To prove or disprove that language is regular

Question

Suppose the language L = {w | w is an integer whose sum of digits is a multiple of 2}. I strongly feel like this is non-regular language, but how should I choose a string to apply the pumping lemma?

Answer 1

The language $L$ is regular as shown by the following "flip-flop" style minimal and deterministic automaton $\mathcal{A}$, the crux is that you need only two states (not an unbounded amount of memory) to check parity. $\mathcal{A} = \langle \{odd,even\}, \{0..9\}, even, \{even\}, \delta \rangle$ which transition function $\delta$ is defined by :

• $\delta(x,even) = even$ when $x\in \{0,2,4,6,8\}$
• $\delta(x,even) = odd$ when $x\in \{1,3,5,7,9\}$
• $\delta(x,odd) = even$ when $x\in \{1,3,5,7,9\}$
• $\delta(x,odd) = odd$ when $x\in \{0,2,4,6,8\}$