Unit 2
Finite Automata
Regular Languages
Reading: Sipser, chapter 1
1
State Diagram
A simplified version of Finite Automaton
2
Automatic door controller
• Let‟s design a controller for the automatic
door in our neighborhood supermarket
front
pad
door
rear
pad
3
• Front pad detects a person about to walk
through.
• Rear pad detects a person standing behind
the door.
• Two states for the door:
– Open
– Close
• Four possible inputs:
–
–
–
–
Front
Rear
Both
Neither
front
pad
rear
pad
4
State Diagram
for automatic door controller
front
closed
neither
rear
both
open
neither
front
rear
both
5
State Transition Table
for automatic door controller
Input signal
state
Neither
Front
Rear
Both
Closed
closed
open
closed
closed
Open
closed
open
open
open
6
Please Welcome
our first computation model:
Finite Automata
7
1
0
0
1
q1
q2
q3
0,1
8
1100
1
0
q1
1100
1
0
q2
1100
q3
1100
0,1
The machine accepts a string if the process
ends in a double circled state
9
Finite Automata (FA)
• Our first formal model of computation.
• Consists of states (nodes) and moves (edges).
• The transition between states is according to
an input word. Each symbol dictates one move.
• Some states are „good‟ (accepting states) and
some are „bad‟ (rejecting states).
• A word is accepted by the Automaton if the
transition it dictates ends in an accepting state.
10
Graphical Representation
•A circle represents a state of the automaton.
•The transition function is represented by
directed and labeled edges between states.
a
•Accepting states have a double circle.
•The start state has an incoming arrow.
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Formal Definition
A Finite Automaton (FA) is a 5-tuple (Q, , , qs, F):
• Q - a finite set called the states.
•  - a finite set called the alphabet.
• :QQ is the transition function.
• qsQ is the start state.
• FQ is the set of accepting states.
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Example:
0
0
1
1
q1
q2
q3
0,1
• A= (Q, , , qs, F) where
– Q={q1, q2, q3}
– ={0, 1}
– q1 is the starting state
– F={q2}
–  is defined as
0
1
q1
q1
q2
q2
q3
q2
q3
q2
q2
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Another Example
a
b
a
b
a
b
• Formal definition: B= (Q, , , qs, F)
– continue in class
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How does an Automaton work?
• Reads one symbol of the input word (i.e. one
letter) in each time slot.
• In each time slot it moves to a new state
(defined in transition function) according to the
current state and the symbol read
• The automaton stops after the last symbol in
the input word is read.
• If the state in which it stops is accepting - the
automaton accepts the input word.
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Formal Definition of Computation
A finite automaton M = (Q,,,qs,F) accepts a
string/word w = w1…wn if and only if there is a
sequence r0…rn of states in Q such that:
1) r0 = qs
2) (ri,wi+1) = ri+1 for all i = 0,…,n–1
3) rn  F
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The Language of an Automaton
• The language of an automaton A is
denoted L(A)
• L(A) consists of all the words that A
accepts, i.e. when reading them it stops
in an accepting state.
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Recognizing the Language of an
Automaton

B:
q0
q1

The language of B:
L(B)= { w | w is over ={0,1} and |w| is even}
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More Examples
• An automaton
accepting the
language *
• an automaton
accepting the empty
language 


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Previous Example
1
0
0
1
q1
q2
q3
0,1
The language of A:
L(A)= { w | w contains at least one 1 and even
number of 0 follows the last 1}
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Designing an Automaton
Construct an automaton B accepting the
following language:
L(B)= { w | w is over ={a,b,c} and |w| is odd}
a,b,c
B:
a,b,c
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Another Example
• Construct an automaton accepting the
following language:
L= { w {0,1}* | |w| mod 4=0}





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Another Example
• Construct an automaton C accepting the
following language:
L(C)= { w over ={a,b,c} | w * and the last letter of w is c}
c
C:
c
a,b
a,b
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Exercise
Construct an automaton accepting the language
L= { w over ={0,1} | |w|=1 or |w|≥3}
Answer: in class
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State Diagram
• A graphical representation of a finite
automaton FA is also called a state diagram.
• Given a formal definition of the FA one can
draw its state diagram.
• Given a state diagram one can. write a formal
definition of the FA.
25
From state diagram to a formal
description
• Consider the finite-state automaton A defined
over {a,b} by the state diagram shown below:
a
b
a
b
a
b
QA={q0,q1,q2}, A={a,b},
start state is q0,
FA ={q0,q1},
A – see table
What is L(A)? What is ~L(A)
A
a
b
q0
q1
q0
q1
q1
q2
q2
q1
q0
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Transition function
• The transition function  is a function
: QQ
• It defines the movement of an automaton
from one state to another.
• Transition function input is an ordered pair:
(current state, current input symbol).
• For each pair of "current state" and "current
input symbol" the transition function produces
as output the next state in the automaton.
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Example
• (q0,a)=q1 means that state q1 is reachable
from state q0 by the transition labeled by the
input symbol a.
a
q0
q1
• where
• q0 is the input state also called “current state”
• a is the an input symbol
• q1 is the output state also called “next state”
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From formal description to a
state diagram
• Draw a state diagram according to given Q, , , q0,
and F.
Q={q0,q1,q2,q3},
={0,1},
q0 initial,
F={q1}.

0
1
q0 q1 q3
q1 q1 q2
q2 q1 q2
q3 q3 q3
• Answer: In class
29
Extended transition function
• Let : QQ a transition function
• We define an extended transition function ‟: Q*Q
• The extended transition function ‟ defines the
movement of an automaton on words.
• Let w= u (w,u- words, - symbol)
• ‟(q,w) = ‟(q,u)=  (‟(q,u), )=  (p, )= r

u
q
p
r
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Extended transition function
Formal definition:
• For all qQ, and for all ,
‟(q,)=(q,) and ‟(q,)=q
• ‟(q,u)=  (‟(q,u), )
’(q,u)
u
q

r
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The Language of an Automaton
• Informally: L(A) is the words that A accepts.
• Formally:
Let A={Q,,,q0,F},
L(A)={x| x*, ‟( q0,x)F}
• We say that A recognizes L(A)
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Regular Languages
A language is called a regular language iff
some finite automaton recognizes it.
33
Two FA Questions
• Given the description of a finite automaton
M = (Q,,,q,F), what is the language L(M)
that it recognizes?
• In general, what kind of languages can be
recognized by finite automata?
(What are the regular languages?)
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The automaton of a language
In-class exercise:
1. Describe formally and graphically an
automaton, A, for the language
L={w over ={0,1} and #1(w) is even}
2. Prove that L(A)=L.
35
More Exercises
Give DFA state diagrams for the following
languages over ={0,1}:
1. { w | w begins with 1 and ends with 0}
2. { w | w does not contain substring 110}
3. {w | w contains 110 as a substring}
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