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DFA Minimization
Strings over  { a,b }  with even number of  a ’s  * Ea Oa Eb Ob Ob Eb b b b b a a a a [Oa,Ob] [Ea,Eb] [Ea,Ob] [Oa,Eb]
Strings over  { a,b }  with even number of  a ’s  * Ea Oa a a [Ea] [Oa] b b
Observation ,[object Object],[object Object]
Equivalent or Indistinguishable States ,[object Object],[object Object],[object Object]
Main Idea qi qn qj qm If qm and qn are distinguishable, then so are qi and qj a a
Example DFA q0 q4 q1 q2 q3 q7 q5 q6 a b a a,b a a,b b b b a b (a  u  b)(a  u  b * ) a,b a
Refinement of State Partitions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Example DFA showing equivalent states q0 q4 q1 q2 q3 q7 q5 q6 a b a a,b a a,b b b b a b (a  u  b)(a  u  b * ) a,b a
Example Minimum DFA q0 q1,q4 q7 q2,q5 q3,q6 a a,b a a,b b a,b b (a  u  b)(a  u  b * )
[object Object]

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DFA Minimization

  • 2. Strings over { a,b } with even number of a ’s  * Ea Oa Eb Ob Ob Eb b b b b a a a a [Oa,Ob] [Ea,Eb] [Ea,Ob] [Oa,Eb]
  • 3. Strings over { a,b } with even number of a ’s  * Ea Oa a a [Ea] [Oa] b b
  • 4.
  • 5.
  • 6. Main Idea qi qn qj qm If qm and qn are distinguishable, then so are qi and qj a a
  • 7. Example DFA q0 q4 q1 q2 q3 q7 q5 q6 a b a a,b a a,b b b b a b (a u b)(a u b * ) a,b a
  • 8.
  • 9. Example DFA showing equivalent states q0 q4 q1 q2 q3 q7 q5 q6 a b a a,b a a,b b b b a b (a u b)(a u b * ) a,b a
  • 10. Example Minimum DFA q0 q1,q4 q7 q2,q5 q3,q6 a a,b a a,b b a,b b (a u b)(a u b * )
  • 11.