Chapter 3 Regular languages and grammars
Section 3.1 regular expressions
A regular expression is
operators:
Examples:
1) A variable name:
2) All even length strings:
3) All strings ending in 00 or 11:
Definition: Let  be an alphabet. Then
1.
2.
3.
Associated with each regular expression r is a language we
denote by L(r).
1.
2.
3.
For regular expressions r1 and r2,
4. L(r1 + r2) =
5. L(r1• r2) =
6. L((r1)) =
7. L(r1*) =
operator precedence:
Examples: Assume the alphabet is  = {a, b}
1) Strings that contain consecutive a’s:
2) The complement of the language in 1)
3) All strings over Σ = {a, b} in which b's occur in runs of even
length. Class exercise Answer:
4) Strings in which the number of a’s is odd.
Class exercise Answer:
5) #7 p. 76 What languages do (*)* and a represent?
6) Strings that contain both aa and bb
Answer:
Class exercise
7) Strings in which aa occurs at least twice
Class exercise
8) Strings that contain both ab and ba
Class Exercise
Answer:
9) Strings over {a, b, c} in which the total number of b’s and c’s is
three
Section 3.2 Connection between Regular Expressions and
Regular Languages
Theorem 3.1 Let r be a regular expression. Then,  an NFA that
accepts L(r). Consequently, L(r) is a regular language.
The NFA constructed in the proof has the following properties:
1.
2.
3.
The induction proof below shows how to construct an NFA from a
regular expression.
Regular expression to NFA construction
Basis: We first construct automata for the three basis cases: Ø,
, and a
Hypothesis:
Induction step:
Finishing the Proof
Case 1:
Case 2:
Case 3
Example: automaton for (a + b)*ab
Regular expressions for regular languages
Basic idea:
generalized transition graph
Example: eliminate vertex q2 in the figure below.
Result:
Theorem 3.2: Let L be a regular language. Then there exists a
regular expression r such that L = L(r).
Proof:
Finishing the proof
Example 3.11 on page 84.
EE OE EE
OO OE OO
EE OE OO
OO OE EE
Thus we have:
Finishing the Example
Now, we need to remove OO
EE OO EE
EO OO EO
EE OO EO
EO OO EE
Final diagram is: