LR Parsing

Challenge the future

Delft University of Technology

Course IN4303 Compiler Construction

Eduardo Souza, Guido Wachsmuth, Eelco Visser

LR ParsingTraditional Parsing Algorithms

lessons learned

LR Parsing

Recap: Traditional Parsing Algorithms

How can we parse context-free languages effectively?

• predictive parsing

Which grammar classes are supported by these algorithms?

• LL(k) grammars, LL(k) languages

How can we generate compiler tools from that?

• implement automaton

• generate parse tables

What are other techniques for implementing top-down parsers?

• Parser Combinators

• PEGs

• ALL(*)

2

today’s lecture

Lexical Analysis

Overview

3

today’s lecture

Lexical Analysis

Overview

efficient parsing algorithms

• LR parsing

• LR parse table generation

• SLR & LALR parse tables

• Generalized LR parsing

• Scannerless Generalized LR parsing

3

LR Parsing

LR Parsing

I

4

idea

LR Parsing

LR parsing

problems with LL parsing

• predicting right rule

• left recursion

LR parsing

• see whole right-hand side of a rule

• look ahead

• shift or reduce

5

example

LR Parsing

LR parsing

6

* 3 $

input

$

stack

7* 37 +

S → E $ E → E * E E → E + E E → Num

grammar

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

* 3 $7+$ E

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

* 3 $7+$ E

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

* 3 $7+$ E

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

* 3 $7+$ E

$ E + E 3* $

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

* 3 $7+$ E

$ E + E 3* $

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

* 3 $7+$ E

$ E + E 3* $

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

* 3 $7+$ E

$ E + E 3* $

$ E + E * E $

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

* 3 $7+$ E

$ E + E 3* $

$ E + E

$ E + E * E $

$

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

* 3 $7+$ E

$ E + E 3* $

$ E + E

$ E + E * E $

$

$ E $

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

* 3 $7+$ E

$ E + E 3* $

$ E + E

$ E + E * E $

$

$ E $

S → E $ E → E * E E → E + E E → Num

example

LR Parsing

LR parsing

6

* 3 $$ 7* 37 +

* 3 $7+$ E 3*

* 3 $7+$ E * E

* 3 $7+$ E

$ E + E 3* $

$ E + E

$ E + E * E $

$

$ E $

$ S

S → E $ E → E * E E → E + E E → Num

LR Parsing

Grammar classes

7

context-free grammars

LL(k)

LL(1)

LL(0)

LR Parsing

Grammar classes

7


LR(0)

LL(k)

LL(1)

LL(0)

LR Parsing

Grammar classes

7


LR(1)

LR(0)

LL(k)

LL(1)

LL(0)

LR Parsing

Grammar classes

7


LR(k)

LR(1)

LR(0)

LL(k)

LL(1)

LL(0)

LR Parsing

Grammar classes

7


LR(k)

LR(1)

SLR

LR(0)

LL(k)

LL(1)

LL(0)

LR Parsing

Grammar classes

7


LR(k)

LR(1)

LALR(1)

SLR

LR(0)

LL(k)

LL(1)

LL(0)

LR Parsing

LR Parse Tables

II

8

parse table

LR Parsing 9

rows

• states of a DFA

columns

• topmost stack symbol

• Σ, N

entries

• reduce, rule number

• shift, goto state

• goto state

• accept state

LR parsing

T1 ... N1 ...

1 s 3

2 g 5

3 r 1

4 r 2 a

5

6 g 1

7 s 1

8

...

items, closure & goto

LR Parsing

LR(0) parse tables

10

S → x S → ( L ) L → S L → L , S

S’ → . S $


LR Parsing

LR(0) parse tables

10

S → x S → ( L ) L → S L → L , S

S’ → . S $


LR Parsing

LR(0) parse tables

10

item

S → x S → ( L ) L → S L → L , S

closure

• for every item A → α . X β

• for every rule X → γ

• add item X → . γ

S’ → . S $


LR Parsing

LR(0) parse tables

10

item

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )

closure

• for every item A → α . X β


• add item X → . γ


LR Parsing

LR(0) parse tables

10

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L )

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

x

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

x

(S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

x

(

S

L → S .

