Formal Definition of a
Nondeterministic Finite
Automaton
Formal Definition of a Nondeterministic Finite Automaton – p.1/28
A comment first
The formal definition of an NFA is similar to that of a
DFA. Both have states, an alphabet, transition function,
one start state, and a collection of accept states
Formal Definition of a Nondeterministic Finite Automaton – p.2/28
A comment first
The formal definition of an NFA is similar to that of a
DFA. Both have states, an alphabet, transition function,
one start state, and a collection of accept states
Differ in one essential way: the transition functions:
Formal Definition of a Nondeterministic Finite Automaton – p.2/28
A comment first
The formal definition of an NFA is similar to that of a
DFA. Both have states, an alphabet, transition function,
one start state, and a collection of accept states
;
In a DFA transition function is
Differ in one essential way: the transition functions:
Formal Definition of a Nondeterministic Finite Automaton – p.2/28
A comment first
The formal definition of an NFA is similar to that of a
DFA. Both have states, an alphabet, transition function,
one start state, and a collection of accept states
;
In a DFA transition function is
In an NFA it is
Differ in one essential way: the transition functions:
Formal Definition of a Nondeterministic Finite Automaton – p.2/28
A comment first
The formal definition of an NFA is similar to that of a
DFA. Both have states, an alphabet, transition function,
one start state, and a collection of accept states
is the collection of subsets of
power set of
;
called the
- where
In a DFA transition function is
In an NFA it is
Differ in one essential way: the transition functions:
Formal Definition of a Nondeterministic Finite Automaton – p.2/28
for
we write
For any alphabet
Notation
Formal Definition of a Nondeterministic Finite Automaton – p.3/28
NFA formal definition (Def. 1.37)
A nondeterministic finite automaton (NFA) is a 5-tuple
where:
Formal Definition of a Nondeterministic Finite Automaton – p.4/28
NFA formal definition (Def. 1.37)
A nondeterministic finite automaton (NFA) is a 5-tuple
where:
is a finite set of states
Formal Definition of a Nondeterministic Finite Automaton – p.4/28
NFA formal definition (Def. 1.37)
is a finite set of states
A nondeterministic finite automaton (NFA) is a 5-tuple
where:
is a finite alphabet
Formal Definition of a Nondeterministic Finite Automaton – p.4/28
NFA formal definition (Def. 1.37)
is a finite alphabet
is a finite set of states
A nondeterministic finite automaton (NFA) is a 5-tuple
where:
is the transition function
Formal Definition of a Nondeterministic Finite Automaton – p.4/28
NFA formal definition (Def. 1.37)
is a finite alphabet
is a finite set of states
A nondeterministic finite automaton (NFA) is a 5-tuple
where:
is the transition function
is the start state
Formal Definition of a Nondeterministic Finite Automaton – p.4/28
NFA formal definition (Def. 1.37)
is the start state
is the transition function
is a finite alphabet
is a finite set of states
A nondeterministic finite automaton (NFA) is a 5-tuple
where:
is the set of accept states
Formal Definition of a Nondeterministic Finite Automaton – p.4/28
Example NFA (Example 1.38)
, Figure 1:
1
,0
1
0,1
0,1
Figure 1: Nondeterministic Finite Automaton
Recall the NFA
Formal Definition of a Nondeterministic Finite Automaton – p.5/28
where:
is given in the table:
3.
2.
1.
The formal definition of
1
is the start set; 5.
4.
0
Formal Definition of a Nondeterministic Finite Automaton – p.6/28
a string over .
be an NFA and
accepts if:
Let
We say that
Computation performed by an NFA
Formal Definition of a Nondeterministic Finite Automaton – p.7/28
,
,
as
We can write
a string over .
be an NFA and
accepts if:
Let
We say that
Computation performed by an NFA
Formal Definition of a Nondeterministic Finite Automaton – p.7/28
exists in
,
A sequence of states
,
as
We can write
a string over .
be an NFA and
accepts if:
Let
We say that
Computation performed by an NFA
s.t:
Formal Definition of a Nondeterministic Finite Automaton – p.7/28
exists in
,
s.t:
A sequence of states
1.
,
as
We can write
a string over .
be an NFA and
accepts if:
Let
We say that
Computation performed by an NFA
Formal Definition of a Nondeterministic Finite Automaton – p.7/28
exists in
,
s.t:
, for
A sequence of states
2.
