In the common model of FSA, the automaton reads the input string once, moving from one state to another after reading each letter in the input string. Epsilon transitions allow moving from one state to another without reading a character. However, the movement over the input string is still unidirectional.
Is there a formal model of a FSA that reads the input string K times, and only then reaches its final state? is this model more powerful than FSA, e.g., can it recognize context-free languages that are not regular?
I think the option of adding an epsilon transition for every transition edge in the FSA means the models are equivalent in power (since we can model K iteration over the input by epsilon transitions from the last state back to the first -- although I am not sure), but if this is so, are there any results regarding the number of states needed in a regular FSA and a FSA with K transitions?