Definition of a language L with alphabet {a} is given as following. L= { a^{nk} | k 0, and n is a positive integer constant} What is the minimum number of states needed in a DFA to recognize L?
A)k+1
B)n+1
C)2^(n+1)
D)2^(k+1)

Question

Definition of a language L with alphabet {a} is given as following. L= { a^{nk} | k > 0, and n is a positive integer constant} What is the minimum number of states needed in a DFA to recognize L?
A)k+1
B)n+1
C)2^(n+1)
D)2^(k+1)

Quizplus · Accepted Answer

The minimum number of states needed in a DFA to recognize the language L is n+1. This is because the DFA needs to be able to distinguish between all strings of length multiple of n (a^n, a^2n, a^3n, ...) and those that are not. To do this, it cycles through n states for each segment of n 'a's and returns to the initial state after counting n 'a's, requiring a total of n+1 states (n states for the cycle and 1 additional state to handle the acceptance condition).

Definition of a Language L with Alphabet {A} Is Given