Parsing Algorithms and LR(0) Parsing for Programming Languages, Slides of Theory of Automata

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Parsers for programming languages

CFG of the java programming language Identifier: IDENTIFIER QualifiedIdentifier:Identifier {. Identifier } Literal: IntegerLiteral FloatingPointLiteralCharacterLiteral StringLiteralBooleanLiteral NullLiteral Expression: Expression1 [AssignmentOperator Expression1]] AssignmentOperator:= +=-= *=/= &=|=

from http://java.sun.com/docs/books/jls /second_edition/html/syntax.doc.html#

Parsing algorithms

• How long would it take to parse this?

• Can we parse faster?

• No! CYK is the fastest known general-purpose

parsing algorithm

exhaustive algorithm about^10 80 years (longer than life of universe)

CYK algorithm about^1 week!

Another way of thinking

Scientist:

Find an algorithm that can parse strings in any grammar

Engineer:

Design your grammar so it has a very fast parsing algorithm

Items

S → •Tc S → T•c S → Tc•

T → •TA

T → T•A

T → TA•

T → •A

T → A•

A → •aTb A → a•Tb A → aT•b A → aTb•

A → •ab A → a•b A → ab•

S → Tc (1)^ T → TA (2)^ T → A (3) A → aTb(4)^ A → ab(5)

Stack (^) Input ε a ab A T Ta

abaabbc baabbc aabbc aabbc aabbc abbc

Action shift shift reduce (5) reduce (3) shift shift

• • • • • • Idea of parsing algorithm: Try to match complete items to top of stack

Some terminology

S → Tc (1) T → TA (2)^ | A (3) A → aTb(4)^ | ab (5)

input: abaabbc

Stack (^) Input

ε a ab A T Ta Taa Taab TaA TaT TaTb TA T Tc S

abaabbc baabbc aabbc aabbc aabbc abbc bbc bc bc bc c c c ε ε

Action shift shift reduce (5) reduce (3) shift shift shift reduce (5) reduce (3) shift reduce (4) reduce (2) shift reduce (1)

handle

valid items: a•Tb, a•b

valid items: T•a, T•c, aT•b

Running the algorithm

Stack Input

S S S R S R

ε a

aa

aab aA aAb A

aabb abb

bb

b b ε ε

A Valid Items

A → •aAb A → •ab A → a•Ab A → a•b A → •aAb A → •ab A → a•Ab A → a•b A → •aAb A → •ab A → ab• A → aA•b A → aAb•

A → aAb | ab A ⇒ aAb ⇒ aabb

Running the algorithm

Stack Input

S S S R S R

ε a

aa

aab aA aAb A

aabb abb

bb

b b ε ε

A Valid Items

A → •aAb A → •ab A → a•Ab A → a•b A → •aAb A → •ab A → a•Ab A → a•b A → •aAb A → •ab A → ab• A → aA•b A → aAb•

A → aAb | ab A ⇒ aAb ⇒ aabb

How to update viable items

• Updating valid items on “reduce β to B”

• First, we backtrack to viable items before reduce
• Then, we apply same rules as for “shift B” (as if B

were a

terminal)

A → α•Bβ is updated to A^ →^ αB•β A → α• X β disappears if^ X^ ≠ B

C → • δ is added for every valid item A → α•Cβ and production C → • δ

Viable item updates by εNFA

• States of εNFA will be items (plus a start state

q 0 )

• For every item S → • α we have a transition

• For every item A → α• X β we have a transition

• For every item A → α•Cβ and production C →

q 0 ε S → • α

A → α X • β

X

A → α• X β

A → α•Cβ^ ε C → • δ

Convert εNFA to DFA

A → •aAb A→ •ab

A → a•Ab A → a•b A → •aAb A → •ab

A → aA•b

A → aAb•

A → ab•

a

b

A b a

states correspond to sets of valid items

transitions are labeled by variables / terminals

die

Attempt at parsing with DFA

Stack Input

S

S

S

R

ε a

aa

aab aA

aabb abb

bb

b b

A DFA state A → •aAb A → •ab A → a•Ab A → a•b A → •aAb A → •ab A → a•Ab A → a•b A → •aAb A → •ab A → ab• A → aA•b

A → aAb | ab A ⇒ aAb ⇒ aabb

LR(0) grammars and deterministic

PDAs

• The parsing procedure can be implemented by

a

deterministic pushdown automaton

• A PDA is deterministic if in every state there is

at

most one possible transition

• for every input symbol and pop symbol, including ε

• Example: PDA for w # wR^ is deterministic, but

PDA for Docsity.com

LR(0) grammars and deterministic

PDAs

• Not every PDA can be made deterministic

• Since PDAs are equivalent to CFLs, LR(0)

parsing algorithm must fail for some CFLs!

• When does LR(0) parsing algorithm fail?