LR(1) & LALR PARSER

LR(1) & LALR PARSER

INTRODUCTION

The LALR ( Look Ahead-LR ) parsing technique is

between SLR and Canonical LR, both in terms of power of parsing grammars and ease of implementation. This method is often used in practice because the tables obtained by it are considerably smaller than the Canonical LR tables, yet most common syntactic constructs of programming languages can be expressed conveniently by an LALR grammar. The same is almost true for SLR grammars, but there are a few constructs that can not be handled by SLR techniques.

CONSTRUCTING LALR PARSING TABLES

A core is a set of LR (0) (SLR) items for the grammar, and an LR (1) (Canonical LR) grammar may produce more than two sets of items with the same core.

The core does not contain any look ahead information. Example: Let s1 and s2 are two states in a Canonical LR

grammar.

S1 – {C ->c.C, c/d; C -> .cC, c/d; C -> .d, c/d}S2 – {C ->c.C, $; C -> .cC, $; C -> .d, $}

These two states have the same core consisting of only the production rules without any look ahead information.

CONSTRUCTION IDEA:

1. Construct the set of LR (1) items.

2. Merge the sets with common core together as one set, if no conflict (shift-shift or shift-reduce) arises.

3. If a conflict arises it implies that the grammar is not LALR.

4. The parsing table is constructed from the collection of merged sets of items using the same algorithm for LR (1) parsing.

ALGORITHM:

• Input: An augmented grammar G’.

• Output: The LALR parsing table actions and goto for G’.

Method:

1. Construct C= {I0, I1, I2,…, In}, the collection of sets of LR(1) items.

2. For each core present in among the set of LR(1) items , find all sets having the core, and replace these sets by their union.

3. Parsing action table is constructed as for Canonical LR.4. The goto table is constructed by taking the union of all sets of

items having the same core. If J is the union of one or more sets of LR (1) items, that is, J=I1 U I2 U … U Ik, then the cores of goto(I1,X), goto(I2,X),…, goto(Ik, X) are the same as all of them have same core. Let K be the union of all sets of items having same core as goto(I1, X). Then goto(J,X)=K.

EXAMPLE

GRAMMAR:

1. S’ -> S

2. S -> CC

3. C -> cC

4. C -> d

SET OF ITEMS:

I0 : S’ -> .S, $ S -> .CC, $ C -> .c C, c /d C -> .d, c /d

I1: S’ -> S., $

I2: S -> C.C, $ C -> .Cc, $ C -> .d, $

I3: C -> c. C, c /d C -> .Cc, c /d

C -> .d, c /d

I4: C -> d., c /d

I5: S -> CC., $

I6: C -> c.C, $ C -> .cC, $ C -> .d, $

I7: C -> d., $

I8: C -> cC., c /d

I9: C -> cC., $

The goto graph:

CANONICAL PARSING TABLE

LALR PARSER

Merge the Cores:

• 3 & 6

• 4 & 7

• 8 & 9

LALR PARSING TABLE

SHIFT-REDUCE CONFLICT

COMPARISON OF LR (1) AND LALR:1. If LR (1) has shift-reduce conflict then LALR will

also have it.2. If LR (1) does not have shift-reduce conflict LALR

will also not have it.3. Any shift-reduce conflict which can be removed by

LR (1) can also be removed by LALR.4. For cases where there are no common cores SLR and

LALR produce same parsing tables.


LR(1) Parser-

How Does a shift-Reduce conflict arises?

IA : { E->α..βc, F->γ. } & FOLLOW(F)={β}

For Eg. I3 : {E->L.=R , R-> L. } & Follow(E)={=}

For Solving this Problem by LALR Parser:

• IA={E->α.βc,ω1 ; F->γ.,δ1}

• IB={E->α.βc,ω2 ; F->γ.,δ2}

• IAB={E->α.βc,ω1|ω2 ; F->γ.,δ1|δ2}


COMPARISON OF SLR AND LALR:1. If SLR has shift-reduce conflict then LALR may or may not

remove it.2. SLR and LALR tables for a grammar always have same number of

states.