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

x

(

S

L → S .

L S → ( L . ) L → L . , S

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

x

(

S

L → S .

L S → ( L . ) L → L . , S

)

S → ( L ) .

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

x

(

S

L → S .

L S → ( L . ) L → L . , S

)

S → ( L ) .

,L → L , . S

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )

L → L , . S S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

x

(

S

L → S .

L S → ( L . ) L → L . , S

)

S → ( L ) .

,

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )

L → L , . S S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

x

(

S

L → S .

L S → ( L . ) L → L . , S

)

S → ( L ) .

,

x

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )

L → L , . S S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

x

(

S

L → S .

L S → ( L . ) L → L . , S

)

S → ( L ) .

,

x

(

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )

L → L , . S S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

x

(

S

L → S .

L S → ( L . ) L → L . , S

)

S → ( L ) .

,

x

(

SL → L , S .

S → x S → ( L ) L → S L → L , S

S’ → . S $ S → . x S → . ( L )

L → L , . S S → . x S → . ( L )


LR Parsing

LR(0) parse tables

10

S’ → S . $

S

x S → x .

( S → ( . L ) L → . S L → . L , S S → . x S → . ( L )

x

(

S

L → S .

L S → ( L . ) L → L . , S

)

S → ( L ) .

,

x

(

SL → L , S .

S → x S → ( L ) L → S L → L , S

12

3

4 6 7

5

8

9

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)x,x(

1

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)x,x

13

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)x,

132

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)x,

13

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)x,

136

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)x,

13

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)x,

135

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)x

1358

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)

13582

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)

1358

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)

13589

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)

13

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$)

135

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$

1357

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$

1

( ) x , $ S L

1 s 3 s 2 g 4

2 r 1 r 1 r 1 r 1 r 1

3 s 3 s 2 g 6 g 5

4 a

5 s 7 s 8

6 r 3 r 3 r 3 r 3 r 3

7 r 2 r 2 r 2 r 2 r 2

8 s 3 s 2 g 9

9 r 4 r 4 r 4 r 4 r 4

result

LR Parsing 11

LR(0) parse tables

S → x S → ( L ) L → S L → L , S

$

14

LR Parsing

Conflict Resolution

III

12

E → T + . E E → . T + E E → . T T → . x

S → . E $ E → . T + E E → . T T → . x

shift-reduce conflicts

LR Parsing

SLR parse tables

13

T → x .

x

E S → E . $

T

T +

E

E → T . + E E → T .

E → T + E .

E → T + E E → T T → xx

E → T + . E E → . T + E E → . T T → . x

S → . E $ E → . T + E E → . T T → . x


LR Parsing

SLR parse tables

13

T → x .

x

E S → E . $

T

T +

E

E → T . + E E → T .

E → T + E .

E → T + E E → T T → x

1

2

3

5 4 6x

E → T + . E E → . T + E E → . T T → . x

S → . E $ E → . T + E E → . T T → . x


LR Parsing

SLR parse tables

13

T → x .

x

E S → E . $

T

T +

E

E → T . + E E → T .

E → T + E .

E → T + E E → T T → x

1

2

3

5 4 6x

x + $ E T1 s 5 g 2 g 32 a3 r 2 ? r 24 s 5 g 6 g 35 r 3 r 3 r 36 r 1 r 1 r 1

E → T + . E E → . T + E E → . T T → . x

S → . E $ E → . T + E E → . T T → . x


LR Parsing

SLR parse tables

13

T → x .

x

E S → E . $

T

T +

E

E → T . + E E → T .

E → T + E .

E → T + E E → T T → x

1

2

3

5 4 6x

x + $ E T1 s 5 g 2 g 32 a3 s 4 r 24 s 5 g 6 g 35 r 3 r 36 r 1

Reduce a production S → …

on symbols k ∈ Σ, k ∈ Follow(S)

look-ahead

LR Parsing

LR(1) parse tables

14

E → T + E E → T T → x

S → . E $ E → . T + E E → . T T → . x

? $ $ + $

E S → E . $ ?

T E → T . + E E → T .

