CSCI 3130: Formal languages
and automata theory
Tutorial 2
Chin
Reminder
• Homework 1 is due at 23:59, today!
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• My office hour: 11:00 – 12:00, Tuesday
Outline
• Regular expression
• Closure properties
• In this (only this) tutorial. Assume S = {0, 1}.
Regular Expressions
• The symbols  and e are regular expressions
• Every a in S is a regular expression
• If R and S are regular expressions, so are R+S,
RS and R*
• Remember for R*, * could be 0
Regular Expressions
• What are their regular expressions?
1. L1 = {w : w is a string of even length}
((0 + 1)(0 + 1))*
2. L2 = {w : w begins with 0 or ends with 0}
0(0 + 1)* + (0 + 1)*0
3. L3 = {w : w contains the pattern 01 at least once}
(0 + 1)* 01 (0 + 1)*
Regular Expressions
• What are their regular expressions?
1. L1 = {w : w is a string of odd length}
2. L2 = {w : w begins with 0 or ends with 0 but not both}
3. L3 = {w : w contains the pattern 01 at least twice}
4. L4 = {w : w contains less 1s than 0s and at most two 0s}
Regular Expressions
1. L1 = {w : w is a string of odd length}
odd means of length 2n + 1
string of length 2n = ((0 + 1)(0 + 1))*
(0 + 1)((0 + 1)(0 + 1))*
Regular Expressions
2. L2 = {w : w begins with 0 or ends with 0 but not both}
w must be of length at least 2
Either starts with 0 and ends with 1, or
starts with 1 and ends with 0
0(0 + 1)*1 + 1(0 + 1)*0
Regular Expressions
3. L3 = {w : w contains the pattern 01 at least twice}
w must contain 01 twice in the expression
can be anything before, after, or between the two
01s
(0 + 1)* 01 (0 + 1)* 01 (0 + 1)*
Regular Expressions
4. L4 = {w : w contains less 1s than 0s and at most two 0s}
There can be at most two 0s
Enumerate all possible cases:
0, 00, 001, 010, 100
OR them using +
0 + 00 + 001 + 010 + 100
Regular Expressions
• What do these regular expressions represent?
1. 0(0 + 1)*0 + 1(0 + 1)*1 + 0 + 1
2. 1*(011*)*
3. (0*10*10*)*
4. ((0 + 1)1)*
Regular Expression
1. 0(0 + 1)*0 + 1(0 + 1)*1 + 0 + 1
First alphabet must be equal to the last alphabet
Regular Expression
2. 1*(011*)*
011* = start with 0 then followed by at least one 1
(011*)* = start with 0 and every 0 must be
followed by at least one 1
1* at the beginning = possibly start with 1s
Every 0 is followed by at least one 1
Regular Expression
3. (0*10*10*)*
There are two 1s inside the brackets
0*10*10* = there can be 0 before, after, or
between the two 1s = there are two 1s
(0*10*10*)* = there are 2n 1s
All strings that contain even number of 1s
Regular Expression
4. ((0 + 1)1)*
(0 + 1)1 = X1, X = 0/1
((0 + 1)1)* = strings of the form X1X1…X1, X = 0/1
Strings of even length and every even position is 1
Closure Properties
• Suppose L and L’ are both regular
1. L is also regular
2. LR is also regular
3. L ∪ L’ is also regular
4. L ∩ L’ is also regular
Closure Properties
• Suppose L and L’ are both regular. Then
L and L’ are also regular
L ∪ L’ is also regular
(L ∪ L’) is also regular
L ∩ L’ = (L ∩ L’) by De Morgan’s law
4. Hence L ∩ L’ is also regular
The construction idea in lecture is also important.
Closure Properties
Let L be a language comprising all strings w such
that
1. w contains an even number of 1s,
2. an odd number of 0s, and
3. no occurrences of the substring 10.
Is L regular?
Closure Properties
• To show a language L is regular,
1. represent L using a DFA / NFA / RE, or
2. use closure properties
Closure Properties
1. L1 = {w : w contains an even number of 1s}
regular
2. L2 = {w : w contains an odd number of 0s}
regular
3. L3 = {w : w contains the substring 10}
regular
Closure Properties
Let L be a language comprising all strings w such
that
1. w contains an even number of 1s,
2. an odd number of 0s
3. no occurrences of the substring 10.
Is L regular? L = L1∩L2∩L3. Hence it is regular.
Regular Expression (Extra)
• L = {1ky : y ∈ {0, 1}* and y contains at least k 1s,
k ≥ 1}
• What is the regular expression of L?
Regular Expression (Extra)
• L = {1ky : y ∈ {0, 1}* and y contains at least k 1s,
k ≥ 1}
1k
y
e.g. 111 0001111 (k = 3) is in L.
Regular Expression (Extra)
• L = {1ky : y ∈ {0, 1}* and y contains at least k 1s,
k ≥ 1}
1k
y
• What about 111 00011?
• Same as 1 1100011. Hence is in L!
• L = Strings that start with 1 and contain at least
two 1s
Non-regular languages
• Homework 1 Problem 4
• What happens if we allow nested parans?
Non-regular languages
•
•
•
•
Consider the case ((((( )))))
same as
L = {(n )n : n ≥ 0}
L’ = {0n1n : n ≥ 0} is NOT regular (lecture)
• More on non-regular languages next week.
The End
• Questions?