Algebraic view of word grammar...

[Ben Goertzel]

Some quasi-mathematical-linguistic musings...

Reviewing a bunch of familiar stuff in my mind, I’m trying to take an
algebraic view of Word Grammar…. This is presumably equivalent to
pregroup grammar under appropriate restrictions but it’s maybe a more
linguistics-ish way to look at it…

Consider a set of words interlinked by ordered dependency links (so
each link has a head corresponding to the parent, and a tail
corresponding to the child). For reasons including those to be
described below, it is useful to consider these dependency links as
typed.

Word Grammar tells us how, given such a set of words and links, to
construct a set of additional (ordered) “landmark links” between the
words.

The rules thereof are as follows…

The parent is the “landmark” of the child.

In some cases a word may have more than one parent. In this case, the
rule is that the landmark is the one that is superordinate to all the
other parents. In the rare case that two words are each others’
parents, then either may serve as the landmark.

A Before landmark is one where the child is before the parent; an
After landmark is one where the child is after the parent.

Rules of “landmark transitivity” are:

* Subordinate transitivity: If A is a Before/After landmark for B, and
B is some kind of landmark for C, then A is a Before/After landmark
for C

* Sister transitivity: If A is a landmark for B, and A is also a
landmark for C, then B is also a landmark for C

* Proxy links: For certain special types T of dependency link, if A
and B are joined by a link of type T, then if A is a landmark for C, B
is also a landmark for C

The “head” of a set of words is a root of the digraph of landmark
links in that set of words

Restricting attention momentarily to the case of phrases with only one
head, one way to look at this is: The landmark transitivity rules tell
what happens when we carry out operations such as

P1 +_T P2

(putting a dependency link between the head of P1 and the head of P2,
with P1 to the left and being at the child end of the link), or

P1 +_T’ P2

(putting a dependency link between the head of P1 and the head of P2,
with P1 to the left and being at the parent end of the link)

noting that this operation is not commutative, and also that the
dependency link may have a type T which may be important (e.g. due to
the existence of proxy links).

These operations at on the space of graphs whose nodes are words and
whose linked are either typed, ordered dependency links, or ordered
landmark links; and for which the landmark links are consistent
according to the rules laid out above.

The landmark transitivity rules tell where the landmark links go in
the combined structures P1 +_T P2 and P1 +_T’ P2, in a way that will
maintain the consistency of the rules regarding landmarks

It is not hard to see that, according to the rules of landmark
transitivity, the free algebra formed by the multiple operations +_T,
+_T’ is distributive, associative, and noncommutative

There is one hole in the above; we haven’t dealt with cases where a
phrase has more than one head, because two words are each others’
parents. The easiest way to look at this formally seems to be to
introduce operations +_Tij, where

P1 +_Tij P2

builds a dependency link of type T from the i’th head of P1 to the
j’th head of P2. We would also have operations of the form

P1 +_Tij’ P2

We can then see that the free algebra formed by the multiple
operations +_T, +_T’, +_Tij, +_T’ij is distributive, associative, and
noncommutative...

A next step would be to make all these links (represented here by +
operators) probabilistically weighted. But I'm out of time just now
so that will be saved for later ;) ...

ben



--
Ben Goertzel, PhD
http://goertzel.org

"I am God! I am nothing, I'm play, I am freedom, I am life. I am the
boundary, I am the peak." -- Alexander Scriabin

[Adam Gwizdala]

this looks to me exactly like a definition for a description logic? except with some constraints on how you define the classes, properties and individuals so that they meet only the conditions of Word Grammar?

eg.

--subordinate trans. looks like: 'x hasParent y' paired with respective inversion 'y isParentOf x'

--sister trans. looks like a subproperty chain: 'x hasParent y isParentOf z'

--proxy links look like another instance of a property and again a subproperty chain eg. 'x hasProxyLinkType1 y hasParent some z'

--the head would a property defined to be the union of valid transitive subproperty chains (including the proxy link rule) to reach the root node?

