I've always been interested in "minimal" grammars, and I've posted
here on the subject from time to time. But it occurred to me that
there might be a way to systematically discover a near-minimal
grammar.

Imagine taking a large corpus of sentences in some natlang such as
English, and analyzing those sentences into component elements. For
example, the sentence:

I see a boy.

Could be analyzed and labelled:

SENT(PRON(I) VERB(see) NPH(DET(a) NOUN(boy)))

From this we could create a lexicon:

DET(a)
NOUN(boy)
PRON(I)
VERB(see)

and a pair of composition rules:

SENT(PRON VERB NPH) - A sentence is composed of pronoun, verb, and noun phrase.
NPH(DET NOUN) - A noun phrase is composed of determiner and noun.

If we expand the lexicon to include:

DET(the)
NOUN(book)
PRON(he)
VERB(read)

Then the same set of composition rules works for:

He read the book.

Now by working our way through a large corpus, sentence by sentence,
we could create an ever-growing lexicon and ever-expanding list of
composition rules like the 3,959 rules Panini collected for Sanskrit
in the 6th century BCE.

But this procedure doesn't get us any closer to a _minimal_ grammar.
The more sentences we analyze, the more our grammar grows, and it's
unlikely that we could ever declare it "complete" in the sense that we
could guarantee that it described every valid English sentence.

But if the goal is specifically to discover a minimal grammar then we
don't want to describe every new sentence we encounter. Instead, we
want to take every new sentence we encounter and be able to paraphrase
it into a form described by the composition rules we already have, and
to add a new composition rule only when it became absolutely necessary
to do so.

We might still never be able to declare that the grammar is minimal
(in the sense that it could not be simplified) or complete (in the
sense that any and all natlang sentences could be paraphrased using
only those rules). We could, however, be "reasonably comfortable" with
a "reasonably small" set of rules if we could, using only those rules,
paraphrase some significant natlang work (e.g. Burroughs' _A Princess
of Mars_, or Melville's _Moby Dick_), and if we could, after some
reasonable effort to do so, find no way to simplify the set of
composition rules.

Note that those rules do not have to conform in any way to English
grammar. As long as the compositional rules convey the desired meaning
of each paraphrased (or translated) sentence, then the rules are
useable. If we paraphrase the English "I see a small boy." to "A boy
small see I." and use that paraphrase to form the basis of our
composition rule, that's just fine, as long as we are consistent
throughout.

Take for example the opening sentence of Edgar Rice Burroughs' _A
Princess of Mars_:

In submitting Captain Carter's strange manuscript to you in book form,
I believe that a few words relative to this remarkable personality
will be of interest.

If we paraphrase this in a way that uses only a few unique rules, we
might get something like this:

To you I submit this strange manuscript which it forms a book and I
believe that you will-have interest which it is-directed-to a-few
words which they are-about this remarkable personality.

This can be analyzed as:

SENT(
	SENT(
		PRPH(
			PRP(To)
			PRON(you)
		)
		SENT(
			PRON(I)
			VERB(submit)
			NPH(
				NPH(
					DET(this)
					ADJ(strange)
					NOUN(manuscript)
				)
				RLPH(
					which
					SENT(
						PRON(it)
						VERB(forms)
						NPH(DET(a) NOUN(book))
					)
				)
			)
		)
	)
	CONJ(and)
	SENT(
		PRON(I)
		VERB(believe)
		OBCL(
			that
			SENT(
				PRON(you)
				VERB(will-have)
				NPH(
					NOUN(interest)
					RLPH(
						which
						SENT[
							PRON(it)
							VERB(is-directed-to)
							NPH(
								NPH(QNT(a-few) NOUN(words))
								RLPH(
									which
									SENT(
										PRON(they)
										VERB(are-about)
										NPH(DET(this) ADJ(remarkable) NOUN(personality))
									)
								)
							)
						)
					)
				)
			)
		)
	)
)

Extracting the rules from this sentence we get these 10 rules:

NPH(DET {ADJ} NOUN)
NPH(NPH RLPH)
NPH(QNT NOUN)
OBCL(that SENT)
PRPH(PREP PRO)
RLPH(which SENT)
SENT(PRON VERB NPH)
SENT(PRON VERB OBCL)
SENT(PRPH SENT)
SENT(SENT CONJ SENT)

Where elements in curly brackets are otional.

This is just a quick, off the cuff example, and I'm sure those rules
could be stated in more generalized form, such as replacing "PRON"
with "SUBJ" and letting "PRON" be one of many different kinds of
"SUBJ". It is also highly likely that a different paraphrase could be
found that used a smaller set of rules.

Choices would also have to be made as to what constituted "VERB" as
opposed to "VRBPH" (verb phrase). Are the verbs fully conjugated so
that "will-have-been-walking" is a single word VERB rather than a
multi-word VRBPH? There are trade-offs: The more complex the
conjugations, the simpler the sentence composition rules, and
vice-versa.

In any case, if the grammar is to be minimal then it must be
absolutely regular. Exceptions require extra rules describing them.

If I were to approach the project with a specific text such as _Moby
Dick_ or _A Princess of Mars_, I would begin by sorting the text into
individual sentences with the shortest sentences first, and working up
to the longest sentences. In this way I would hope that the longer
sentences would be found to be compositions of patterns already
discovered in the shorter sentences.

Thus I could start with the shortest sentences (from _A Princess of Mars):

Yes.
Come!
Look!
I am done.
But I did.
You may go.
We await it.
It is custom.
I have spoken.
It is the end.

and work my way up to the longest sentences (from _A Princess of Mars_):

You will be treated by us as a Tharkian chieftain, but you must not
forget that every chief who ranks you is responsible for your safe
delivery to our mighty and most ferocious ruler.

Zad first attempted to rush me down as a bull might a wolf, but I was
much too quick for him, and each time I side-stepped his rushes he
would go lunging past me, only to receive a nick from my sword upon
his arm or back.

Once the composition rule SENT(SENT CONJ SENT) is collected these
longer sentences can be simplified before being paraphrased and
analyzed:

I was much too quick for him.
You will be treated by us as a Tharkian chieftain.
Zad first attempted to rush me down as a bull might a wolf.
and so on...

--gary