Combinatory Categorial Grammar is equivalent to Link Grammar

[Linas Vepstas]

The relationship between different grammar formalisms is almost always cloudy and opaque. Sometimes, it's how the grammar is formalized, sometimes, it's the notation.

In the case of Combinatory Categorial Grammar, it's the notation. After a minor, almost trivial restructuring of the notation, it can be seen to be equivalent to Link Grammar.  See attached PDF.

Thanks to Adam Vandervorst, who raised this in an OpenCog Discord chat discussion. I've long known of this equivalence, having sensed it by gut-feel. However, having to actually write it out, in detail, to make it convincing to others, was ... educational.

-- Linas

--

Patrick: Are they laughing at us?

Sponge Bob: No, Patrick, they are laughing next to us.

 

[Linas Vepstas]

Alas, I attached a PDF that contains errors, instead of a URL to a PDF where the latest and greatest version can be found. So -- a mistake is fixed, and additional clarification is provided. The presentation is now much stronger. Here:

https://github.com/opencog/atomspace/raw/master/opencog/sheaf/docs/ccg.pdf

[Ben Goertzel]

Nice work Linas! Indeed this correspondence was always conceptually
clear, but it's good to see it worked out in detail for the first
time...

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--
Ben Goertzel, PhD
b...@goertzel.org

"My humanity is a constant self-overcoming" -- Friedrich Nietzsche

[Linas Vepstas]

Thank you, Ben!

The part I found most interesting was about type-raising being link-crossing. Type-raising seemed very mysterious, at first. And link-crossing in LG seemed impossible until the day I realized that non-planar electrical circuit diagrams are drawn  on flat pieces of paper all the time.

There's a trick: an electric wire arrives, out of the blue, on one side, jumps over, and leaves into the yonder on the other side. Written with types, this has the distinctive signature  W -> T- & W & T+  where the wire to be crossed is W, and the one doing the crossing is T.  It's mysterious only in that one thinks "what the heck is T and where did it come from?" and the answer is "it doesn't matter, T came from somewhere out there, and then it left again, leaving us untouched, as we were before."  Once you see this pattern, it is not hard to spot.

It's also a great way to do long-range coordination in LG. It shows how something distant can affect local behavior: this distant thing just hops over whatever  is in the way, until it arrives in the local area.

-- Linas

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