Chapter 2 (Part a) Finite Automata

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Chapter 2 Chapter 2

Finite AutomataFinite Automata(part a)(part a)

Hokkaido, Japan

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OutlineOutline

(part a --- in this PPT)(part a --- in this PPT)

2.0 Introduction2.0 Introduction

2.1 An Informal Picture of Finite Automata 2.1 An Informal Picture of Finite Automata

2.2 Deterministic Finite automata2.2 Deterministic Finite automata

2.3 Nondeterministic Finite Automata2.3 Nondeterministic Finite Automata

(part b --- in another PPT)(part b --- in another PPT)

2.4 An Application: Text Search2.4 An Application: Text Search

2.5 Finite Automata with Epsilon-Transitions2.5 Finite Automata with Epsilon-Transitions

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2.0 Introduction2.0 Introduction

Two types of finite automata (FA):Two types of finite automata (FA):– Deterministic FA (DFA)Deterministic FA (DFA)

– Nondeterministic FA (NFA)Nondeterministic FA (NFA)

Both types are of the same power, Both types are of the same power, – The former is easier for hardware and software The former is easier for hardware and software

implementationsimplementations..

– The latter is more efficient for The latter is more efficient for descriptionsdescriptions of of applications of FA.applications of FA.

44

2.1 An Informal Picture of Finite 2.1 An Informal Picture of Finite Automata Automata

A complete application example of A complete application example of

finite automata for protocol design finite automata for protocol design

and verificationand verification

– Read by yourself!Read by yourself!

– Involving a concept of “product of two Involving a concept of “product of two

automata”automata”

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2.2 Deterministic Finite Automata2.2 Deterministic Finite Automata

Recall the example of a vending machine Recall the example of a vending machine selling 20-dolllar food in Chapter 0selling 20-dolllar food in Chapter 0

$0StartStart

$5 $5

$10 $10

$5

$20

$15

$5

$10

$5

$10

$10$10

$20 transition transition diagramdiagram

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2.2.1 Definition of DFA2.2.1 Definition of DFA– A DFA A DFA AA consists of 5-tuples (1/2): consists of 5-tuples (1/2):

a finite (nonempty) set of states a finite (nonempty) set of states QQ;; a finite (nonempty) set of input symbols a finite (nonempty) set of input symbols ;; a (state) transition function a (state) transition function such that such that

qq, , aa) = ) = pp

meansmeans

“ “automaton automaton AA, in state , in state qq, takes input , takes input aa and enters state and enters state pp;”;”

77


2.2.1 Definition of DFA2.2.1 Definition of DFA

– A DFA A DFA AA consists of 5-tuples (2/2): consists of 5-tuples (2/2):

a start state a start state qq00;;

a set of (nonempty) a set of (nonempty) finalfinal or or acceptingaccepting

states states FF..

– AA may be written as a “5-tuple” may be written as a “5-tuple”

AA = ( = (QQ, , , , , , qq00, , FF))..

88


2.2.1 Definition of DFA2.2.1 Definition of DFA– graphic model for a DFAgraphic model for a DFA

0 11 0 1 0

有限

控制器

讀取頭

磁帶

…

圖 4.2 一部有限自動機的圖形模式。

0

tape

tape reader

finite

control

99


2.2.2 How a DFA processes strings2.2.2 How a DFA processes strings

– Given an input string Given an input string xx = = aa11aa22……aann, if , if

((qq00, , aa11) = ) = qq11, , ((qq11, , aa22) = ) = qq22, …, , …,

qqii11, , aaii) = ) = qqii, ..., , ..., ((qqnn11, , aann) = ) = qqnn, ,

and and qqnn FF,,

then then xx is “ is “acceptedaccepted”; otherwise, “”; otherwise, “rejectedrejected.”.”

– Every transition is Every transition is deterministicdeterministic..

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Example 2.1Example 2.1– Design an FA Design an FA AA to accept the language to accept the language

LL = { = {xx0101yy | | xx and and yy are any strings of 0’s and 1’s} are any strings of 0’s and 1’s}..