,
as
We can write
1.
a string over .
be an NFA and
accepts if:
Let
We say that
Computation performed by an NFA
Formal Definition of a Nondeterministic Finite Automaton – p.7/28
exists in
s.t:
, for
,
3.
A sequence of states
2.
,
as
We can write
1.
a string over .
be an NFA and
accepts if:
Let
We say that
Computation performed by an NFA
Formal Definition of a Nondeterministic Finite Automaton – p.7/28
Interpretation
Condition 1 says that the machine starts its
computation in the start state
Formal Definition of a Nondeterministic Finite Automaton – p.8/28
Interpretation
Condition 1 says that the machine starts its
computation in the start state
Condition 2 says that state
is one of the allowable
. Note
new states when is in state and reads
is a set
that
Formal Definition of a Nondeterministic Finite Automaton – p.8/28
Interpretation
Condition 1 says that the machine starts its
computation in the start state
Condition 2 says that state
is one of the allowable
. Note
new states when is in state and reads
is a set
that
Condition 3 says that the machine accepts the input if
the last state is in the accept state set.
Formal Definition of a Nondeterministic Finite Automaton – p.8/28
An important property
Any NFA can be converted to an equivalent NFA
has a single accept state
that
Formal Definition of a Nondeterministic Finite Automaton – p.9/28
An important property
that
be any NFA. Then
Proof: by construction. Let
where:
Any NFA can be converted to an equivalent NFA
has a single accept state
Formal Definition of a Nondeterministic Finite Automaton – p.9/28
An important property
that
be any NFA. Then
where
1.
Proof: by construction. Let
where:
Any NFA can be converted to an equivalent NFA
has a single accept state
is a new state
Formal Definition of a Nondeterministic Finite Automaton – p.9/28
An important property
that
be any NFA. Then
where
is a new state
2.
1.
Proof: by construction. Let
where:
Any NFA can be converted to an equivalent NFA
has a single accept state
Formal Definition of a Nondeterministic Finite Automaton – p.9/28
An important property
that
be any NFA. Then
where
is a new state
3.
2.
1.
Proof: by construction. Let
where:
Any NFA can be converted to an equivalent NFA
has a single accept state
Formal Definition of a Nondeterministic Finite Automaton – p.9/28
An important property
that
be any NFA. Then
is a new state
where
3.
2.
1.
Proof: by construction. Let
where:
Any NFA can be converted to an equivalent NFA
has a single accept state
;
or
.
and
if
if
4.
Formal Definition of a Nondeterministic Finite Automaton – p.9/28
An important property
that
be any NFA. Then
is a new state
where
3.
2.
1.
Proof: by construction. Let
where:
Any NFA can be converted to an equivalent NFA
has a single accept state
.
and
if
;
5.
or
if
4.
Formal Definition of a Nondeterministic Finite Automaton – p.9/28
Equivalence of NFA and DFA
DFAs and NFAs recognize the same class of
languages
Formal Definition of a Nondeterministic Finite Automaton – p.10/28
Equivalence of NFA and DFA
DFAs and NFAs recognize the same class of
languages
This equivalence is both surprising and useful
Formal Definition of a Nondeterministic Finite Automaton – p.10/28
Equivalence of NFA and DFA
DFAs and NFAs recognize the same class of
languages
This equivalence is both surprising and useful
1. It is surprising because NFAs appears to have more power
than DFA, so we might expect that NFA recognizes more
languages
Formal Definition of a Nondeterministic Finite Automaton – p.10/28
Equivalence of NFA and DFA
DFAs and NFAs recognize the same class of
languages
This equivalence is both surprising and useful
1. It is surprising because NFAs appears to have more power
than DFA, so we might expect that NFA recognizes more
languages
2. It is useful because describing an NFA for a given language
sometimes is much easier than describing a DFA
Formal Definition of a Nondeterministic Finite Automaton – p.10/28
Theorem 1.39
Every nondeterministic finite automation (NFA) has an
equivalent deterministic finite automaton (DFA).
Proof idea: Convert NFA recognizing a language into a DFA
that recognizes the same language by simulating the NFA
Formal Definition of a Nondeterministic Finite Automaton – p.11/28
Theorem 1.39
Every nondeterministic finite automation (NFA) has an
equivalent deterministic finite automaton (DFA).
