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Theory of Computation
Nondeterminism & Nondeterministic
Finite Automata (NFA), Epsilon Closure,
NFA with & w/o Epsilon Transitions,
NFA vs. DFA, Subset Construction
Vladimir Kulyukin
Department of Computer Science
Utah State University
http://www.vkedco.blogspot.com/
Outline
• Nondeterminism & Nondeterministic
Finite Automata (NFA)
• Epsilon Closure
• Equivalence of NFAs with & without
Epsilon Transitions
• NFAs vs. DFAs
• Subset Construction: Equivalence of NFAs
& DFAs
Nondeterminism
&
Nondeterministic Finite Automata (NFA)
Nondeterminism
• Given an input, there can be more than
one legal sequence of steps to process
the input
• The input is accepted if at least one legal
sequence of moves ends up in an
accepting state
• The input is accepted if a final state is
reached on it
Nondeterministic Example
q0 q1 q2
a,b
a a
a,b
a b
q0 {q0, q1} {q0}
q1 {q2} { }
q2 {q2} {q2}
On the input aa this NFA
can reach q0, q1, and q2
Nondeterministic Example
q0 q1 q2
a,b
a a
a,b
2100
* ,,, qqqaaq
Power Sets
. of size theis whereelements, 2 has general,In
.3,2,1,3,2,3,1,2,1,3,2,1,
Then .3,2,1Let
.,,,,Then .,Let
:Examples
.
is ofset power The set. a be Let
SnSP
SP
S
babaSPbaS
SR|R P(S)
SS
n
NFA: Definition
states. accepting ofset theis
state;start theis
;:
symbols; ofset finite a i.e. alphabet,an is
states; ofset finite a is
where,,,,, tuple-5 a is NFA An
0
0
QF
QPQ
Q
FqQMM
Instantaneous Descriptions (IDs)
. with ends and with startsthat
relations moreor zero of sequence a is thereif
only and if i.e. IDs,on relation a is
., ifonly and if ,
,or any and stringany For
input. theofpart unread theis and
statecurrent theis where,,pair a is
NFA an for IDn descriptio ousinstantaneAn
**
*
*
JI
JI
aqrr,xaxq
aax
x
Qqxq
NFAs & Their Languages
. then NFA,an is If
.on NFA by the reached becan that
states all ofset the;,,|,
0
**
F,xq|δΣx MLM
x
rxqrxq
**
Epsilon Closure
Spontaneous Transitions
• Unlike DFAs, NFAs are allowed to make
transitions without reading any input
symbols
• When such spontaneous transitions (aka
ε–transitions) occur, the NFA is said to
read ε
• Note: when a spontaneous transition
occurs, the NFA may change a state
without reading any input
Example: Spontaneous Transition from q0
q0
q1
q2
ε
ε
a
b
.**
baML
Epsilon Closures
• When we deal with NFAs with ε-
transitions, we have to keep track of
where we can get to not only on symbols
but also on ε
• ε-closure(q) = {r | there is a path from
q to r consisting of only ε-transitions}
Epsilon Closure Example
q0 q1 q2
0 1
2
0 1 2 ε
q0 {q0} { } { } {q1}
q1 { } {q1} { } {q2}
q2 { } { } {q2} { }
ε ε
Epsilon Closure Example
.,,
closure
closure
0,0,0,closure
0,,,closure
0,,closure0, .4
.,,, .3
.,closure .2
.,,closure .1
210
0
0
210
210
0
*
0
2100
*
211
2100
qqq
q
q
qqq
qqq
qqqq
qqq
qqqq
Equivalence of NFAs with ε-transitions
and NFAs w/o ε-transitions
• Theorem: If L is accepted by an NFA with
ε-transitions, L is accepted by an NFA
without ε-transitions.
• Proof: Use ε-closure to get rid of ε-
transitions.
Epsilon Closure Example
q0 q1 q2
0 1
2
0 1 2 ε
q0 {q0} { } { } {q1}
q1 { } {q1} { } {q2}
q2 { } { } {q2} { }
ε ε
http://www.vkedco.blogspot.com/
Epsilon Closure Example
.,, are states final The
.0,closure2,
.1,closure1,
.0,closure0,
.2,closure2,
.,1,closure1,
.0,closure0,
.2,closure2,
.,1,closure1,
.,,0,closure0,
. from obtain We
s.transition that w/ofunction ion transitnew
thedenote Let s.transition of ridget We
210
222
'
22
'
22
'
211
'
2111
'
11
'
200
'
2100
'
21000
'
'
'
qqq
qqq
qqq
qqqq
qqq
qqqq
qqqqq
Epsilon Closure Example
q0 q1 q2 0,1 1,2
0,1,2
0 1 2
Practical Implication
If an NFA has ε-transitions, we can always
get rid of them before constructing an
equivalent DFA. So we can, without loss
of generality, assume that NFAs do not
contain ε-transitions.
