This document defines and provides examples of deterministic finite automata (DFAs). It begins by defining a DFA as a finite state machine where the transition function is deterministic. Formally, a DFA is defined as a 5-tuple consisting of a finite set of states, a finite input alphabet, a transition function, an initial state, and a set of final/accepting states. The document provides examples of representing DFAs using state diagrams and transition tables and discusses how to represent the extended transition function. It concludes by posing several problems about designing DFAs with certain language definitions.

1. Deterministic
Finite
Automaton
(DFA)
-Sampath Kumar S,
AP/CSE, SECE

2. Deterministic Finite Automaton (DFA)
 In DFA, for each input symbol, one can determine
the state to which the machine will move. Hence, it
is called Deterministic Automaton.
 As it has a finite number of states, the machine is
called Deterministic Finite
Machine or Deterministic Finite Automaton.
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3. Formal definition of DFA:
A DFA can be represented by a 5-tuples (Q, ∑, δ, q0,
F) where −
 Q is a finite set of states.
 ∑ is a finite set of symbols, called the alphabet of
the automaton.
 δ is the transition function.
 q0 is the initial state from where any input is
processed (q0 ∈ Q).
 F is a set of final state/states of Q (F ⊆ Q).
Note: δ is called Delta and ∑ is called Sigma
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4. Graphical Representation of a DFA
A DFA is represented by digraphs called state
diagram or Transition Diagram .
 The vertices represent the states.
 The arcs labeled with an input alphabet show the
transitions.
 The initial state is denoted by an empty single
incoming arc.
 The final state is indicated by double circles.
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5. Transition Table:
 It is a tabular representation of a function like δ that
takes 2 arguments and returns a value.
 The rows of the table correspond to the state and
the columns correspond to the inputs.
 The entry for the row corresponding to state q and
the column corresponding to input a is the state
δ(q, a)
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6. Example
Let a deterministic finite automaton be →
 Q = {a, b, c},
 ∑ = {0, 1},
 q0={a},
 F={c}, and
 Transition function δ as shown by the following
table −
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Present State Next State for Input 0 Next State for Input 1
->a a b
b c a
c* b c

7. Example Cont..,
Its graphical representation would be as follows:
Can you guess the language of above DFA?
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8. Extended Transition Function:
 Describes what happens when we start in any state
and follow any sequence of inputs
 If δ is our transition function, then the extended
transition function is denoted by δ*
 The extended transition function is a function that
takes a state q and a string w and returns a state p
(the state that the automaton reaches when starting
in state q and processing the sequence of inputs
w)
 Formal definition :δ*(q, ǫ) = q
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9. Problems to Discuses:
1. Design DFA to accept string over ∑={0,1} with 2
consecutive 0’s.
2. Construct a DFA that accepts all strings on
∑={0,1} except those contains substring 101.
3. L= { w|w is of even length and begins with 01
over ∑={0,1} }.
4. Give DFA accepting the string starting with
substring 101 over ∑={0,1}
5. Design DFA to accept string contain number of
1’s is in multiples of 3 over ∑={0,1}.
6. Design DFA to accept string contain number of
1’s is not in multiples of 3 over ∑={0,1}.
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10. Problems (Cont..,):
7. Construct a DFA that accepts all strings on
∑={0,1} that contain exactly 4 zeros.
8. Construct a DFA over ∑={a,b} where the number
of b’s is divisible by 3.
9. Construct a DFA that accepts even binary
numbers on ∑={0,1}.
10. Construct a DFA that accepts odd binary numbers
on ∑={0,1}.
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12. நன்றி
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