1. Properties of Regular Languages
Draw DFA for L={e}
for a regular language, L={e, 01}
for the r.l., L={e, 01, 0011}
for the r.l., L={e, 01, 0011, } …
now, try for the r.l., L={0k1k | 0≤k ≤ infinity}

2. Pumping Lemma for Regular Languages
Pumping Lemma relates the size of string accepted
with the number of states in a DFA
What is the largest string accepted by a DFA with n states,
presuming there is no loop?
Now, if there is a loop,
what type of strings are accepted via the loop(s)?

3. Pumping Lemma for Regular Languages
Lemma: (the pumping lemma)
Let M be a DFA with |Q| = n states.
If there exists a string x in L(M), such that |x|  n,
then there exists a way to write it as x = uvw,
where u,v, and w are all in Σ* and:
1 |uv|  n
|v|  1
such that, the strings uviw are also in L(M), for all i  0

4. Pumping Lemma for Regular Languages
Proof:
Let x = a1a2 … am where m  n, x is in L(M), and δ(q0, a1a2 … ap) = qjp
a1 a2 a3 … am
qj0 qj1 qj2 qj3… qjm m  n and qj0 is q0
Consider the first n symbols, and first n+1 states on the above path:
a1 a2 a3 … an
qj0 qj1 qj2 qj3… qjn
Since |Q| = n, it follows from the pigeon-hole principle that js = jt for some 0  s<t  n, i.e., some state appears on this path twice (perhaps many states appear more than once, but at least one does).

5. … … … same loop q0 qj1 qjs qjt qjn q0 qjs = qjt qjn a1 a2 as as+1 at
… … …
same
loop q0
qj1
qjs
qjt a1 a2 as
as+1 at
at+1
qjn an q0
qjs = qjt
qjn
a1…as
at+1…an
as+1…at

6. Since 0  s<t  n and uv = a1…at it follows that:
Let:
u = a1…as
v = as+1…at
Since 0  s<t  n and uv = a1…at it follows that:
1  |v| and therefore 1  |uv|
|uv|  n and therefore 1  |uv|  n
In addition, let:
w = at+1…am
It follows that uviw = a1…as(as+1…at)iat+1…am is in L(M), for all i  0. 
In other words, when processing the accepted string x, the loop was traversed once, but could have been traversed as many times as desired, and the corresponding strings would be accepted.

7. q0 0/1 q4 q2 q1 q3 1 Example: n = 5 x = 0001000 is in L(M)
q0 q1 q2 q3 q1 q2 q3 q4
first n+1 states
u = 0
v = 001
w = uviw is in L(M), i.e., 0(001)i000 is in L(M), for all i  0 q0
0/1 q4 q2 q1 q3 1

8. Note that this does not mean that every string accepted by the DFA has this form:
001 is in L(M) but is not of the form 0(001)i000
Similarly, this does not mean that every long string accepted by the DFA has this form:
is in L(M), is very long, but is not of the form 0(001)i000
Note, however, in this latter case could be similarly decomposed. q0
0/1 q4 q2 q1 q3 1

9. Note: It may be the case that no x in L(M) has |x|  n.q0
0/1 q1 q2

10. Example:
n = 4
x = bbbab is in L(M) b b b a b
|x| = q0 q0 q0 q0 q1 q3
u = ε u = b u = bb
v = b or v = b v = b
w = bbab w = bab w = ab
(b)ibbab is in L(M), for all i  0 b(b)ibab is in L(M), for all i  0 a b a q0 q1 b b a q2 q3
a/b

11. NonRegularity Example
Theorem: The language:
L = {0k1k | k  0} (1)
is not regular.
Proof: (by contradiction) Suppose that L is regular. Then there exists a DFA M such that:
L = L(M) (2)
We will show that M accepts some strings not in L, contradicting (2).
Suppose that M has n states, and consider a string x=0m1m, where m>>n.
By (1), x is in L.
By (2), x is also in L(M), note that the machine accepts a language not just a string

12. Since |x| = m >> n, it follows from the pumping lemma that:
x = uvw
1  |uv|  n
1  |v|, and
uviw is in L(M), for all i  0
Since 1  |uv|  n and n<<m, it follows that 1  |uv| < m.
Also, since x = 0m1m it follows that uv is a substring of 0m.
In other words v=0j, for some j  1.
Since uviw is in L(M), for all i  0, it follows that 0m+cj1m is in L(M), for all c  1.
But by (1) and (2), 0m+cj1m is not in L(M), for any c  1, a contradiction. 
Note that L basically corresponds to balanced parenthesis.

