After working through some examples of proving the non-regularity of languages I encountered this language

$$ L = \{(ab)^{i}a^{j} | i \geq j, i,j \in \mathbb{N}\} $$

Where $a^{k}$ = a repeated k times. Is this language regular? I believe not since there are infinitely many strings $(01)^{i}$ with $i \geq j$, so a DFA or NFA cannot accept the language


1 Answer 1


A regular language can be recognised by a finite state machine.

After processing (ab)^i and (ab)^i’, i ≠ i', the state machine must be in different states: If say i < i’, and j = i’, then (ab)^i a^j is not in the language, but (ab)^i’ a^j is, so processing a^j from both states must end in different states, one not accepting, one accepting.

Since i, i’ were arbitrary there cannot be a finite number of states.


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