$ $

T → x .

x

+ $

T +

x

E → T + . E E → . T + E E → . T T → . x

$ $ $ + $ E E → T + E . $

closure

• for every item A → α . X β, z


• for every w ∈ First(βz)

• add item X → . γ, w

look-ahead

LR Parsing

LR(1) parse tables

14

E → T + E E → T T → x

S → . E $ E → . T + E E → . T T → . x

? $ $ + $

E S → E . $ ?

T E → T . + E E → T .

$ $

T → x .

x

+ $

T +

x

E → T + . E E → . T + E E → . T T → . x

$ $ $ + $ E E → T + E . $

x + $ E T1 s 5 g 2 g 32 a3 s 4 r 24 s 5 g 6 g 35 r 3 r 36 r 1

closure

• for every item A → α . X β, z


• for every w ∈ First(βz)

• add item X → . γ, w

state space reduction

LR Parsing

LALR(1) parse tables

unify states

• with same items

• and same outgoing transitions

• but different look-ahead sets

might introduce new conflicts

15


LR Parsing


16

S’ → S $ S → a E c S → a F d S → b F c S → b E d E → e F → e

S' → . S $ S → . a E c S → . a F d S → . b F c S → . b E d

? $ $ $ $


LR Parsing


17



? $ $ $ $

S → a . E c S → a . F d E → . e F → . e

$ $ c d

a


LR Parsing


18



? $ $ $ $

S → b . F c S → b . E d E → . e F → . e

$ $ d c

a

b

S → a . E c S → a . F d E → . e F → . e

$ $ c d


LR Parsing


19



? $ $ $ $

E → e . F → e .

c d

a

b e

S → b . F c S → b . E d E → . e F → . e

$ $ d c

S → a . E c S → a . F d E → . e F → . e

$ $ c d


LR Parsing


20



? $ $ $ $

E → e . F → e .

c d

a

b e

eE → e . F → e .

d c

S → b . F c S → b . E d E → . e F → . e

$ $ d c

S → a . E c S → a . F d E → . e F → . e

$ $ c d


LR Parsing


21



? $ $ $ $

E → e . F → e .

c d

a

b e

eE → e . F → e .

d c

S → b . F c S → b . E d E → . e F → . e

$ $ d c

S → a . E c S → a . F d E → . e F → . e

$ $ c d


LR Parsing


22



? $ $ $ $

S → a . E c S → a . F d E → . e F → . e

$ $ c d

S → b . F c S → b . E d E → . e F → . e

$ $ d c

E → e . F → e .

{c, d} {c, d}

a

b e

e


LR Parsing


23



? $ $ $ $

E → e . F → e .

{c, d} {c, d}

a

b e

e Reduce/Reduce conflict!

S → a . E c S → a . F d E → . e F → . e

$ $ c d

S → b . F c S → b . E d E → . e F → . e

$ $ d c

LR Parsing

Generalized-LR Parsing

IV

24

Lexical Analysis

Generalized Parsing

• Parse all interpretations of the input, therefore it can handle ambiguous grammars.

• Parsers split whenever finding an ambiguous interpretation and act in (pseudo) parallel.

• Multiple parsers can join whenever they finish parsing an ambiguous fragment of the input.

• Some parsers may "die", if the ambiguity was caused by a lack of lookahead.

25

Lexical Analysis

Generalized LR

• Multiple parsers are synchronized on shift actions.

• Each parser has its own stack, and as they share states, the overall structure becomes a graph (GSS).

• If two parsers have the same state on top of their stack, they are joined into a single parser.

• Reduce actions affect all possible paths from the top of the stack.

26

Lexical Analysis

Generalized LR

27

S → E $ E → E + E E → E * E E → a

SLR table

Lexical Analysis

Generalized LR

28

S → E $ E → E + E E → E * E E → a

S → . E $ E → . E + E E → . E * E E → . a

0

SLR table

Lexical Analysis

Generalized LR

29

S → E $ E → E + E E → E * E E → a

S → . E $ E → . E + E E → . E * E E → . a

S → E . $ E → E . + E E → E . * E

E → a .

E

a

10

2

SLR table

Lexical Analysis

Generalized LR

30

S → E $ E → E + E E → E * E E → a

S → . E $ E → . E + E E → . E * E E → . a

S → E . $ E → E . + E E → E . * E

E → a .

