MELJUN CORTES -- AUTOMATA THEORY LECTURE - 14

7/31/2019 MELJUN CORTES -- AUTOMATA THEORY LECTURE - 14

1/24

CSC 3130: Automata theory and formal languages

Andrej Bogdanov

http://www.cse.cuhk.edu.hk/~andrejb/csc3130

LR(k) grammars

Fall 2008


2/24

LR(0) example from last time

A aAb

Aab

A aAb

A ab

A aAbA ab

A aAb

A aAb

A ab

a

b

bAa

1

2

3

4

5

A aAb | ab


3/24

LR(0) parsing example revisited

Stack Input

S

S

SRS

R

1

1a2

1a2a2

1a2a2b3

1a2A4

1a2A4b5

1A

aabb

abb

bb

b

b

A S

A aAb | ab A aAb aabb

1

2

2

3

4

5

A

A aAb

Aab A aAb

A ab

A aAb

A ab

A aAb A aAb

A ab

a

b

b

A

a1

2

3

4 5

Aa b

a b


4/24

Meaning of LR(0) items

a

A

A aXb

NFA transitions to:

Xg

X b

focus

shift focus tosubtree rooted atX(ifXis nonterminal)

A aXb

move past subtreerooted atX


5/24

Outline of LR(0) parsing algorithm

Algorithm can perform two actions:

What if:

no completeitem

is valid

there is one valid item,and it is complete

shift (S) reduce (R)

some valid itemscomplete, some

not

more than one validcomplete item

S / R conflict R / R conflict


6/24

Definition of LR(0) grammar

A grammar is LR(0) if S/R, R/R conflicts neveroccur

LR means parsing happens left to right and produces arightmost derivation

LR(0) grammars are unambiguous and have afast

parsing algorithm

Unfortunately, they are not expressive enough

to describe programming languages


7/24

context-free grammarsparse using CYK algorithm (slow)

LR() grammars

Hierarchy of context-free grammars

LR(1) grammars

LR(0) grammarsparse using LR(0) algorithm

javaperl

python


8/24

A grammar that is not LR(0)

S A(1)

| Bc(2)

A aA(3) | a(4)

B a(5) | ab(6)

input: a


9/24

A grammar that is not LR(0)

S A(1)

| Bc(2)

A aA(3) | a(4)

B a(5) | ab(6)

A

S

A B

A

aA

a a

A

a a

S S

ca

input:

possibilities:

shift (3), reduce (4)

reduce (5), shift (6)

valid LR(0) items:A aA, A a

B a, B ab,

A aA, A a

a

S/R, R/R conflicts!


10/24

Lookahead

S A(1)

| Bc(2)

A aA(3) | a(4)

B a(5) | ab(6)

A

S

A B

A

aA

a a

A

a a

S S

ca

input:

apeek inside!


B a, B ab,

A aA, A a


11/24

Lookahead

S A(1)

| Bc(2)

A aA(3) | a(4)

B a(5) | ab(6)

input: a apeek inside!


B a, B ab,

A aA, A a

A

A

a a

S

parse tree mustlook like this

action: shift


12/24

Lookahead

S A(1)

| Bc(2)

A aA(3) | a(4)

B a(5) | ab(6)

input: a a apeek inside!


A aA, A a


A

A

aA

a

S

action: shift


13/24

Lookahead

S A(1)

| Bc(2)

A aA(3) | a(4)

B a(5) | ab(6)

input: a a a


A aA, A a


action: reduce

A

A

aA

a a

S


14/24

LR(0) items vs. LR(1) items

A

A

a b

a b

Aa b

A aAb | ab

A aAb

A

A

a b

a b

Aa b

[A aAb, b]

LR(0) LR(1)


15/24

LR(1) items

LR(1) items are of the form

to represent this state in the parsing

[A ab, x] [A ab, ]or

a b x

A

a b

A


16/24

Outline of LR(1) parsing algorithm

Step 1: Build NFA that describes valid itemupdates

Step 2: Convert NFA to DFA

As in LR(0), DFA will have shift and reduce states

Step 3: Run DFA on input, using stack to

remembersequence of states

Use lookahead to eliminate wrong reduce items


17/24

Recall NFA transitions for LR(0)

States of NFA will be items (plus a start state q0) For every item S a we have a transition

For every itemA aXb we have a transition

For every itemA aCb and production C d

S aq0

A aXbX

A aXb

C dA aCb


18/24

NFA transitions for LR(1)

For every item [S a, ]we have a transition

For every itemA aXb we have a transition

For every item [A aCb, x] and production C

d

for everyyin FIRST(bx)

[S a, ]q0

[A aXb, x]X

[A aXb, x]

[C d,y]

[A aCb, x]


19/24

FIRST sets

Example

FIRST(a) is the set of terminals that occuron the left in some derivation startingfrom a

S A(1) | cB(2)

A aA(3) | a(4)

B

a

(5)

| ab

(6)

FIRST(a) = {a}

FIRST(A) = {a}

FIRST(S) = {a, c}

FIRST(bAc) = {b}FIRST(BA) = {a}

FIRST() =


20/24

Explaining the transitions

[A aXb, x]X

[A aXb, x]

[C d,y]

[A aCb, x]

a

A

C b x

a

A

X b x a

A

X b x

yFIRST(bx)

y

C b

d


21/24

Example

S

A

(1)

| Bc

(2)

A aA(3) | a(4)

B a(5) | ab(6)

[S A, ]

q0

[S Bc, ]

[S A, ]

A[A aA, ]

[B a,c]

[S Bc, ]

[B ab,c]

. . .

B

[A a, ]


22/24

Convert NFA to DFA

Each DFA state is a subset of LR(1) items, e.g.

States can contain S/R, R/R conflicts

But lookahead can always resolve such conflicts

[A aA, ] [A a, ]

[B a, c] [B ab, c]

[A aA, ] [A a, ]


23/24

Example

S A(1)

| Bc(2)

A aA(3) | a(4)

B a(5) | ab(6)

stack input

a

abB

Bc

S

abc

bc

cc

A valid items

[S A, ] [S Bc, ] [A aA, ][A a, ] [B a, c] [B ab, c]

S

SRSR

[A aA, ] [A a, ] [B a, c]

[B ab, c] [A aA, ] [A a, ]

[B ab, c][S Bc, ]

[S Bc, ]

look ahead!


24/24

LR(k) grammars

A context-free grammar is LR(1) if all S/R, R/Rconflicts can be resolved with one lookahead

More generally, LR(k) grammars can resolve allconflicts with k lookahead symbols

Items have the form [A ab, x1...xk]

LR(1) grammars describe the semantics of mostprogramming languages

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MELJUN CORTES -- AUTOMATA THEORY LECTURE - 14