Complement - operations on languages, Theory of Computation

The fact that SL2 is closed under intersection but not under union implies that it is not closed under complement since, by DeMorgan's Theorem

L1 ∩ L2 = 412_lema.png

We know that the intersection of SL2 languages is also SL2. If the complement of SL2 languages was also necessarily SL2, then 412_lema.png would be SL2 contradicting the fact that there are SL2 languages whose union are not SL2.

Lemma The class of strictly 2-local languages is not closed under complement .

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We got the class LT by taking the class SL and closing it under Boolean operations. We have observed that LT ⊆ Recog, so certainly any Boolean combination of LT languages will also