NFA vs. DFA
Equivalence of NFAs and DFAs
• Basic insight: A DFA can keep track of the
states that the equivalent NFA may be in after
reading each symbol of the input
• Since the NFA may be in more than one state
after reading a symbol, each state of the DFA
must correspond to a subset of the NFA’s
states
• The construction of an equivalent DFA from an
NFA is called subset construction
Subset Construction Example
NFA
q0 q1 q2
0,1
1 0,1
DFA? a NFA to isconvert th wedo How
.1 is of symbollast next to the|1,0*
xxML
State q0; Input 0
If the NFA is in q0 and reads 0, it returns back to
q0. So, on 0, the DFA moves from {q0} back to
{q0}:
{q0} 0
State q0; Input 1
If the NFA is in q0 and reads 1, it goes to q0 or
q1, i.e., {q0, q1}. So, on 1, the DFA moves from
{q0} to {q0, q1}:
{q0} 0 {q0, q1} 1
State {q0, q1}; Input 0
Suppose that the NFA is in one of {q0, q1} and reads 0. If it is in q0, it goes to q0. If it is in q1, it goes to q2. So, on 0, the DFA goes from {q0, q1} to {q0, q2}:
{q0} 0 {q0, q1} 1
{q0, q2} 0
http://www.vkedco.blogspot.com/
State {q0, q1}; Input 1
Suppose that the NFA is in one of {q0, q1} and reads 1. If it is in q0, it goes to q0 or q1, i.e., {q0, q1}. If it is in q1, it goes to q2. So, on 1, the DFA goes from {q0, q1} to {q0, q1, q2}:
{q0} 0 {q0, q1} 1
{q0, q2} 0 {q0, q1, q2}
1
State {q0, q2}; Input 0
Suppose that the NFA is in one of {q0, q2} and reads 0. If it is in q0, it goes to q0. If it is in q2, it goes nowhere. So, on 0, the DFA from {q0, q2} goes to {q0}:
{q0} 0 {q0, q1} 1
{q0, q2} 0 {q0, q1, q2}
1 0
State {q0, q2}; Input 1
Suppose that the NFA is in one of {q0, q2} and reads 1. If it is in q0, it goes to q0 or q1, i.e., {q0, q1}. If it is in q2, it goes nowhere. So, on 1, the DFA goes from {q0, q2} to {q0, q1}:
{q0} 0 {q0, q1} 1
{q0, q2} 0 {q0, q1, q2}
1 0
1
State {q0, q1, q2}; Input 0
Suppose that the NFA is in one of {q0, q1, q2} and reads 0. If it is in q0, it will go to q0. If it is in q1, it will go to q2. If it is in q2, it will go nowhere. So, on 0, the DFA goes from {q0, q1, q2} to {q0, q2}:
{q0} 0 {q0, q1} 1
{q0, q2}
0
{q0, q1, q2}
1 0
1
0
State {q0, q1, q2}; Input 1 Suppose that the NFA is in one of {q0, q1, q2} and reads 1. If it is in q0, it will go to q0 or q1, i.e., {q0, q1}. If it is in q1, it goes to q2. If it is in q2, it goes nowhere. So, on 1, the DFA goes from {q0, q1, q2} back to {q0, q1, q2}:
{q0} 0 {q0, q1} 1
{q0, q2}
0
{q0, q1, q2}
1 0
1
0 1
DFA’s Final States
What are the final states in the new DFA? Any subset that has a non-empty intersection with the NFA’s set of final states. So, the DFA has two final states: {q0, q2} and {q0, q1, q2}.
{q0} 0 {q0, q1} 1
{q0, q2}
0
{q0, q1, q2}
1 0
1
0 1
Subset Construction Theorem
. allfor ,,, on,constructiBy
.| .4
; .3
;,,,, .2
; 1.
where,,,,DFA new a
Construct .,,,,Let :Proof
.such that DFA a is Then there
ansitions.epsilon tr NFA w/o a be Let :Theorem
***
*
xxqxq
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aQRaraR
QPQ
FqQD
FqQN
DLNLD
N
NNDD
NDD
ND
D
Rr
ND
ND
DDDD
NNNN
NFA vs. DFA: Relative Advantages
& Disadvantages
• NFAs are simpler to write, because, in
general, they have fewer states and allow
for spontaneous transitions with epsilon-
moves
• However, they are not more powerful
than DFAs, i.e. they accept the same
regular languages as DFAs
• For every NFA, one can construct a DFA
that accepts the same language
References
• A. Brooks Weber. Formal Language: A
Practical Introduction. Franklin, Beedle &
Associates.
• M. Davis, R. Sigal, E. Weyuker.
Computability, Complexity, & Languages:
Fundamentals of Theorectical Computer
Science. Academic Press.
