Closure Properties of Regular Languages

1
Closure Properties of Regular Languages

• L1 and L2 are regular.
• How about L1?L2, L1?L2 , L1L2 , L1, L1 ?

2
Theorem 1

• If L1 and L2 are regular, then so are L1?L2,
  L1?L2 , L1L2 , L1, L1.
• (The family of regular languages is closed under
  intersection, union, concatenation, complement,
  and star-closure.)

3
Proof

• L1 L(r1) L2 L(r2)
• L(r1 r2) L(r1)?L(r2)
• L(r1 . r2) L(r1)L(r2)
• L(r1) (L(r1))

4
Proof

• M (Q, ?, ?, q0, F) accepts L1.
• M (Q, ?, ?, q0, Q F) accepts L1.

5
Proof

• M1 (Q, ?, ?1, q0, F1) accepts L1.
• M2 (P, ?, ?2, p0, F2) accepts L2.

a1
an
q0
qf
?1(qi, a) qk
a1
an
p0
pf
?2(pj, a) pl
?1((qi, pj), a) (qk, pl)
6
Proof

• Example
• L1 abn n ? 0
• L2 anb n ? 0
• L1?L2 ab

7
Theorem 2

• The family of regular languages is closed under
  reversal
• If L is regular, then so is LR.

8
Proof

• ?

9
Homomorphism

• Suppose ? and ? are alphabets.
• h ? ? ?
• is called a homomorphism

10
Homomorphism

• Extended definition
• w a1a2 ... an
• h(w) h(a1)h(a2) ... h(an)

11
Homomorphism

• If L is a language on ?, then its homomorphic
  image is defined as
• h(L) h(w) w ? L

12
Example

• ? a, b
• ? a, b, c
• h(a) ab h(b) bbc
• h(aba) abbbcab
• L aa, aba ? h(L) abab, abbbcab

13
Homomorphism

• If L is the language on ? of a regular expression
  r,
• then the regular expression for h(L) is obtained
  by
• applying the homomorphism to each ? symbol of r.

14
Example

• ? a, b
• ? b, c, d
• h(a) dbcc h(b) bdc
• r (a b)(aa) L L(r)
• h(r) (dbcc (bdc))(dbccdbcc)

15
Theorem 3

• The family of regular languages is closed under
  homomorphism
• If L is regular, then so is h(L).

16
Proof

• Let L(r) L for some regular expression r.
• Prove h(L(r)) L(h(r)).

17
Homework

• Exercises 2, 4, 6, 8, 9, 11, 18, 22 of Section
  4.1 - Linzs book.
• Presentation Section 4.3 - Linzs book
  (pigeonhole principle pumping lemma and
  examples).

18
Right Quotient

• Let L1 and L2 be languages on the same alphabet.
• Then the right quotient of L1 with L2 is defined
  as
• L1/L2 x xy ? L1 and y ? L2

19
Example

• L1 anbm n ? 1, m ? 0?ba
• L2 bm m ? 1
• L1/L2 ?

20
Example

• L1 anbm n ? 1, m ? 0?ba
• L2 bm m ? 1
• L1/L2 anbm n ? 1, m ? 0

21
Example
L1 anbm n ? 1, m ? 0?ba
b
a
b
a
q0
q1
q2
b
a
q3
q5
a
a, b
a, b
q4
22
Example
L1 anbm n ? 1, m ? 0?ba
L2 bm m ? 1
b
a
b
a
q0
q1
q2
?(q0, x) qi ?(qi, y) ? F and y ? L2
b
a
q3
q5
a
a, b
a, b
q4
23
Example
L1 anbm n ? 1, m ? 0?ba
L2 bm m ? 1
b
a
b
a
q0
q1
q2
?(q0, x) qi ?(qi, y) ? F and y ? L2
b
a
q3
q5
a
a, b
a, b
q4
24
Theorem

• The family of regular languages is closed under
  right quotient
• If L1 and L2 are regular, then so is L1/L2.

25
Proof

• M (Q, ?, ?, q0, F) accepts L1.
• M (P, ?, ?, q0, F) accepts L1/L2.
• If y ? L2 and ?(qi, y) ? F ? add qi to F

26
Proof

• M (Q, ?, ?, q0, F) accepts L1.
• M (P, ?, ?, q0, F) accepts L1/L2.
• If y ? L2 and ?(qi, y) ? F ? add qi to F
• Mi (P, ?, ?, qi, F) and L(Mi) ? L2 ? ?.

27
Example

• L1 L(abaa)
• L2 L(ab)
• L1/L2 ?

28
Example
L1 L(abaa)
L1 L(ab)
a
a
a
b
L(M0) ? L2 ? L(M1) ? L2 a L(M2) ? L2
a L(M3) ? L2 ?
q0
q1
q2
b
b
q3
a, b
29
Example
L1 L(abaa)
L1 L(ab)
a
a
a
b
L(M0) ? L2 ? L(M1) ? L2 a L(M2) ? L2
a L(M3) ? L2 ?
q0
q1
q2
b
b
q3
a, b
30
Questions about RLs

• Given a regular language L on ? and any w ? ?,
• is there an algorithm to determine whether or
  not
• w ? L?.

31
Questions about RLs

• Given a regular language L on ? and any w ? ?,
• is there an algorithm to determine whether or
  not
• w ? L?.
• Yes.

32
Questions about RLs

• Is there an algorithm to determine whether or not
• a regular language is empty, finite, or
  infinite?.

33
Questions about RLs

• Is there an algorithm to determine whether or not
• a regular language is empty, finite, or
  infinite?.
• Yes.

34
Questions about RLs

• Given two regular languages L1 and L2, is there
  an algorithm to determine whether or not L1
  L2?.

35
Questions about RLs

• Given two regular languages L1 and L2, is there
  an algorithm to determine whether or not L1
  L2?.
• Yes.

36
Homework

• Exercises 1, 2, 3, 5, 9 of Section 4.2 - Linzs
  book.
• Exercises 3, 4, 5, 6, 8, 10, 12 of Section 4.3 -
  Linzs book.