Understanding Nondeterministic Finite Automata (NFA) | Theory of Computation 📘

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Nondeterministic Finite Automata NFA Theory of Computation, Automata Theory, in hindi, lectures, iit, tutorial,Video lecture for gate exam preparation. What is the difference between deterministic and non-deterministic finite-state automata?
In deterministic finite automata, there's exactly one state transition for every input symbol-state pair. There are also no epsilon transitions, meaning that you're not allowed to change states without consuming anything from the input.
In non-deterministic finite automata, there can be 0 or more state transitions for every input-state pair. You can also have epsilon transitions. When there's no state transition for a given input-state pair, then we say that the automata had crashed, meaning that it can't proceed processing the input, and therefore it doesn't accept the input. When there's more than one choice for a state transition for a given input-state pair, then the machine can just follow all possible paths (think of it as parallel computation), if one of the paths ends up in an accept state, then we say that the automata accepted the input string.
Both automata are equivalent in terms of power though. It may seem that a non-deterministic automata is more powerful, but both automata are proven to be equivalent, meaning that they recognize the same class of languages called regular languages. The proof of equivalence is by construction in which you show that given a DFA, you can construct an equivalent NFA, and vice versa. The proof can be found in any textbook on theory of computation.
Definition of non deterministic finite automaton
Let Q be a finite set and let be a finite set of symbols. Also let be a function from Q to 2Q , let q0 be a state in Q and let A be a subset of Q. We call the elements of Q a state, the transition function, q0 the initial state and A the set of accepting states.
Then a non deterministic finite automaton is a 5-tuple ( Q , , q0 , , A )
Notes on the definition
As in the case of DFA the set Q in the above definition is simply a set with a finite number of elements. Its elements can be interpreted as a state that the system (automaton) is in.
The transition function is also called a next state function . Unlike DFAs an NFA moves into one of the states given by (q, a) if it receives the input symbol a while in state q. Which one of the states in (q, a) to select is determined nondeterministically.
Note that is a function. Thus for each state q of Q and for each symbol a of (q, a) must be specified. But it can be the empty set, in which case the NFA aborts its operation.
As in the case of DFA the accepting states are used to distinguish sequences of inputs given to the finite automaton. If the finite automaton is in an accepting state when the input ends i.e. ceases to come, the sequence of input symbols given to the finite automaton is "accepted". Otherwise it is not accepted.
Note that any DFA is also a NFA.

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