Is the language L generated by 'Fibonacci Strings' (as given in the desciption) regular? If not, disprove by Pumping Lemma

Question

The Fibonacci strings are defined as follows: S1=a, S2=b and Sk=S k-1S k-2 for k>2 . For example S3=ba , S4=bab etc . Let L be the language generated by the Fibonacci strings. Is the language' L ' Regular? If not, disprove by Pumping Lemma.

Answer 1

Consider the lengths of strings in this language. They are the Fibonacci numbers themselves. Consider S(i), the ith string in this set. It has length F(i), where F(i) is the ith Fibonacci number.

Now consider S(i+1), the (i+1)th string in the language with length F(i+1). What strings can we append to this language to get another string in the language? We can, of course, append the empty string. The next smallest string we can append is S(i) to get S(i+2). S(i) has length F(i). Therefore, the second shortest string we can append to any given string in the language to get another string in the language is unique for every S(i); they are therefore all distinguishable with respect to the Myhill-Nerode theorem, and a minimal DFA for the language would require infinitely many states to distinguish them. Since DFAs cannot have infinitely many states, there is no DFA for this language, and since there is no DFA for the language, the language is not regular.