class2

11

Finite AutomataFinite Automata

22

FiniteFinite AutomatonAutomatonInput

String

Output

String

FiniteAutomaton

A string over a given alphabet written on an input file

A control unit consisting of a finite number of internal states

The internal state at the next time step is determined by the transition function

33

FiniteFinite AccepterAccepter

Input

“ Accept” or“Reject”

String

FiniteAutomaton

Output

44

Input String of

Internal states


““ Accept”Accept”

oror

““Reject”Reject”

q0

q1

q2

q3

q4

q5

Initial state

final state

55

FormalitiesFormalitiesDeterministic Finite Accepter (DFA)

(p.38)

FqQM ,,,, 0Q

0q

F

: set of internal states

: input alphabet

: Q × Q transition function

: initial state

: set of final states

66

configuration

Input String

Internal states



q0

q1

q2

q3

q4

q5

abba

a ab b

Leftmost position

initial state

: a particular status of the input file and internal stateInitial

×

77

Internal states


q0

q1

q2

q3

q4

q5

a ab b 10 , qaq Transition function

current state current input symbol

move to the next state

Next state

At q0 now check for current symbol-

×

88

Internal states


q0

q1

q2

q3

q4

q5

a ab b 21, qbq Transition function


Next state

move to the next state


×

99

Internal states


q0

q1

q2

q3

q4

q5

a ab b 32 , qbq Transition function


×

1010

Internal states


q0

q1

q2

q3

q4

q5

a ab b 43 , qaq Transition function


×

1111

Internal states


q0

q1

q2

q3

q4

q5

a ab b No symbol left and it stops at final state

The string abba is accepted by the machine.


×

1212

Transition functions are the most important part for a finite accepter.

We use a transition graph to represent a finite accepter such that Node state, and Edge with label transition function

(current symbol)

Transition Graph Transition Graph GGMM

1313

32 , qbq Transition Graph forTransition Graph for


Next state

q2 q3b

Initial state

Final state qf

q0

1414

Transition Graph

initialstate

final

state

“accept”statetransition

abba –Finite Accepter

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

32 , qbq It is important to have a clear and intuitive picture to work with DFA. Transition graph: vertices > states, edges > transitions

1515

Initial ConfigurationInitial Configuration

Input Stringa b b a

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

1616

Reading the InputReading the Input


5q

a a bb

ba,

ba,

One at a time from left to right

a b b a

1717

a b b a


5q

a a bb

ba,

ba,

a b b a

1818

a b b a


5q

a a bb

ba,

ba,

a b b a

1919

a b b a


5q

a a bb

ba,

ba,

a b b a

2020

Output: “accept”

a b b a


5q

a a bb

ba,

ba,

Input finished

a b b a

2121

Another Example: Another Example: abaaba

a b a

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

2222

a b a

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

Initial ConfigurationInitial Configuration

a b a

2323

a b a


5q

a a bb

ba,

ba,

a b a

2424

a b a


5q

a a bb

ba,

ba,

a b a

2525

a b a


5q

a a bb

ba,

ba,

a b a

2626

Output:“reject”

a b a


5q

a a bb

ba,

ba,

Input finished

a b a

2727

Another Input StringAnother Input String

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

2828

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

Output:“ reject”

End of input!

2929


5q

a a bb

ba,

ba,

A Trap StateA Trap State

GGMM

It is important to have a clear and intuitive picture to work with DFA. Transition graph: vertices > states, edges > transitions

3030

What are Q, , , q0, F?

A Deterministic Finite Accepter or DFA is defined as (Def. 2.1 p.38)

FqQM ,,,, 0


5q

a a bb

ba,

ba,

GGMM


3131

InputInput AlphabetAlphabet

ba,


5q

a a bb

ba,

ba,

3232

SetSet ofof StatesStates

Q

543210 ,,,,, qqqqqqQ


5q

a a bb

ba,

ba,

3333

InitialInitial StateState

0q

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

3434

SetSet ofof FinalFinal StatesStates

F

4qF

0q 1q 2q 3qa b b a

5q

a a bb

ba,

ba,

4q

3535

TransitionTransition FunctionFunction

QQ :


5q

a a bb

ba,

ba,

By Transition Graph

GM

3636

What does “deterministic” mean?What does “deterministic” mean?

