1. CS2303-THEORY OF COMPUTATION
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CS2303-THEORY OF COMPUTATION
Chapter:
Minimization of Automata

2. Minimized DFA Myhill-Nerode Theorem
DFA with minimum number of states that will solve the problem
There may be states which are equivalent and not necessary

3. Equivalent States

4. Distinguishable States

6. By default final states of a DFA are distinguishable from non final states
Hence in the given DFA, state C is distinguishable from other states
To determine whether two states are equivalent we have to analyze the states reachable from the states for various strings from length 0 to some ‘n’
Difficult hence we go for table filling algorithm

7. Table filling algorithm
Build a (n - 1) x (n - 1) table where n is the number of states in the given DFA
Each column is named by the name of the states starting with the first state till n-1 states
Each row is named with name of the states starting from the second state
Duplicate cells are eliminated (i.e.) only one cell is used to determine whether two states are equal

8. Table filling algorithm
Each cell in the table represents a pair of states in the DFA.
Cells are marked with a “x” as soon as the two states represented by it are decided to be distinguishable
At the end of the process, the pairs of states corresponding to the cells of the table without the mark are concluded as equivalent states

18. Testing equivalence of two FA
To test whether the given two FAs M1 and M2 are equivalent it is imagined as if the two FAs are combined.
The imaginary FA will contain m + n states in it where m is the number of states in M1 and n is the number of states in M2.
The imaginary FA contains two initial states, initial state of M1 and M2 which is not allowed in real case.

19. Testing equivalence of two FA
Table filling algorithm is used for the imaginary DFA. A table of size (m + n-1) x (m + n-1) is used and the two initial states are tested for equivalence.
If they are equivalent then it may be concluded that both the DFAs represent same language.