In a deterministic finite automaton (DFA), the number of states required to recognize a given string of length n depends on the specifics of the string and the behavior of the DFA.

Here are a few examples to illustrate how the number of states might vary:

If the DFA is designed to recognize a string of n zeros, it will need at least n+1 states, since it will need one state for each zero and one additional state to indicate that the string has been fully recognized.


If the DFA is designed to recognize a string of n ones, it will also need at least n+1 states for the same reason.


If the DFA is designed to recognize two different strings of length n, such as 0001 and 1111, it will need at least 2n+1 states. This is because it will need one state for each combination of a zero or one at each position in the string, plus one additional state to indicate that the string has been fully recognized.


If the DFA is designed to recognize a string of n zeros or n ones, it will need at least n+2 states. This is because it will need one state for each zero or one at each position in the string, plus one additional state to indicate that the string has been fully recognized and another state to indicate that the string consists of either all zeros or all ones.