Chapter 2
Finite Automata
(part a)
Hokkaido, Japan
1
Outline
(part a --- in this PPT)
2.0 Introduction
2.1 An Informal Picture of Finite Automata
2.2 Deterministic Finite automata
2.3 Nondeterministic Finite Automata
(part b --- in another PPT)
2.4 An Application: Text Search
2.5 Finite Automata with Epsilon-Transitions
2
2.0 Introduction
Two types of finite automata (FA):
– Deterministic FA (DFA)
– Nondeterministic FA (NFA)
Both types are of the same power,
– The former is easier for hardware and software
implementations.
– The latter is more efficient for descriptions of
applications of FA.
3
2.1 An Informal Picture of Finite
Automata
A complete application example of
finite automata for protocol design
and verification
– Read by yourself!
– Involving a concept of “product of two
automata”
4
2.2 Deterministic Finite Automata
Recall the example of a vending machine
selling 20-dolllar food in Chapter 0
$10
$5
Start
$0
$15
$5
$5
$10
$10
$5 $10
$10
$20
$5
transition diagram
5
2.2 Deterministic Finite Automata
2.2.1 Definition of DFA
– A DFA A consists of 5-tuples (1/2):
a finite (nonempty) set of states Q;
a finite (nonempty) set of input symbols S;
a (state) transition function d such that
d(q, a) = p
means

“automaton A, in state q, takes input a
and enters state p;”
6
2.2 Deterministic Finite Automata
2.2.1 Definition of DFA
– A DFA A consists of 5-tuples (2/2):
a start state q0;
a set of (nonempty) final or accepting
states F.
– A may be written as a “5-tuple”
A = (Q, S, d, q0, F).
7
2.2 Deterministic Finite Automata
2.2.1 Definition of DFA
– graphic model for a DFA
tape
磁帶
0 1 1 0 1 0 0 …
讀取頭
tape reader
有限
finite
control
控制器
圖 4.2
一部有限自動機的圖形模式。
8
2.2 Deterministic Finite Automata
2.2.2 How a DFA processes strings
–
Given an input string x = a1a2…an, if
d(q0, a1) = q1, d(q1, a2) = q2, …,
d(qi1, ai) = qi, ..., d(qn1, an) = qn,
and qn  F,
then x is “accepted”; otherwise, “rejected.”
–
Every transition is deterministic.
9
2.2 Deterministic Finite automata
Example 2.1
– Design an FA A to accept the language
L = {x01y | x and y are any strings of 0’s and 1’s}.
– Examples of strings in L:
01, 11010, 100011…
10
2.2 Deterministic Finite automata
Example 2.1
– Transitions:
d(q0, 1) = q0, d(q0, 0) = q2, d(q2, 1) = q1,
d(q2, 0) = q2, d(q1, 0) = q1, d(q1, 1) = q1
– 5-Tuple:
A = ({q0, q1, q2}, {0, 1}, d, q0, {q1})
11
2.2 Deterministic Finite automata
2.2.3 Simpler Notations for DFA’s
– Transition diagram --1
Start
q0
0
0
q2
1
q1
0, 1
12
2.2 Deterministic Finite automata
2.2.3 Simpler Notations for DFA’s
– Transition table --0
1
 q0
q2
q0
*q1
q1
q1
q2
q2
q1
: initial state; *: final state
13
dˆ
2.2 Deterministic Finite automata
2.2.4 Extending transition function to
strings
– Extended transition function dˆ
If x = a1a2…an and d is such that
d(p, a1) = q1, d(q1, a2) = q2, …,
d(qi1, ai) = qi, ..., d(qn1, an) = q,
then we define dˆ to be
dˆ (p, x) = q.
14
dˆ
2.2 Deterministic Finite automata
2.2.4 Extending Transition Function to
Strings
– Also may be defined recursively as in the textbook.
– Recursive definition for dˆ
Basis:dˆ (q, e) = q.
Induction: if w = xa (a is the last symbol of w), then
dˆ (q, w) = d(dˆ(q, x), a).
15
dˆ
2.2 Deterministic Finite automata
2.2.4 Extending Transition Function to
Strings
A graphic diagram for the following concept:
Induction:
if w = xa (a is the last symbol of w), then
dˆ(q, w) = d (dˆ(q, x), a).
…
x
dˆ (q, x)
a
d
0, 1
1 w=xa)
dˆq(q,
16
dˆ
2.2 Deterministic Finite automata
2.2.5 The Language of a DFA
– The language of a DFA A is defined as
L(A) = {w | dˆ(q0, w) is in F}.
– If L is L(A) for some DFA A, then we say L is a
regular language.
17
2.3 Nondeterministic Finite
Automata
2.3.1 An informal view of NFA’s
– Review of a previous example of DFA (of
Example 2.1)
1
Start
q0
0
0
q2
1
q1
$20
0, 1
– An NFA version of above DFA (more intuitive!)
0, 1
Start
q0
0
q2
1
q1
$20
0, 1
18
2.3 Nondeterministic Finite
Automata
Some properties of NFA’s
0, 1
Start
q0
0
q2
1
q1
$20
0, 1
– Some transitions may “die,” like d(q2, 0).
– Some transitions have multiple choices, like
d(q0, 0) = q0 and q2.
19
2.3 Nondeterministic Finite
Automata
Example 2.6
– Design an NFA accepting the following language
L = {w | w{0, 1}* and ends in 01}.
0, 1
Start
q0
0
q1
1
q2
$20
– Nondeterminism creates many transition paths, but
if there is one path leading to a final state, then the
input is accepted.
20
2.3 Nondeterministic Finite
Automata
Example 2.6 (cont’d)
0, 1
0
q0
Start
q1
1
q2
$20
– When input x = 00101, the NFA processes x in the
following way:
q0
Fig. 2.10
0
q0
q0
q1
Stuck!
q1
q0
q0
q0
q1
q2 Stuck!
0
1
0
1
q2
Accept!
21
2.3 Nondeterministic Finite
Automata
2.3.2 Definition of NFA
– An NFA A is a 5-tuple A = (Q, S, d, q0, F) where
Q = a finite (nonempty) set of states;
S = a finite (nonempty) set of input symbols;
q0 = a start state;
F = a set of (nonempty) final or accepting states;
22
2.3 Nondeterministic Finite
Automata
2.3.2 Definition of NFA
– An NFA A is a 5-tuple A = (Q, S, d, q0, F) where
d = a (state) transition function such that
d(q, a) = {p1, p2, …, pm}
which means