13. NonRegularity Example
Theorem: The language:
L = {0k1k2k | k  0} (1)
is not regular.
Proof: (by contradiction) Suppose that L is regular. Then there exists a DFA M such that:
L = L(M) (2)
We will show that M accepts some strings not in L, contradicting (2).
Suppose that M has n states, and consider a string x=0m1m2m, where m>>n.
By (1), x is in L.
By (2), x is also in L(M), note that the machine accepts a language not just a string

14. Since |x| = m >> n, it follows from the pumping lemma that:
x = uvw
1  |uv|  n
1  |v|, and
uviw is in L(M), for all i  0
Since 1  |uv|  n and n<<m, it follows that 1  |uv|  m.
Also, since x = 0m1m2m it follows that uv is a substring of 0m.
In other words v=0j, for some j  1.
Since uviw is in L(M), for all i  0, it follows that 0m+cj1m2m is in L(M), for all c  1.
But by (1) and (2), 0m+cj1m2m is not in L(M), for any c  1, a contradiction. 
Note that the above proof is almost identical to the previous proof.

15. NonRegularity Example
Theorem: The language:
L = {0m1n2m+n | m,n  0} (1)
is not regular.
Proof: (by contradiction) Suppose that L is regular. Then there exists a DFA M such that:
L = L(M) (2)
We will show that M accepts some strings not in L, contradicting (2).
Suppose that M has n states, and consider a string x=0m1n2m+n, where m>>n.
By (1), x is in L.
By (2), x is also in L(M).

16. Since |x| = m >> n, it follows from the pumping lemma that:
x = uvw
1  |uv|  n
1  |v|, and
uviw is in L(M), for all i  0
Since 1  |uv|  n and n<<m, it follows that 1  |uv|  m.
Also, since x = 0m1n2m+n it follows that uv is a substring of 0m.
In other words v=0j, for some j  1.
Since uviw is in L(M), for all i  0, it follows that 0m+cj1m2m+n is in L(M), for all c  1. In other words v can be “pumped” as many times as we like, and we still get a string in L(M).
But by (1) and (2), 0m+cj1n2m+n is not in L(M), for any c  1, because the acceptable expression should be 0m+cj1n2m+cj+n, a contradiction. 
Note that the above proof is almost identical to the previous proof.

17. Theorem: Let M = (Q, Σ, δ, q0, F) be a DFA
Theorem: Let M = (Q, Σ, δ, q0, F) be a DFA. Then L(M) is finite iff |x| < |Q| for all x in L(M).
Proof:
(if) Suppose that |x| < |Q| for all x in L(M). Since the number of states |Q| and the number of input symbols |Σ| are both fixed, it follows that there are at most a finite number of strings of length less than |Q|. It follows that L(M) is finite (exercise: give an upper bound on the number of such strings).
(only if) By contradiction. Suppose that L(M) is finite, but that |x|  |Q| for some x in L(M). From the pumping lemma it follows that x=uvw, |v|  1 and uviw is in L(M) for all i  0. But then L(M) would be infinite, a contradiction. 

18. Theorem: Let M = (Q, Σ, δ, q0, F) be a DFA
Theorem: Let M = (Q, Σ, δ, q0, F) be a DFA. Then L(M) is infinite iff there exists an x in L(M) such that |x|  |Q|.
Proof:
(if) Suppose there exists an x in L(M) such that |x|  |Q|. From the pumping lemma it follows that x=uvw, |v|  1 and uviw is in L(M) for all i  0. Therefore L(M) is infinite.
(only if) By contradiction. Suppose that L(M) is infinite, but that there is no x in L(M) such that |x|  |Q|. It follows that each x in L(M) has length less than |Q|. Since the number of states |Q| and the number of input symbols |Σ| are both fixed, it follows that there are at most a finite number of strings of length less than |Q|. It follows that L(M) is finite. A contradiction. 
Note that the above also follows directly from the previous theorem.

19. Theorem: Let M = (Q, Σ, δ, q0, F) be a DFA
Theorem: Let M = (Q, Σ, δ, q0, F) be a DFA. Then L(M) is non-empty iff there exists an x in L(M) such that |x| < |Q|.
Proof:
(if) Suppose there exists a string x in L(M) such that |x| < |Q|. Then clearly L(M) is non-empty.
(only if) By contradiction. Suppose that L(M) is non-empty, but that there exists no string x in L(M) such that |x| < |Q|. It follows that |y|  n, where n = |Q|, for all y in L(M). Let z be a string of shortest length in L(M). Then |z|  n and, by the pumping lemma z=uvw, |v|  1 and uviw is in L(M) for all i  0. But then uv0w = uw is in L(M) and:
|uw| = |z| - |v|
 |z| - 1
< |z|
Since uw is in L(M), it follows that z is not a string of shortest length in L(M), a contradiction.

