I'm having today a test on formal language theory, and I've seen a question about it I'm having hard time solving. The question is: Give an example of 2 languages, L,M which are non-context-free but LM(the concatenation) is a context-free language but not a regular language. Thanks in advance


Let $1$ denote the empty word. Let $$ S = \{a^nb^n \mid n \geqslant 0\},\quad T= 1 + \{c^{2^n}\mid n \geqslant 0\},\quad L = ST,\quad M = 1 + (c^* - T). $$ Observe that $$c^* = T \cup M \subseteq (1+T)(1 + M) = TM$$ and hence $TM = c^*$. Now the languages $L$ and $M$ are not context-free but $LM = Sc^*$ is context-free but not regular.

  • $\begingroup$ What is the meaning $1 + (c^* - T)$ and so on? $\endgroup$ – vonbrand Feb 11 '20 at 18:09
  • $\begingroup$ $+$ denotes union and $-$ set difference. $\endgroup$ – J.-E. Pin Feb 11 '20 at 18:16

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