Prove that the following language is not regular aibaj i

Prove that the following language is not regular: { aⁱba^j | i >= 2j >= 0 } SHOW ALL WORK, THANKS!

Solution

Following language is not regular: { aⁱba^j | i >= 2j >= 0 }

Because it cannot be recognized with a finite automaton, since a finite automaton has finite memory and it cannot remember the exact number of a\'s.

Example:

if i = j = 0 then string we get is b

if j=1 then i>=2 means we get string aaba, aaaba, aaaaba, aaaaaba,.........

so for all this finite automata need to remember the exact value of j and that is not possible in this case so the given language is not regular.