The document discusses the theory of automata and regular expressions. It provides examples of writing regular expressions over alphabets to represent certain languages. It also discusses finite automata, how to construct them from regular expressions, and important properties of regular languages and finite automata.

1. Theory of Automata
By
Amjad Khan Khalil
amjad@aup.edu.pk

2. Regular Expression Practice
Continue
 Write down the regular expression defined over Σ={a,b} where
each string ends with double a’s or double b’s.
 Write down the regular expression defined over Σ={a,b} Where
each string start with double b’s and end with double a’s.
 Write down the regular expression for the language defined
over Σ={a,b} where each string have exactly one double b’s.
 Write down the regular expression for the language defined
over Σ={a,b} where each string have exactly one triple a’s.
 Write down the regular expression defined over Σ={0,1} which
consist of even number of 0’s and even number of 1’s.
 Write down a regular expression for the language defined over
Σ={a,b} where each string consist of odd number of a’s and odd
number of b’s.

3. Practice continue
 Find the shortest string that is in the language represented by the
regular expression a*(ab)*b*.
 Find the shortest string that is not in the language represented by the
regular expression a*(ab)*b*.
 Consider the language S*, where S={aa, b}
How many words this language have of length 2? of length 3?
 Construct a regular expression defining each of the following languages
over the alphabet Σ = {a, b}
 All strings that end in double letter
 All strings that do not end in double letter
 All words that contain exactly two b’s or exactly three b’s, not more
 Show that the following pairs of regular expressions define the same language
over the alphabet Σ = {a, b} or not???
 (ab)*a and a(ba)*
 (a*+b*) and (a+b)*
 (a*+b*)* and (a+b)*

4. Practice Continue
 Write RE of strings of a and b, which
must end on triple b’s
 Write RE of strings of a and c followed
by 3 b’s
 Write RE of strings of a and c, followed
by any numbers of b’s

5. Practice Continue
A language is defined over Σ={0,1} , you need to Write a regular
expression for each of the following sets of binary strings.
 contains at least three consecutive 1's
 Does not contains two consecutive 0’s
 Contains at most two 0’s
 contains the substring 101
 contains the substring 1101100
 Write down the regular expression for the language defined over
Σ={a,b} that does not contain aab

6. Equivalent Regular Expression
 Two or more regular expressions are said to be
equivalent if they generate the same language.
 Exp: Write down the regular expression for the
language defined over Σ={a,b} where each string
ends with aa or bb.
then
R1= (a|b)*(aa|bb)
R2=(a|b)*aa|(a|b)*bb
As both the above mentioned regular expression
defined the same language so both are equivalent.

7. Con’t
 Now consider the following two regular expression
R1= (a*|b*)
R2= (a|b)*
Then both of the regular expressions are
equal??????????????
The answer is NO!
Because the R1 does not generate the string which is
the concatenation of a and b while the R2 generate
such string which is the concatenation of a and b.

8. Con’t
 If R1=(aa|bb) and R2=(a|b) then show that the
following are equal or not?
 R1|R2
 R1.R2
 R1*
 We need to check that all the above Regular
Expression generate the same language or
not.

9. Some important identities
 ϵ.R=R. ϵ=R
 ϵ*=ϵ
 Φ*= ϵ
 R+R=R|R=R
 R*.R*=R*
 (R*)*=R*
 R+ Φ =R
 (ϵ+R)*=(ϵ|R)*=R*
 (R+R)S*=(R|R)S*=RS*

10. What is Regular language?
 The language generated by any regular expression is
called a regular language.
 It is to be noted that if R1, R2 are regular expressions,
corresponding to the languages L1 and L2 then the
languages generated by
R1| R2
R1R2
R2R1 and
R1*( or R2*) are also regular languages.

11. Example
 If R1 = (aa+bb) and R2 = (a+b) then the language of
strings generated by R1+R2, is also a regular
language, expressed by (aa+bb) + (a+b)
 If R1 = (aa+bb) and R2 = (a+b) then the language of
strings generated by R1R2, is also a regular
language, expressed by (aa+bb)(a+b)
 If R = (aa+bb) then the language of strings generated
by R*, is also a regular language, expressed by
(aa+bb)*

12. All finite languages are regular
 Consider the following regular expression
R.E=(aa|ab)
Thus the language generated by this regular
expression is
L={aa,ab}
Which is a language of length 2 where each
string start from a.
Thus L is a regular language.

13. What is finite automaton.(FA)
 It is the fourth way of defining the language and can
be defined as the collection of the following.
 Finite number of states, having one initial and having none,
one or many final states.
 Finite set of input letters (Σ) from which input strings are
formed.
 Finite set of transitions i.e. for each state and for each input
letter there is a transition showing how to move
from one state to another.
It should be noted that for one language there may two or more
FA’s which all be correct but for a single FA there will be exactly
one language.

14. Points to be noted
 We should take care of the following points
while constructing an FA.
 A state may be denoted by any name.
 A transition is denoted by an arrow initiates from one stated
and terminates on another.
 The label(letter or digit) above the arrow shows the inputs.
 If an arrow initiate from a state and terminate on the same
state is called loop.
 The loop represent any combination of the label including
NULL.
 An FA is also called finite state machine or transition
diagram.

15. Points to be noted
 It may be noted that to indicate the initial state an
arrow head can also be placed before that state and
the final state can be indicated by double circle. It is
also to be noted while expressing the FA by its
transition diagram the labels of states are not
necessary.

16. Example
 A language is defined over Σ={a,b} then
construct a finite state machine that
accept any combination of b’s but must
end with a.

17. Example
Σ = {a,b}
States: x, y, z where x is an initial state and z is
final state.
Transitions:
1. At state x reading a, go to state z
2. At state x reading b, go to state y
3. At state y reading a, b go to state y
4. At state z reading a, b go to state z

18. Con’t
 These transition can be expressed by
the table called transition table as
under:

19. Con’t
 The finite state machine or transition diagram
will be as under:
The above transition
diagram is an FA
accepting the language of strings, defined over
Σ = {a, b}, starting with a.
It may be noted that this language may be expressed by
the regular expression a(a + b)*