minimization of dfa algo

 

1. Start with your original DFA, which has states, transitions, start state, and accept states.

2. Divide the states into two initial groups: one group containing the final/accepting states, and another group containing the non-final/rejecting states. These groups represent the initial equivalence classes.

3. Create a new set called "Partition" and put these two initial groups into it.

4. Repeat the following steps until the Partition set stops changing:

   4.1. Create an empty set called "NewPartition".

   4.2. For each group of states (called a "partition") in the Partition set:

   4.2.1. If the partition contains only one state, add it to the NewPartition set.

   4.2.2. Otherwise, divide the states in the partition based on their transitions. For each input symbol in the DFA's alphabet, see which states lead to different partitions based on their transitions. Create new partitions for each unique set of transitions and add them to the NewPartition set.

   4.3. If the NewPartition set is the same as the Partition set, stop the loop. Otherwise, replace the Partition set with the NewPartition set and repeat the loop.

5. The final Partition set represents the minimized equivalent DFA. Each group of states in the Partition set represents a state in the minimized DFA.

6. Create a new DFA using the states, transitions, start state, and accept states based on the Partition set.

The algorithm works by distinguishing between states based on their transitions and repeatedly refining the groups of equivalent states until they no longer change. The resulting Partition set represents the equivalent states in the minimized DFA.

Please note that this is a simplified explanation, and the algorithm can be further optimized and expanded upon. Other algorithms like Hopcroft's algorithm and Moore's algorithm also exist for DFA minimization.