Consider the following two statements:

For S1: {02n | n ≥ 1}

• This generates strings with an even number of zeros (length ≥ 2): L1 = {00, 0000, 000000, .....}.

• Since it only requires checking if the count of $0$s is even or odd, it can be mapped by a simple regular expression 00(00)* or a 3-state Finite Automaton.

• Therefore, S1 is Regular.

For S2: {0m 1n 0m+n |m ≥ 1 and n ≥ 1}

• Here, the number of trailing zeros must exactly equal the sum of the leading zeros (m) and intermediate ones (n).

• A finite automaton has no memory to store or count arbitrarily large values of m and n to verify this matching condition later in the string. (It requires a stack, making it a Context-Free Language).

• Therefore, S2 is Non-Regular.