The Question (with details and generality removed)
With proper capitalization, how many grammatical English sentences could "buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo" possibly represent?
Motivation
The English word 'buffalo' has several definitions:
- (noun) any of several wild bovids: such as water buffalo or cape buffalo, or [long sciencey definition]
- (verb, transitive) see also: bewilder, baffle, bamboozle ['intimidates' is also a common]
- (proper noun) A city in New York.
- (adjective) A demonym for something from Buffalo, New York.
The first three are directly from Merriam-Webster; the last one I actually can't find a reputable source for, but I've picked it up by cultural osmosis.
This has led to the observation that, ignoring capitalization, you can put any number $n$ of copies of the word 'buffalo' to make a valid English sentence. (or, to use the internal source...)
Examples
The usual example here is the $n=8$ case, which is usually capitalized
Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo.
and means
The New York bovids that New York bovids intimidate, themselves intimdate New York bovids.
However, $n=8$ can also be capitalized differently:
Buffalo buffalo buffalo Buffalo buffalo Buffalo buffalo buffalo
which means
New York boivds intimidate the New York bovids that New York bovids intimidate.
This is not only a different grammatical parsing, but is in fact a different semantic meaning. In the second sentence, there could be three groups of bovids, one of which is intimidated by the other two. This is impossible with the first sentence— if there are three groups of bovids, they intimidate each other sequentially.
There is yet another interpretation:
Buffalo buffalo buffalo buffalo Buffalo buffalo buffalo buffalo.
which can mean
The bovids that bovids intimidate, themselves intimidate the New York bovids that bovids intimidate.
This sentence is clearly different, since there could be as many as four groups of bovids, and only one of them need be from New York.
However, this is not what I meant when I wrote the sentence. I meant that
The New York bovids— namely, those that the bovids who intimidate New York (!) do intimidate— intimidate bovids.
This is also trivially different from any of the other sentences above; never before have we encountered bovids that intimidate an entire city! This example shows that even with the same capitalization, we can get different semantic meaning.
The Question
Using the four definitions from the motivation, and assuming that you are allowed to capitalize at your leisure, how many grammatical English sentences are given by simply writing down the word 'buffalo' $n$ times (and a period)?
Feel free to call this number the Buffalo number $B(n)$, and say it counts the number of valid parsings :) ... or alternatively, that it counts Buffalo functions, i.e. $f:\{1,2,\cdots, n\}\to\{1,2,3,4\}$ satisfying... whatever they're supposed to.
Caveats and Variations
- The Wikipedia page that I cited above permits the sentence "Buffalo.", a command meaning "Hey, you! Intimidate them!". English permits this kind of obnoxiousness, where you have both an implied subject by using the imperative mood, and an implied direct object for a transitive verb.
Such sentences should be allowed by this count, but I wouldn't mind too much if you removed this restriction.
"Buffalo buffalo buffalo?" either means either "Do bovids actually intimidate other bovids?" (i.e., a reasonable complaint about this entire setup) or "Do New York bovids actually intimidate things?". These are both different from the plausible meanings of "Buffalo buffalo buffalo."
"Buffalo, buffalo buffalo." is another possibility (in the imperative mood) but with different punctuation, in which the governor of New York is suggesting that the city should intimidate the bovids in their midst, perhaps to avoid having to come up with ways to describe these unlikely scenarios.
Such sentences should not be allowed by this count, but I would be impressed if you had a method of counting under which it was easier to count them.
I would be rather surprised if there is a nice answer to this question which doesn't essentially give a generating function (on how many copies of each buffalo definition are being used) and certainly wouldn't mind if you expressed the answer in that language.
Tiny Computations
$B(1)=1$. The example above shows $B(1)\geq 1$, and any English sentence requires a verb. QED.
$B(2)\geq 2$. One of the two words must be a verb. The other cannot be an adjective, since any adjective requires a noun. Therefore $B(2)\leq 4$, and the following are valid parsings:
- Hey, you! Intimidate bovids!
- Hey, you! Intimidate New York!
- Bovids intimidate things.
Therefore $B(2)=3$, since the verb in "Buffalo$^3$ buffalo$^2$ is not correctly conjugated ("Buffalo buffalos." would work, but it's not what we're counting...)
$B(3)\geq 4$, and I think I understand a proof that equality holds, but I'm having a hard time writing it.
- Hey you! Intimidate New York bovids!
- Bovids intimidate bovids.
- Bovids intimidate New York.
- New York bovids intimidate things.
(Some thoughts on the proof: I am pretty sure that there cannot be two verbs in the sentence; the only one that I'm not sure of is if you can't do something funky with implied objects on "buffalo$^X$ buffalo$^2$ buffalo$^2$. New York cannot intimidate anything, since the conjugation is wrong.)
Disclaimer:
In case you've not figured this out by now, this is all for entertainment value only. I believe that the question is now (mostly) unambiguous, although of course I'm happy to specify something if need be.
I am aware that, even if it is properly disambiguated, the question is maybe too hard, and maybe should be split into a couple different questions. But I would be happy with asymptotics, or even nontrivial one-sided bounds.