Let L be the language consisting of all strings of the form a m b n , where m and n are positive integers and m ≥ n .

Show that there is no finite-state automaton that accepts L.

To prove:

That there does not exist any finite-state automaton which accepts all the strings in the form ambn.

Given information:

The language L consisting only the strings in the form ambn where m and n are positive integers and m≥n

Proof:

Suppose there is a finite-state automaton such that accept the string apbq where m=p and n=q

Then, we can observe that from the initial state, the automaton accept p number of a ’s as inputs in a row and next q number of b ’s in a row and then it reaches to the accepting state.

Notice that, p+1 and q+1 are integers and p+1≥q+1