Re: Universal grammar

In article <1161246509.794397.244600@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Rob
Freeman" <groups@xxxxxxxxxxxxxxxxxxx> wrote:

I am interested in the parallel you get if you think of a "statement",
or a rule, as an ordering over a set. If ordering A is true (that is, a
valid ordering over the set), but incompatible ordering B is also true,
then A and not A can be true, depending on the truth of B.

This happens all the time in the real world. For instance if you try to
order people w.r.t. height and IQ at the same time.

This does not have anything to do with the Goedel theorems, which arise,
roughly, by the assumption that there are infinities that can be treated
as objects.

What I want to know is what this says about our ability to resolve a
distribution down to some parametrization in terms of rules. Is it
always possible to find a consistent set of rules which completely
describes all the patterns in a distribution?

This is somewhat too diffuse to be possible to be pinned down in
mathematics (see below, though). You might try the newsgroups comp.theory
and comp.ai. Particular the idea how humans learn, and trying to emulate
that in computers seem to be an AI problem - which means it is hard.

This is relevant to language, because it is exactly the problem you get
if you try to learn natural language grammar from distributions of word
associations.

In fact I have read it was inconsistencies of this kind which first led
Chomsky to reject the idea language could be learned from examples and
propose it must be parametrized innately. A "grammar" which because
innate, must be universal. Hence a Universal Grammar, of which the
important property was that it could _not_ be observed in distributions
of word associations.

I think one knows pretty well that although the ability to learn languages
is innate, languages itself are not, as there have been a few cases of
children not getting the chance of learning it early in life, and when
they grow up, they cannot learn it, or very little. If language somehow
was innate, one should have it with the genes. So language is learned, and
one is probably learning a collection of structural patterns, just as with
all else.

In fact, the definition of a formal grammar closely follows this. (The
description I use is in a book by Waite & Goos, "Compiler Construction",
chapter 5, but it rather hard core math. And learning about parsers, there
is the newsgroup comp.compilers - but it does not deal much about parsing
of natural languages. In the Unicode List, "John D. Burger" <john AT mitre
DOT org> mentioned he had been working with natural language processing
for two decades, so perhaps he can give some more inputs.)

Anyway, a formal language L is a subset of the set V*, the set of finite
sequences (or strings) of a set V called the vocabulary (mathematicians
call this the free monoid on the set V). Each member of L is called a
sentence. So already here, one can define a language by just writing down
all legal sentences. (A formal sentence could though be anything legal in
the language, like a whole book, if one so sets the language definition.)
The grammar comes into play only by the attempt to find a more structured
description of L. The problem between learning a language naturally and
finding a formal description of it arises from the inability to read off
how the human brain structures the information it contains.

Then in order to pin down some types of languages, one uses a grammar. A
production is a pair (s, p), where s, p in V*, and is written s -> p (and
some write it s <- p or s: p). Now, if a string m in V* contains s as a
substring, it can be replaced by p to get a new string n, and one then
says that m is directly rewritable into n by the rule s -> p. And if one
has a set P of productions, and a string m is directly rewritable by a
sequence of members of P into n, then one says that m is rewritable into n
by P.

A formal grammar G is then a quadruple (T, N, P, Z), where T is the set of
terminals (or constants), corresponding to the words in a language L, N is
the set of non-terminals, which are grammar symbols (or variables) like
"noun", verb", etc. in natural languages. The vocabulary V is the disjoint
union of T and N. P is a set of productions on V. And Z is the sentence
(or start) symbol. The language L(G) derived as a function of the grammar
G is then the set of members of T* that can be derived by rewriting Z in
V* using the set of productions P.

So, from this, you can see that it is quite easy to define productions and
add them to get a larger language, which answers part of your question.
The problem, though, is that your language might become easily too large,
if your rules are too unspecific. So there, the theory has the problem
that one has to play around with the extra grammar variables N, to get
exactly the right language.

Now, the type of grammar used the most in implementing computer languages
are the so called context free grammars (Chomsky hierarchy type 1), where
each production (or rule) in P, has the form A -> x, where A in N, and x
in V*. This does not admit implementing say the construction of
definitions, widely used in computer languages:

I could decide to introduce, in my computer language, the construct
  define <name> ...
and then <name> will subsequently behave syntactically according to this
definition. In a natural language parsing computer language, might have
the construct:
  define flies := noun.
and then this word should behave like that grammatically. Such constructs
can be handled using so called attribute grammars. But a parser attaches
actions to the productions, so one way is to make a table with the name
"flies", and whenever this word appears in the input, the lexer returns
the grammar variable "noun" to the parser.

Now, in an unambiguous language, once "flies" has been defined, either a
new definition of it must be illegal, or override the old definition, say
in a limited scope {...} which is often called an "environment"
in computer lingo. But in a natural language, it is legal to have
  define flies := verb
admitting the ambiguous "Time flies like an arrow". This leads to a parser
parsing several possibilities in parallel.

So, this answers in part your question (I hope). One can admit adding
rules, but there will be ambiguities, and if the rules added are too
narrow, the grammar will admit sentences which are not legal sentences in
the language one wants to capture.

--
Hans Aberg
.

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