“automaton A, in state q, takes input a and enters
one of the states p1, p2, …, pm.”
23
2.3 Nondeterministic Finite
Automata
Example 2.7
– Transition table of NFA of the last example --0, 1
Start
0
q0
q1
1
q2
$20
0
1
{q0, q1}
{q0}
q1

{q2}
*q2


 q0
24
2.3 Nondeterministic Finite
Automata
2.3.3 Extended Transition Function
– Recursive definition of dˆ (with string as input)
Basis: dˆ(q, e) = {q}.
Induction: if w = xa (a is the last symbol of w),
dˆ(q, x) = {p1, p2, …, pk}, and
k
U d(pi, a) = {r1, r2, …, rm}
then
i=1
dˆ (q, w) = {r1, r2, …, rm}.
25
dˆ
2.3 Nondeterministic Finite
Automata (supplemental)
2.3.3 Extended Transition Function
– A graphic diagram for the following concept:
Induction: if w = xa (a is the last symbol of w),
dˆ (q, x) = {p1, p2, …, pk}, and d(pi, a) = {r1, r2, …, rm}
then dˆ (q, w) = {r1, r2, …, rm}.
dˆ (q, x)=
…
x
{ p1
p2
.
.
.
pk}
a
a
a
d
dˆ (q, w=xa)
= {r1
r2
.
.
.
rm}
26
2.3 Nondeterministic Finite
Automata
Example 2.8 --–
For the input w = 00101, we have
1. dˆ(q0, e) = {q0}.
2. dˆ(q0, e 0) = d(q0, 0) = {q0, q1}.
3. dˆ(q0, 00) = d(q0, 0)∪d(q1, 0) = {q0, q1}∪ ={q0, q1}
0, 1
…
Start
q0
0
q1
1
q2
$20
27
2.3 Nondeterministic Finite
Automata
2.3.4 The Language of an NFA
–
Definition --If A = (Q, S, d, q0, F) is an NFA, then the language
accepted by A is
L(A) = {w d| ˆ (q0, w)∩F  }.
–
That is, as long as a final state is reached, which is
among possibly many, the input is accepted.
28
2.3 Nondeterministic Finite
Automata
2.3.5 Equivalence of DFA and NFA
– NFA’s are more intuitive to construct for many
languages.
– DFA’s are easier for use in software and hardware
implementations.
29
2.3 Nondeterministic Finite
Automata
2.3.5 Equivalence of DFA and NFA
– Every NFA N has an equivalent DFA D,
i.e.,
L(D) = L(N).
(Definition: “Two models are said to be equivalent
if they have identical languages.”)
– The proof needs “subset construction.”
30
2.3 Nondeterministic Finite
Automata
2.3.5 Equivalence of DFA and NFA
– Subset construction: each state of DFA is a subset
of NFA
– Given an NFA N = (QN, S, dN, q0, FN), we
construct a DFA D = (QD, S, dD, q0', FD) such that:
The two alphabets are identical;
QD is the power set of QN (i.e., set of all subsets of QN);
Inaccessible states in QD should be deleted;
(continued in the next page)
31
2.3 Nondeterministic Finite
Automata
2.3.5 Equivalence of DFA and NFA
– (cont’d)
q0' = {q0};
Each subset S of QN is in final state set FD if S
includes at least one accepting state in FN (i.e.,
S∩FN  ).
For each subset S of QN and for each input
symbol a, define
dD(S, a) = U dN(p, a)
pS
32
2.3 Nondeterministic Finite
Automata
Example 2.10 --– For the NFA of Example 2.6 below:
0, 1
Start
q0
0
q1
1
q2
$20
– The power set of QN = {q0, q1, q2} is
QD = {, {q0}, {q1}, {q2}, {q0, q1}, {q0, q2},
{q1, q2}, {q0, q1, q2}}.
33
2.3 Nondeterministic Finite
Automata
Example 2.10 (cont’d)
0, 1
Start
q0
0
– The transition table is
q1
1

