ECS 120 Lesson 4 – Closure Properties of
Regular Languages, Pt. 1
Oliver Kreylos
Friday, April 6th, 2001
1
Operations on Languages
We defined a (formal) language
L
over an alphabet Σ as a set of words:
L
⊂
Σ
*
. Today we introduce operations on languages, and how the class of
regular languages behaves under such operations. Since languages are sets
of words, the first operations we look at are the set operations complement,
intersection and union.
•
Let
A
⊂
Σ
*
be a language. Then we define the
complement
¯
A
of
A
as
¯
A
:=
w
∈
Σ
*
w /
∈
A
, or, equivalently,
¯
A
= Σ
*
\
A
.
•
Let
A, B
⊂
Σ
*
be languages over the same alphabet. Then we define the
intersection
A
∩
B
of
A
and
B
as
A
∩
B
:=
w
∈
Σ
*
w
∈
A
∧
w
∈
B
,
the usual set intersection; and we define the
union
A
∪
B
of
A
and
B
as
A
∪
B
:=
w
∈
Σ
*
w
∈
A
∨
w
∈
B
, the usual set union.
The next two operations are specific to sets of words:
Concatenation
and Kleene Star. Before we introduce them, we have to formally define the
concatenation of two words.
Let
x
1
, x
2
∈
Σ
*
be two words over the same
alphabet. Then we define the
concatenation
x
1
x
2
of
x
1
and
x
2
recursively as
follows:
1. If
x
1
∈
Σ
0
, i. e.,
x
1
=
, then we define
x
1
x
2
:=
x
2
.
2. If
x
1
∈
Σ
*
\ { }
, i. e.,
x
1
is not the empty word, we can split
x
1
into
a character
a
∈
Σ and a word
x
1
∈
Σ
*
:
x
1
=
ax
1
.
Then we define
x
1
x
2
= (
ax
1
)
x
2
:=
a
(
x
1
x
2
).
1
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Less formally, two words are concatenated by writing their characters as a sin
gle word: If
x
1
=
a
1
a
2
. . . a
n
and
x
2
=
b
1
b
2
. . . b
m
, then
x
1
x
2
=
a
1
a
2
. . . a
n
b
1
b
2
. . . b
n
.
From the formal definition of concatenation, we can derive its following
two properties:
•
Associativity: If
a, b, c
∈
Σ
*
are words over the same alphabet, then
a
(
bc
) = (
ab
)
c
. In other words, concatenating
a
with the result of con
catenating
b
and
c
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 Spring '07
 Filkov
 Naive set theory, Formal language, Regular expression, Regular language

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