NFA with epsilon

NFA with ∈-Moves

NFA with ∈ moves is exactly same as NFA without ∈ moves. 
But differece exist in the transition function δ. 
δ must include information about ∈ transitions. 

NFA with ∈-Moves has 6 tuples (Q, Σ, δ, q0, F).
Where, 

Q = finite set of states. 

Σ = finite set input symbols.

δ = transition function that maps Q × (Σ ∪{∈}) to 2Q.

q0 = initial state.

F = set of final states.


The non-deterministic finite automaton can be extended to include the transitions on null/empty input ∈.

For example,
NFA with 

In this NFA with epsilon, 
It accept an input string 'aabc'. 
Or string as number of a's followed by number of b's followed by number of c's. 
The string 'aabc' is accepted by the NFA by following the path with labels a, a, ∈, b, ∈, c.

Transition table for above NFA with ∈NFA's with ∈-transitions are closely related to regular expressions and useful in proving the equivalence between the classes of languages accepted by finite automata and regular expressions.


∈-closure

∈-closure of a state q is a set of states following by all transitions of q that are labeled as ∈.

∈-closure (q0) = (q0, q1, q2) 
∈-closure (q1) = (q1, q2) 
∈-closure (q2) = (q2)

NFA with ∈ to NFA without ∈

NFA with ∈
Transition diagram

Transition table NFA with ∈ 

First find out ∈ closure:
∈-closure (q0) = (q0, q1, q2) 
∈-closure (q1) = (q1, q2) 
∈-closure (q2) = (q2)
Transition table NFA without ∈ 
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