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?

  • $\begingroup$ Remember that a language is regular iff it is in SPACE(O(1)). $\endgroup$
    – quicksort
    Oct 21, 2017 at 9:04

1 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 $x\in \{0,2,4,6,8\}$
  • $\begingroup$ Why is it that the automaton would fail in the case you mentioned? Also, what do you mean by fail? I think I must be overlooking something. $\endgroup$
    – Odo Frodo
    Oct 21, 2017 at 17:50
  • $\begingroup$ The last sentence was added. I dont understand it either. $\endgroup$
    – Romuald
    Oct 21, 2017 at 17:57
  • $\begingroup$ As far as I can see, the added sentence was wrong. I deleted it. You should always feel free to undo edits to your own posts that you think are wrong. $\endgroup$ Oct 21, 2017 at 19:56

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