1 Chapter 3 Scanning - Theory and Practice Prof Chung. 10/8/2015.

1

Chapter 3 Scanning - Theory and Practice

Prof Chung.

04/21/23

22

Outlines 3.1 Overview 3.2 Regular Expressions 3.4 Finite Automata and Scanners 3.5 Using a Scanner Generator

LEX --- Introduce in TA course: LEX introduction 3.7 Practical Considerations 3.8 Translating Regular Expressions into Finite

Automata 3.9 Summary

Modify form http://www.cs.ualberta.ca/~amaral/courses/680/

04/21/23

Overview(1) Formal notations

For specifying the precise structure of tokens are necessary

Quoted string in Pascal Can a string split across a line? Is a null string allowed? Is .1 or 10. ok? The 1..10 problem

Scanner generators Tables, Programs

What formal notations to use?

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Overview(2) Lexical analyzer (scanner) role

Produce a sequence of (tokens) for parser Stripe out comments and whitespaces Associate a line number with each error message Expand macros

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LexicalAnalyzer

Parser

SymbolTable

sourceprogram

to semanticanalysis

token

getNextToken

Regular Expressions (1) Tokens

built from symbols of a finite vocabulary.

Structures of tokens use regular expressions to define

Set Definition The sets of strings defined by regular expressions are termed is a regular expression denoting the empty set

is a regular expression denoting the set that contains only the empty string

A string s is a regular expression denoting a set containing only s

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Regular Expression (2) if A and B are regular expressions, so are

A | B (alternation)

A regular expression formed by A or B

(a)|(b) = {a, b}

AB or A•B (concatenation)

A regular expression formed by A followed by B

(a)(b) = {ab}

A* (Kleene closure)

A regular expression formed by zero or more repetitions of A

a* = {, a, aa, aaa, …}

04/21/236

More Complex Example (a|b|c)* = {, a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc …}

Regular Expression (3) Some notational convenience

P+

PP* (at least one)

Not(A) V - A

Not(S) V* - S

AK

AA …A (k copies)

A?

Optional, zero or one occurrence of A

04/21/237

More Complex Example Let D = (0 | 1 | 2 | 3 | 4 | ... | 9 ) Let L = (A | B | ... | Z | a | b |... | z)

comment = -- not(EOL)* EOL ex: --hello12_34 \n

decimal = D+ · D+

ex: 123.456

ident = L (L | D | _)* ex: A1a5_6

comments = ((#| ) not(#))* ex:#A435#3

Regular Expressions (4) Is regular expression as power as CFG? { [i ]i | i1}

Regular grammar

04/21/238

Aa

AaB

Aa

ABaor

Finite Automata and Scanners (1) Finite automaton (FA)

can be used to recognize the tokens specified by a regular expression

A FA consists of A finite set of states S A set of input symbols (the input symbol alphabet) A set of transitions (or moves) from one state to

another, labeled with characters in V

A special start state s0 (only one) A set of final, or accepting, states F

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FA = {S, , s0, F, move }

Finite Automata and Scanners (2)

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is a transition

is a state

is a final state

is the start state

Example at next page….

Example A transition diagram

This machine accepts (abc+)+


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a

a b c

c

( a b c + ) +

Finite Automata and Scanners (4) Other Example

(0|1)*0(0|1)(0|1)

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1 42 30 0,1 0,1

5

0,1 0

0,1 (0|1)*0(0|1)(0|1)


ID = L(L|D)*(_(L|D)+)*

A data structure can be translated for many REs or FAs

04/21/2313

L -

L | D

L | D

(_(L|D)+)*L (L|D)*

Final for two * symbol

What difference?Answer : “_” by times

item 2 = item 3


RealLit = (D+(|.))|(D*.D+)

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Two kinds of FA: Deterministic: next transition is unique Non-deterministic: otherwise


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...a

a

Which path we should select?

...

