Parsing IV Bottom-up Parsing. Parsing Techniques Top-down parsers (LL(1), recursive descent) Start...

Parsing IVBottom-up Parsing

Parsing Techniques

Top-down parsers (LL(1), recursive descent)

• Start at the root of the parse tree and grow toward leaves

• Pick a production & try to match the input• Bad “pick” may need to backtrack• Some grammars are backtrack-free (predictive

parsing)

Bottom-up parsers (LR(1), operator precedence)

• Start at the leaves and grow toward root• As input is consumed, encode possibilities in an internal

state• Start in a state valid for legal first tokens• Bottom-up parsers handle a large class of grammars

Bottom-up Parsing (definitions)

The point of parsing is to construct a derivation

A derivation consists of a series of rewrite stepsS 0 1 2 … n–1 n sentence

• Each i is a sentential form If contains only terminal symbols, is a sentence in L(G) If contains ≥ 1 non-terminals, is a sentential form

• To get i from i–1, expand some NT A i–1 by using A Replace the occurrence of A i–1 with to get i

In a leftmost derivation, it would be the first NT A i–1

A left-sentential form occurs in a leftmost derivationA right-sentential form occurs in a rightmost derivation

Bottom-up Parsing

A bottom-up parser builds a derivation by working fromthe input sentence back toward the start symbol S

S 0 1 2 … n–1 n sentence

To reduce i to i–1 match some rhs against i then

replace with its corresponding lhs, A. (assuming the production A)

In terms of the parse tree, this is working from leaves to root

• Nodes with no parent in a partial tree form its upper fringe

• Since each replacement of with A shrinks the upper fringe, we call it a reduction.

The parse tree need not be built, it can be simulated|parse tree nodes| = |words| + |reductions|

bottom-up

Finding Reductions

Consider the simple grammar

And the input string abbcde

The trick is scanning the input and finding the next reduction

The mechanism for doing this must be efficient

Finding Reductions (Handles)

The parser must find a substring of the tree’s frontier that matches some production A that occurs as one step in the rightmost derivation ( A is in

RRD)

Informally, we call this substring a handle

Formally,A handle of a right-sentential form is a pair <A,k> whereA P and k is the position in of ’s rightmost symbol.

If <A,k> is a handle, then replacing at k with A produces the right sentential form from which is derived in the rightmost derivation.

Because is a right-sentential form, the substring to the right of a handle contains only terminal symbols

the parser doesn’t need to scan past the handle (very far)

Finding Reductions (Handles)

Critical Insight (Theorem?)

If G is unambiguous, then every right-sentential form has a unique handle.

If we can find those handles, we can build a derivation !

Sketch of Proof:

1 G is unambiguous rightmost derivation is unique

a unique production A applied to derive i from i–

1

a unique position k at which A is applied a unique handle <A,k> This all follows from the definitions

Example (a very busy slide)

The expression grammar Handles for rightmost derivation of x – 2 * y

This is the inverse of Figure 3.9 in EaC

Handle-pruning, Bottom-up Parsers

The process of discovering a handle & reducing it to the appropriate left-hand side is called handle pruning

Handle pruning forms the basis for a bottom-up parsing method

To construct a rightmost derivationS 0 1 2 … n–1 n w

Apply the following simple algorithmfor i n to 1 by –1 Find the handle <Ai i , ki > in i

Replace i with Ai to generate i–1

This takes 2n steps

Handle-pruning, Bottom-up Parsers

One implementation technique is the shift-reduce parser

push INVALIDtoken next_token( )repeat until (top of stack = Goal and token = EOF) if the top of the stack is a handle A then // reduce to A pop || symbols off the stack push A onto the stack else if (token EOF) then // shift push token token next_token( ) else // need to shift, but out of input

report an error

Figure 3.7 in EAC

How do errors show up?

