notes06 Regular Expressions - CS 373 Theory of Computation...

CS 373: Theory of Computation
Gul Agha Mahesh Viswanathan
Fall 2010
1

1 Operations on Languages
Operations on Languages
• Recall: A language is a set of strings
• We can consider new languages derived from operations on given languages
– e.g., L
1
∪ L
2
, L
1
∩ L
2
,
1
2
L , . . .
• A simple but powerful collection of operations:
– Union, Concatenation and Kleene Closure
Union is a familiar operation on sets. We define and explain the other two operations below.
Concatenation of Languages
Definition 1. Given languages L
1
and L
2
, we define their concatenation to be the language L
1
◦
L
2
= { xy | x ∈ L
1
, y ∈ L
2
}
Example 2 . • L
1
= { hello } and L
2
= { world } then L
1
◦ L
2
= { helloworld }
• L
1
= { 00 , 10 } ; L
2
= { 0 , 1 } . L
1
◦ L
2
= { 000 , 001 , 100 , 101 }
• L
1
= set of strings ending in 0; L
2
= set of strings beginning with 01. L
1
◦ L
2
= set of strings
containing 001 as a substring
• L ◦ { } = L . L ◦ ∅ = ∅ .
Kleene Closure
Definition 3.
L
n
=
(
{ } if n = 0
L
n - 1
◦ L otherwise
L
*
=
[
i ≥ 0
L
i
i.e., L
i
is L ◦ L ◦ · · · ◦ L (concatenation of i copies of L ), for i > 0.
L
*
, the Kleene Closure of L : set of strings formed by taking any number of strings (possibly
none) from L , possibly with repetitions and concatenating all of them.
• If L = { 0 , 1 } , then L
0
= { } , L
2
= { 00 , 01 , 10 , 11 } . L
*
= set of all binary strings (including
).
• ∅
0
= { } . For i > 0, ∅
i
= ∅ . ∅
*
= { }
• ∅ is one of only two languages whose Kleene closure is finite. Which is the other? { }
*
= { } .
2

2 Regular Expressions
2.1 Definition and Identities
Regular Expressions
A Simple Programming Language
Figure 1: Stephen Cole Kleene
A regular expression is a formula for representing a (complex) language in terms of “elementary”
languages combined using the three operations union, concatenation and Kleene closure.
Regular Expressions
Formal Inductive Definition
Syntax and Semantics
A regular expression over an alphabet Σ is of one of the following forms:
Syntax Semantics
∅ L ( ∅ ) = {}
Basis L ( ) = { }
a L ( a ) = { a }
( R
1
∪ R
2
) L (( R
1
∪ R
2
)) = L ( R
1
) ∪ L ( R
2
)
Induction ( R
1
◦ R
2
) L (( R
1
◦ R
2
)) = L ( R
1
) ◦ L ( R
2
)
( R
*
1
) L (( R
*
1
)) = L ( R
1
)
*