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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 research...
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Some thoughts on classic optimality theory(Prince & Smolensky 1993/2004)

[This is a term paper of mine, I'm not that familar with OT, thus I cannot guarantee that similar proposals had not been made before , comments are welcome .] Basic structure of classic optimality theory   Generation Function : G(input) = {C1, C2,..., Cn}  Evaluation Function :  E({C1, C2, C3,..., Cn}, Con) = Ck (1<=k<=n);                                        Ck = <In_k , G(in_k)>, In_k is the kth input, 1<=k<=n         To be more precise, G-function could be considered as a function of the form P(g(x), x), in which x is input, g is a generation function that generates an infinite number of outputs,  and P is a pairing function which generates an order pair called candidate, in which an input is paired with its corresponding output. The G-function thus generates an infinite number of candidates, which are inputs to the E-function...
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A Rough Picture of Various Forms of Syntactic Merge

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Merge is the core computational device in Minimalism, below I draw a tree of various forms of merge.  Figure 1
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