• failure to find a handle

• hitting EOF & needing to shift (final else clause)

Either generates an error

Back to x - 2 * y

Stack Input Handle Action$ id – num * id none shift$ id – num * id

1. Shift until the top of the stack is the right end of a handle2. Find the left end of the handle & reduce

Back to x - 2 * y

Stack Input Handle Action$ id – num * id none shift$ id – num * id 9,1 red. 9$ Factor – num * id 7,1 red. 7$ Term – num * id 4,1 red. 4$ Expr – num * id


Back to x - 2 * y

Stack Input Handle Action$ id – num * id none shift$ id – num * id 9,1 red. 9$ Factor – num * id 7,1 red. 7$ Term – num * id 4,1 red. 4$ Expr – num * id none shift$ Expr – num * id none shift$ Expr – num * id


Back to x - 2 * y

Stack Input Handle Action$ id – num * id none shift$ id – num * id 9,1 red. 9$ Factor – num * id 7,1 red. 7$ Term – num * id 4,1 red. 4$ Expr – num * id none shift$ Expr – num * id none shift$ Expr – num * id 8,3 red. 8$ Expr – Factor * id 7,3 red. 7$ Expr – Term * id


Back to x - 2 * y

Stack Input Handle Action$ id – num * id none shift$ id – num * id 9,1 red. 9$ Factor – num * id 7,1 red. 7$ Term – num * id 4,1 red. 4$ Expr – num * id none shift$ Expr – num * id none shift$ Expr – num * id 8,3 red. 8$ Expr – Factor * id 7,3 red. 7$ Expr – Term * id none shift$ Expr – Term * id none shift$ Expr – Term * id


Back to x – 2 * y

Stack Input Handle Action$ id – num * id none shift$ id – num * id 9,1 red. 9$ Factor – num * id 7,1 red. 7$ Term – num * id 4,1 red. 4$ Expr – num * id none shift$ Expr – num * id none shift$ Expr – num * id 8,3 red. 8$ Expr – Factor * id 7,3 red. 7$ Expr – Term * id none shift$ Expr – Term * id none shift$ Expr – Term * id 9,5 red. 9$ Expr – Term * Factor 5,5 red. 5$ Expr – Term 3,3 red. 3$ Expr 1,1 red. 1$ Goal none accept


5 shifts + 9 reduces + 1 accept

Example

Goal

<id,x>

Term

Fact.

Expr –

Expr

<id,y>

<num,2>

Fact.

Fact.Term

Term

*

Shift-reduce Parsing

Shift reduce parsers are easily built and easily understood

A shift-reduce parser has just four actions• Shift — next word is shifted onto the stack• Reduce — right end of handle is at top of stack

Locate left end of handle within the stack Pop handle off stack & push appropriate lhs

• Accept — stop parsing & report success• Error — call an error reporting/recovery routine

Accept & Error are simpleShift is just a push and a call to the scannerReduce takes |rhs| pops & 1 pushIf handle-finding requires state, put it in the stack 2x

work

Handle finding is key• handle is on stack• finite set of handles use a DFA !

An Important Lesson about Handles

To be a handle, a substring of a sentential form must have two properties: It must match the right hand side of some rule A There must be some rightmost derivation from the goal

symbol that produces the sentential form with A as the last production applied

• Simply looking for right hand sides that match strings is not good enough

• Critical Question: How can we know when we have found a handle without generating lots of different derivations? Answer: we use look ahead in the grammar along with

tables produced as the result of analyzing the grammar. LR(1) parsers build a DFA that runs over the stack & finds

them

An Important Lesson about Handles

• To be a handle, a substring of a sentential form must have two properties: It must match the right hand side of some rule A There must be some rightmost derivation from the goal

symbol that produces the sentential form with A as the last production applied

• We have seen that simply looking for right hand sides that match strings is not good enough

• Critical Question: How can we know when we have found a handle without generating lots of different derivations? Answer: we use look ahead in the grammar along with

tables produced as the result of analyzing the grammar. o There are a number of different ways to do this.o We will look at two: operator precedence and LR

parsing

LR(1) Parsers

• LR(1) parsers are table-driven, shift-reduce parsers that use a limited right context (1 token) for handle

recognition• LR(1) parsers recognize languages that have an LR(1)

grammar

Informal definition:A grammar is LR(1) if, given a rightmost derivation

S 0 1 2 … n–1 n sentence

We can 1. isolate the handle of each right-sentential form i, and

2. determine the production by which to reduce,

by scanning i from left-to-right, going at most 1 symbol beyond

the right end of the handle of i

LR(1) Parsers

A table-driven LR(1) parser looks like

Tables can be built by handHowever, this is a perfect task to automate

ScannerTable-driven

Parser

ACTION & GOTOTables

ParserGenerator

sourcecode

grammar

IR

LR(1) Skeleton Parser

stack.push(INVALID); stack.push(s0); not_found = true;token = scanner.next_token();do while (not_found) { s = stack.top(); if ( ACTION[s,token] == “reduce A” ) then {

stack.popnum(2*||); // pop 2*|| symbols s = stack.top(); stack.push(A); stack.push(GOTO[s,A]);

