# Finite automaton that recognizes the empty language $\emptyset$

Since the language $L = \emptyset$ is regular, there must be a finite automaton that recognizes it. However, I’m not exactly sure how one would be constructed. I feel like the answer is trivial. Can someone help me out?

#### Solutions Collecting From Web of "Finite automaton that recognizes the empty language $\emptyset$"

One state, non-accepting, and no transitions. (That’s an NFA; if you want a DFA, have one transition from the state to itself for each letter of whatever alphabet is specified.)

You have only one state $s$ that is initial, but not accepting with loops $s \overset{\alpha}{\rightarrow} s$ for any letter $\alpha \in \Sigma$ (with non-deterministic automaton you can even skip the loops, i.e. the transition relation would be empty).

I hope this helps 😉

Given language is “empty language”.We have to construct a finite automata for this language.In general we consider “the construction of finite automata” as “the construction of DFA”.So….{Let us assume input symbol as ‘a’ and ‘b’}

(a)If we take one state(initial state) and don’t show any transition of any input symbol over this state,then this structure will not be a DFA because in a DFA there should be a transition of all input symbol over each state.
(b)If we take one state(initial state) and show the transition of both input symbol ‘a’ and ‘b’ over this state, then also this will not be a DFA because there should be a final state.
(c)If we take one state(initial state) and show the transition of both input symbol ‘a’ and ‘b’ over this state,and making this state final also then this FA will not be acceptor of “empty language”.
(d)If we take one initial state ‘A'(not making final it) showing the transition of both input symbol over ‘A’ itself AND taking one another state ‘B'(as final)showing the transition of both input symbol over the “transion edge” from final state ‘B’ to initial state ‘A'(‘B’is UNREACHABLE STATE here).Then this structure will be a DFA but not minimal DFA because in minimal DFA we remove UNREACHABLE STATE.
(e)similarly we cannot take concept of dead state in construction of minimal DFA.

SO NOW THE EXACT SOLUTION IS :-

” TAKE ONE INITIAL STATE ‘A'(not making final it) and ONE ANOTHER STATE ‘B'(making it final) and SHOW transition of both input symbol ‘a’ and ‘b’ over both state A’ and ‘B’. BUT don’t connect both states with any transition edge.
This is the desired minimal DFA which accepts “empty language”.

This is the DFA that iterates over (0|1)* but does not accept anything (not even empty string).