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Minimization of DFA using Myhill-Nerode Theorem

Minimization of DFA using Myhill-Nerode Theorem :
Minimization of DFA is Required to obtain the minimal and equivalent version of any DFA which consists of minimum number of states possible. Myhill-Nerode theorem can be used to convert a DFA to its equivalent DFA with minimum no of states. This method of minimization is also called Table filling method. There is also another method called Partitioning Method or Equivalence Method for the minimization of DFA ( please visit  https://www.geeksforgeeks.org/minimization-of-dfa/amp/ to know about Equivalence/Partitioning Method).

Steps for the Minimization of DFA :

  1. Create the pairs of all the states involved in the given DFA.
  2. Mark all the pairs (Qa,Qb) such a that Qa  is Final state and Qb is Non-Final State.
  3. If there is any unmarked pair (Qa,Qb) such a that δ(Qa,x) and δ(Qb,x) is marked, then mark (Qa,Qb). Here x is a input symbol. Repeat this step until no more marking can be made.
  4. Combine all the unmarked pairs and make them a single state in the minimized DFA.

Example

Consider the following DFA,

Following is the transition table for the above DFA

Minimizing the above DFA using Myhill-Nerode Theorem :

Step-1: Create the pairs of all the states involved in DFA.

Step-2: Mark all the pairs (Qa,Qb) such a that Qa  is Final state and Qb is Non-Final State.

Step-3: If there is any unmarked pair (Qa,Qb) such a that δ(Qa,x) and δ(Qb,x) is marked, then mark (Qa,Qb). Here x is a input symbol. Repeat this step until no more marking can be made.

Step-4: Combine all the unmarked pairs and make them as a single state in the minimized DFA.

Following is the Minimized DFA with Q1Q2 and Q3Q4 as the combined states.

Transition table for Minimized DFA

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