Hi Linas,
Your general comments are all sensible; but by CCG I mean a specific
approach to syntax and semantics as elaborated by the literature
linked from http://groups.inf.ed.ac.uk/ccg/ .... There might indeed
be other ways to model natural language using category theory but
that's not what I was talking about...
For instance, in the CCG approach, very roughly/crudely (just to give
you a quick flavor)
--- Determiners may be viewed as functions from the category of noun
phrases into the category of noun phrases
-- Verb phrases may be viewed as functions from the category of noun
phrases into the category of sentences
A "parse" of a sentence in CCG basically consists of a proof that the
sequence of words in question has a type that reduces to the type
"Sentence" ...
So to map link grammar in to CCG of this sort, I suggest, one has to
come up with **sets of link combinations** corresponding to categories
like "noun phrase", "verb phrase", "determiner", etc. Then one can
construct types corresponding to mappings between these categories,
mapping between mappings between these categories, etc. These would
be function types (often higher order function types), not the same as
the link parser link types, though derived therefrom.
I think we can have a better conversation about this at some future
point when you read the basic papers on CCG.... But I realize you're
busy with other more urgent matters at the moment ;)
-- Ben
On Tue, Jun 24, 2014 at 6:47 AM, Linas Vepstas <[log in to unmask]> wrote:
> Well, lets start by recalling that category theory is just the theory of
> arrows pointing at types. Since dependency grammars have arrows pointing at
> types ...
>
> The word "categorial" in "categorial grammar" refers to category theory.
> Wikipedia (see below) reminds me that categorial grammar was invented by
> Adjukiewicz, a category theorist.
>
> The "combinatory" presumably means that CCG works only for those categories
> that have combinators (I'm not clear which categories those might be,
> because I don't know if combinators can be extended beyond lambda calculus.
> Recall that (simply typed) lambda calculus is the "internal language" of the
> cartesian closed categories.
>
> Recall that all monoidal categories have an "internal language". The
> categories are the "types" of the language. Thus, we have "type theory"...
>
> I'm not sure which categories pair with which internal languages. I know
> that "linear logic" aka "quantum logic" pairs with the closed categories.
> (recall tensor calculus is not cartesian-closed)
>
> If you read Stedman's intro article, you'll see references to Adjukiewicz,
> Lambek, who were category theorists, not linguists.
>
> Hmm ...wikipedia credits https://en.wikipedia.org/wiki/Kazimierz_Ajdukiewicz
> with the invention of categorial grammar,
>
> I quote wikipedia below:
> Historical notes[edit]
> The basic ideas of categorial grammar date from work by Kazimierz
> Ajdukiewicz (in 1935) and Yehoshua Bar-Hillel (in 1953). In 1958, Joachim
> Lambek introduced a syntactic calculus that formalized the function type
> constructors along with various rules for the combination of functions. This
> calculus is a forerunner of linear logic in that it is a substructural
> logic. Montague grammar uses an ad hoc syntactic system for English that is
> based on the principles of categorial grammar. Although Montague's work is
> sometimes regarded as syntactically uninteresting, it helped to bolster
> interest in categorial grammar by associating it with a highly successful
> formal treatment of natural language semantics. More recent work in
> categorial grammar has focused on the improvement of syntactic coverage. One
> formalism which has received considerable attention in recent years is
> Steedman and Szabolcsi's combinatory categorial grammar which builds on
> combinatory logic invented by Moses Schönfinkel and Haskell Curry.
>
> ===========
> So, my gut sense is that the reason that you can convert english-language
> CCG into quasi-first-order logic is because of curry-howard correspondance.
> The details remain vague to me.
>
> I've studied enough topos theory to know that curry-howard correspondance
> extends to a more general setting than proofs/programs.
>
> What I don't have is the big giant atlas that says "this category has that
> internal language, which curry-howard-corresponds to this-n-such logic and
> that-n-such type system."