– Examples of strings in Examples of strings in LL::

0101, 11, 1101010, 1000, 10001011…1…

1111


Example 2.1Example 2.1

– Transitions: Transitions:

qq00, 1) = , 1) = qq00, , ((qq00, 0) = , 0) = qq22, , ((qq22, 1) = , 1) = qq11, ,

((qq22, 0) = , 0) = qq22, , qq11, 0) = , 0) = qq11, , qq11, 1) = , 1) = qq11

– 5-Tuple:5-Tuple:

AA = ({ = ({qq00, , qq11, , qq22}, {0, 1}, }, {0, 1}, , , qq00, {, {qq11})})

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2.2.3 Simpler Notations for DFA’s2.2.3 Simpler Notations for DFA’s

– Transition diagram ---Transition diagram ---


1 0

0, 1q0StartStart q2 0 q1

1

1313

2.2.3 Simpler Notations for DFA’s2.2.3 Simpler Notations for DFA’s

– Transition table ---Transition table ---

: initial state; *: final state


0 1

q0 q2 q0

*q1 q1 q1

q2 q2 q1

1414


2.2.4 Extending transition function to strings2.2.4 Extending transition function to strings– Extended transition functionExtended transition function

If If xx = = aa11aa22……aann and and is such thatis such that

((pp, , aa11) = ) = qq11, , ((qq11, , aa22) = ) = qq22, …, , …,

qqii11, , aaii) = ) = qqii, ..., , ..., ((qqnn11, , aann) = ) = qq,,

then we define to bethen we define to be

((pp, , xx) = ) = qq..

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2.2.4 Extending Transition Function to Stri2.2.4 Extending Transition Function to Stringsngs

– Also may be defined Also may be defined recursively recursively as in the textbook.as in the textbook.– Recursive definition for Recursive definition for

BasisBasis: (: (qq, , ) = ) = qq. .

InductionInduction: if : if ww = = xaxa ( (aa is the last symbol of is the last symbol of ww), then ), then

((qq, , ww) = ) = ( (( (qq, , xx), ), aa).).

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2.2.4 Extending Transition Function to Str2.2.4 Extending Transition Function to Stringsings

A graphic diagram for the following concept:A graphic diagram for the following concept:

InductionInduction: :

if if ww = = xaxa ( (aa is the last symbol of is the last symbol of ww), then ), then

((qq, , ww) = ) = ( (( (qq, , xx), ), aa).).

0, 1…… q1

a (q, x) x

(q, w=xa)

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2.2.5 The Language of a DFA2.2.5 The Language of a DFA– The language of a DFA The language of a DFA AA is defined as is defined as

LL((AA) = {) = {ww | ( | (qq00, , ww) is in ) is in FF}.}.

– If If LL is is LL((AA) for some DFA ) for some DFA AA, then we say , then we say LL is a is a

regular languageregular language. .

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0, 1

2.3 Nondeterministic Finite 2.3 Nondeterministic Finite AutomataAutomata

2.3.1 An informal view of NFA’s2.3.1 An informal view of NFA’s– Review of a previous example of DFA (of ExamplReview of a previous example of DFA (of Exampl

e 2.1)e 2.1)

– An NFA version of above DFA (An NFA version of above DFA (more intuitivemore intuitive!)!)

q0StartStart q2

0$20 q1

1

1 0

0, 1

q0StartStart q2

0$20 q1

1 0, 1

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Some properties of NFA’sSome properties of NFA’s

– Some transitions may “Some transitions may “diedie,” like ,” like qq22, 0), 0)..

– Some transitions have Some transitions have multiplemultiple choices, like choices, like

((qq00, 0) = , 0) = qq00 and and qq22..

q0StartStart q2

0$20 q1

1

0, 1

0, 1

2020


Example 2.6 Example 2.6 – Design an NFA accepting the following languageDesign an NFA accepting the following language

LL = { = {ww | | ww{0, 1}* and ends in 01}.{0, 1}* and ends in 01}.

