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Given a $\Sigma$ I define a regular language as one of the folllows:

  • $\emptyset$
  • $\left\{ \sigma \right\}$ for any $\sigma \in \Sigma$
  • $L_1 \cup L_2$ for regular $L_1, L_2$
  • $L_1 \cdot L_2 $ for regular $L_1,L_2$
  • $L^*$ for regular $L$

Now, I would like to show that for a given regular language $L$, its complement $\overline{L}$ is regular as well. I want to do so based only on this definition (i.e. without proving its equivalent to DFA, NFA or reg expression).

However, I'm having a bit of trouble here, not sure how to do so. Maybe I want to express the complement of $L$ as a sort of unions/contactinations? Not really sure how to go for it. Any ideas?

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Your definition is the same as the usual definition with regular expressions. While regular expressions also admit the empty word $\epsilon$, it is expressible as $\emptyset^*$.

There is no simple rule for complementing regular expressions, and it is known that such complementation can result in an exponential blow-up (for example, the language of all words not containing all alphabet symbols has a regular expression of size $O(|\Sigma|^2)$, but its complement requires a regular expression of length exponential in $|\Sigma|$).

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  • $\begingroup$ So there's no way of proving $\overline{L}$ is regular without a DFA/NFA? $\endgroup$
    – Eric_
    Nov 2 '21 at 12:21
  • $\begingroup$ None that I’m aware of. $\endgroup$ Nov 2 '21 at 12:23

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