>
> What I do know is the the links of link-grammar are types in the sense of
> type theory, and we can make a (weak) isomoprhism between link-grammar and
> at least some forms of categorial grammar. However, I don't know exactly
> which one(s)... nor do I know what it curry-howard-corresponds to. I am
> very easily distracted in the search for such answers. I can glimpse the big
> picture, but haven't mapped out the details, of which there are vast
> quantities of...
>
> -- Linas
>
>
>
> On 22 June 2014 20:28, Ben Goertzel <[log in to unmask]> wrote:
>>
>> I don't think that New Scientist paper is about CCG, is it? It's
>> about a different approach involving category theory and grammar...
>>
>> For CCG, check out
>>
>> http://groups.inf.ed.ac.uk/ccg/
>>
>> including links to their OSS code,
>>
>> http://groups.inf.ed.ac.uk/ccg/software.html
>>
>> I will write out some specific examples of the mapping I mean when I
>> find the time, which will likely be in a couple months...
>>
>> -- Ben
>>
>> On Mon, Jun 23, 2014 at 5:30 AM, Linas Vepstas <[log in to unmask]>
>> wrote:
>> > Hmm.
>> >
>> > Well, I've stated in several forums that I believe that LG is
>> > more-or-less
>> > isomorphic to CCG (including in the language-learning paper). Perhaps
>> > the
>> > most obvious way to see this is in the figure at the bottom of page 3 of
>> >
>> > http://www.cs.ox.ac.uk/people/bob.coecke/NewScientist.pdf
>> >
>> > Note BTW, that diagram is not an 'accident'; physicists have a detailed
>> > diagrammatic calculus with specific rules on how to draw appropriate
>> > formulas as diagrams; the fact that the diagram ends up looking
>> > essentially
>> > like link-grammar is not an accident either .. there are multiple papers
>> > from Coecke as well as others including John Baez, that detail the
>> > diagrammatic approach. Its also used in knot theory and string theory,
>> > since these all have the same kind of structure. (See
>> > http://math.ucr.edu/home/baez/rosetta.pdf if interested.)
>> >
>> > Anyway, I did not really understand the steps 1..4 below, but that's OK.
>> > I
>> > think the easiest way to do this would be to work through some specific
>> > examples. That is, pick a specific CCG you want to use (there are
>> > assorted
>> > notational differences between them, and other quirks) and then get some
>> > sentences parsed in that CCG (e.g. from a journal article) If you sit
>> > them
>> > next to the corresponding LG parse, I think it will slowly become clear
>> > exactly how to write some code that does the conversion from one to the
>> > other.
>> >
>> > If there is some specific, dominant CCG that is in wide-spread use (I
>> > don't
>> > know) then perhaps writing up the formal correspondence would be
>> > worthwhile,
>> > and I'd volunteer for that.
>> >
>> > --linas
>> >
>> >
>> >
>> > On 22 June 2014 09:32, Ben Goertzel <[log in to unmask]> wrote:
>> >>
>> >> Hmmm...
>> >>
>> >> I was thinking a bit about how to map link / word grammar [the two are
>> >> almost the same] into Combinatory Categorial Grammar (CCG)...
>> >>
>> >> Why would one want to do this? Because, among other reasons,
>> >>
>> >> -- CCG presents syntax in a way that maps very naturally into
>> >> semantics, in a sense going beyond link/word grammar. Specifically it
>> >> embodies the type structure of syntax/semantics in an interesting way,
>> >> e.g. noting that a determiner is a function mapping a noun into a noun
>> >> phrase; or a verb is a function mapping a noun phrase into a sentence;
>> >> etc. The (often higher-order) type relationships it identifies seem
>> >> often to span the syntax and semantics level, which is interesting.
>> >>
>> >> -- I have a suspicion that link / word grammar is more easily
>> >> learnable via unsupservised statistical corpus (or embodied
>> >> experience) analysis, as Linas and I have conjectured in
>> >> http://arxiv.org/abs/1401.3372
>> >>
>> >> ...
>> >>
>> >> The fact that such a mapping exists is clear, in that there is a weak
>> >> equivalence between TAG (tree adjoining grammar) and CCG, and also
>> >> between TAG and dependency grammars (which are formally similar to
>> >> link/word grammar). But I want to explore a mapping that is
>> >> intuitively straightforward and practically usable.