– NondeterminismNondeterminism creates many transition paths, but creates many transition paths, but if there is if there is one path leading to a final stateone path leading to a final state, then the , then the input is accepted.input is accepted.

q0StartStart q1

0$20 q2

1

0, 1

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Example 2.6 (cont’d)Example 2.6 (cont’d)

– When input When input xx = 00101, the NFA processes = 00101, the NFA processes xx in the in the following way:following way:

0 0 1 0 10 0 1 0 1

q0StartStart q1

0$20 q2

1

0, 1

q0

q1

Stuck!

q0

q1

q2 Stuck!

q0 q0

q1

q2

Accept!

q0 q0q0 q0 q0

q1

q2

Fig. 2.10

2222


2.3.2 Definition of NFA2.3.2 Definition of NFA

– An NFA An NFA AA is a 5-tuple is a 5-tuple AA = ( = (QQ, , , , , , qq00, , FF)) where where

Q = Q = a finite (nonempty) set of states;a finite (nonempty) set of states;

a finite (nonempty) set of input symbols;a finite (nonempty) set of input symbols;

qq00 = a start state; = a start state;

F = F = a set of (nonempty) final or accepting states;a set of (nonempty) final or accepting states;

2323


2.3.2 Definition of NFA2.3.2 Definition of NFA

– An NFA An NFA AA is a 5-tuple is a 5-tuple AA = ( = (QQ, , , , , , qq00, , FF)) where where

a (state) transition functiona (state) transition functionsuch that such that

qq, , aa) = ) = {{pp11, , pp22, …, , …, ppmm} }

which which meansmeans

“ “automaton automaton AA, in state , in state qq, takes input , takes input aa and enters and enters

one of the states one of the states pp11, , pp22, …, , …, ppmm.”.”

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Example 2.7 Example 2.7 – Transition table of NFA of the last example ---Transition table of NFA of the last example ---

q0StartStart q1

0$20 q2

1

0, 1

0 1

q0 {q0,

q1}

{q0}

q1 {q2}

*q2

2525


2.3.3 Extended Transition Function2.3.3 Extended Transition Function– Recursive definition of Recursive definition of (with string as input)(with string as input)

BasisBasis: (: (qq, , ) = {) = {qq}. }.

InductionInduction: if : if ww = = xaxa ( (aa is the last symbol of is the last symbol of ww), ), ((qq, , xx) = {) = {pp11, , pp22, …, , …, ppkk},}, and and

((ppii, , aa) = {) = {rr11, , rr22, …, , …, rrmm}}

then then

((qq, , ww) = {) = {rr11, , rr22, …, , …, rrmm}.}.

1

k

iU

2626

2.3 Nondeterministic Finite 2.3 Nondeterministic Finite Automata Automata (supplemental)(supplemental)

2.3.3 Extended Transition Function2.3.3 Extended Transition Function– A graphic diagram for the following concept:A graphic diagram for the following concept:

InductionInduction: if : if ww = = xaxa ( (aa is the last symbol of is the last symbol of ww), ), ((qq, , xx) = {) = {pp11, , pp22, …, , …, ppkk},}, and and ((ppii, , aa) = {) = {rr11, , rr22, …, , …, rrmm}}

then then ((qq, , ww) = {) = {rr11, , rr22, …, , …, rrmm}.}.

……

a (q, x)= { p1 p2 . . . pk}

x

a

a

(q, w=xa) = {r1 r2 . . . rm}

2727


Example 2.8 ---Example 2.8 ---– For the input For the input ww = 00101, we have = 00101, we have

1.1. ((qq00, , ) = {) = {qq00}.}.

2. (2. (qq00, , 0) = 0) = ((qq00, , 00) = {) = {qq00, , qq11}.}.

