Problem B. Following is an incomplete non-regularity proof that uses the minimum state lemma. Give an
appropriate answer for each blank in the given incomplete proof. For each blank, write a string precisely using the
alphabet \{a,b,c\} and possibly integer variables and constants, for example $ca^{x+99}b^{3y+k}$, except for blank (2), where a
conditional expression involving the variables used in blank (1) is expected. Type your answer for each blank on
an answer sheet.
The language $L_2 = \{c^{2n}a^ib^i \mid n, i \ge 0\}$ is not regular as proved below. Consider the infinite set
$D = \{(1) \mid (2)\}$. For an arbitrary pair of strings (3) and (4) from $D$ such that $x \ne y$, the string (5)
is a distinguisher because the string (6) $\in L_2$, but the string (7) $\notin L_2$. Hence, the strings in $D$ are pairwise
distinguishable with respect to $L_2$. Thus, by the minimum state lemma, $L_2$ is not regular.