E → E + . E E → . E + E E → . E * E E → . a

E → E * . E E → . E + E E → . E * E E → . a

S → E $ .E$

a

*

+

10 3

2

4

5

SLR table

Lexical Analysis

Generalized LR

31

S → E $ E → E + E E → E * E E → a

S → . E $ E → . E + E E → . E * E E → . a

S → E . $ E → E . + E E → E . * E

E → a .

E → E + . E E → . E + E E → . E * E E → . a

E → E * . E E → . E + E E → . E * E E → . a

E → E * E . E → E . + E E → E . * E

S → E $ .E$

a

*

+ E

a

10 3

2

4

5

6

SLR table

Lexical Analysis

Generalized LR

32

S → E $ E → E + E E → E * E E → a

S → . E $ E → . E + E E → . E * E E → . a

S → E . $ E → E . + E E → E . * E

E → a .

E → E + . E E → . E + E E → . E * E E → . a

E → E * . E E → . E + E E → . E * E E → . a

E → E + E . E → E . + E E → E . * E

E → E * E . E → E . + E E → E . * E

S → E $ .E$

a

*

+ E

E

a

a

10 3

2

4

5

7

6

SLR table

Lexical Analysis

Generalized LR

33

S → E $ E → E + E E → E * E E → a

S → . E $ E → . E + E E → . E * E E → . a

S → E . $ E → E . + E E → E . * E

E → a .

E → E + . E E → . E + E E → . E * E E → . a

E → E * . E E → . E + E E → . E * E E → . a

E → E + E . E → E . + E E → E . * E

E → E * E . E → E . + E E → E . * E

S → E $ .E$

a

*

+ E

*

E

+a

a

10 3

2

4

5

7

6

SLR table

Lexical Analysis

Generalized LR

34

S → E $ E → E + E E → E * E E → a

S → . E $ E → . E + E E → . E * E E → . a

S → E . $ E → E . + E E → E . * E

E → a .