3. (3. (qq00, , 0000) = ) = ((qq00, , 0)0)∪∪((qq11, , 00)) = {= {qq00, , qq11}}∪∪{{qq00, , qq11}}

… …

q0StartStart q1

0$20 q2

1

0, 1

2828


2.3.4 The Language of an NFA2.3.4 The Language of an NFA– Definition ---Definition ---

If If AA = ( = (QQ, , , , , , qq00, , FF)) is an NFA, then the language is an NFA, then the language

accepted by accepted by AA is is

LL((AA) = {) = {ww | ( | (qq00, , ww))∩∩FF }.}.

– That is, as long as a final state is reached, which is That is, as long as a final state is reached, which is among possibly many, the input is accepted.among possibly many, the input is accepted.

2929


2.3.5 Equivalence of DFA and NFA2.3.5 Equivalence of DFA and NFA

– NFA’s are more intuitive to construct for many lanNFA’s are more intuitive to construct for many lan

guages.guages.

– DFA’s are easier for use in software and hardware DFA’s are easier for use in software and hardware

implementations.implementations.

3030



– Every NFA Every NFA NN has an has an equivalentequivalent DFA DFA DD, ,

i.e.,i.e., LL((DD) = ) = LL((NN).).

((DefinitionDefinition: “Two models are said : “Two models are said to be to be equivalentequivalent

if they have identical languages.”)if they have identical languages.”)

– The proof needs “The proof needs “subset constructionsubset construction.”.”

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– Subset constructionSubset construction: each state of DFA is a : each state of DFA is a subsetsubset

of NFAof NFA

– Given an NFA Given an NFA NN = ( = (QQNN, , , , , , qq00, , FFNN),), we construc we construc

t a DFA t a DFA DD = ( = (QQDD, , , , DD, , qq00'', , FFDD) such that:) such that:

The two alphabets are identical;The two alphabets are identical;

QQDD is the power set of is the power set of QQN N (i.e., set of all subsets of (i.e., set of all subsets of QQNN););

InaccessibleInaccessible states in states in QQDD should be deleted; should be deleted;

(continued in the next page)(continued in the next page)

3232


2.3.5 Equivalence of DFA and NFA2.3.5 Equivalence of DFA and NFA– (cont’d)(cont’d)

qq00'' = { = {qq00}};;

Each subset Each subset SS of of QQN N is in final state set is in final state set FFDD if if SS i i

ncludes at least one accepting state in ncludes at least one accepting state in FFNN (i.e., (i.e., SS

∩∩FFNN ). ).

For each subset For each subset SS of of QQN N and for each input symand for each input sym

bol bol aa, define , define

DD((S, aS, a)) = = NN((p, ap, a))p SU

3333


Example 2.10 ---Example 2.10 ---– For the NFA of Example 2.6 below:For the NFA of Example 2.6 below:

– The power set of The power set of QQN N = = {{qq00,, q q11, , qq22} is } is

QQDD == , {, {qq00}, {}, {qq11}, {}, {qq22}, {}, {qq00, , qq11}, {}, {qq00, , qq22}, },

{{qq11, , qq22}, {}, {qq00,, q q11, , qq22}}}}..

q0StartStart q1

0$20 q2

1

0, 1

3434


Example 2.10 (cont’d)Example 2.10 (cont’d)

– The transition table isThe transition table is

q0StartStart q1

0$20 q2

1

0, 1

0 1

{q0}

{q1}

*{q2}

{q0, q1}

*{q0, q2}

*{q1, q2}

*{q0, q1, q2}

{q0, q1}

{q0, q1}

{q0, q1}

{q0, q1}

{q0}

{q2}

{q0, q2}

{q0}

{q2}

{q0, q2}

3535


Example 2.10 (cont’d)Example 2.10 (cont’d)– Change names of the states, we getChange names of the states, we get

0 1

AB

C*D

E*F*G*H

AEAAEEAE

ABDAFBDF

3636


Example 2.10 (cont’d)Example 2.10 (cont’d)– Checking Checking accessibleaccessible states from the start state as states from the start state as

redred ones; the others are inaccessible. ones; the others are inaccessible.