>> >>
>> >> Part of my goal here is to think about if what we're now doing in
>> >> OpenCog via "RelEx2Logic" -- formulating rules mapping link parser
>> >> output (well, RelEx outputs derived therefrom) into logic formulas,
>> >> http://wiki.opencog.org/w/RelEx2Logic_Rules -- could be done in a
>> >> fully automated way, without need for hand-coded mapping rules, via
>> >> inferring a CCG that would contain much of the logical semantics
>> >> embedded in it...
>> >>
>> >> Here is my thinking... (phrased in terms of link grammar, but applies
>> >> to word grammar too)
>> >>
>> >> 1)
>> >> Define a "linkage" as a connected graph of (typed link parser) links,
>> >> which can exist within a legal parse of some sentence
>> >>
>> >> 2)
>> >> Define the "lexicon entry" for a linkage L as a
>> >> link-grammar-dictionary-style disjunct, expressing which links L can
>> >> or must have to the left or to the right.... (For a linkage L to have
>> >> a link l1 to the right, means that some node within L has a link l1 to
>> >> the right, and the right target of l1 is to the right of every word in
>> >> L.
>> >>
>> >> 3)
>> >> Define a linkage class, as a set of linkages whose lexicon entries all
>> >> contain some non-trivial disjunct D as a sub-disjunct. I.e. D
>> >> represents the syntactic commonality between all the linkages in the
>> >> class.
>> >>
>> >> 4)
>> >> Assign (perhaps higher-order) functions to linkage classes as follows.
>> >> Let fL denote the function corresponding to a particular linkage, and
>> >> let fL1 denote the class of functions corresponding to linkages in
>> >> class L1.
>> >>
>> >> If it happens that: connecting a linkage of class L1 to a linkage of
>> >> class L2, from the left, with a link of type l1, results in a
>> >> composite linkage of class L3 ... then we may say
>> >>
>> >> fL1 : fL2 --> fL3
>> >>
>> >> That is: the categories of CCG are being modeled as linkage classes
>> >> (classes of linkages with the same linking requirements and options to
>> >> the left and right); and the function application operation is being
>> >> modeled as linkage itself (between linkage structures).
>> >>
>> >> ...
>> >>
>> >> In this way, from a corpus parsed via link grammar, one could
>> >> potentially infer a CCG automatically. What this CCG would look like
>> >> remains to be seen. In theory, then the inferred CCG could be
>> >> mapped directly into higher-order logic relations representing
>> >> semantics. How much of the semantic mapping captured by RelEx2Logic
>> >> would get captured in this way, also remains to be seen...
>> >>
>> >> Apologies for the compact/not-so-expository nature of the above... if
>> >> time permits I will write the idea up in more detail at some point...
>> >>
>> >> -- Ben
>> >>
>> >>
>> >>
>> >>
>> >>
>> >>
>> >>
>> >>
>> >>
>> >> --
>> >> Ben Goertzel, PhD
>> >> http://goertzel.org
>> >>
>> >> "In an insane world, the sane man must appear to be insane". -- Capt.
>> >> James T. Kirk
>> >>
>> >> "Emancipate yourself from mental slavery / None but ourselves can free
>> >> our minds" -- Robert Nesta Marley
>> >
>> >
>> > --
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>>
>>
>>
>> --
>> Ben Goertzel, PhD
>> http://goertzel.org
>>
>> "In an insane world, the sane man must appear to be insane". -- Capt.
>> James T. Kirk
>>
>> "Emancipate yourself from mental slavery / None but ourselves can free
>> our minds" -- Robert Nesta Marley
>
>
> --
> You received this message because you are subscribed to the Google Groups
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> To unsubscribe from this group and stop receiving emails from it, send an
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--
Ben Goertzel, PhD
http://goertzel.org
"In an insane world, the sane man must appear to be insane". -- Capt.
James T. Kirk
"Emancipate yourself from mental slavery / None but ourselves can free
our minds" -- Robert Nesta Marley
|