A transition diagram

A transition table

4321


04/21/2316

1/ /

Not(Eol)

3 42Eol

State Character

- Eol a b …

3

3

2

4 3 3 3

Finite Automata and Scanners (9) Any regular expression

can be translated into a DFA that accepts the set of strings denoted by the regular expression

The transition can be done Automatically by a scanner generator : LEX (TA course) Manually by a programmer :

Coding the DFA in two form 1. Table-driven, commonly produced by a scanner

generator 2. Explicit control, produced automatically or by

hand

04/21/2317

Finite Automata and Scanners (10) Scanner Driver Interpreting a Transition

Table/* Note: CurrentChar is already set to the current input character. */

State = StartState;

while (TRUE) {

NextSate = T[State , CurrentChar];

if (NextSate == ERROR)

break;

State = NextState;

CurrentChar = getchar();

}

If(is_final_state(State))

/* Return or process valid token. */

else

lexical_error(CurrentChar);

04/21/2318

Table-driven

Finite Automata and Scanners (11) Scanner with Fixed Token Definitionif (CurrentChar == ‘/') {


if (CurrentChar == ‘/') {

do {


} while (CurrentChar != '\n');

} else {

ungetc(CurrentChar, stdin);


}

}

else


/* Return or process valid token. */

04/21/2319

Explicit control

Finite Automata and Scanners (12) Transducer

We may perform some actions during state transition.

A scanner can be turned into a transducer by the appropriate insertion of actions based on state transitions

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2121

Using a Scanner Generator By TA….

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Practical Considerations (1) Reserved Words

Usually, all keywords are reserved in order to simplify parsing.

In Pascal, we could even write begin begin; end; end; begin; end if else then if = else;

The problem with reserved words is that they are too numerous. COBOL has several hundreds of reserved words!

ZEROS ZERO ZEROES

04/21/2322

Practical Considerations (2) Compiler Directives and Listing Source Lines

Compiler options e.g. optimization, profiling, etc. handled by scanner or semantic routines Complex pragmas are treated like other statements.

Source inclusion e.g. #include in C handled by preprocessor or scanner

Conditional compilation e.g. #if, #endif in C useful for creating program versions

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Practical Considerations (3) Entry of Identifiers into the Symbol Table

Who is responsible for entering symbols into symbol table?

Scanner?

Consider this example: { int abc; … { int abc; } }

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Practical Considerations (4) How to handle end-of-file?

Create a special EOF token. EOF token is useful in a CFG

Multicharacter Lookahead Blanks are not significant in Fortran DO 10 I= 1,100

Beginning of a loop

DO 10 I = 1.100 An assignment statement DO 10I=1.100

A Fortran Scanner can determine whether the O is the last character of a DO

token only after reading as far as the comma

04/21/2325

Practical Considerations (5) Multicharacter Lookahead (Cont’d)

In Ada and Pascal To scan 10..100

There are three token 10 .. 100

Two-character (..) lookahead after the 10

It is easy to build a scanner that can perform general backup.

If we reach a situation in which we are not in final state and

cannot scan any more characters, we extract characters from the right end of the buffer and queue them fir rescanning

Until we reach a prefix of the scanned characters flagged as a valid token

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Example at next page

Practical Considerations (6) An FA That Scans Integer and Real Literals

and the Subrange Operator

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D ●

D

D

D

● ●Buffered

TokenToken Flag

1 Integer Literal

12 Integer Literal

12. Invalid

12.3 Real Literal

12.3e Invalid

12.3e+ Invalid

Detail Operation of each case at next page



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D

1

Buffered Token Token Flag

Integer Literal1

Input Token

Input string: 12.3e+q



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D

1


Integer Literal1

Input Token

D

2 2




04/21/2330

D

1


Invalid1

Input Token

D

2 2. .

●




04/21/2331

D

1


Real Literal1

Input Token

D

2 2. .

●

3 3

D




04/21/2332

D

1


Invalid1

Input Token

D

2 2. .

●

3 3

D

e e

?


Backup is invoked!



04/21/2333

D

1


Invalid1

Input Token

D

2 2. .

●

3 3

D

e e

?


Backup is invoked!



04/21/2334

D

1


Invalid1

Input Token

D

2 2. .

●

3 3

D

e e

?

+ +

?


Practical Considerations (13)cannot scan any more characters, and not in

accept state Backup is invoked !

04/21/2335

D

1


Invalid1

Input Token

D

2 2. .

●

3 3

D

e e

?

+ +

?


