realization is that the "jigsaw pieces"
> are exactly the same thing as the local sections of a sheaf, and so I am
> busy data-mining those.
>
>
> Interesting, need to understand more there...
>
On the one hand, I feel like I'm giving away my best secret. On the other
hand, its "obvious", and you will recognize it immediately.
I'm repeating again, because I really want the other readers on the mailing
list to grok the ideas.
By example/analogy:
A common, almost canonical way to describe a graph is to list all of the
vertexes, and all of the edges. This gives a "global" description of the
graph. You get all of it, in one big gulp.
Another way to describe it, uncommon but perfectly valid, is to list the
vertexes, and, associated to each vertex, a set of edges coming off of it.
(it is convenient, sometimes, to describe "half of an edge", to be used for
this purpose).
Each pair (vertex, {set of edges originating/terminating on that vertex})
is a local section of the graph. Glue these together, you get the whole
graph.
In link-grammar, these pairs are called "disjuncts" and are the basic
entries in the LG dictionary.
In the puzzle-piece analogy, these pairs are puzzle-pieces, and snapping
them together in a legal fashion is the act of parsing a sentence.
Each "tab" on a jigsaw puzzle piece is called a "connector" in LG.
In CCG, each "connector" has (for example) the form of NP\S or S/NP\VP and
so on. In a pregroup grammar, I guess you would write it as "sat cat_L^-1
mat_R^-1" or something like that (for the verb in "the cat sat on the
mat"). In LG, these rather verbose expressions are replaced by a simple
label, called the "link type". As it happens, the link type is more-or-less
the same thing as a type in type theory.
So, really, for language, I should write (word, {partially-ordered set of
connectors}) instead of (vertex, {set of edges originating/terminating on
that vertex})
Anyway, each pair (vertex, {set of edges originating/terminating on that
vertex}) can be recognized as a section, in the sense of algebraic
topology, where you have all this machinery for when you can glue them
together, & etc. In sheaves, I guess you can call them sections, and a
dictionary entry would then be a germ or a stalk, or thereabouts. Although
this analogy/terminology holds, its perhaps a bit over-blown, as I have not
yet seen any reason to push hard on it; I have not yet had any insight that
would require any of the fancier machinery of algebraic topology. So its a
casual observation.
Similarly, you could call this (vertex, {set of edges
originating/terminating on that vertex}) thingy a "diagram" in category
theory, but that isn't quite the suitable analogy. It could be forced into
working but would be clunky. (so e.g. verbs are kind-of-ish like spans but
that's goofy because there's no obvious limit to go with that. or no
obvious reason to even invoke to the concept of a limit. So why am I doing
so now? I dunno.)
So, roughly that's the correspondence and the general idea.
--linas
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