Regular language and Regular expression

Automata Theory and Logic
Regular Language & Regular Expression
A TUTORIAL
BY
ANIMESH CHATURVEDI
AT
INDIAN INSTITUTE OF TECHNOLOGY INDORE (IIT-I)

DFA, NFA, Regular Expression (RegEx)
and Regular Language (RegLang)
A DFA represent a Regular Expression language

Regular Expression Regular Language
(0 + 10∗)
(0∗ 10∗)
(0 + ε)(1 + ε)
(a + b)*
(a + b)* abb
(11)*
(aa)*(bb)*b
(aa + ab + ba +bb)*
Language of given Regular Expression?
https://www.tutorialspoint.com/automata_theory/regular_expressions.htm

(0 + 10∗) L = { 0, 1, 10, 100, 1000, 10000, … }
(0∗ 10∗) L = {1, 01, 10, 010, 0010, …}
(0 + ε)(1 + ε) L = {ε, 0, 1, 01}
(a + b)*
(a + b)* abb
(11)*
(aa)*(bb)*b
(aa + ab + ba +bb)*

(0 + 10∗) L = { 0, 1, 10, 100, 1000, 10000, … }
(0∗ 10∗) L = {1, 01, 10, 010, 0010, …}
(0 + ε)(1 + ε) L = {ε, 0, 1, 01}
(a + b)* Set of strings of a’s and b’s of any length including the null string. So L = { ε, a, b,
aa , ab , bb , ba, aaa…….}
(a + b)* abb Set of strings of a’s and b’s ending with the string abb. So L = {abb, aabb, babb,
aaabb, ababb, …………..}
(11)*
(aa)*(bb)*b
(aa + ab + ba +bb)*

(0 + 10∗) L = { 0, 1, 10, 100, 1000, 10000, … }
(0∗ 10∗) L = {1, 01, 10, 010, 0010, …}
(0 + ε)(1 + ε) L = {ε, 0, 1, 01}
(a + b)* Set of strings of a’s and b’s of any length including the null string. So L = { ε, a, b,
aa , ab , bb , ba, aaa…….}
(a + b)* abb Set of strings of a’s and b’s ending with the string abb. So L = {abb, aabb, babb,
aaabb, ababb, …………..}
(11)* Set consisting of even number of 1’s including empty string, So L= {ε, 11, 1111,
111111, ……….}
(aa)*(bb)*b Set of strings consisting of even number of a’s followed by odd number of b’s , so L = {b, aab, aabbb, aabbbbb, aaaab,
aaaabbb, …………..}
(aa + ab + ba +bb)* String of a’s and b’s of even length can be obtained by concatenating any
combination of the strings aa, ab, ba and bb including null, so L = {aa, ab, ba, bb, aaab, aaba, ………..}

Number of states for a given language
Definition of a language L with alphabet {a} is given as following
L = {ank | k > 0, and n is a positive integer constant}
What is the minimum number of states needed in a DFA to recognize L?
Computer Science GATE 2011

Number of states for a given language
Definition of a language L with alphabet {a} is given as following
L = {ank | k > 0, and n is a positive integer constant}
What is the minimum number of states needed in a DFA to recognize L?
n+1 states are needed in a DFA to recognize L
Let n = 3 and k=1
3+1 = 4 states

Minimal DFA for a given language
Draw a minimal DFA which accepts the same language as a DFA with alphabet ∑ = {0, 1} is given
below {0, 1} ∪ {x ∈ {0, 1}∗ | len(x) ≥ 3}
Konrad Slind Notes on Computation Theory, September 21, 2010

Minimal DFA for a given language
Draw a minimal DFA which accepts the same language as a DFA with alphabet ∑ = {0, 1} is given
below {0n | ∃k. n = 3k + 1}

Convert NFA to DFA for a given RegEx
Construct DFA to accept 00(0+1)*
Dr. Ding-Zhu Du http://www.utdallas.edu/~dzdu/cs4384/lect7.ppt

Construct DFA to accept (0+1)*11

Construct DFA to accept 00(0+1)*11

Convert NFA to DFA for a given RegLang
Construct DFA to accept L(M)=ε

Construct DFA to accept L(M)=Ǿ

Construct DFA to accept L(M)=(0+1)*

Regular Expression
What is the language accepted by the NFA for one literal {a} show below?
Assume ε is the empty string

Regular Expression
What is the language accepted by the NFA for one literal {a} show below?
Assume ε is the empty string
Language accepted by NFA is a+, so complement of this language is {є}

What is the DFA & Regular expression?
30Lexical Analysis by Prof. O. Nierstrasz
Example: the set of strings containing an even number of zeros and an even number of ones

The RE is (0011)*((0110)(0011)*(0110)(0011)*)*

Draw NFA for given regular expressions
For the RE (a|b)*abb ?

35
State s0 has multiple transitions on a! It is a non-deterministic finite automaton
Lexical Analysis by Prof. O. Nierstrasz

36
A language is regular if it is recognized by
a deterministic finite automaton
Whether L = { w | w contains 001} is regular or not?
Check it by building an automaton accepts all and only those strings that contain 001
Steven Rudich: www.cs.cmu.edu/~rudich rudich0123456789

37
q q00
1 0
1
q0 q001
0 0 1
0,1
Whether L = { w | w contains 001} is regular or not?
Check it by building an automaton accepts all and only those strings that contain 001

Whether L = { w | w has an even number of 1s} is regular or not?
Check it by building an automaton accepts all and only those strings that has an even number
of 1s

q0
q1
0 0
1
1
Whether L = { w | w has an even number of 1s} is regular or not?
Check it by building an automaton accepts all and only those strings that has an even number
of 1s

References
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University
http://www.cs.tau.ac.il/~orilahav/CompModelFall10/Compute2-print.pdf
Slide by: Dr. Harry H. Porter http://web.cecs.pdx.edu/~harry/compilers/slides/LexicalPart3.pdf
Slide by Dr. Ding-Zhu Du http://www.utdallas.edu/~dzdu/cs4384/lect7.ppt
Prof. Shunichi Toida http://www.cs.odu.edu/~toida/nerzic/390teched/regular/fa/nfa-2-dfa.html
Lexical Analysis by Prof. O. Nierstrasz
GATE (Graduate Aptitude Test of Engineering) jointly conducted by IIT’s

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