Hence, LALR parsing is the most suitable for parsing general programming languages.The table size is quite small as compared to LR (1) , and by carefully designing the grammar it can be made free of conflicts. For example, in a language like Pascal LALR table will have few hundred states, but a Canonical LR will have thousands of states. So it is more convenient to use an LALR parsing.

REDUCE-REDUCE CONFLICT

The Reduce-Reduce conflicts still might just remain .This claim may be better comprehended if we take the example of the following grammar:

• IA={ A->γ1.,δ1 ; ->γ2.,δ2}If δ1≠δ2 then NO reduce-reduce conflict.(mutual exclusive sets)• IA={ A->γ1.,δ1 ; ->γ2.,δ2}• IB={ A->γ1.,δ3 ; B->γ2.,δ4}• IAB={ A->γ1.,δ1|δ3 ; B->γ2.,δ2|δ4}Conflicts arises when δ1=δ4 or δ2=δ3.

Example of R-R Conflict

• S'-> S

• S -> aAd

• S -> bBd

• S -> aBe

• S -> bAe

• A -> c

• B -> c

Generating the LR (1) items for the above grammar

• I0 : S’-> .S , $• S-> . aAd, $• S-> . bBd, $ • S-> . aBe, $• S-> . bAe, $• I1: S’-> S ., $ • I2: S-> a . Ad, $• S-> a . Be, $ • A-> .c, d • B->.c, e • I3: S-> b . Bd, $• S-> b . Ae, $ • A->.c, e• B->.c,d• I4: S->aA.d, $

Generating the LR (1) items for the above grammar

• I5: S-> aB.e,$• I6: A->c. , d ; B->c. , e • I7: S->bB.d, $ • I8: S->bA.e, $• I9: B->c. ,d ; A->c. , e • I10: S->aAd. , $ • I11: S->aBe., $• I12: S->bBd., $• I13: S->aBe., $

• The underlined items are of our interest. We see that when we make the Parsing table for LR (1), we will get something like this…

The LR (1) Parsing Table

The LALR Parsing table

LR(1) Parsing table on reduction to the LALR parsing table

Conclusion

• So, we find that the LALR gains reduce-reduce conflict whereas the corresponding LR (1) counterpart was void of it. This is a proof enough that LALR is less potent than LR (1).

• But, since we have already proved that the LALR is void of shift-reduce conflicts (given that the corresponding LR(1) is devoid of the same), whereas SLR (or LR (0)) is not necessarily void of shift-reduce conflict, the LALR grammar is more potent than the SLR grammar

Conclusion

SHIFT-REDUCE CONFLICT present in

SLR (Some of them are solved in….)

LR (1) (All those solved are preserved in…)

LALR

So, we have answered all the queries on LALR that we raised intuitively.

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Shift-Reduce Parsers

• Reviewing some technologies:– Phrase– Simple phrase– Handle of a sentential form

S

A b C

b C a C

b C b a C

A sentential form

handle Simple phrase

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Shift-reduce parser

• A parse stack– Initially empty, contains symbols already parsed

• Elements in the stack are not terminal or nonterminal symbols

– The parse stack catenated with the remaining input always represents a right sentential form

– Tokens are shifted onto the stack until the top of the stack contains the handle of the sentential form

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Shift-reduce parser

• Two questions1. Have we reached the end of handles and how

long is the handle?

2. Which nonterminal does the handle reduce to?

• We use tables to answer the questions– ACTION table– GOTO table

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Shift-reduce parser

• LR parsers are driven by two tables:– Action table, which specifies the actions to take

• Shift, reduce, accept or error

– Goto table, which specifies state transition

• We push states, rather than symbols onto the stack

• Each state represents the possible subtree of the parse tree

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Shift-reduce parser

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31

<program>

begin <stmts> end $

SimpleStmt ; <stmts>

SimpleStmt ; <stmts>

R 4

R 2

R 2

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LR Parsers• LR(1):

– left-to-right scanning– rightmost derivation(reverse)– 1-token lookahead

• LR parsers are deterministic– no backup or retry parsing actions

• LR(k) parsers – decide the next action by examining the tokens

already shifted and at most k lookahead tokens– the most powerful of deterministic bottom-up

parsers with at most k lookahead tokens.

