Minimization of a Deterministic Finite Automaton (DFA).pptx

Minimization of a Deterministic Finite Automaton (DFA) is a process used to reduce the number of states in the DFA while maintaining the same language recognition capability. The resulting minimized DFA will have the smallest possible number of states.

Steps for Minimization of DFA:
Remove Unreachable States:

Identify and remove states that cannot be reached from the initial state. This helps in simplifying the DFA before further minimization.
Identify Distinguishable States:

Two states are distinguishable if there exists at least one string that, when processed from these states, leads to one state ending in an accepting state and the other in a non-accepting state.
Table Filling Method (Distinguishable State Identification):

Step 1: Create a table to record pairs of states.
Step 2: Mark pairs of states (p, q) as distinguishable if one is a final state and the other is not.
Step 3: Iteratively mark pairs of states (p, q) as distinguishable if, for some input symbol, the resulting states from p and q are already marked as distinguishable.
Step 4: Continue until no new pairs are marked distinguishable.
Form Equivalence Classes:

States that are not distinguishable are considered equivalent. Group these equivalent states together.
Construct the Minimized DFA:

Create a new DFA where each state corresponds to an equivalence class of the original DFA's states.
Define the transitions between these new states based on the transitions of the original DFA.
Set the initial state of the new DFA as the equivalence class containing the original initial state.
The accepting states of the new DFA are the equivalence classes that contain any of the original accepting states.
Result:

The resulting DFA after applying these steps is the minimized DFA, which is equivalent to the original DFA but with the fewest possible states.

1. CSE322 Formal Languages and Automation Theory Topic: Minimization of DFA

2. Minimization of DFA-   The process of reducing a given DFA to its minimal form is called as minimization of DFA.   It contains the minimum number of states. The DFA in its minimal form is called as a  Minimal DFA .

3. How To Minimize DFA?   The two popular methods for minimizing a DFA are-    

4. Minimization of DFA Using Equivalence Theorem-   Step-01:   Eliminate all the dead states and inaccessible states from the given DFA (if any).   Dead State   All those non-final states which transit to itself for all input symbols in ∑ are called as dead states.   Inaccessible State   All those states which can never be reached from the initial state are called as inaccessible states.

5. Step-02:   Draw a state transition table for the given DFA. Transition table shows the transition of all states on all input symbols in Σ.   Step-03:   Now, start applying equivalence theorem. Take a counter variable k and initialize it with value 0. Divide Q (set of states) into two sets such that one set contains all the non-final states and other set contains all the final states. This partition is called P .  

6. Step-04:   Increment k by 1. Find P k  by partitioning the different sets of P k-1  . In each set of P k-1  , consider all the possible pair of states within each set and if the two states are distinguishable, partition the set into different sets in P k .   Two states q 1  and q 2  are distinguishable in partition P k  for any input symbol ‘a’, if δ (q 1 , a) and δ (q 2 , a) are in different sets in partition P k-1 .  

7. Step-05:   Repeat step-04 until no change in partition occurs. In other words, when you find P k  = P k-1 , stop.   Step-06:   All those states which belong to the same set are equivalent. The equivalent states are merged to form a single state in the minimal DFA.   Number of states in Minimal DFA = Number of sets in P k

8. PRACTICE PROBLEMS BASED ON MINIMIZATION OF DFA-   Problem-01 :   Minimize the given DFA-

9. Step-01 :   The given DFA contains no dead states and inaccessible states.   Step-02:   Draw a state transition table-   States a b → q0 q1 q2 q1 q1 q3 q2 q1 q2 q3 q1 *q4 *q4 q1 q2

10. Step-03:   Now using Equivalence Theorem, we have- P  = { q  , q 1  , q 2  , q 3  } { q 4  } P 1  = { q  , q 1  , q 2  } { q 3  } { q 4  } P 2  = { q  , q 2  } { q 1  } { q 3  } { q 4  } P 3  = { q  , q 2  } { q 1  } { q 3  } { q 4  }   Since P 3  = P 2 , so we stop. From P 3 , we infer that states q  and q 2  are equivalent and can be merged together.

11. So , Our minimal DFA is-  

12. Problem-02 :   Minimize the given DFA-

13. Solution-   Step-01:   State q 3  is inaccessible from the initial state. So, we eliminate it and its associated edges from the DFA.   The resulting DFA is-

14. Step-02 :   Draw a state transition table- a b → q0 *q1 q0 *q1 *q2 *q1 *q2 *q1 *q2

15. Step-03:   Now using Equivalence Theorem, we have- P  = { q  } { q 1  , q 2  } P 1  = { q  } { q 1  , q 2  }   Since P 1  = P , so we stop. From P 1 , we infer that states q 1  and q 2  are equivalent and can be merged together.

16. So , Our minimal DFA is-  

17. Problem # 3

18. Solution of Problem # 3

19. Problem # 4

20. Solution of Problem No 4