Automata and Formal Language (chapter 3) ppt

Regular Language and
Regular Grammar
Objectives
•
Regular Expression and Regular
Language
•
Regular Expression vs Regular Language
•
Regular Grammar
Regular Expression
Alphabet Σ
1. ∅, λ, a∈Σ are regular expressions
(known as primitive regular
expressions).
2. If r
1
and r
2
are regular expressions,
so are r
1
+ r
2
, r
1
. r
2
, r
1
*

, and (r
1
).
Operator Precedence
•
parentheses
•
star-closure (*)
•
concatenation (.)
•
union (+)
Languages Associated with
Regular Expressions
•
Each regular expression stands for a set
of strings of symbols in Σ  each regular
expression represents a language, called
regular language
•
r L(r)
Example 3.1
•
L(a) = {a}
•
L((a + b.c)* ) = {λ, a, bc, aa, abc, bca,
bcbc, aaa, aabc, }
•
L(a + b +)  syntax error
Regular Languages

1. L(∅) = {}
2. L(λ) = {λ}
3. L(a) = {a}
4. L(r
1
+ r
2
) = L(r
1
)∪L(r
2
)
5. L(r
1
. r
2
) = L(r
1
)L(r
2
)
6. L(r
1
*
) = (L(r
1
))
*
7. L((r
1

)) = L(r
1
)
Example 3.2
L(a
*
. (a + b))
= L(a
*
) L(a + b)
= (L(a))
*
(L(a)∪L(b))
= {λ, a, aa, aaa, }.{a, b}
= {a, aa, aaa, , b, ab, aab, }
Example 3.3
r = (a + b)
*
(a + bb)
L(r) = ?
Example 3.3
r = (a + b)
*
(a + bb)
L(r) = {w| w ends with a or bb}
Example 3.5
r = (aa)
*
(bb)
*

b
L(r) = ?
Example 3.5
r = (aa)
*
(bb)
*
b
L(r) = {a
2n
b
2m+1
: n ≥ 0, m ≥ 0}
Example 3.6
L(r) = {w∈{0, 1}
*
| w has at least one
pair of consecutive zeros}
r =?
Example 3.6
L(r) = {w∈{0, 1}
*
| w has at least one
pair of consecutive zeros}
r = (0 + 1)
*
00 (0 + 1)
*
Example 3.7
L(r) = {w∈{0, 1}

*
| w has no pair of
consecutive zeros}
r = ?
Example 3.7
L(r) = {w∈{0, 1}
*
| w has no pair of
consecutive zeros}
r = (1 + 01)
*
(0 + λ)
Equivalent Regular Expression
r
1
and r
2
are equivalent iff L(r
1
) = L(r
2
)
Example 3.8
r
1
= a . (b + c) r
2
= a . b + a . c
L(r
1

) = L(r
2
) = {ab, ac}
Regular Expressions and
Languages
Given a regular expression r, there exists
an NFA that accepts L(r)  L(r) is a
regular language
Primitive NFAs
q
0
q
1
NFA that accepts ∅
q
0
q
1
NFA that accepts {λ}
λ
q
0
q
1
NFA that accepts {a}
a
Primitive NFAs (cont’d)
NFA that accepts L(r
1
+ r

2
)
M(r
1
)
M(r
2
)
λ
λ λ
λ
Primitive NFAs (cont’d)
M(r
1
) M(r
2
)
λλ λ
Primitive NFAs (cont’d)
M(r
1
) M(r
2
)
λλ λ
Primitive NFAs (cont’d)
NFA that accepts L(r
1
*
)

M(r
1
)
λ λ
λ
λ
Example 3.9
b
λ
b bλ
λ
λ
λ
λ
a
λ
λ
λ
λ
a
λ
λ
λ
λ
λ
λ
λ
λ
λ
λ

r = (a + bb)*(ba* + λ)