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LR(0) Parsing

• A production has the form – AX1X2…Xj

• By adding a dot, we get a configuration (or an item)– A•X1X2…Xj

– AX1X2…Xi • Xi+1 … Xj

– AX1X2…Xj •

• The • indicates how much of a RHS has been shifted into the stack.

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LR(0) Parsing

• An item with the • at the end of the RHS

– AX1X2…Xj •

– indicates (or recognized) that RHS should be reduced to LHS

• An item with the • at the beginning of RHS– A•X1X2…Xj

– predicts that RHS will be shifted into the stack

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LR(0) Parsing

• An LR(0) state is a set of configurations– This means that the actual state of LR(0)

parsers is denoted by one of the items.

• The closure0 operation:– if there is an configuration B • A in the set

then add all configurations of the form A • to the set.

• The initial configuration– s0 = closure0({S • $})

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LR(0) Parsing

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LR(0) Parsing• Given a configuration set s, we can compute its

successor, s', under a symbol X– Denoted go_to0(s,X)=s'

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LR(0) Parsing• Characteristic finite state machine (CFSM)

– It is a finite automaton, p.148, para. 2.– Identifying configuration sets and successor operation with CFSM

states and transitions

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LR(0) Parsing• For example, given grammar G2

S'S$SID|

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LR(0) Parsing• CFSM is the goto table of LR(0) parsers.

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LR(0) Parsing• Because LR(0) uses no lookahead, we must

extract the action function directly from the configuration sets of CFSM

• Let Q={Shift, Reduce1, Reduce2 , …, Reducen}– There are n productions in the CFG

• S0 be the set of CFSM states

– P:S02Q

• P(s)={Reducei | B • s and production i is

B } (if A • a s for a Vt Then {Shift} Else )

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LR(0) Parsing• G is LR(0) if and only if s S0 |P(s)|=1

• If G is LR(0), the action table is trivially extracted from P– P(s)={Shift} action[s]=Shift

– P(s)={Reducej}, where production j is the augmenting production, action[s]=Accept

– P(s)={Reducei}, ij, action[s]=Reducei

– P(s)= action[s]=Error

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• Consider G1

SE$EE+T | TTID|(E)

CFSM for G1

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LR(0) Parsing

• Any state s S0 for which |P(s)|>1 is said to be inadequate

• Two kinds of parser conflicts create inadequacies in configuration sets– Shift-reduce conflicts

– Reduce-reduce conflicts

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LR(0) Parsing• It is easy to introduce inadequacies in

CFSM states– Hence, few real grammars are LR(0). For

example,• Consider -productions

– The only possible configuration involving a -production is of the form A •

– However, if A can generate any terminal string other than , then a shift action must also be possible (First(A))

• LR(0) parser will have problems in handling operator precedence properly

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LR(1) Parsing

• An LR(1) configuration, or item is of the form– AX1X2…Xi • Xi+1 … Xj, l where l Vt{}

• The look ahead commponent l represents a possible lookahead after the entire right-hand side has been matched

• The appears as lookahead only for the augmenting production because there is no lookahead after the endmarker

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LR(1) Parsing

• We use the following notation to represent the set of LR(1) configurations that shared the same dotted production

AX1X2…Xi • Xi+1 … Xj, {l1…lm}

={AX1X2…Xi • Xi+1 … Xj, l1}

{AX1X2…Xi • Xi+1 … Xj, l2}

…

{AX1X2…Xi • Xi+1 … Xj, lm}

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LR(1) Parsing

• There are many more distinct LR(1) configurations than LR(0) configurations.

• In fact, the major difficulty with LR(1) parsers is not their power but rather finding ways to represent them in storage-efficient ways.

