1. COP46– Programming Language Translators Dr. Manuel E. Bermudez
DFA-> Minimum DFA
COP46– Programming Language Translators Dr. Manuel E. Bermudez

2. DFA-> Minimum DFA
RGR RGL Minimum
DFA
RE NFA DFA
Done
Soon
Now

3. DFA-> Minimum DFA Given a DFA M, there exists an equivalent minimal
DFA M’: no other equivalent DFA exists with fewer states than M’.
State Minimization:
Will build MDFA from DFA (not “surgery”).
Start with two states: final and non-final.
Split states only if necessary: minimum # of states.

4. DFA-> Minimum DFA
Definition: A partition of a set S is a set of subsets
of S so that every element of S appears in exactly one
of the subsets.
Example: S = {1, 2, 3, 4, 5}
Π1 = { {1, 2, 3, 4}, {5} }
Π2 = { {1, 2, 3,}, {4}, {5} }
Π3 = { {1, 3}, {2}, {4}, {5} }
Π2 is a refinement of Π1 , and Π3 is a refinement of Π2.

5. DFA-> Minimum DFA Minimization Algorithm:
Remove all undefined transitions by introducting a TRAP state, from which no final state is reachable.
Partition all states into two groups (final states and non-final states).
Complete the “Next State” table for each group, by specifying transitions from group to group.
Form the next partition: split groups in which Next State table entries differ.
Repeat 3,4 until no further splitting is possible.
Determine start and final states.

6. Example: Π0 = { {1, 2, 3, 4}, {5} } St. a b 1234 1234 1234 5
DFA-> Minimum DFA b a 1 2 3 5 4
Example:
Π0 = { {1, 2, 3, 4}, {5} }
St. a b 1 2 3 4 5
Split {4} from partition {1,2,3,4}:
Π1 = { {1, 2, 3},{4}, {5} }

7. DFA-> Minimum DFA Π1 = { {1, 2, 3},{4}, {5} } St. a b 123 123 123 4 1 2 3 4 5
Split {2} from partition {1,2,3}
Π2 = { {1, 3}, {2}, {4}, {5} }

8. DFA-> Minimum DFA Π2 = { {1, 3},{2},{4}, {5} } St. a b 2 13 2 4 2 5
2 13
2 4
2 5 5 13 4 2 a b 1 3 2 4 5
No more splitting !
Minimal DFA:

9. DFA-> Minimum DFA
RGR RGL Minimum
DFA
RE NFA DFA
Done !!

10. summary Summary of Regular Languages
Smallest class in the Chomsky hierarchy.
Appropriate for lexical analysis.
Four representations: RGR , RGL , RE and FSA.
All four are equivalent; algorithms to perform transformations among them.
Various advantages and disadvantages among these four, for language designer, implementer, and user.
FSA’s can be made deterministic, and minimal.