}

else if ( ACTION[s,token] == “shift si” ) then {stack.push(token); stack.push(si);token scanner.next_token();

} else if ( ACTION[s,token] == “accept”

& token == EOF )then not_found = false;

else report a syntax error and recover;} report success;

The skeleton parser

•uses ACTION & GOTO tables

•does |words| shifts

•does |derivation| reductions •does 1 accept

•detects errors by failure of 3 other cases

To make a parser for L(G), need a set of tables

The grammar

The tables

LR(1) Parsers (parse tables)

Example Parse 1

The string “baa”

Example Parse 1


Example Parse 1


Example Parse 1


Example Parse 2

The string “baa baa ”

Example Parse 2


Example Parse 2


Example Parse 2


LR(1) Parsers

How does this LR(1) stuff work?• Unambiguous grammar unique rightmost derivation

• Keep upper fringe on a stack All active handles include top of stack (TOS) Shift inputs until TOS is right end of a handle

• Language of handles is regular (finite) Build a handle-recognizing DFA ACTION & GOTO tables encode the DFA

• To match subterm, invoke subterm DFA & leave old DFA’s state on stack

• Final state in DFA a reduce action New state is GOTO[state at TOS (after pop), lhs] For SN, this takes the DFA to s1

S0

S3

S2

S1

baa

baa

SN

Control DFA for SN

Reduce action

Reduce action

Building LR(1) Parsers

How do we generate the ACTION and GOTO tables?• Use the grammar to build a model of the DFA

• Use the model to build ACTION & GOTO tables• If construction succeeds, the grammar is LR(1)

The Big Picture• Model the state of the parser• Use two functions goto( s, X ) and closure( s )

goto() is analogous to move() in the subset construction closure() adds information to round out a state

• Build up the states and transition functions of the DFA

• Use this information to fill in the ACTION and GOTO tables

Terminal or non-terminal

LR(k) items

The LR(1) table construction algorithm uses LR(1) items to represent valid configurations of an LR(1) parser

An LR(k) item is a pair [P, ], where

P is a production A with a • at some position in the rhs

is a lookahead string of length ≤ k (words or EOF)

The • in an item indicates the position of the top of the stack[A•,a] means that the input seen so far is consistent with the

use of A immediately after the symbol on top of the stack[A •,a] means that the input sees so far is consistent with the

use of A at this point in the parse, and that the parser has already recognized .

[A •,a] means that the parser has seen , and that a lookahead symbol of a is consistent with reducing to A.

LR(1) Items

The production A, where = B1B1B1 with lookahead a, can give rise to 4 items

[A•B1B2B3,a], [AB1•B2B3,a], [AB1B2•B3,a], & [AB1B2B3•,a]

The set of LR(1) items for a grammar is finite

What’s the point of all these lookahead symbols?• Carry them along to choose the correct reduction, if

there is a choice

• Lookaheads are bookkeeping, unless item has • at right end Has no direct use in [A•,a] In [A•,a], a lookahead of a implies a reduction by A For { [A•,a],[B•,b] }, a reduce to A; FIRST() shift

Limited right context is enough to pick the actions

High-level overview Build the canonical collection of sets of LR(1) Items, I

a Begin in an appropriate state, s0

[S’ •S,EOF], along with any equivalent items Derive equivalent items as closure( s0 )

b Repeatedly compute, for each sk, and each X, goto(sk,X) If the set is not already in the collection, add it Record all the transitions created by goto( )

This eventually reaches a fixed point

2 Fill in the table from the collection of sets of LR(1) items

The canonical collection completely encodes the transition diagram for the handle-finding DFA

LR(1) Table Construction

Back to Finding Handles

Revisiting an issue from last class

Parser in a state where the stack (the fringe) wasExpr – Term With lookahead of *

How did it choose to expand Term rather than reduce to Expr?