3636

Outlines 3.1 Over View 3.2 Regular Expression 3.3 Finite Automata and Scanners 3.4 Using a Scanner Generator 3.5 Practical Considerations 3.6 Translating Regular Expressions into Finite

Automata Creating Deterministic Automata Optimizing Finite Automata

3.7 Tracing Example

04/21/23

Translating Regular Expressions into Finite Automata(1) Regular expressions are equivalent to FAs

The main job of a scanner generator To transform a regular expression definition into an

equivalent FA

04/21/2337

A regular expression

Nondeterministic FA

Deterministic FA

Optimized Deterministic FA

minimize # of states

Importance in NFA->DFA

Translating Regular Expressions into Finite Automata(2) We can transform any regular expression

into an NFA with the following properties: There is an unique final state The final state has no successors Every other state has at least one successors

Example : A Nondeterministic Finite Automaton (NFA) Input : babb Regular Expressions : (a|b)*abb

04/21/2338

Unique final state

Final S has no successor

0a

a

2b b

b

31

either one or two successors

Translating Regular Expressions into Finite Automata(3) We need to review the definition of regular

expression Item 1:

It is null string

Item 2: a It is a char of the vocabulary

Item 3 : | It is “or” operation. Example : A|B

Item 4 : ● It is the operation of catenation Example : AB

Item 4 : * It is the operation of repetition Example : A*

04/21/2339

More Example at Next Page

Translating Regular Expressions into Finite Automata(4) NFA : (null string)

NFA : a (1string) A char of the vocabulary

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a

Processing Token

aProcessing Token

NFA :

NFAFor A

Translating Regular Expressions into Finite Automata(5)

04/21/2341

NFAFor B

Processing Token

NFA : ●


04/21/2342

NFAFor A

NFAFor B

Processing Token ●

NFA :


04/21/2343

NFAFor A

Processing Token

= 0 times

> 1 times

Construct an NFA for Regular Expression

01* | 1 (0(1*)) |1


04/21/2344

1 *Processing Token

Start


01*|1 (0(1*)) |1


04/21/2345

Processing Token 1*

Start

0

For Connection


01*+1 (0(1*))+1


04/21/2346

Processing Token 1*

0 | 1

Start

What’s problem about NFA? Ans: It may be ambiguous that difficult to

program!!!

A Nondeterministic Finite Automaton (NFA):(a|b)*abb


04/21/2347

2b

3Start 0a

1b

a

b

Input : babbProcessing Tokenb a

Ambiguous!!!

Which one should we select?

What’s problem about NFA? Ans: It may be ambiguous that difficult to

program!!!

A deterministic Finite Automaton (NFA): b*abb


04/21/2348

2b

3Start 0a

1b

b

Input : babbProcessing Tokenb a

No Ambiguous!!!

It have unique path!

b b

Creating Deterministic Automata(1) The transformation

from an NFA N to an equivalent DFA M works by what is sometimes called the subset construction

An Example for each step… Initial NFA : 01*|1 (0(1*)) |1

04/21/2349

Start

4

652

3

1 10

7

8 9

More Detail operation at next page…

Creating Deterministic Automata(2) Step 1:

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Start

4 1

652

0

3

110

7

8 9 1

1. -closure(1) ={1, 2, 8}2. -closure(2) ={2}3. -closure(3) ={3 ,4 ,5 ,7 ,10}4. -closure(4) ={4 ,5 ,7 ,10}5. -closure(5) ={5}

6. -closure(6) ={5 ,6 ,7 ,10}7. -closure(7) ={5, 7, 10}8. -closure(8) ={8}9. -closure(9) ={9 ,10}10. -closure(10) ={10}

More Detail operation at next page…


04/21/2351

Start

4 1

652

0

3

110

7

8 9 1


6. -closure(6) ={5 ,6 ,7 ,10}7. -closure(7) ={5, 7 ,10}8. -closure(8) ={8}9. -closure(9) ={9 ,10}10. -closure(10) ={10}


04/21/2352

2

0

1

8 1

1. -closure(1) ={1, 2, 8}


04/21/2353

Start

4 1

652

0

3

110

7

8 9 1




04/21/2354

2

0

2. -closure(2) ={2}


04/21/2355

Start

4 1

652

0

3

110

7

8 9 1




04/21/2356

4 1

53

10

7

3. -closure(3) ={3 ,4 ,5 ,7 ,10}


04/21/2357

Start

4 1

652

0

3

110

7

8 9 1




04/21/2358

4 1

5

10

7

4. -closure(4) ={4 ,5 ,7 ,10}


04/21/2359

Start

4 1

652

0

3

110

7

8 9 1




04/21/2360

15

5. -closure(5) ={5}


04/21/2361

Start

4 1

652

0

3

110

7

8 9 1




04/21/2362

1

65

10

7

6. -closure(6) ={5 ,6 ,7 ,10}

This point line not be computed!!


