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 clumps of even
length.
4) Strings in which the number of a’s is odd.
#6 p. 76 What languages do (*)* and a represent?
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.
Proof:
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 3.10 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:
Section 3.3 regular grammars
Definition 3.3 A grammar G = (V, T, S, P) is said to be right-linear
if
Grammars to Automata and vice versa
Right linear grammars generate regular languages
Grammar over {a, b} that generates all strings of odd length
ending in b.
Sb
S  bA | aA
A  aS | bS
Algorithm to construct an NFA from the grammar
Note: this discussion differs from that in the book.
1.
2.
3.
4.
5.
Theorem 3.3: Let G = (V, T, S, P) be a right-linear grammar. Then
L(G) is a regular language.
Proof sketch—by induction on the length of a derivation
Basis:
Hypothesis: If w  L(G) and w can be derived in n or fewer steps,
then w  L(M) and if w  L(M) and |w|  n, then S derives w in n or
fewer steps.
Induction: Suppose S derives w in n + 1 steps.
Finishing the grammar – NFA proof
Right linear grammars for regular languages.
The grammar has four parts
N = set of variables
T = set of terminals
P = set of productions or rules
S = start symbol
automaton move
(q0, a) = q1
(q0, b) = q2
(q1, a) = q0
(q1, b) = q0
(q2, a) = q0
(q2, b) = q0
{A, B, C}
{a, b}
{A}
grammar production
Finishing the construction
A  aB | bC
B  aA | bA
C  aA | bA
Productions needed to terminate derivations
Theorem 3.4 If L is a regular language on the alphabet  then  a
right-linear grammar G = (V, , S, P) such that L = L(G).
V=
T=
S=
P is defined as follows:
Finally, we have theorem 3.6: A language L is regular iff there
exists a regular grammar G such that L = L(G).
Another example of machine to grammar construction
Let’s let S be EE, A be OE, B be OO and C be EO