Your problem where words may be parents of each other, is a cyclic ontology feature, which messes up uniform interpolation eg. forgetting  eg. can't represent it finitely, eg. non-terminating. Your operation solution looks similar to the fixpoint operators in the literature, but I'm getting out of my depth there

So to my reckoning, you definition matches:

SHIQ description logic where...
ontology model where roles/individuals are words in the phrase(s)
you have two properties: landmarks, proxy. where transitivity/inversions are allowed in certain cases
you dont use classes/subclasses because you don't need them

As for your weighted links... they don't fit anywhere :-)

[Ben Goertzel]

The mapping into description logic looks to make sense...

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[Richard Hudson]

Dear Ben,

Thanks very much for your musings, and sorry for the long delay at my end. I'm sure it's time to sort out the formal properties of WG structures, and I'm equally sure that you're better at this kind of thinking than I am. What I can report is a major rethink on WG word order rules, which is summarised in an article just published in the Journal of Linguistics. A not-quite-final version is at http://dickhudson.com/papers/#2017 , and it's about pied-piping, which is another exception to  the simple picture (as in "the book in which I saw it"). The exceptional feature here is that "in" takes its position from "which", in spite of the fact that "which" depends on "in". Here's what I now think:

1. We need to distinguish between constraints on word order and actual order. The mechanism of landmarks is for constraints, but actual order is much more concrete, with a 'next' link from one word to the next. The difference is clearest in a language with free order:  every order of sister dependencies is grammatical, so there's no need for a distinction between before/after, but 'next' links are still essential because we're using a network, and networks don't have a left-right dimension. If we use > as the 'next' link, we'd distinguish John>snores from snores>John, but they would both have the same 'landmark' structure. (The 'landmark' system is still needed even in a free-order language, because I gather that clauses never scramble freely; so in "She warned Mary John snores", the landmark for John would be snores, not warned.)
2. The distinction between before/after landmarks doesn't work in default inheritance because there's no mechanism whereby 'before' can override 'after' (or vice versa). In my 2010 book I introduced a new basic relation 'or' specially to handle this choice, but I couldn't think of any other good use for 'or', so it was very very suspect. Instead, I now have a much simpler solution: every word (indeed, every object in the mental universe) has a position, and (like every other property) one position can override another; and it's positions, not words, that are related by 'before' and 'after' links, so 'landmark transitivity' is replaced by 'position transitivity': if A is-before B and B is-before C, then A is-before C, etc. For example, suppose you want to say that a dependent D follows its parent P:
1. D's landmark is P.
2. Each word has a position pos(x).
3. pos(D) is-after pos(P).
   
3. As before, the landmark of a word W is typically its parent, but
1. if W has more than one parent linked by a chain of dependencies p1-p2..., where pi+1 depends on pi, the landmark is typically p1
1. this defines 'raising', but 'lowering' is also possible, as in German partial VP fronting (eine Concorde gelandet ist hier nie = a Concorde landed has here never = a C has never landed here).
2. in a pied-piping construction, W may have a 'pipee' (e.g. "in" is the pipee of "which") which inherits the landmark relations of W.
   
4. Returning to #1, the constraints apply to the actual order:
   
1. 'Position transitivity' applies so that a word's position has the same relation as its landmark's position to that of the landmark's landmark.
   
2. Given a pair of adjacent words W1 > W2, each word has a position: pos(W1) is-before pos(W2).
3. This order is grammatical provided it's compatible with the constraints and position transitivity.
   
5. I don't think you need to bring 'proxies' into the picture, because they essentially add extra dependencies and only affect word order indirectly.
6. Your 'sister transitivity' sounds more like a statement about actual order than about constraints. There may be a constraint on the order of sisters (e.g. I gave the students high marks vs: *I gave high marks the students), but it's not necessary, whereas they obviously have to be in some actual order. (Sister constraints are an outstanding gap in the theory, I'm afraid.)
7. Mutual dependency (your phrases with two heads) is already covered, I think, because both dependencies lead to the same actual ordering.
   

I hope that covers it. That was very hard work for a non-mathematician!

Best wishes, Dick

-- 
Richard Hudson (dickhudson.com)

[Ben Goertzel]

Wow, Dick, thanks a lot for the detailed and thoughtful response!

Now I'm afraid it will be my turn to delay a bit, as I'll be traveling
and such for the next week and a half, and may not get time to dig
into this again till my return. But I'm eager to figure out how to
adapt my preliminary formalization to account for the added subtleties
you note ;)

-- Ben

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