04/21/2363

Start

4 1

652

0

3

110

7

8 9 1




04/21/2364

10

7

7. -closure(7) ={5, 7 ,10}

1

5


04/21/2365

Start

4 1

652

0

3

110

7

8 9 1




04/21/2366

81

8. -closure(8) ={8}


04/21/2367

Start

4 1

652

0

3

110

7

8 9 1




04/21/2368

10

9

9. -closure(9) ={9 ,10}


04/21/2369

Start

4 1

652

0

3

110

7

8 9 1




04/21/2370

10

10. -closure(10) ={10}


04/21/2371

Start

4 1

652

0

3

110

7

8 9 1



Total closures, but…..


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Start

4 1

652

0

3

110

7

8 9 1



Delete Sub Set ...


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Start

4 1

652

0

3

110

7

8 9 1


6. -closure(6) ={5 ,6 ,7 ,10}7. -closure(7) ={7 ,10}8. -closure(8) ={8}9. -closure(9) ={9 ,10}10. -closure(10) ={10}

Now Closures, No Sub Set ...

State 3 state 3state 4state 5,7state 10empty


The initial state of M is the set of states reachable from the initial state of N by -transitions

Usually called l-closure or ε-closure

04/21/2374

Algorithm for example at upside


To create the successor states Take any state S of M and any character c, and compute S’s

successor under c S is identified with some set of N’s states, {n1, n2,…}

Find all possible successor states to {n1, n2,…} under c Obtain a set {m1, m2,…}

T=close({m1, m2,…})

04/21/2375

S T{n1, n2,…} close({m1, m2,…})


void make_deterministic( nondeterministic_fa N , deterministic *M){ set_of_fa_states T; M->initial_state = SET_OF(N.initial_state) ; close (& M->initial_state ); Add M-> initial_state to M->states; while( states or transitions can be added) { choose S in M->states and c in Alphabet; T=SET_OF (y in N . states SUCH THAT x->y under c for some x in S); close(& T); if(T not in M->states) add T to M->states;

Add the transition to M->transitions: S->T under c; } M->final_states = SET_OF(S in M->states SUCH_THAT N.final_state in S);}

04/21/2376

Example at next page…


First Re-Number for simplifying the work flow1. -closure(1) ={1, 2, 8} A = {1, 2, 8}

3. -closure(3) ={3 ,4 ,5 ,7 ,10} B = {3 ,4 ,5 ,7 ,10}

6. -closure(6) ={5 ,6 ,7 ,10} C = {5 ,6 ,7 ,10}

9. -closure(9) ={9 ,10} D = {9, 10}

04/21/2377

Start

4 1

652

0

3

1 10

7

8 9 1

More Operation at next page ……

Creating Deterministic Automata(6)

04/21/2378

{1,2,8}

{3 ,4 ,5 ,7 ,10}

{9, 10}

{5 ,6 ,7 ,10

}

Start

4 1

652

0

3

1 10

7

8 9 1

A : {1, 2, 8} B : {3 ,4 ,5 ,7 ,10} C : {5 ,6 ,7 ,10}D : {9, 10}

A

B C

D

Start

0

1

1

1

NoOut-

Degree

Final


void make_deterministic( nondeterministic_fa N , deterministic *M){ set_of_fa_states T; M->initial_state = SET_OF(N.initial_state) ; close (& M->initial_state ); Add M-> initial_state to M->states; while( states or transitions can be added) { choose S in M->states and c in Alphabet; T=SET_OF (y in N . states SUCH THAT x->y under c for some x in S); close(& T); if(T not in M->states) add T to M->states;

Add the transition to M->transitions: S->T under c; } M->final_states = SET_OF(S in M->states SUCH_THAT N.final_state in S);}

04/21/2379

Example at next page…

Optimizing Finite Automata(1) Minimize number of states

Every DFA has a unique smallest equivalent DFA Given a DFA M

we use Transition Table to construct the equivalent minimal DFA.

Initially, we draw a transition table from DFA diagram.

04/21/2380

Start

1

A

DFA

D

1

B C

Table

StateCharacter

0 1

A B D

B C

C C

D

A: Start StateB,C,D: Final State

Optimizing Finite Automata(2) Minimize number of states

04/21/2381

StateCharacter

0 1

A B D

B C

C C

D

Start

1

A

DFA

D

1

B C

OptimizeB is

equal C

StateCharacter

0 1

A {B, C} D

{B, C} {B, C}

D

New DFAStart A

D

1B,C

A: Start StateB,C,D: Final StateSpecial :

B can merge into C, Because the B and C are final state.

