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Deterministic Finite A u t o m a t o n
Finite Automaton can be classified into two types −
Deterministic Finite Automaton (DFA)
Non-deterministic Finite Automaton (NDFA / NFA)
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.
Formal Definition of a DFA
A DFA can be represented by a 5-tuple (Q, ∑, δ,q0, F) where −
Q is a finite set of states.
∑ is a finite set of symbols called the alphabet.
δ is the transition function where δ:Q × ∑ → Q
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).
Graphical Representation of a DFA
A DFA is represented by digraphs called state 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.
Example
Let a deterministic finite automaton be →
Q = {a, b, c},
∑ = {0, 1},
q0 = {a},
F = {c}, and
, lOMoARcPSD|16248954
Transition function δas shown by the following table −
Present State Next State for Input 0 Next State for Input 1
a a b
b c a
c b c
− Its graphical representation would be as follows
, No n -d e t e rmi n i s t i c Finite A u t o m a t o n
In NDFA, for a particular input symbol, the machine can move to any combination of the states in
the machine. In other words, the exact state to which the machine moves cannot be determined.
Hence, it is called Non-deterministic Automaton. As it has finite number of states, the machine
is called Non-deterministic Finite Machine or Non-deterministic Finite Automaton.
Formal Definition of an NDFA
An NDFA can be represented by a 5-tuple (Q, ∑, δ,q0, F) where −
Q is a finite set of states.
∑ is a finite set of symbols called the alphabets.
δ is the transition function where δ:Q × ∑ → 2Q
(Here the power set of Q (2Q) has been taken because in case of NDFA, from a state,
transition can occur to any combination of Q states)
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).
Graphical Representation of an NDFA: (same a s DFA)
An NDFA is represented by digraphs called state 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.
Example
Let a non-deterministic finite automaton be →
Q = {a, b, c}
∑ = {0, 1}
q0 = {a}
F = {c}
The transition function δas shown below −
, lOMoARcPSD|16248954
Present State Next State for Input 0 Next State for Input 1
a a, b b
b c a, c
c b, c c
Its graphical representation would be as follows −
DFA vs NDFA
The following table lists the differences between DFA and NDFA.
DFA NDFA
The transition from a state is to a single The transition from a state can be to multiple
particular next state for each input symbol. next states for each input symbol. Hence it is
Hence it is called deterministic. called non-deterministic.
Empty string transitions are not seen in DFA. NDFA permits empty string transitions.
Backtracking is allowed in DFA In NDFA, backtracking is not always possible.
Requires more space. Requires less space.
A string is accepted by a DFA, if it transits to a A string is accepted by a NDFA, if at least one of
final state. all possible transitions ends in a final state.
Acceptors, Classifiers, and Transducers
Acceptor (Recognizer)
An automaton that computes a Boolean function is called an acceptor. All the states of an
acceptor is either accepting or rejecting the inputs given to it.
Classifier
A classifier has more than two final states and it gives a single output when it terminates.
Transducer
Deterministic Finite A u t o m a t o n
Finite Automaton can be classified into two types −
Deterministic Finite Automaton (DFA)
Non-deterministic Finite Automaton (NDFA / NFA)
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.
Formal Definition of a DFA
A DFA can be represented by a 5-tuple (Q, ∑, δ,q0, F) where −
Q is a finite set of states.
∑ is a finite set of symbols called the alphabet.
δ is the transition function where δ:Q × ∑ → Q
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).
Graphical Representation of a DFA
A DFA is represented by digraphs called state 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.
Example
Let a deterministic finite automaton be →
Q = {a, b, c},
∑ = {0, 1},
q0 = {a},
F = {c}, and
, lOMoARcPSD|16248954
Transition function δas shown by the following table −
Present State Next State for Input 0 Next State for Input 1
a a b
b c a
c b c
− Its graphical representation would be as follows
, No n -d e t e rmi n i s t i c Finite A u t o m a t o n
In NDFA, for a particular input symbol, the machine can move to any combination of the states in
the machine. In other words, the exact state to which the machine moves cannot be determined.
Hence, it is called Non-deterministic Automaton. As it has finite number of states, the machine
is called Non-deterministic Finite Machine or Non-deterministic Finite Automaton.
Formal Definition of an NDFA
An NDFA can be represented by a 5-tuple (Q, ∑, δ,q0, F) where −
Q is a finite set of states.
∑ is a finite set of symbols called the alphabets.
δ is the transition function where δ:Q × ∑ → 2Q
(Here the power set of Q (2Q) has been taken because in case of NDFA, from a state,
transition can occur to any combination of Q states)
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).
Graphical Representation of an NDFA: (same a s DFA)
An NDFA is represented by digraphs called state 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.
Example
Let a non-deterministic finite automaton be →
Q = {a, b, c}
∑ = {0, 1}
q0 = {a}
F = {c}
The transition function δas shown below −
, lOMoARcPSD|16248954
Present State Next State for Input 0 Next State for Input 1
a a, b b
b c a, c
c b, c c
Its graphical representation would be as follows −
DFA vs NDFA
The following table lists the differences between DFA and NDFA.
DFA NDFA
The transition from a state is to a single The transition from a state can be to multiple
particular next state for each input symbol. next states for each input symbol. Hence it is
Hence it is called deterministic. called non-deterministic.
Empty string transitions are not seen in DFA. NDFA permits empty string transitions.
Backtracking is allowed in DFA In NDFA, backtracking is not always possible.
Requires more space. Requires less space.
A string is accepted by a DFA, if it transits to a A string is accepted by a NDFA, if at least one of
final state. all possible transitions ends in a final state.
Acceptors, Classifiers, and Transducers
Acceptor (Recognizer)
An automaton that computes a Boolean function is called an acceptor. All the states of an
acceptor is either accepting or rejecting the inputs given to it.
Classifier
A classifier has more than two final states and it gives a single output when it terminates.
Transducer