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LR(1) Parsing• Parsing begins with the configuration

– closure1({S • $, {}})

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LR(1) Parsing• Consider G1

SE$EE+T | TTID|(E)

• closure1(S • E$, {})

S • E$, {}

E • E+T, {$}

E • T, {$}

E • E+T, {+}

E • T, {+}

T • ID, {+}

T • (E), {+}T • ID, {$}

T • (E), {$}

closure1(S • E$, {})={S • E$, {};E • E+T, {$+}E • T, {$+}T • ID, {$+}T • (E), {$+}}

How many configures?

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LR(1) Parsing• Given an LR(1) configuration set s, we

compute its successor, s', under a symbol X– go_to1(s,X)

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LR(1) Parsing

• We can build a finite automation that is analogue of the LR(0) CFSM – LR(1) FSM, LR(1) machine

• The relationship between CFSM and LR(1) macine– By merging LR(1) machine’s configuration

sets, we can obtain CFSM

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• G3

SE$

EE+T|T

TT*P|P

PID|(E)

Is G3 an LR(0)Grammar?

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LR(1) Parsing• The go_to table used to drive an LR(1) is

extracted directly from the LR(1) machine

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LR(1) Parsing• Action table is extracted directly from the

configuration sets of the LR(1) machine

• A projection function, P– P : S1Vt2Q

• S1 be the set of LR(1) machine states

• P(s,a)={Reducei | B •,a s and

production i is B } (if A • a,b s Then {Shift} Else )

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LR(1) Parsing• G is LR(1) if and only if

s S1 a Vt |P(s,a)|1

• If G is LR(1), the action table is trivially extracted from P– P(s,$)={Shift} action[s][$]=Accept

– P(s,a)={Shift}, a$ action[s][a]=Shift

– P(s,a)={Reducei}, action[s][a]=Reducei

– P(s,a)= action[s][a]=Error

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SLR(1) Parsing

• LR(1) parsers are the most powerful class of shift-reduce parsers, using a single lookahead– LR(1) grammars exist for virtually all

programming languages– LR(1)’s problem is that the LR(1) machine

contains so many states that the go_to and action tables become prohibitively large

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SLR(1) Parsing

• In reaction to the space inefficiency of LR(1) tables, computer scientists have devised parsing techniques that are almost as powerful as LR(1) but that require far smaller tables

1. One is to start with the CFSM, and then add lookahead after the CFSM is build– SLR(1)

2. The other approach to reducing LR(1)’s space inefficiencies is to merger inessential LR(1) states– LALR(1)

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SLR(1) Parsing

• SLR(1) stands for Simple LR(1)– One-symbol lookahead – Lookaheads are not built directly into

configurations but rather are added after the LR(0) configuration sets are built

– An SLR(1) parser will perform a reduce action for configuration B • if the lookahead symbol is in the set Follow(B)

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SLR(1) Parsing

• The SLR(1) projection function, from CFSM states, – P : S0Vt2Q

– P(s,a)={Reducei | B •,a Follow(B) and

production i is B } (if A • a s for a Vt Then {Shift} Else )

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SLR(1) Parsing• G is SLR(1) if and only if

s S0 a Vt |P(s,a)|1

• If G is SLR(1), the action table is trivially extracted from P– P(s,$)={Shift} action[s][$]=Accept




• Clearly SLR(1) is a proper superset of LR(0)

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SLR(1) Parsing• Consider G3

– It is LR(1) but not LR(0)

– See states 7,11

– Follow(E)={$,+,)}

SE$

EE+T|T

TT*P|P

PID|(E)

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SE$

EE+T|T

TT*P|P

PID|(E)

G3 is both SLR(1) and LR(1).

