1. Summary Showing regular Showing non-regular construct DFA, NFA
construct regular expression
show L is the union, concatenation, intersection (regular operations) of regular languages.
Showing non-regular
pumping lemma
assume regular, apply closure properties of regular languages and obtain a known non-regular language.

2. Equivalence & Minimization of Automata

3. Outline Equivalence of a state in a DFA regular languages
Minimization of DFA
smallest possible DFA for a given regular language

4. Equivalent States. Example
Consider the accept states c and g. They are both sinks meaning that any string which ever reaches them is guaranteed to be accepted later.
Q: Do we need both states?
0,1 b c 1
0,1 1 a d e
0,1 1 1 f g

5. Example (cont) A: No, they can be unified as illustrated below.
Q: Can any other states be unified because any subsequent string suffixes produce identical results? b
0,1 1 1 a d
0,1 e cg 1 1 f

6. Example (cont) A: Yes, b and f. Notice that if you’re in b or f then:
if string ends, reject in both cases
if next character is 0, forever accept in both cases
if next character is 1, forever reject in both cases
So unify b with f. b
0,1 1 1 a d
0,1 e cg 1 1 f

7. Example (cont)
Intuitively two states are equivalent if all subsequent behavior from those states is the same.
Come up with a formal characterization of state equivalence.
Q: Any other ways to simplify the automaton?
0,1 1 a d
0,1 e cg
0,1 1 bf

8. Example (cont) A: Get rid of d.
i.e. Getting rid of unreachable useless states doesn’t affect the accepted language.
0,1 a
0,1 e cg
0,1 1 bf

9. Equivalent States. Definition
Two states q and q’ in a DFA M = (Q, Σ, δ, q0, F ) are said to be equivalent (or indistinguishable) if for all strings u  Σ*, the states on which u ends on when read from q and q’ are both accept, or both non-accept.
Equivalent states may be glued together without affecting M’ s behavior.

10. Minimization Algorithm. Goals
DEF: An automaton is irreducible if
it contains no useless states, and
no two distinct states are equivalent.
The goal of minimization algorithm is to create irreducible automata from arbitrary ones. Later: remarkably, the algorithm actually produces smallest possible DFA for the given language, hence the name “minimization”.

11. Minimization of DFA’s Example: The DFA has equivalence classes
{{A,E}, {B,H}, {C}, {D,F}, {G}}

12. Equivalence Class Transitivity: If p ≡ q and q ≡ r, then p ≡ r
Proof: Suppose to the contrary that p ≢ r
Then
or vice versa. Now is either accepting
or not If
Otherwise
The vice versa case is proved symmetrically

13. Minimization of DFA’s (cont)

14. Example
Can be minimized to

15. Cannot apply the TF-algo to NFA’s
For Example, to minimize
Simply remove state C
However, A ≢ C

16. Minimized DFA Cannot be Beaten
Let B be the minimized DFA obtained by applying the TF-Algo to DFA A. We know that L(A) = L(B)
What if there existed a DFA C with L(C) = L(B) and fewer states than B?
If we run the TF-Algo on B “union” C
Also
If we could distinguish successors then

17. Minimized DFA Cannot be Beaten (cont)
Claim: For each state p in B there is at least one state q in C, s.t.
Proof: There are no inaccessible states, so Now and
Since C has fewer states than B, there must be two states r & s of B s.t For some state t of C. But CONTRADICTION!!