20. Corollary: Let M = (Q, Σ, δ, q0, F) be a DFA
Corollary: Let M = (Q, Σ, δ, q0, F) be a DFA. Then there is an algorithm to determine if L(M) is empty.
Proof:
From the theorem it follows that if L(M) is non-empty then there exists a string x where |x| < n and n = |Q| such that M accepts x. We can try running M on each string of length < n to see if any are accepted. If one is accepted then L(M) is non-empty, otherwise L(M) is empty. Since the number of states |Q| and the number of input symbols |Σ| are both fixed, it follows that there are at most a finite number of strings (of length less than |Q|) that need to be tested.
Note that Dijkstra’s algorithm works here too…

21. Theorem: Let M = (Q, Σ, δ, q0, F) be a DFA and let n = |Q|
Theorem: Let M = (Q, Σ, δ, q0, F) be a DFA and let n = |Q|. Then L(M) is infinite iff there exists an x in L(M) such that n  |x| < 2n.
Proof: (left as an exercise; similar to the previous theorem).
(Contradiction) Suppose there is no string of length between n and 2n in the language accepted by DFA M with n states.
Suppose a string z is the shortest string accepted by the machine M, such that |z|  2n. Since |z| > n, by pumping lemma z = uvw, and 1  |v|  n, because |uv|  n. Then z’=uw must be accepted by the machine M. So, |z’| = |z| - |v| cannot be < n. As per assumption, |z’|  2n is not possible as z is the shortest such string and |z|>|z’|. And we presumed that there is no string of size between n and 2n, so, n  |z’|  2n cannot be true either. Contradiction.
Corollary: Let M = (Q, Σ, δ, q0, F) be a DFA. Then there is an algorithm to determine if L(M) is infinite.
Proof: (left as an exercise).

22. Closure Properties of Regular Languages
Consider various operations on languages:
= {x | x is in Σ* and x is not in L}
= {x | x is in L1 or L2}
= {x | x is in L1 and L2}
= {x | x is in L1 but not in L2}
L1L2 = {xy | x is in L1 and y is in L2}
L* = Li = L0 U L1 U L2 U…
L+ = Li = L1 U L2 U…

23. Theorem: Let M = (Q, Σ, δ, q0, F) be a DFA
Theorem: Let M = (Q, Σ, δ, q0, F) be a DFA. Then M’ = (Q, Σ, δ, q0, Q-F) is a DFA where L(M’) = Σ* - L(M).
Proof: (left as an exercise).
Corollary: The regular languages are closed under complementation, i.e., if L is a regular language then so is .
Does the above construction work for NFAs?

24. Theorem: The regular languages are closed with respect to union, concatenation and Kleene closure.
Proof: Let L1 and L2 be regular languages. By definition there exist regular expressions r1 and r2 such that L(r1) = L1 and L(r2) = L2. But then r1 + r2 is a regular expression representing Similarly for concatenation and Kleene closure.

25. Lemma: Let L1 and L2 be subsets of Σ*. Then .
Theorem: Let L1 and L2 be regular languages. Then is a regular language.
Proof: Let L1 and L2 be regular languages. Then:
is regular
But by the lemma: 

26. A direct construction of a DFA M such that :
Let:
M1 = (Q1, Σ, δ1,p0,F1), where Q1 = {p0, p1,…}
M2 = (Q2, Σ, δ2,q0,F2), where Q2 = {q0, q1,…}
where L(M1) = L1 and L(M2) = L2.
Construct M where:
Q = Q1 x Q2 Note that M has a state for each
= {[p0, q0], [p0, q1], [p0, q2],…} pair of states in M1 and M2
Σ = as with M1 and M2
F = F1 x F2
start state = [p0, q0]
δ([pi, qj], a) = [δ1(pi, a), δ2(qj, a)] for all [pi, qj] in Q and a in Σ

27. p0 p1 1 1 q0 q1 1 q2 0/1 Example: The construct gives: Q = Q1 x Q2
= {[p0, q0], [p0, q1], [p0, q2], [p1, q0], [p1, q1], [p1, q2]}
Σ = {0, 1}
F = {[p1, q2]}
start state = [p0, q0]
δ – Some examples: δ([p0, q2], 0) = [δ1(p0, 0), δ2(q2, 0)] = [p1, q2]
δ([p1, q2], 1) = [δ1(p1, 1), δ2(q2, 1)] = [p1, q2] p0 p1 1 1 q0 q1 1 q2
0/1

28. Question: What if Σ1 does not equal Σ2?
Final Result:
Question: What if Σ1 does not equal Σ2?
Exercise: Prove that is a regular language. 1 1 1
p0, q0
p0, q1
p0, q2
p1, q0
p1, q1
p1, q2 1 1 1