0 1

AB

C*D

E*F*G*H

AEAAEEAE

ABDAFBDF

Figure 2.13

3737


Example 2.10 (cont’d) Example 2.10 (cont’d) (supplemental)(supplemental)– solved by “solved by “Lazy evaluationLazy evaluation” using table: ” using table:

starting from the start state, only accessible states are evaluated, in order to avoid generation of inaccessible states.

– The transition table isThe transition table is

q0StartStart q1

0$20 q2

1

0, 1

0 1

{q0}

{q1}

*{q2}

{q0, q1}

*{q0, q2}

*{q1, q2}

*{q0, q1, q2}

{q0, q1}

{q0, q1}

{q0, q1}

{q0, q1}

{q0}

{q2}

{q0, q2}

{q0}

{q2}

{q0, q2}

0 1

{q0}

{q1}

*{q2}

{q0, q1}

*{q0, q2}

*{q1, q2}

*{q0, q1, q2}

{q0, q1}

{q0, q1}

{q0, q1}

{q0, q1}

{q0}

{q2}

{q0, q2}

{q0}

{q2}

{q0, q2}

0 1

{q0}

{q1}

*{q2}

{q0, q1}

*{q0, q2}

*{q1, q2}

*{q0, q1, q2}

{q0, q1}

{q0, q1}

{q0, q1}

{q0, q1}

{q0}

{q2}

{q0, q2}

{q0}

{q2}

{q0, q2}

3838

1


{q0}StartStart

0

0 1{q0, q1}

{q0, q2}

0

1

Example 2.10 (cont’d) --- Example 2.10 (cont’d) ---

– solved by “solved by “Lazy evaluationLazy evaluation” using diagram: ” using diagram: starting from the start state, only accessible states are

evaluated, in order to avoid generation of inaccessible states.

Figure 2.14

3939


Theorem 2.11Theorem 2.11

If DFA If DFA DD = ( = (QQDD, , , , DD, {, {qq00}, }, FFDD) is constructed fr) is constructed fr

omom NFA NFA NN = ( = (QQNN, , , , , , qq00, , FFNN)) by subset constr by subset constr

uction, then uction, then LL((DD) = ) = LL((NN).).

– Proof. See the textbook.Proof. See the textbook.

4040


Theorem 2.12 Theorem 2.12 (equivalence of DFA and (equivalence of DFA and

NFA):NFA):

A language A language LL is accepted by some DFA is accepted by some DFA if and only ifif and only if

LL is accepted by some NFA. is accepted by some NFA.

– Proof. See the textbook. Proof. See the textbook.

4141


2.3.6 A Bad Case of Subset Construction2.3.6 A Bad Case of Subset Construction

– The DFA constructed from an NFA usually has The DFA constructed from an NFA usually has the the samesame number of accessible states of the NFA. number of accessible states of the NFA.

– But the worse case is But the worse case is mm = = 22nn

wherewherem m is the number of states in the DFA.is the number of states in the DFA.

nn is the number of states in the NFA. is the number of states in the NFA.

4242


2.3.7a Regarding NFA’s as DFA’s2.3.7a Regarding NFA’s as DFA’s

– In each state of a DFA, for each input there In each state of a DFA, for each input there mustmust bb

e ae a next state. next state.

– In an NFA, for a certain input there In an NFA, for a certain input there mightmight bebe nono ne ne

xt state. So the following is an NFA. xt state. So the following is an NFA.

startstart t

t h

th e

the n

then

4343


2.3.7a Regarding NFA’s as DFA’s2.3.7a Regarding NFA’s as DFA’s

– But it is But it is deterministicdeterministic in nature --- either getting int in nature --- either getting int

o a next state or becoming “o a next state or becoming “deaddead” (getting into a “” (getting into a “

dead” state).dead” state).

– Such a kind of NFA has Such a kind of NFA has at most one transition out at most one transition out

of any state on any symbolof any state on any symbol, and may be regarded a, and may be regarded a

s a DFA.s a DFA.

startstart t

t h

th e

the n

then

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Chapter 2 (Part a) Finite Automata