COSC 3340: Introduction to Theory of
Computation
University of Houston
Dr. Verma
Lecture 2
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Lecture 2
UofH - COSC 3340 - Dr. Verma
1st model -- Deterministic Finite
Automaton (DFA)
Read only Head
Finite Control
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UofH - COSC 3340 - Dr. Verma
DFA (contd.)
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The DFA has:
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–
–
–
–
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a finite set of states
1 special state - initial state
0 or more special states - final states
input alphabet
transition table containing
(state, symbol) -> next state
Lecture 2
UofH - COSC 3340 - Dr. Verma
Informally -- How does a DFA work?
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An input string is placed on the tape (left-justified).
DFA begins in the start state.
Head placed on leftmost cell.
DFA goes into a loop until the entire string is read.
–
In each step, DFA consults a transition table and changes
state based on (s, ) where
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s - current state
 - symbol scanned by head
Lecture 2
UofH - COSC 3340 - Dr. Verma
How does a DFA work? (contd.)
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After reading input string,
– if DFA state final, input accepted
– if DFA state not final, input rejected
Language of DFA -- set of all strings
accepted by DFA.
Lecture 2
UofH - COSC 3340 - Dr. Verma
Pictorial representation of DFA
(q,σ) -> q'
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Lecture 2
UofH - COSC 3340 - Dr. Verma
Example: Diagram of DFA
L = {a2n + 1 | n >= 0}
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Answer:
L = {a, aaa, aaaaa, ...}
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Lecture 2
UofH - COSC 3340 - Dr. Verma
JFLAP Step-By-Step
aaa
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UofH - COSC 3340 - Dr. Verma
JFLAP Step-By-Step
aaa
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UofH - COSC 3340 - Dr. Verma
JFLAP Step-By-Step
aaa
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Lecture 2
UofH - COSC 3340 - Dr. Verma
JFLAP Step-By-Step
aaa
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Lecture 2
UofH - COSC 3340 - Dr. Verma
Formal definition of DFA
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DFA M = (Q, , , s, F)
Where,
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Q is finite set of states
 is input alphabet
s  Q is initial state
F  Q is set of final states
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: Q X  -> Q
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–
–
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Lecture 2
UofH - COSC 3340 - Dr. Verma
Formal definition of L(M)
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L(M) - Language accepted by M
Define *:
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*(q, ) = q
*(q, wσ) = (*(q,w),σ)
Definition: L(M) = { w in * | * (s,w) in F }.
Lecture 2
UofH - COSC 3340 - Dr. Verma
Example: L(M) = {w in {a,b}* | w
contains even no. of a's}
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Lecture 2
UofH - COSC 3340 - Dr. Verma
JFLAP Step-By-Step
aa
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Lecture 2
UofH - COSC 3340 - Dr. Verma
JFLAP Step-By-Step
aa
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Lecture 2
UofH - COSC 3340 - Dr. Verma
JFLAP Step-By-Step
aa
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Lecture 2
UofH - COSC 3340 - Dr. Verma
JFLAP Step-By-Step
ab
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Lecture 2
UofH - COSC 3340 - Dr. Verma
JFLAP Step-By-Step
ab
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Lecture 2
UofH - COSC 3340 - Dr. Verma
JFLAP Step-By-Step
ab
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Lecture 2
UofH - COSC 3340 - Dr. Verma
Given a language, how to define DFA?
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Creative process requiring you to:
(i) Put yourself in DFA's shoes
(ii) Find finite amount of info, based on
which string accepted/rejected
(iii) From step (ii), determine number of states and
then transitions.
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Lecture 2
UofH - COSC 3340 - Dr. Verma
Regular Languages
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Definition: A Language is regular iff there is a DFA
that accepts it.
Examples:
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
{}
*
{w in {0,1}* | second symbol of w is a 1}
Exercise
{w in {0,1}* | second last symbol of w is a 1} Exercise
{w in {0,1}* | w contains 010 as a substring} - (importance?)
Lecture 2
UofH - COSC 3340 - Dr. Verma