$$
L = \{\, a^{n} b^{m} c^{k} \mid n,m,k > 0 \text{ and } k = |n-m| \,\}
$$
Is it dcfl or cfl ?
my approach
We can split the condition \(k = |n - m|\) into two cases:
Case 1:\(n \ge m\), so \(k = n - m\):
\[
L_1 = \{\, a^{n} b^{m} c^{\,n-m} \mid n \ge m > 0 \,\} => dcfl
\]
Case 2: \(m \ge n\), so \(k = m - n\):
\[
L_2 = \{\, a^{n} b^{m} c^{\,m-n} \mid m \ge n > 0 \,\} => dcfl
\]
Hence,
\[
L = L_1 \cup L_2. => dcfl \cup dclf is \ not \ closed \ under \ dclf => cfl
\]
