1. More on DFA minimization and DFA equivalence
The Chinese University of Hong Kong
Fall 2008
CSC 3130: Automata theory and formal languages
More on DFA minimization and DFA equivalence
Andrej Bogdanov

2. Example
Construct a DFA over alphabet {0, 1} that accepts those strings that end in 111
This is big, isn’t there a smaller DFA for this?
q000 1
q00 1
q001 q0 1 …
q01 … qe 1
q101
q10 … 1 q1 … 1
q11 1
q111 1

3. Smaller DFA
Yes, we can do it with 4 states: 1 q0 q1 q2 q3

4. No smaller DFA! Last time we saw that
In fact, it is also true that
There is no DFA with 3 states for L
This is the unique 4-state DFA for L 1 q0 q1 q2 q3

5. Theorem
The DFA M found by the minimization algorithm is the unique minimal DFA for L
Proof outline
Suppose there are two minimal DFAs M, M’ for L
In both M and M’, every pair of states is distinguishable M M’ q0
q0’

6. Proof outline for uniqueness
If M and M’ are not the same, there must be states q1, q2 of M and q’ of M’ like this:
For some strings u and v M M’ u q1 q’ q0
q0’
u, v v q2

7. Proof outline for uniqueness
Since q1 and q2 are distinguishable, there is a string w such that
But in M’ this cannot happen! M’
M’’ u q1 w q’ w ?
u, v v q2 w

8. Testing equivalence of DFAs
Two DFAs are called equivalent if they accept the same language
q00 q0 1 1
q01 qe 1
q10 1 1 q1 1
q11 1 1 1 qA qB qC 1

9. Testing equivalence of DFAs
How can we tell if M and M’ are equivalent?
Algorithm for testing DFA equivalence:
Run DFA minimization algorithm on M
Run DFA minimization algorithm on M’
Check that M and M’ are the same

10. Example Do the following two NFAs accept the same language? 0,1 0,1 11 q0 q1 q0 1 1 q1

11. Answer We cannot run minimization procedure on NFAs
But we can convert both NFAs to DFA and test equivalence
regular expression
NFA
DFA