Equivalence of DFA



  • Every DFA is a NFA, but not vice versa.

  • But there is an equivalent DFA for every NFA

  • For every NFA, there exists a DFA that simulates the characteristics of NFA

Both NFA and DFA has five tuples ; < Q, Σ, q0, qf, 𝜹 > 

where, 

Q denotes the set of states

Σ, denotes input alphabet

q0 indicates initial state

qf indicates final state

the transition function of NFA is defined as 𝜹 : Q x Σ → 2Q

the transition function of DFA is defined as 𝜹 : Q x Σ → Q

By replacing Q with 2Q  the equivalent NFA of DFA can be obtained.

Example 1:

Draw the Equivalent DFA for the transition table given

 

a

b

q0

q0, q1

q2

q1

q0

q1

q2*

ø

q0, q1

Solution:

 

 

a

b

[q0]

[q0, q1]

[q2]

q2

ø

[q0, q1]

[q0, q1]

[q0, q1]

[q1, q2]

[q1, q2]*

[q0]

[q0, q1]

Example 2:

State

0

1

q0

q0, q1

q0

q1

q2

q1

q2

q3

q3

q3*

ø

q2

Solution:

State

0

1

[q0]

[q0, q1]

[q0]

[q0, q1]

[q0, q1, q2]

[q0, q1]

[q0, q1, q2]

[q0, q1, q2,q3]

[q0,q1,q3]

[q0,q1,q3]*

[q0, q1, q2]

[q0, q1,q2]

[q0, q1, q2,q3]*

[q0, q1, q2,q3]

[q0, q1, q2,q3]