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.

∈-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 ∈ 

NFA without ∈
Transition diagram

Reference:

  • Introduction to the Theory of Computation” by Michael Sipser.