Additional Simplifying rules (removing

parentheses) “*” has highest precedence and is left associative Concatenation has 2nd highest precedence and is left

associative “| “has lowest precedence and is left associative E.g., (a)|((b)*(c)) == a|b*c

04/21/2382

8383

Outlines 3.1 Over View 3.2 Regular Expression 3.3 Finite Automata and Scanners 3.4 Using a Scanner Generator 3.5 Practical Considerations 3.6 Translating Regular Expressions into Finite

Automata 3.7 Tracing Example

Modify form http://www.cs.ualberta.ca/~amaral/courses/680/

04/21/23

Tracing Example(1) Review Steps of Scanner Generator

04/21/2384

A regular expression

Nondeterministic FA

Deterministic FA

Optimized Deterministic FA


Importance in NFA->DFA

Tracing Example(2) Regular Expression

IF and IFA

04/21/2385

if {return IF;}

[a - z] [a – z|0 - 9 ] * {return ID;}

[0 - 9] + {return NUM;}

. {error ();}

Tracing Example(3) Translate from RE to NFA

04/21/2386

A regularexpression

Nondeterministic FA

Deterministic FA

OptimizedDeterministic FA


Tracing Example(4)

04/21/2387

The NFA for a symbol i is: i1 2start

The NFA for the regular expression if is:

f 31start 2i

The NFA for a symbol f is: f 2start 1

IF

if {return IF;}

Tracing Example(5)

04/21/2388

a-z1start

[a-z] [a-z|0-9 ] * {return ID;}

42 3a-z

0-9

ID

Tracing Example(6)

04/21/2389

54320-91start

NUM

[0 – 9]+ {return NUM;}

0-9

Tracing Example(9)

04/21/2390

NUM

21

any but \n

error

IDIF

1 2

i f

3 a-z1 42 3a-z

0-9

54320-91 0-9

Tracing Example(10) Translate from NFA to DFA

04/21/2391

A regularexpression

Nondeterministic FA

Deterministic FA



Tracing Example(11)

04/21/2392

2 3 84 5 6 7

139 10 11 1214 15

1

a-z

0-90-9

a-z

0-9i

f

IF

error

NUM

ID

anycharacter

Full NFA Diagram

Special case :Handle in Final

Tracing Example(12)

04/21/239393

2 3 84 5 6 7

914

1

a-z

0-90-9

a-z

0-9i

f

IF

error

NUM

ID

anycharacter

1. -closure(1) ={ 1, 4, 9, 14}

2. -closure(2) ={ 2}

3. -closure(3) ={ 3}

4. -closure(5) ={ 5, 6, 8}5. -closure(7) ={ 7, 8}

6. -closure(8) ={ 6, 8}

7. -closure(10) ={ 10, 11, 13}

9. -closure(13) ={11, 13}

8. -closure(12) ={12, 13}

10 11 12 13

10. -closure(15) ={15}

15

Tracing Example(13)

04/21/2394

DFA States = {1-4-9-14}

1-4-9-14

Now we need to compute:

move(1-4-9-14,a-h) = {5,15}

-closure({5,15}) = {5,6,8,15}

a-h 5-6-8-15

2 3 84 5 6 7

914

1

a-z

0-90-9

a-z

0-9i

f

IF

error

NUM

ID

anycharacter

10 11 12 1315

Tracing Example(16)

04/21/2395

DFA States = {1-4-9-14}move(1-4-9-14, i) =

1-4-9-14

a-h {2,5,15}

-closure({2,5,15}) = {2,5,6,8,15}

2-5-6-8-15i

2 3 84 5 6 7

914

1

a-z

0-90-9

a-z

0-9i

f

IF

error

anycharacter

10 11 12 1315

5-6-8-15

Tracing Example(21)

04/21/2396

DFA States = {1-4-9-14}move(1-4-9-14, j-z) =

-closure({5,15}) =1-4-9-14

a-h 5-6-8-15

2-5-6-8-15i

j-z{5,15}

{5,6,8,15}

2 3 84 5 6 7

914

1

a-z

0-90-9

a-z

0-9i

f

IF

error

NUM

ID

anycharacter

10 11 12 1315

Tracing Example(22)