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Limitations of the SLR(1) Technique• The use of Follow sets to estimate the

lookaheads that predict reduce actions is less precise than using the exact lookaheads incorporated into LR(1) configurations– Consider G4

Elem(List, Elem)

ElemScalar

ListList,Elem

List Elem

Scalar ID

Scalar(Scalar)

Fellow(Elem)={“)”,”,”,….}

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Fellow(Elem)={“)”,”,”,….}

LR(1) lookahead for ElemScalar • is “,”

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LALR(1)

• LALR(1) parsers can be built by first constructing an LR(1) parser and then merging states– An LALR(1) parser is an LR(1) parser in which

all states that differ only in the lookahead components of the configurations are merged

– LALR is an acronym for Look Ahead LR

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The core of a configuration

• The core of the above two configurations is the same

EE+T

T T*P

T P

P id

P (E)

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States Merge• Cognate(s)={c|cs, core(s)=s}

EE+T

T T*P

T P

P id

P (E)

s

Cognate( ) =

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LALR(1)• LALR(1) machine

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LALR(1)

• The CFSM state is transformed into its LALR(1) Cognate – P : S0Vt2Q

– P(s,a)={Reducei | B •,a Cognate(s) and

production i is B } (if A • a s Then {Shift} Else )

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LALR(1) Parsing• G is LALR(1) if and only if

s S0 a Vt |P(s,a)|1

• If G is LALR(1), the action table is trivially extracted from P– P(s,$)={Shift} action[s][$]=Accept




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LALR(1) Parsing• Consider G5

<stmt>ID

<stmt><var>:=<expr>

<var> ID<var> ID [<expr>] <expr><var>

• Assume statements are separated by ;’s, the grammar is not SLR(1) because; Follow(<stmt>) and

; Follow(<var>), since <expr><var>

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LALR(1) Parsing

• However, in LALR(1), if we use <var> ID the next symbol must be :=so action[ 1, := ] = reduce(<var> ID)

action[ 1, ; ] = reduce(<stmt> ID)

action[ 1,[ ] = shift

• There is no conflict.

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LALR(1) Parsing• A common technique to put an LALR(1) grammar

into SLR(1) form is to introduce a new nonterminal whose global (i.e. SLR) lookaheads more nearly correspond to LALR’s exact look aheads– Follow(<lhs>) = {:=}

<stmt>ID

<stmt><var>:=<expr>

<var> ID<var> ID [<expr>] <expr><var>

<stmt>ID

<stmt><lhs>:=<expr>

<lhs> ID<lhs> ID [expr>]

<var> ID<var> ID [expr>] <expr><var>

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LALR(1) Parsing• At times, it is the CFSM itself that is at fault.

S(Exp1)S[Exp1]S(Exp2]S[Exp2)<Exp1>ID<Exp2>ID

• A different expression nonterminal is used to allow error or warning diagnostics

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Building LALR(1) Parsers

• In the definition of LALR(1)– An LR(1) machine is first built, and then its

states are merged to form an automaton identical in structure to the CFSM

• May be quite inefficient

– An alternative is to build the CFSM first.• Then LALR(1) lookaheads are “propagated” from

configuration to configuration

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• Propagate links:– Case 1: one configuration is created from

another in a previous state via a shift operation

A •X , L1 A X• , L2

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• Propagate links:– Case 2: one configuration is created as the

result of a closure or prediction operation on another configuration

B •A , L1

A • , L2

L2={ x|xFirst(t) and t L1 }

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Building LALR(1) Parsers• Step 1: After the CFSM is built, we can create all

the necessary propagate links to transmit lookaheads from one configuration to another

• Step 2: spontaneous lookaheads are determined– By including in L2, for configuration A,L2, all

spontaneous lookaheads induced by configurations of the form B A,L1

• These are simply the non- values of First()

• Step 3: Then, propagate lookaheads via the propagate links– See figure 6.25

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87

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• A number of LALR(1) parser generators use lookahead propagation to compute the parser action table– LALRGen uses the propagation algorithm

– YACC examines each state repeatedly

• An intriguing alternative to propagating LALR lookaheads is to compute them as needed by doing a backward search through the CFSM– Read it yourself. P. 176, Para. 3

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• An intriguing alternative to propagating LALR lookaheads is to compute them as needed by doing a backward search through the CFSM– Read it yourself. P. 176,

Para. 3

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Calling Semantic Routines in Shift-Reduce Parsers

• Shift-reduce parsers can normally handle larger classes of grammars than LL(1) parsers, which is a major reason for their popularity