Other than using Other than using transition graph transition graph GGMM to represent a to represent a

transition function, there are two common ways to transition function, there are two common ways to represent a transition function.represent a transition function.

is a total function


5q

a a bb

ba,

ba,GM

FqQM ,,,, 0


3737



5q

a a bb

ba,

ba,

10 , qaq 50 , qbq

51, qaq 21, qbq

52 , qaq 32 , qbq

43 , qaq 53 , qbq

54 , qaq 54 , qbq

55 , qaq 55 , qbq

3838



5q

a a bb

ba,

ba,

a b

0q

1q

2q

3q

4q

5q

1q 5q

5q 2q

5q 3q

4q 5q

5q5q5q5q

3939

ExtendedExtended Transition Transition Function Function

*QQ *:*

QQ :Transition Transition functionfunction

including including


5q

a a bb

ba,

ba,

4040

20 ,* qabq

3q 4qa b b a

5q

a a bb

ba,

ba,

0q 1q 2q

QQ *:*

4141

40 ,* qabbaq


5q

a a bb

ba,

ba,

4242

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

50 ,* qabbbaaq

4343

Extended Transition Extended Transition Function Function

)),,(*(),(*

,),(*

awqwaq

qq

Recursive DefinitionRecursive Definition

for all for all qq Q, w Q, w*, a*, a

QQ *:*

4444

RecursiveRecursive DefinitionDefinition

)),,(*(,*

,*

awqwaq

qq

q qwa

)),,(*(,* awqwaq

qwaq ,*If

1

1

,*

),(,*

qwq

and

aqwaq

then a state, say q1, such that

aq qw1q

4545


5q

a a bb

ba,

ba,

0,

2q

q

1 ,bq

, ,ba0q , 0 ,,* baq

* b 0 ),,( aq * ab

)),,(*(,*

,*

wqwq

qq

4646

qwq ,*kw ...21

In the DFA M, there iswith

q q1 2 k

There is a walk in GM

from to with labelq q

q qw

kw ...21

4747

q q1 2 k



q qw

kw ...21

qwq ,*In the DFA M, there iswith kw ...21

4848

q q1 2 k



q qw

kw ...21

qwq ,*In the DFA M, there iswith kw ...21

Theorem 2.1Theorem 2.1

4949

50 ,* qabbbaaq

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

There is a walk from to with label

0qabbbaa

5q

If and only if

5050

LanguagesLanguages AcceptedAccepted byby DFAsDFAs

Take DFA

Definition: (p.40)

The language contains all input strings accepted by

= { strings that drive to a final state}

M

MLM

ML

= { = { w w * : * : *(*(qq00, , ww)) FF } }

FqQM ,,,, 0

5151

ExampleExample


5q

a a bb

ba,

ba,

abbaML

M

acceptaccept

back to complement Ex. p.51

5252

AnotherAnother ExampleExample

abbaabML ,,

M

acceptaccept


5q

a a bb

ba,

ba,

acceptacceptacceptaccept

5353

Formally, for a DFA Formally, for a DFA

Since M is DFA, every w*, corresponding to a unique walk in the transition graph of M.

If

If

qwq ,* 0

FqQM ,,,, 0

0q qwFq )(MLw

Fq )(MLw

qw0q

5454

ObservationObservation Language rejected by :

FwqwML ,*:* 0

M

Fq

What is the complement of language on example (p.48)?Example on p.48

qw0q

5555

Examples on Constructing Examples on Constructing DFAsDFAs

}0:{ nbaML n

a

b ba,

ba,

0q 1q 2q

accept trap state

5656

= { all strings with prefix }ab ML

accept

a b

ba,

0q 1q 2q

ba,3q

ab

5757

= { all strings without substring 001} ML

001 0 00

1

0

1

10

0 1,0

What if consider DFA for {strings with What if consider DFA for {strings with 001}?001}?

5858

RegularRegular LanguagesLanguages

A language is regular if there is a DFA such that

All regular languages form a language family

LM MLL

5959

abba abbaab,,

}0:{ nban

{ all strings with prefix }{ all strings with prefix }ab

{ all strings without substring }{ all strings without substring }001

Examples of regular languages:

There exist automata that accept theseLanguages (see previous slides).

6060

abbaIs the complement of also regular?

How can you show that the complement of a regular language is regular? (referring to Hw# 3, 4 on p.47)

6161


The language is regular

*,: bawawaL

a

a

b a

b

b

0q 2q 3q

ba,

4q

MLL

since we can find a DFA M such that

L*=L {}

6262


The language is regular. How about L2 ?

*,: bawawaL

Can you show LL* is a regular language?Try Hw # 20, 21, 22

6363

Is Is L?L?

If there is a DFA M for L, can you tell if L?

If L is regular, then how about L-{}?

Try hw #17 & 18 p.48

How can you construct a DFA for L-{How can you construct a DFA for L-{} } fromfrom the DFA of L? the DFA of L? How can you construct a DFA for LHow can you construct a DFA for L{{} } from from the DFA of L? the DFA of L?

6464

Homework 2.1 Homework 2.1 p.47

Try: 1 ~7, 9, 11~25 Hand in: 2bd, 4, 6, 7e, 9bf, 18

For hw18, state the general method then use the example to illustrate the method where L {a,b}*L= {w: w= or w contains a substring “ab”}

class2

Documents

transition graphgm

transition functions

q q transition function

vertices states

set of states

set of final states

dfastake dfa definition

input strings