I am having confusion in understanding the concept of the word "powerful" in automata theory.
Even the confusion arises that if I have a DFA with single final state ,so then is it more or less powerful than another DFA with more than one final states?
And can we make a similar argument for NFA ?
Machine $M1$ is more powerful than machine $M2$ if, intuitively, anything that $M2$ can do, $M1$ can do as well. Formally, $M1$ is more powerful than $M2$ if, given any specification of a problem and a solution of it on $M2$, there exists a solution of the same problem on $M1$. One may further demand that the complexity (either time or space) does not increase when implementing on $M1$, so things can become a bit more complicated. Typically, one shows that machine $M1$ is more powerful than machine $M2$ by showing that any $M2$ type machine can be converted to an $M1$ type machine, which solves the exact same problem. In the case of DFA with 1 final state vs. multiple final states, it is obvious then anything that can be done with 1 final state can also be done with multiple final states. However, it is very easy to convert any FDA with multiple final states into a DFA with 1 final state without altering the accepted language; indeed, simply add a new state, label it the unique final state, and link to it precisely from all the previously final states. So, DFA's with 1 final state is as powerful as DFA's with multiple final states.