E → E + . E E → . E + E E → . E * E E → . a

E → E * . E E → . E + E E → . E * E E → . a

E → E + E . E → E . + E E → E . * E

E → E * E . E → E . + E E → E . * E

S → E $ .E$

a

*

+ E

*

*

+

E

+a

a

10 3

2

4

5

7

6

SLR table

Lexical Analysis

Generalized LR

35

Nonterminal

Nullable First Follow

S

E

StateAction Goto

a + * $ S E

0

1

2

3

4

5

6

7

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

SLR table

Lexical Analysis

Generalized LR

36

Nonterminal


S no a -

E no a +, *, $

StateAction Goto

a + * $ S E

0

1

2

3

4

5

6

7

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

SLR table

Lexical Analysis

Generalized LR

37

Nonterminal


S no a -

E no a +, *, $

StateAction Goto

a + * $ S E

0 s2 1

1 s4 s5 s3

2 r3 r3 r3

3 acc

4 s2 7

5 s2 6

6 s4/r2 s5/r2 r2

7 s4/r1 s5/r1 r1

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

SLR table

Lexical Analysis

Generalized LR

38

Parsing

State Action Gotoa + * $ S E

0 s2 11 s4 s5 s32 r3 r3 r33 acc4 s2 75 s2 66 s4/r2 s5/r2 r27 s4/r1 s5/r1 r1

a + a * a

0

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

Lexical Analysis

Generalized LR

39

Parsing



+ a * a

0

2a

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

synchronize on shifts

0

2a

Lexical Analysis

Generalized LR

40

Parsing



+ a * a

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

Lexical Analysis

Generalized LR

41

Parsing



+ a * a

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

0

2a

1a : E

0

2a

1a : E 4+

Lexical Analysis

Generalized LR

42

Parsing



a * a

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

synchronize

Lexical Analysis

Generalized LR

43

Parsing



a * a

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

01a : E 4

+

Lexical Analysis

Generalized LR

44

Parsing



* a

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

01a : E 4

+2a

synchronize

01a : E 4

+2a

Lexical Analysis

Generalized LR

45

Parsing



* a

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

01a : E 4

+2a

a : E 7

Lexical Analysis

Generalized LR

46

Parsing



* a

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

Lexical Analysis

Generalized LR

47

Parsing* a

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

01a : E 4

+2a

a : E 71a + a : E



01a : E 4

+2a

a : E 71a + a : E

5

**

Lexical Analysis

Generalized LR

48

Parsinga

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

synchronize



Lexical Analysis

Generalized LR

49

Parsing



(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

01a : E 4

+ a : E 7

1a + a : E

5*

*

a

01a : E 4

+ a : E 7

1a + a : E

5*

*

2a

Lexical Analysis

Generalized LR

50

Parsing



(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

synchronize

01a : E 4

+ a : E 7

1a + a : E

5*

*

2a

Lexical Analysis

Generalized LR

51

Parsing

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a



01a : E 4

+ a : E 7

1a + a : E

5*

*

6a : E

Lexical Analysis

Generalized LR

52

Parsing

(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a



01a : E 4

+ a : E 7

1a + a : E

5*

*

6a : E

Lexical Analysis

Generalized LR

53

Parsing



(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

01a : E 4

+ a : E 7

1a + a : E

5*

*

6a : E

7a * a : E

Lexical Analysis

Generalized LR

54

Parsing



(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

01a : E 4

+ a : E 7

1a + a : E

5*

*

6a : E

7a * a : E

Lexical Analysis

Generalized LR

55

Parsing



(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

01a : E 4

+ a : E 7

1a + a : E

5*

*

6a : E

7a * a : E

1[a + a] * a : E

Lexical Analysis

Generalized LR

56

Parsing



(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

01a : E 4

+ a : E 7

1a + a : E

5*

*

6a : E

7a * a : E

1[a + a] * a : E

Lexical Analysis

Generalized LR

57

Parsing



(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

[a + a] * a : Eor

a + [a * a] : E

01a : E 4

+ a : E7

1a + a : E

5*

*

6a : E

7

a * a : E

1

Lexical Analysis

Generalized LR

58

Parsing



(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

[a + a] * a : Eor

a + [a * a] : E

01a : E 4

+ a : E7

1a + a : E

5*

*

6a : E

7

a * a : E

1 3$

Lexical Analysis

Generalized LR

59

Parsing



(0) S → E $ (1) E → E + E (2) E → E * E (3) E → a

synchronize

accept with trees on the link to initial state

LR Parsing

Scannerless Generalized-LR Parsing

V

60

Lexical Analysis

Scannerless Generalized LR

• Integrates scanning + parsing into a single GLR algorithm.

• Normalization separates lexical and context-free symbols.

• Crucial for avoiding conflicts when composing languages.

• Introduces lexical ambiguities. Solution:

‣Follow Restrictions (implement longest match).

‣Reject Rules (implement reserved keywords).

61

Lexical Analysis


62

Normalization

context-free syntax

Exp.Add = <<Exp> + <Exp>> {left} Exp.Inc = <<Exp>++> Exp.ID = ID

lexical syntax

ID = [a-zA-Z] IDRest* IDRest = [a-zA-Z0-9] LAYOUT = [\ \t\n\r] ID = "let" {reject}

context-free syntax


lexical syntax


Lexical Analysis


63

Normalization

Exp-CF.Add = Exp-CF LAYOUT?-CF "+" LAYOUT?-CF Exp-CF {left} Exp-CF.Inc = Exp-CF LAYOUT?-CF "++" Exp-CF.ID = ID-CF ID-LEX = [a-zA-Z] IDRest*-LEX IDRest*-LEX = [a-zA-Z0-9] ID-LEX = "let" {reject} LAYOUT-CF = LAYOUT-LEX

Separate lexical from context-free symbols and separate symbols in context-free syntax by optional layout.

Lexical Analysis


64

Normalization

LAYOUT-CF = LAYOUT-LEX IDRest-CF = IDRest-LEX IDRest*-CF = IDRest*-LEX ID-CF = ID-LEX

context-free syntax


lexical syntax


Create injections from lexical to context-free symbols

Lexical Analysis


65

Normalization

"+" = [\43] "++" = [\43] [\43] "let" = [\108] [\101] [\116] ID-LEX = [\65-\90\97-\122] IDRest*-LEX IDRest*-LEX = [\48-\57\65-\90\97-\122] LAYOUT-LEX = [\9-\10\13\32]

context-free syntax


lexical syntax


Normalize literals symbols and character classes.