Proof idea: Convert NFA recognizing a language into a DFA
that recognizes the same language by simulating the NFA
Formal Definition of a Nondeterministic Finite Automaton – p.11/28
More on proof idea
The approach is to pretend to be an DFA simulating an
NFA. Questions to be answered:
Formal Definition of a Nondeterministic Finite Automaton – p.12/28
More on proof idea
The approach is to pretend to be an DFA simulating an
NFA. Questions to be answered:
How would you simulate an NFA if you were pretending
to be a DFA?
Formal Definition of a Nondeterministic Finite Automaton – p.12/28
More on proof idea
The approach is to pretend to be an DFA simulating an
NFA. Questions to be answered:
How would you simulate an NFA if you were pretending
to be a DFA?
What does one need to keep track of as the input is
processed?
Formal Definition of a Nondeterministic Finite Automaton – p.12/28
Note
In the examples of NFA we kept track of various
branches of the computation by placing tokens on
each state that could be activated at given points in the
input
Formal Definition of a Nondeterministic Finite Automaton – p.13/28
Note
In the examples of NFA we kept track of various
branches of the computation by placing tokens on each
state that could be activated at given points in the input
Tokens were updated by moving, adding, and
removing them according to the way NFA operates
Formal Definition of a Nondeterministic Finite Automaton – p.13/28
Note
In the examples of NFA we kept track of various
branches of the computation by placing tokens on each
state that could be activated at given points in the input
Tokens were updated by moving, adding, and
removing them according to the way NFA operates
Hence, all we needed to keep track of was the set of
states with tokens on them.
Formal Definition of a Nondeterministic Finite Automaton – p.13/28
An NFA with
states has
Conclusion
subsets of states
Formal Definition of a Nondeterministic Finite Automaton – p.14/28
An NFA with
states has
Conclusion
subsets of states
Each subset corresponds to one of the possibilities
that DFA must remember. So, the DFA simulating an
NFA with states has states
Formal Definition of a Nondeterministic Finite Automaton – p.14/28
An NFA with
states has
Conclusion
subsets of states
Each subset corresponds to one of the possibilities
that DFA must remember. So, the DFA simulating an
NFA with states has states
The only thing we need now is to figure out which are
the start state, the accept states, and transition
function
Formal Definition of a Nondeterministic Finite Automaton – p.14/28
guage
Let
The formal proof
be the NFA recognizing the lan-
. We construct the DFA
recognizing
. Be-
fore doing the full construction, consider first the easier case
has no transitions.
when
Formal Definition of a Nondeterministic Finite Automaton – p.15/28
1.
; every state of
Constructing
is a set of states of
Formal Definition of a Nondeterministic Finite Automaton – p.16/28
is a set of states of
let
and
2. For
; every state of
1.
Constructing
Formal Definition of a Nondeterministic Finite Automaton – p.16/28
let
is a set of states of
- Note:
and
2. For
; every state of
1.
Constructing
Formal Definition of a Nondeterministic Finite Automaton – p.16/28
let
is a set of states of
;
starts computing in the same state as
3.
- Note:
and
2. For
; every state of
1.
Constructing
Formal Definition of a Nondeterministic Finite Automaton – p.16/28
let
4.
starts computing in the same state as
is a set of states of
;
3.
- Note:
and
2. For
; every state of
1.
Constructing
Formal Definition of a Nondeterministic Finite Automaton – p.16/28
let
- that is, the machine
accepts if one of the possible states that
could be in at this point is an accept state
4.
starts computing in the same state as
is a set of states of
;
3.
- Note:
and
2. For
; every state of
1.
Constructing
Formal Definition of a Nondeterministic Finite Automaton – p.16/28
Consider the transitions
Notation: for any
define
to be the collection of
states that can be reached from by going only along
transitions, including the members of themselves.
Formally:
Formal Definition of a Nondeterministic Finite Automaton – p.17/28
Consider the transitions
Notation: for any
define
to be the collection of
states that can be reached from by going only along
transitions, including the members of themselves.
Formally:
Formal Definition of a Nondeterministic Finite Automaton – p.17/28
Consider the transitions
Notation: for any
define
to be the collection of
states that can be reached from by going only along
transitions, including the members of themselves.
Formally:
Note: any
can be reached from
by traveling
along 0 or more arrows.
