I've started learning automata recently, and am struggling to figure out how to find a finite automaton that satisfies the following:
Given DFA $A_1 = (Q, \Sigma, \delta, q_0, F)$, find a finite automaton $A_2$ such that $$L(A_2) = \{s \mid (s\cdot s) \in L(A_1)\}$$
Where $s\cdot s$ is the concatenation of $s$ onto itself.
I can't seem to grasp how to get the automaton to "remember" $s$ so that I can check it twice, any tips on how to find such an automaton?