• Shift-reduce parsers are not predictive, so we cannot always be sure what production is being recognized until its entire right-hand side has been matched– The semantic routines can be invoked only after a

production is recognized and reduced• Action symbols only at the extreme right end of a right-hand

side

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• Two common tricks are known that allow more flexible placement of semantic routine calls

• For example,<stmt>if <expr> then <stmts> else <stmts> end if

• We need to call semantic routines after the conditional expression else and end if are matched– Solution: create new nonterminals that generate

<stmt>if <expr> <test cond>

then <stmts> <process then part>

else <stmts> end if

<test cond><process than part>

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• If the right-hand sides differ in the semantic routines that are to be called, the parser will be unable to correctly determine which routines to invoke– Ambiguity will manifest. For example,

<stmt>if <expr> <test cond1> then <stmts> <process then part> else <stmts> end if;

<stmt>if <expr> <test cond2> then <stmts> <process then part> end if;

<test cond1><test cond2><process than part>

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• An alternative to the use of –generating nonterminals is to break a production into a number of pieces, with the breaks placed where semantic routines are required

<stmt><if head><then part><else part>

<if head>if <expr>

<then part>then <stmts>

<else part>then <stmts> end if;

– This approach can make productions harder to read but has the advantage that no –generating are needed

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Uses (and Misuses) of Controlled Ambiguity

• Research has shown that ambiguity, if controlled, can be of value in producing efficient parsers for real programming languages.

• The following grammar is not LR(1)Stmt if Expr then Stmt

Stmt if Expr then Stmt else Stmt

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• The following grammar is not ambiguous<stmt><matched>|<unmatched>

<matched> if <logic_expr> then <matched> else <matched>

| <any_non-if_statement>

<unmatched> if <logic_expr> then <stmt>

| if <logic_expr> then <matched> else <unmatched>

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• In LALRGen, when conflicts occur, we may use the option resolve to give preference to earlier productions.

• In YACC, conflicts are solved in– 1. shift is preferred to reduce;– 2. earlier productions are preferred.

• Grammars with conflicts are usually smaller.• Application: operator precedence and

associativity.

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• We no longer specify a parser purely in terms of the grammar it parses, but rather we explicitly include auxiliary rules to disambiguate conflicts in the grammar.– We will present how to specify auxiliary rules

in yacc when we introduce the yacc.

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• The precedence values assigned to operators resolve most shift-reduce conflicts of the formExpr Expr OP1 Expr

Expr Expr OP2 Expr

• Rules:– If Op1 has a higher precedence than Op2, we Reduce.– If Op2 has a higher precedence, we shift.– If Op1 and Op2 have the same precedence, we use the

associativity definitions.• Right-associative, we shift. Left-associative, we reduce. • No-associative, we signal, and error.

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EE+EEE*EEID

EE+EEE*E

EID

EE+EEE+EEE*EEID

EE*EEE+EEE*EEID

EE+EEE+EEE*E

EE*EEE+EEE*E

EID

EID

ID

E

E

E

+

*

ID

ID

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Uses (and Misuses) of Controlled Ambiguity (Cont’d)

• More example: Page 234 of “lex&yacc”

%nonassoc LOWER_THAN_ELSE%nonassoc ELSE%%

stmt: IF expr stmt %prec LOWER_THAN_ELSE | IF expr stmt ELSE stmt;

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Uses (and Misuses) of Controlled Ambiguity (Cont’d)

• If your language uses a THEN keyword (like Pascal does):

%nonassoc THAN%nonassoc ELSE%%

stmt: IF expr THAN stmt | IF expr THAN stmt ELSE stmt ;

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Optimizing Parse Tables

• A number of improvements can be made to decrease the space needs of an LR parse– Merging go_to and action tables to a single

table

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• Encoding parse table– As integers

• Error entries as zeros

• Reduce actions as positive integers

• Shift actions as negative integers

• Single reduce states– E.g. state 4 in Figure 6.35

• State 5 can be eliminated

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• See figure 6.36

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LR(1) & LALR PARSER