Formal Definition of a Nondeterministic Finite Automaton – p.17/28
Modifications
Modify to place additional tokens on all states that
can be reached by going along arrows after every
by
step. This effect is achieved by replacing
in definition of . That is
Formal Definition of a Nondeterministic Finite Automaton – p.18/28
Modifications
Modify to place additional tokens on all states that
can be reached by going along arrows after every
by
step. This effect is achieved by replacing
in definition of . That is
Formal Definition of a Nondeterministic Finite Automaton – p.18/28
Modifications
Modify to place additional tokens on all states that
can be reached by going along arrows after every
by
step. This effect is achieved by replacing
in definition of . That is
Modify the start state of
to put tokens initially on all
states that can be reached from along transitions,
i.e.,
Formal Definition of a Nondeterministic Finite Automaton – p.18/28
Note
thus constructed obviously simulates
Formal Definition of a Nondeterministic Finite Automaton – p.19/28
Note
thus constructed obviously simulates
At every step in the computation of
it clearly enters
a state that corresponds to the subset of states that
could be in at that point
Formal Definition of a Nondeterministic Finite Automaton – p.19/28
Conclusion
Since every NFA can be converted into a DFA by the procedure discussed above, NFA provides an alternative way of
characterizing regular sets
Formal Definition of a Nondeterministic Finite Automaton – p.20/28
Corollary 1.40
A language is regular iff some NFA recognizes it
Formal Definition of a Nondeterministic Finite Automaton – p.21/28
Corollary 1.40
A language is regular iff some NFA recognizes it
Proof:
Formal Definition of a Nondeterministic Finite Automaton – p.21/28
Corollary 1.40
A language is regular iff some NFA recognizes it
Proof:
If a language is recognized by an NFA then
the DFA equivalent, hence, is regular
is recognized by
Formal Definition of a Nondeterministic Finite Automaton – p.21/28
Corollary 1.40
A language is regular iff some NFA recognizes it
Proof:
If a language is recognized by an NFA then
the DFA equivalent, hence, is regular
is recognized by
If a language is regular, it means that it is recognized by a DFA.
But any DFA is also an NFA hence, the language is recognized by
an NFA
Formal Definition of a Nondeterministic Finite Automaton – p.21/28
described in Figure 2:
b
a
a,b
a
Figure 2:
The NFA
Consider the NFA
Example conversion NFA to DFA
Formal Definition of a Nondeterministic Finite Automaton – p.22/28
where
b
a
is given in the table:
Formal description of
Formal Definition of a Nondeterministic Finite Automaton – p.23/28
’s set of states is
)
Since
Construction of
Formal Definition of a Nondeterministic Finite Automaton – p.24/28
’s set of states is
)
: the set of states reachable from 1
) traveling by :
The start state of
(the start state of
Since
Construction of
Formal Definition of a Nondeterministic Finite Automaton – p.24/28
’s set of states is
)
: the set of states reachable from 1
) traveling by :
that
The accept state of : are those states of
contain the accept states of : i.e.,
The start state of
(the start state of
Since
Construction of
Formal Definition of a Nondeterministic Finite Automaton – p.24/28
we have
and
, where for each
Transition function
Formal Definition of a Nondeterministic Finite Automaton – p.25/28
, i.e.,
on both
State : goes to
’s transitions
Constructing
Formal Definition of a Nondeterministic Finite Automaton – p.26/28
and to
on
on
: goes to
, i.e.,
on both
State
State : goes to
’s transitions
Constructing
Formal Definition of a Nondeterministic Finite Automaton – p.26/28
on
and to
on
on
and to
: goes to
on
, i.e.,
on both
: goes to
State
State
State : goes to
’s transitions
Constructing
Formal Definition of a Nondeterministic Finite Automaton – p.26/28
on
and to
on
on
and to
on
on
: goes to
: goes to
and to
, i.e.,
on
State
State
on both
: goes to
State
State : goes to
’s transitions
Constructing
Formal Definition of a Nondeterministic Finite Automaton – p.26/28
on
and to
on
on
and to
on
on
: goes to
: goes to
and to
, i.e.,
on
State
State
on both
: goes to
State
State : goes to
’s transitions
Constructing
And so on, for each state
Formal Definition of a Nondeterministic Finite Automaton – p.26/28
b
a
equivalent to NFA
DFA
a
equivalent to NFA
Figure 3:
a
b
b
a
a
b
a,b
a
b
b
a,b
Figure 3 shows the resulting DFA
Putting all together
Formal Definition of a Nondeterministic Finite Automaton – p.27/28
Simplifying
a,b
b
a
a
The automaton
may be simplified by observing that no
and
. Removing these
arrows point at the states
states we obtain the automaton in Figure 4
a
b
The simplified version of
Figure 4:
a
a
b
b b
Formal Definition of a Nondeterministic Finite Automaton – p.28/28