• Lookahead symbol is the key• With lookahead of + or –, parser should reduce to Expr• With lookahead of * or /, parser should shift• Parser uses lookahead to decide • All this context from the grammar is encoded in the

handle recognizing mechanism

Back to x – 2 * y

Stack Input Handle Action$ id – num * id none shift$ id – num * id 9,1 red. 9$ Factor – num * id 7,1 red. 7$ Term – num * id 4,1 red. 4$ Expr – num * id none shift$ Expr – num * id none shift$ Expr – num * id 8,3 red. 8$ Expr – Factor * id 7,3 red. 7$ Expr – Term * id none shift$ Expr – Term * id none shift$ Expr – Term * id 9,5 red. 9$ Expr – Term * Factor 5,5 red. 5$ Expr – Term 3,3 red. 3$ Expr 1,1 red. 1$ Goal none accept

1. Shift until TOS is the right end of a handle2. Find the left end of the handle & reduce

shift here

reduce here

Computing FIRST Sets

Define FIRST as• If * a, a T, (T NT)*, then a FIRST()• If * , then FIRST()

Note: if = X, FIRST() = FIRST(X)

To compute FIRST

• Use a fixed-point method• FIRST(A) 2(T )

• Loop is monotonic Algorithm halts

Computation of FIRST uses FOLLOW, so build FOLLOW sets before FIRST sets

Computing FOLLOW Sets

FOLLOW(S) {EOF }for each A NT, FOLLOW(A) Ø

while (FOLLOW sets are still changing) for each p P, of the form A12 … k

FOLLOW(k) FOLLOW(k) FOLLOW(A) TRAILER FOLLOW(A) for i k down to 2

if FIRST( i ) then

FOLLOW(i-1 ) FOLLOW(i-1) { FIRST(i ) – { } } TRAILER

else FOLLOW(i-1 ) FOLLOW(i-1) FIRST(i )

TRAILER Ø

For SheepNoise:FOLLOW(Goal ) = { EOF }FOLLOW(SN) = { baa, EOF }

Computing FIRST Sets

for each x T, FIRST(x) { x }for each A NT, FIRST(A) Ø

while (FIRST sets are still changing) for each p P, of the form A, if is then FIRST(A) FIRST(A) { } else if is B1B2…Bk then begin

FIRST(A) FIRST(A) ( FIRST(B1) – { } )

for i 1 to k–1 by 1 while FIRST(Bi )

FIRST(A) FIRST(A) ( FIRST(Bi +1) – { } )

if i = k–1 and FIRST(Bk )

then FIRST(A) FIRST(A) { } end

for each A NTif FIRST(A) then

FIRST(A) FIRST(A) FOLLOW(A)

For SheepNoise:FIRST(Goal) = { baa }FIRST(SN ) = { baa }FIRST(baa) = { baa }

on SN baaon G SN

Computing Closures

Closure(s) adds all the items implied by items already in s

• Any item [AB,a] implies [B,x] for each production with B on the lhs, and each x FIRST(a)

• Since B is valid, any way to derive B is valid, too

The algorithm

Closure( s ) while ( s is still changing ) items [A •B,a] s productions B P b FIRST(a) // might be if [B • ,b] s then add [B • ,b] to s

Classic fixed-point method Halts because s ITEMS

Worklist version is faster

Closure “fills out” a state

Example From SheepNoise

Initial step builds the item [Goal•SheepNoise,EOF]and takes its closure( )

Closure( [Goal•SheepNoise,EOF] )

So, S0 is { [Goal • SheepNoise,EOF], [SheepNoise • SheepNoise baa,EOF],

[SheepNoise• baa,EOF], [SheepNoise • SheepNoise baa,baa],

[SheepNoise • baa,baa] }

It em From

[Goal•SheepNoise,EOF] Original item

[SheepNoise•SheepNoise baa,EOF] 1, a is EOF

[SheepNoise • baa,EOF] 1, a is EOF

[SheepNoise•SheepNoise baa,baa] 2, a is baa EOF

[SheepNoise • baa,baa] 2, a is baa EOF

Remember, this is the left-recursive SheepNoise; EaC shows the right-recursive version.

