Some Sidenotes upon My Tentative Proof of Chomsky(1956)'s Claim concerning FSG.

The Claim: if S has an m-termed dependency set, then at least 2^m states are necessary for the FSG that generates L.

Here is The proof. It's proven based on the Myhill-Nerode Theorem. 

I'm working on this simply because: 

It's a crucial point in understanding the whole motivation of the development of transformational grammar. 

Simply put, for a FSL contains m dependencies, the FSG that could generate it would need at least 2^m such states. In other words, a FSG with 2^m states could at most generate a FSL with m dependencies. More generally,  for any finite state grammar, the language it generates shall contain finite number of dependencies. Natural language, however, has no such upper bound. 

It is for this reason that natural language could not be adequately described by the type of FSG defined in Chomsky(1956). 

My interactions with people in the field (of linguistics) informed me that there is systematic ignorance of the researches of mathematical psychology done by Noam Chomsky and his collegues (Marcel P. Schutzenberger, George A. Miller) in the 50s-60s among linguists these days. 

And the lack of understanding of these historical roots renders them incapable of appreciating the mathematico-psychological essence of generative linguistics, and more importantly, prevents them from seeing the larger scientific context. And some of their contemporaries with different knowledge backgrounds, not aware of this history for more obvious reasons, demonstrated a even more hostile attitude towards this line of research. 

In other words, there is a culture problem in linguistics too: students learn about linguistics from their teachers, and teachers learn about it from their teachers, so this lack of understanding and appreciation for the mathematico-psychological nature of generative linguistics in our culture replicates itself indefinitely. [adapted from Lockhart 2009] 

References

[1] Lockhart, P. (2009). A mathematician's lament: How school cheats us out of our most fascinating and imaginative art form. Bellevue literary press.

[2] Chomsky, N. (1956). Three models for the description of language. IRE Transactions on information theory, 2(3), 113-124.