1. Deterministic Finite Automaton (DFA)
Cpt S 317: Spring 2009
Deterministic Finite Automaton (DFA)
Sampath Kumar S,
AP/CSE, SECE
School of EECS, WSU

2. Deterministic Finite Automaton (DFA)
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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.

3. Formal definition of DFA:
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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

4. Graphical Representation of a DFA
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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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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)

6. Example Let a deterministic finite automaton be → Q = {a, b, c},
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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 −
Present State
Next State for Input 0
Next State for Input 1
->a a b c c*

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Example Cont..,
Its graphical representation would be as follows: Can you guess the language of above DFA?

8. Extended Transition Function:
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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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Problems to Discuses:
Design DFA to accept string over ∑={0,1} with 2 consecutive 0’s.
Construct a DFA that accepts all strings on ∑={0,1} except those contains substring 101.
L= { w|w is of even length and begins with 01 over ∑={0,1} }.
Give DFA accepting the string starting with substring 101 over ∑={0,1}
Design DFA to accept string contain number of 1’s is in multiples of 3 over ∑={0,1}.
Design DFA to accept string contain number of 1’s is not in multiples of 3 over ∑={0,1}.

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Problems (Cont..,):
Construct a DFA that accepts all strings on ∑={0,1} that contain exactly 4 zeros.
Construct a DFA over ∑={a,b} where the number of b’s is divisible by 3.
Construct a DFA that accepts even binary numbers on ∑={0,1}.
Construct a DFA that accepts odd binary numbers on ∑={0,1}.

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