Computing Gotos

Goto(s,x) computes the state that the parser would reach if it recognized an x while in state s

• Goto( { [AX,a] }, X ) produces [AX,a] (obviously)

• It also includes closure( [AX,a] ) to fill out the state

The algorithm

Goto( s, X ) new Ø items [A•X,a] s new new [AX•,a]

return closure(new)

Not a fixed-point method!

Straightforward computation

Uses closure ( )

Goto() moves forward

Example from SheepNoise

S0 is { [Goal • SheepNoise,EOF], [SheepNoise • SheepNoise

baa,EOF], [SheepNoise • baa,EOF], [SheepNoise • SheepNoise

baa,baa], [SheepNoise • baa,baa] }

Goto( S0 , baa )

• Loop produces

• Closure adds nothing since • is at end of rhs in each item

In the construction, this produces s2

{ [SheepNoisebaa •, {EOF,baa}]}

I tem From

[SheepNoise baa•, EOF] I tem 3 in s0

[SheepNoise baa•, baa] I tem 5 in s0

New, but obvious, notation for two distinct items

[SheepNoisebaa •, EOF] & [SheepNoisebaa •, baa]


S0 : { [Goal • SheepNoise, EOF], [SheepNoise • SheepNoise baa, EOF],

[SheepNoise• baa, EOF], [SheepNoise • SheepNoise baa, baa],

[SheepNoise • baa, baa] }

S1 = Goto(S0 , SheepNoise) =

{ [Goal SheepNoise •, EOF], [SheepNoise SheepNoise • baa, EOF],

[SheepNoise SheepNoise • baa, baa] }

S2 = Goto(S0 , baa) = { [SheepNoise baa •, EOF], [SheepNoise baa •, baa] }

S3 = Goto(S1 , baa) = { [SheepNoise SheepNoise baa •, EOF],

[SheepNoise SheepNoise baa •, baa] }

Building the Canonical Collection

Start from s0 = closure( [S’S,EOF ] )

Repeatedly construct new states, until all are found

The algorithm

s0 closure ( [S’S,EOF] )S { s0 }k 1

while ( S is still changing ) sj S and x ( T

NT ) sk goto(sj,x) record sj sk on x if sk S then

S S sk

k k + 1

Fixed-point computation Loop adds to S S 2ITEMS, so S is finite

Worklist version is faster


Starts with S0





Starts with S0




Iteration 1 computesS1 = Goto(S0 , SheepNoise) =





Starts with S0








Iteration 2 computes S3 = Goto(S1 , baa) = { [SheepNoise SheepNoise baa •, EOF],



Starts with S0








Iteration 2 computes S3 = Goto(S1 , baa) = { [SheepNoise SheepNoise baa •, EOF],


Nothing more to compute, since • is at the end of every item in S3 .

Example (grammar & sets)

Simplified, right recursive expression grammar

Goal ExprExpr Term – ExprExpr TermTerm Factor * Term Term FactorFactor ident

Example (building the collection)

Initialization Step

s0 closure( { [Goal •Expr , EOF] } )

{ [Goal • Expr , EOF], [Expr • Term – Expr , EOF], [Expr • Term , EOF],

[Term • Factor * Term , EOF], [Term • Factor * Term , –],

[Term • Factor , EOF], [Term • Factor , –],

[Factor • ident , EOF], [Factor • ident , –], [Factor • ident , *] }

S {s0 }

Example (building the collection)

Iteration 1

s1 goto(s0 , Expr)

s2 goto(s0 , Term)

s3 goto(s0 , Factor)

s4 goto(s0 , ident )

Iteration 2

s5 goto(s2 , – )

s6 goto(s3 , * )

Iteration 3

s7 goto(s5 , Expr )

s8 goto(s6 , Term )

Example (Summary)S0 : { [Goal • Expr , EOF], [Expr • Term – Expr , EOF], [Expr • Term , EOF],