Lexical Analysis


66

Normalization

LAYOUT?-CF = LAYOUT-CF LAYOUT?-CF = IDRest+-LEX = IDRest-LEX IDRest+-LEX = IDRest+-LEX IDRest-LEX IDRest*-LEX = IDRest*-LEX = IDRest+-LEX IDRest+-CF = IDRest+-LEX

context-free syntax


lexical syntax


Normalize regular expressions.

Lexical Analysis


67

Normalization

context-free start-symbols

Exp

<START> = LAYOUT?-CF Exp-CF LAYOUT?-CF <Start> = <START> [\256]

Define extra rules for the start symbols.

Lexical Analysis


68

Parse Table Generation

• Generation is based on SLR(1) item-sets.

• Uses character classes instead of tokens.

• Follow sets and goto actions are calculated for productions instead of non-terminals.

Lexical Analysis


69

Lexical Disambiguation

• A reject rule is of the form: ID = "let" {reject}

• When SGLR does a reduction with a reject rule, it marks the link as rejected.

• Further action on a stack is forbidden whenever all links to it are rejected.

• Follow restrictions are implemented as filters on the follow sets of productions.

Lexical Analysis

Generalized LR

70

• Priority rules: applied at parse table generation.

E → E * . E E → . E + E E → . E * E E → . a

multiplication has higher priority than addition!

Context-free Disambiguation

Lexical Analysis

Generalized LR

71


multiplication is left associative!

E → E * . E E → . E + E E → . E * E E → . a


Lexical Analysis

Generalized LR

72


• Disambiguation filters: applied after parsing (prefer and avoid).

multiplication is left associative!

E → E * . E E → . E + E E → . E * E E → . a


LR Parsing

Summary

VI

73

lessons learned

LR Parsing

Summary

74

lessons learned

LR Parsing

Summary

How can we generate LR parse tables?

• items, closure, goto

74

lessons learned

LR Parsing

Summary



How can we improve LR(0) parse table generation?

• SLR: consider FOLLOW sets to avoid shift-reduce conflicts

• LR(1): consider look-ahead in states

• LALR(1): unify LR(1) states to reduce state space

74

lessons learned

LR Parsing

Summary



How can we improve LR(0) parse table generation?

• SLR: consider FOLLOW sets to avoid shift-reduce conflicts

• LR(1): consider look-ahead in states

• LALR(1): unify LR(1) states to reduce state space

How can we handle conflicts in the parse table?

• generalized parsing - supports all class of context free grammars

• scannerless generalized LR - allow for proper language composition.

74

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LR Parsing

Literature

75

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LR Parsing

Literature

LR parsing

Andrew W. Appel, Jens Palsberg: Modern Compiler Implementation in Java, 2nd edition. 2002

Alfred V. Aho, Ravi Sethi, Jeffrey D. Ullman, Monica S. Lam: Compilers: Principles, Techniques, and Tools, 2nd edition. 2006

75

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LR Parsing

Literature

LR parsing

Andrew W. Appel, Jens Palsberg: Modern Compiler Implementation in Java, 2nd edition. 2002

Alfred V. Aho, Ravi Sethi, Jeffrey D. Ullman, Monica S. Lam: Compilers: Principles, Techniques, and Tools, 2nd edition. 2006

Generalised LR parsing

Eelco Visser: Syntax Definition for Language Prototyping. PhD thesis 1997

M.G.J. van den Brand, J. Scheerder, J.J. Vinju, and E. Visser: Disambiguation Filters for Scannerless Generalized LR Parsers. CC 2002

75

LR Parsing

copyrights

76

LR Parsing 77

http://creativecommons.org/licenses/by-nc-sa/3.0/

copyrights

LR Parsing

Pictures

Slide 1: Book Scanner by Ben Woosley, some rights reserved

Slide 19: Ostsee by Mario Thiel, some rights reserved

78

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http://www.flickr.com/people/thiemadotcom/

Software

LR Parsing