04/21/2397

DFA States = {1-4-9-14}move(1-4-9-14, 0-9) =

1-4-9-14

a-h 5-6-8-15

2-5-6-8-15i

j-z10-11-13-15

0-9

{10,15}

-closure({10,15}) ={10,11,13,15}

2 3 84 5 6 7

914

1

a-z

0-90-9

a-z

0-9i

f

IF

error

NUM

ID

anycharacter

10 11 12 1315

Tracing Example(23)

04/21/239898

DFA States = {1-4-9-14}move(1-4-9-14, other) =

1-4-9-14

a-h 5-6-8-15

2-5-6-8-15i

j-z10-11-13-15

0-9

15other

{15}

-closure({15}) = {15}

2 3 84 5 6 7

914

1

a-z

0-90-9

a-z

0-9i

f

IF

error

anycharacter

10 11 12 1315

NUM

ID

Tracing Example(24)

04/21/2399

DFA states = {1-4-9-14}

The analysis for 1-4-9-14is complete. We mark it andpick another state in the DFAto analysis. (Practice)

2 3 84 5 6 7

139 10 11 1214 15

1

a-z

0-90-9

a-z

0-9i

f

IF

error

NUM

ID

anycharacter

1-4-9-14

a-h 5-6-8-15

2-5-6-8-15i

j-z10-11-13-15

0-9

15other

Tracing Example(25)

04/21/23100

5-6-8-15

2-5-6-8-15

10-11-13-15

3-6-7-8

11-12-13

6-7-8

15

1-4-9-14

a-e, g-z, 0-9

a-z,0-9

a-z,0-9

0-9

0-9

f

i

a-h

j-z

0-9

other

ID

ID

NUM NUM

IF

error

ID

a-z,0-9

See pp. 118 of Aho-Sethi-Ullmanand pp. 29 of Appel.

Tracing Example(26)

04/21/23101

A regularexpression

Nondeterministic FA

Deterministic FA



Minimize DFA

Tracing Example(27)

04/21/23102

State

character

0-9 a-e f g-h i j-z other

A D C C C B C E

B G G F G G G -

C G G G G G G -

D H - - - - - -

E - - - - - - -

F G G G G G G -

G G G G G G G -

H H - - - - - -

A

B

C

D

E

F

G

HTransition Table

DFA

Tracing Example(28)

04/21/23103

State

character

0-9 a-e f g-h i j-z other

A D C C C B C E

B G G F G G G -

C G G G G G G -

D H - - - - - -

E - - - - - - -

F G G G G G G -

G G G G G G G -

H H - - - - - -

A

B

C

D

E

F

G

HTransition Table

DFA

State

character

0-9

a-e f g-

h i j-zother

A D C C C B C E

B C C C C C C -

C C C C C C C -

D D - - - - - -

E - - - - - - -

New Transition Table-1

Tracing Example(29)

04/21/23104

A

B

C

D

E

F

G

H

DFA

State

character

0-9

a-e f g-

h i j-zother

A D C C C B C E

B C C C C C C -

C C C C C C C -

D D - - - - - -

E - - - - - - -


State

character

0-9

a-e f g-

h i j-zother

A D B B B B B E

B B B B B B B -

D D - - - - - -

E - - - - - - -


Tracing Example(30)

04/21/23105

A

B

C

D

E

F

G

H

DFA

B

D

E

A

0-9

a-z

0-9

other

IF

ID

NUM

error

a-z,0-9

B=C=F=GD=H

State

character

0-9

a-e f g-

h i j-zother

A D B B B B B E

B B B B B B B -

D D - - - - - -

E - - - - - - -


i

fNew DFA

IF can be handled by look-ahead programming

Chapter 3 End

Any Question?

04/21/23106

隨堂考試 (1+) What is the optimized DFA for 1+?

1. -closure(1) ={1, 2}2. -closure(2) ={2}3. -closure(3) ={3 ,4 ,2}4. -closure(4) ={4 ,2}

1 42 3

1*

= 1 . 1+ (Can use this method)

1. -closure(1) ={1, 2} A3. -closure(3) ={3 ,4 ,2} B

StateCharacter

0 1

A B

B B

A

B

{1,2}

AStart

{3,4,2}

1B 1

Can Not Optimized, (Merge)For A is Start State, B is Final State!

1 Chapter 3 Scanning - Theory and Practice Prof Chung. 10/8/2015.

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Transcript of 1 Chapter 3 Scanning - Theory and Practice Prof Chung. 10/8/2015.