1 Minimizing DFA’s By Partitioning

2 Minimizing DFA’s Lots of methods All involve finding equivalent states: –States that go to equivalent states under all inputs (sounds recursive) We will use the Partitioning Method

3 Minimizing DFA’s by Partitioning Consider the following dfa (from Forbes Louis at U of KY): Accepting states are yellow Non-accepting states are blue Are any states really the same?

4 S 2 and S 7 are really the same: Both Final states Both go to S6 under input b Both go to S3 under an a S 0 and S 5 really the same. Why? We say each pair is equivalent Are there any other equivalent states? We can merge equivalent states into 1 state

5 Partitioning Algorithm First –Divide the set of states into Final and Non-final states Partition I Partition II a b S0S0 S1S1 S4S4 S1S1 S5S5 S2S2 S3S3 S3S3 S3S3 S4S4 S1S1 S4S4 S5S5 S1S1 S4S4 S6S6 S3S3 S7S7 *S 2 S3S3 S6S6 *S 7 S3S3 S6S6

6 Partitioning Algorithm Now –See if states in each partition each go to the same partition S 1 & S 6 are different from the rest of the states in Partition I (but like each other) We will move them to their own partition a b S0S0 S 1 I S 4 I S1S1 S 5 I S 2 II S3S3 S 3 I S4S4 S 1 I S 4 I S5S5 S 1 I S 4 I S6S6 S 3 I S 7 II *S 2 S 3 I S 6 I *S 7 S 3 I S 6 I

7 Partitioning Algorithm a b S0S0 S1S1 S4S4 S5S5 S1S1 S4S4 S3S3 S3S3 S3S3 S4S4 S1S1 S4S4 S1S1 S5S5 S2S2 S6S6 S3S3 S7S7 *S 2 S3S3 S6S6 *S 7 S3S3 S6S6

8 Partitioning Algorithm Now again –See if states in each partition each go to the same partition –In Partition I, S 3 goes to a different partition from S 0, S 5 and S 4 –We’ll move S3 to its own partition a b S0S0 S 1 III S 4 I S5S5 S 1 III S 4 I S3S3 S 3 I S4S4 S 1 III S 4 I S1S1 S 5 I S 2 II S6S6 S 3 I S 7 II *S 2 S 3 I S 6 III *S 7 S 3 I S 6 III

9 Partitioning Algorithm Note changes in S 6, S 2 and S 7 a b S0S0 S 1 III S 4 I S5S5 S 1 III S 4 I S4S4 S 1 III S 4 I S3S3 S 3 IV S1S1 S 5 I S 2 II S6S6 S 3 IV S 7 II *S 2 S 3 IV S 6 III *S 7 S 3 IV S 6 III

10 Partitioning Algorithm Now S 6 goes to a different partition on an a from S 1 S 6 gets its own partition. We now have 5 partitions Note changes in S 2 and S 7 a b S0S0 S 1 III S 4 I S5S5 S 1 III S 4 I S4S4 S 1 III S 4 I S3S3 S 3 IV S1S1 S 5 I S 2 II S6S6 S 3 IV S 7 II *S 2 S 3 IV S 6 V *S 7 S 3 IV S 6 V

11 Partitioning Algorithm All states within each of the 5 partitions are identical. We might as well call the states I, II III, IV and V. a b S0S0 S 1 III S 4 I S5S5 S 1 III S 4 I S4S4 S 1 III S 4 I S3S3 S 3 IV S1S1 S 5 I S 2 II S6S6 S 3 IV S 7 II *S 2 S 3 IV S 6 V *S 7 S 3 IV S 6 V

12 Partitioning Algorithm a b I III I *II IVV III I II IV V II Here they are:

13 V a a a a a b b b b b b