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CSC 3130: Automata theory and formal languages

LR(k) grammars

Fall 2008ELJUN P. CORTES MBA MPA BSCS ACS

MELJUN CORTES

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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

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LR(0) parsing example revisited

Stack Input

S

S

S

R

S

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

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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

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Outline of LR(0) parsing algorithm

Algorithm can perform two actions:

What if:

no complete

item

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

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Definition of LR(0) grammar

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

LR means parsing happens leftto right and produces a

rightmost derivation

LR(0) grammars are unambiguous and have a

fast

parsing algorithm

Unfortunately, they are not expressive enough

to describe programming languages

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context-free grammarsparse using CYK algorithm (slow)

LR() grammars

Hierarchy of context-free grammars

LR(1) grammars

LR(0) grammarsparse using LR(0) algorithm

java

perl

python

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A grammar that is not LR(0)

S A(1)

| Bc(2)

A aA(3)| a(4)

B a(5)| ab(6)

input: a

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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!

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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!

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

B a, B ab,

A aA, A a

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Lookahead

S A(1)

| Bc(2)

A aA(3)| a(4)

B a(5)| ab(6)

input: a apeek inside!

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

B a, B ab,

A aA, A a

A

A

a a

S

parse tree must

look like this

action: shift

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Lookahead

S A(1)

| Bc(2)

A aA(3)| a(4)

B a(5)| ab(6)

input: a a apeek inside!

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

A aA, A a

parse tree must

look like this

A

A

aA

a

S

action: shift

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Lookahead

S A(1)

| Bc(2)

A aA(3)| a(4)

B a(5)| ab(6)

input: a a a

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

A aA, A a

parse tree must

look like this

action: reduce

A

A

aA

a a

S

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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)

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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

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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 shiftand reducestates

Step 3: Run DFA on input, using stack to

remembersequence of states

Use lookaheadto eliminate wrong reduce items

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Recall NFA transitions for LR(0)

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

For every itemA aXbwe have a transition

For every itemA aCband production C d

S aq0

A aXbX

A aXb

C dA aCb

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NFA transitions for LR(1)

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

For every itemA aXbwe 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]

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FIRSTsets

Example

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

from 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() =

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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

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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,]

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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 lookaheadcan always resolve such conflicts

[A aA, ] [A a, ]

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

[A aA, ] [A a, ]

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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

SR

S

R

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

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

[B ab, c][S Bc, ]

[S Bc, ]

look ahead!

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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 klookahead symbols

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

LR(1) grammars describe the semantics of most

programming languages