Finite state grammar is one that is “capable of generating an infinite set of sentences by means of a finite number of recursive rules operating upon a finite vocabulary” (Lyons 1970:51). By ‘recursive’ it is meant applicable more than once in the generation of the same sentence. According to finite state grammar, sentences are generated as a result of choices made from left to right. To put it more specifically, the leftmost element of a sentence is it initial state, analogously, the rightmost element of a sentence is its final state. Thus, sentences are formed by a process of shifting from an initial state to a final state through a finite number of internal states engaged in the creation of sentences. The very process may be represented graphically by a ‘state diagram’ as follows:
fig1
Fig. 1 State diagram (a) (Chomsky 1957:19)
The above state diagram represents the grammar which produces only two sentences: “The man comes” and “The men come”. This grammar can be extended to produce an infinite number of sentences by implementing closed loops, as presented in the state diagram below:
fig2 Fig. 2 State diagram (b) (Chomsky 1957:19)
In the place of a loop one can insert any possible string of words. The number of and the length of closed loops may of course vary depending on how complex sentences one wishes to create. Thus, to sum up, according to a finite state grammar theory, sentences are generated by a process of moving from initial state of a sentence to a final state of a sentence through a number of internal states, each of which limits the scope of words that may be implemented further on within the very process due to the grammatical restrictions posed by internal states.

Chomsky (1957) claims that English is not a finite state language since it seizes to construct a finite state grammar that would account for all and only grammatical sentences. As he further states, finite state grammar fails to explain certain processes of sentence formation in English. It is possible that some no-adjacent words of a sentence will hold certain amount of dependency and such a dependency may be further separated by a phrase or clause containing string of non-adjacent interdependent words. This, in turn, results in a sentence with ‘mirror image properties’. Considering a sentence with a following structure a + b + c … x + y + z, there is dependency between the outermost constituents (a and z), between the next outermost (b and y) and so forth. Any language that contains an indefinitely large number of sentences with ‘mirror image properties’ similar to the above example is outside the scope of the finite state grammar. Chomsky did not, however, reject finite state grammar theory as being inapplicable to the study of language, but he rather claimed that ‘left to right’ and ‘word by word’ generation of sentences would not account for some of the constructions in English. Thus, he provided more powerful tool for the study of language, phrase structure grammar, which comprised of all what finite state grammar was capable of doing – and much more.
References:
Chomsky, N. 1957. Syntactic Structures. The Hague: Mouton.
Lyons, John. 1970. Chomsky. Fontana.