Head, Dept' of Computer Science

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A.M. Padma Reddy
Head, Dept. of Computer Science Information
Science
R.N.S Institute of Technology
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What is Finite Automata?

• Study of Abstract machines or computing devices

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ADVANTAGES OF FA

• Checking and design the behavior of the digital
• circuits using software

• Compiler Design (LEX and YACC)

• Designing a software for identifying words,
• phrases and other patterns in large bodies
• of text (such as collection of web pages)

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ADVANTAGES OF FA Theory

• Study the computers without worrying
• about the intricacies of a particular
• architecture

• Study the computers without worrying
• about the limitations imposed by a specific
• hardware

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TYPES OF FA
Deterministic Finite Automata (DFA)
Non Deterministic Finite Automata (NFA)
Non Deterministic Finite Automata with e-moves
(e-NFA)
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Mathematical Concepts
What is A x B?
What is a power set?
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Definitions
What is an alphabet?
An alphabet is a finite non-empty set of symbols
and is denoted by S
S 0,1 S a, b, c
What is a string?
A string is a sequence of symbols from S. For
example 00, 01, 10 and 11. ? - stands for an
empty string
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Definitions
What is a language?
A language over an alphabet S is a set of string
over S. A language can be finite or infinite
For example 00, 01, 10 and 11 L wwR
w ?a,b
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FINITE AUTOMATA
Q q0, q1, q2, q3
The transition table
S a, b
q0 start state
b
a
d
F q3
q0
q1
q2
q1
q0
q3
q2
q3
q0
q3
q2
q1
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Representation of a DFA
d(q0,a) q1
d(q0,b) q2
d(q1,a) q0
b
a
d
d(q1,b) q3
d(q2,a) q3
q0
q1
q2
d(q2,b) q0
q1
q0
q3
d(q3,a) q2
q2
q3
q0
d(q3,b) q1
q3
q2
q1
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What is a DFA?
, q0
)
(
Q
, S
, d
, F
M
Q- Set of finite states
S - Set of input alphabets
d transition from Q x S to Q
q0 start state
F set of final states
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DFA to accept strings of as and bs ending
with abb
a
b
b
q3
q0
q1
q2
q3
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DFA to accept strings of as and bs except
those ending with abb
a
b
b
q3
q0
q1
q2
q3
q3
q0
q1
q2
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DFA to accept strings of as and bs which has
even number of as and bs
a
q0
q0
q1
q0
q1
q1
a
b
b
b
b
a
q2
q2
q3
q2
q3
a
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Non Deterministic Automata
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Definition of NFA

Set of states, i.e.
Input aplhabet, i.e.
Transition function
Initial state
Final states
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NFA vs. DFA

• Difference is in the definition of d
• NFAs and DFAs recognize the same set of
  languages
• regular languages
• DFAs are faster to execute
• There are no choices to consider
• For a given language, NFA can be simple than DFA
  and we can easily construct NFA
• DFA can be exponentially larger than NFA

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NFA
Alphabet
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NFA
Alphabet
Two choices
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NFA
Alphabet
Two choices
No transition
No transition
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First Choice
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First Choice
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First Choice
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First Choice
accept
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Second Choice
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Second Choice
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Second Choice
No transition the automaton hangs
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Second Choice
reject
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Observation
An NFA accepts a string if there is a computation
of the NFA that accepts the string
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Example
is accepted by the NFA
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Lambda Transitions
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(read head doesnt move)
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accept
String is accepted
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Language accepted
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Another NFA Example
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accept
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Another String
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accept
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Language accepted
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Another NFA Example
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Language accepted
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Transition Function
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Extended Transition Function

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Formally
It holds
if and only if
there is a walk from to with label
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The Language of an NFA

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Formally

• The language accepted by NFA is
• where

(final state)
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Equivalence of NFAs and DFAs

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Equivalence of Machines

• For DFAs or NFAs
• Machine is equivalent to machine
• if

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Example
NFA

DFA
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• Since
• machines and are equivalent

NFA
DFA
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Equivalence of NFAs and DFAs
Question NFAs DFAs ?
Same power? Accept the same languages?
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Equivalence of NFAs and DFAs
Question NFAs DFAs ?
YES!
Same power? Accept the same languages?
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We will prove
Languages accepted by NFAs
Languages accepted by DFAs
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We will prove
Languages accepted by NFAs
Languages accepted by DFAs
NFAs and DFAs have the same computation power
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Step 1
Languages accepted by NFAs
Languages accepted by DFAs
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Step 1
Languages accepted by NFAs
Languages accepted by DFAs
Proof
Every DFA is also an NFA
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Step 1
Languages accepted by NFAs
Languages accepted by DFAs
Proof
Every DFA is also an NFA
A language accepted by a DFA is also accepted by
an NFA
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Step 2
Languages accepted by NFAs
Languages accepted by DFAs
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Step 2
Languages accepted by NFAs
Languages accepted by DFAs
Any NFA can be converted to an equivalent DFA
Proof
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Step 2
Languages accepted by NFAs
Languages accepted by DFAs
Any NFA can be converted to an equivalent DFA
Proof
A language accepted by an NFA is also accepted by
a DFA
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NFA to DFA
NFA

DFA
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NFA to DFA
NFA

DFA
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NFA to DFA
NFA

DFA
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NFA to DFA
NFA

DFA
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NFA to DFA
NFA

DFA
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NFA to DFA
NFA

DFA
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NFA to DFA
NFA

DFA
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NFA to DFA Remarks

• We are given an NFA
• We want to convert it
• to an equivalent DFA
• With

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• If the NFA has states
• the DFA has states in the powerset

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Procedure NFA to DFA

• 1. Initial state of NFA
• Initial state of DFA

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Example
NFA

DFA
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Procedure NFA to DFA

• 2. For every DFAs state
• Compute in the NFA
• Add transition

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Exampe
NFA

DFA
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Procedure NFA to DFA

• Repeat Step 2 for all letters in alphabet,
• until
• no more transitions can be added.

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Example
NFA

DFA
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Procedure NFA to DFA

• 3. For any DFA state
• If some is a final state in the NFA
• Then,
• is a final state in the DFA

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Example
NFA

DFA
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Theorem
Take NFA

Apply procedure to obtain DFA
Then and are equivalent

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Finally
We have proven
Languages accepted by NFAs
Languages accepted by DFAs
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We have proven
Languages accepted by NFAs
Languages accepted by DFAs
Regular Languages
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We have proven
Languages accepted by NFAs
Languages accepted by DFAs
Regular Languages
Regular Languages
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How Does Lex Work?
Flex
RE ? NFA NFA ? DFA Optimize DFA
REs for Tokens
DFA Simulation
Character Stream
Token stream (and errors)
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Regular Expression to NFA

• Its possible to construct an NFA from a regular
  expression
• Thompsons construction algorithm
• Build the NFA inductively
• Define rules for each base RE
• Combine for more complex REs

s
f
E
general machine