[Term • Factor * Term , EOF], [Term • Factor * Term , –],

[Term • Factor , EOF], [Term • Factor , –],

[Factor • ident , EOF], [Factor • ident , –], [Factor • ident, *] }

S1 : { [Goal Expr •, EOF] }

S2 : { [Expr Term • – Expr , EOF], [Expr Term •, EOF] }

S3 : { [Term Factor • * Term , EOF],[Term Factor • * Term , –], [Term Factor •, EOF],

[Term Factor •, –] }

S4 : { [Factor ident •, EOF],[Factor ident •, –], [Factor ident •, *] }

S5 : { [Expr Term – • Expr , EOF], [Expr • Term – Expr , EOF], [Expr • Term , EOF],

[Term • Factor * Term , –], [Term • Factor , –],

[Term • Factor * Term , EOF], [Term • Factor , EOF],

[Factor • ident , *], [Factor • ident , –], [Factor • ident , EOF] }

Example (Summary)

S6 : { [Term Factor * • Term , EOF], [Term Factor * • Term , –],

[Term • Factor * Term , EOF], [Term • Factor * Term , –], [Term • Factor , EOF], [Term • Factor , –], [Factor • ident , EOF], [Factor • ident , –], [Factor • ident , *] }

S7: { [Expr Term – Expr •, EOF] }

S8 : { [Term Factor * Term •, EOF], [Term Factor * Term •, –] }

Example (Summary)

The Goto Relationship (from the construction)

State Expr Term Factor - * Ident

0 1 2 3 4

1

2 5

3 6

4

5 7 2 3 4

6 8 3 4

7

8

Filling in the ACTION and GOTO Tables

The algorithm

Many items generate no table entry Closure( ) instantiates FIRST(X) directly for [A•X,a ]

set sx S item i sx

if i is [A •a,b] and goto(sx,a) = sk , a T then ACTION[x,a] “shift k” else if i is [S’S •,EOF] then ACTION[x ,a] “accept” else if i is [A •,a] then ACTION[x,a] “reduce A”

n NT if goto(sx ,n) = sk

then GOTO[x,n] k

x is the state number

Example (Filling in the tables)

The algorithm produces the following table

ACTION GOTO

Ident - * EOF Expr Term Factor

0 s 4 1 2 31 acc2 s 5 r 33 r 5 s 6 r 54 r 6 r 6 r 65 s 4 7 2 36 s 4 8 37 r 28 r 4 r 4

What can go wrong?

What if set s contains [A•a,b] and [B•,a] ?

• First item generates “shift”, second generates “reduce”

• Both define ACTION[s,a] — cannot do both actions

• This is a fundamental ambiguity, called a shift/reduce error• Modify the grammar to eliminate it (if-then-

else)

• Shifting will often resolve it correctly

What is set s contains [A•, a] and [B•, a] ?

• Each generates “reduce”, but with a different production

• Both define ACTION[s,a] — cannot do both reductions

• This is a fundamental ambiguity, called a reduce/reduce conflict

• Modify the grammar to eliminate it (PL/I’s overloading of (...))

In either case, the grammar is not LR(1)

Shrinking the Tables

Three options:

• Combine terminals such as number & identifier, + & -, * & / Directly removes a column, may remove a row For expression grammar, 198 (vs. 384) table entries

• Combine rows or columns Implement identical rows once & remap states Requires extra indirection on each lookup Use separate mapping for ACTION & for GOTO

• Use another construction algorithm Both LALR(1) and SLR(1) produce smaller tables Implementations are readily available

LR(k) versus LL(k) (Top-down Recursive Descent )

Finding ReductionsLR(k) Each reduction in the parse is detectable with

the complete left context, the reducible phrase, itself, and the k terminal symbols to its right

LL(k) Parser must select the reduction based on The complete left context The next k terminalsThus, LR(k) examines more context

“… in practice, programming languages do not actually seem to fall in the gap between LL(1) languages and deterministic languages” J.J. Horning, “LR Grammars and Analysers”, in Compiler Construction, An Advanced Course, Springer-Verlag, 1976

Summary

Advantages

Fast

Good locality

Simplicity

Good error detection

Fast

Deterministic langs.

Automatable

Left associativity

Disadvantages

Hand-coded

High maintenance

Right associativity

Large working sets

Poor error messages

Large table sizes

Top-down

recursive

descent

LR(1)

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