{q0}
{q1}
*{q2}
{q0, q1}
*{q0, q2}
*{q1, q2}
*{q0, q1, q2}
q2
$20
0
1

{q0, q1}


{q0, q1}
{q0, q1}

{q0, q1}

{q0}
{q2}

{q0, q2}
{q0}
{q2}
{q0, q2}
34
2.3 Nondeterministic Finite
Automata
Example 2.10 (cont’d)
– Change names of the states, we get
A
B
C
*D
E
*F
*G
*H
0
A
E
A
A
E
E
A
E
1
A
B
D
A
F
B
D
F
35
2.3 Nondeterministic Finite
Automata
Example 2.10 (cont’d)
– Checking accessible states from the start state as
red ones; the others are inaccessible.
Figure 2.13
A
B
C
*D
E
*F
*G
*H
0
A
E
A
A
E
E
A
E
1
A
B
D
A
F
B
D
F
36
2.3 Nondeterministic Finite
Automata
Example 2.10 (cont’d) (supplemental)
– solved by “Lazy evaluation” using table:
starting from the start state, only accessible states are
evaluated, in order to avoid generation of inaccessible states.
0, 1
Start
q0
0
q1
1
q2
$20
– The transition table is

{q00}
{q11}
*{q22}
{q00, q11}
*{q00, q22}
*{q11, q22}
*{q00, q11, q22}
0

{q00, q11}


{q00, q11}
{q00, q11}

{q00, q11}
1

{q00}
{q22}

{q00, q22}
{q00}
{q22}
{q00, q22}
37
2.3 Nondeterministic Finite
Automata
Example 2.10 (cont’d) ---
– solved by “Lazy evaluation” using diagram:
starting from the start state, only accessible states are
evaluated, in order to avoid generation of inaccessible
states.
1
Start
{q0}
0
0
{q0, q1}
1
{q0, q2}
0
Figure 2.14
1
38
2.3 Nondeterministic Finite
Automata
Theorem 2.11
If DFA D = (QD, S, dD, {q0}, FD) is constructed
from NFA N = (QN, S, dN, q0, FN) by subset
construction, then L(D) = L(N).
– Proof. See the textbook.
39
2.3 Nondeterministic Finite
Automata
Theorem 2.12 (equivalence of DFA and
NFA):
A language L is accepted by some DFA if and only if
L is accepted by some NFA.
– Proof. See the textbook.
40
2.3 Nondeterministic Finite
Automata
2.3.6 A Bad Case of Subset
Construction
– The DFA constructed from an NFA usually has the
same number of accessible states of the NFA.
– But the worse case is m = 2n
where
m is the number of states in the DFA.
n is the number of states in the NFA.
41
2.3 Nondeterministic Finite
Automata
2.3.7a Regarding NFA’s as DFA’s
– In each state of a DFA, for each input there must
be a next state.
– In an NFA, for a certain input there might be no
next state. So the following is an NFA.
start
t
t
h
th
e
the
n
then
42
2.3 Nondeterministic Finite
Automata
2.3.7a Regarding NFA’s as DFA’s
– But it is deterministic in nature --- either getting
into a next state or becoming “dead” (getting into a
“dead” state).
start
t
t
h
th
e
the
n
then
– Such a kind of NFA has at most one transition out
of any state on any symbol, and may be regarded
as a DFA.
43