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cs466(Prasad) L8Norm 2

Normal Forms

Chomsky Normal Form

Griebach Normal Form

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• Language preserving transformations• Improve parsing efficiency• Prove properties about languages and derivations

λ|SSS → λ→S

aSSS |→ aaSS |→Shorter derivations

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Elimination of λ−rules

aaaS

abSL

bBL

bBB

aBSS

7

**)()(

*)(

|

|

⇒==

→→

λλ

Reduces the length of the derivation

aaaS

abSL

bBL

bbBB

aBSaSS

4

**)()(

)(

|

||

⇒=

+=

→→ λ

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• Aim: Restrict the grammar such that

• Approach:– Introduce S’

)'()(

)'},'{,},'{('

),,,(

GLGL

SSSPSVG

SPVG

=→∪Σ∪=

Σ=

L(G) iff Rules ∈∈→ λλS

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– Add rules to capture the effect of λ-rules to be deleted.

–

(Ensures non-contracting rules)

bbBB

bbBB

bBB

|

||

|

→→→

λλ

.' add then)( If λλ →∈ SGL

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Example

λλ

|

|

aAA

bBB

ABS

→→→

aaAA

bbBB

BAABS

SS

||

||

||

'

λλ

→→→→ aaAA

bbBB

BAABS

SS

|

|

||

|'

→→→→ λ

++++ ∪∪∪

=

baba

ba

λ **

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Determination of nullable non-terminals

AbCBcCC

CaAAACS

→→→→

|

|

λ

},,{},{}{ CASCAC

Bottom-up flow of information

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Algorithm Nullable Nonterminals

NULL := {A | A->λ ε P};

repeat

PREV := NULL;

foreach A ε V do

if there is an A-rule A->w

and w ε PREV*

then NULL := NULL U {A}

until NULL = PREV;

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Proof of correctness

• Soundness– If A ε NULL(final) then A=>* λ.

• Induction on the number of iterations of the loop.

• Completeness– If A=>* λ then A ε NULL(final).

• Induction on the minimal derivation of the null string from a non-terminal.

• Termination• Bounded by the number of non-terminals.

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cC

cbbBB

cbbBaaAA

→→→

||

||||

Elimination of Chain rules

cC

CbbBB

BaaAA

→→→

||

||

Removing renaming rules: redundant procedure calls.

Top-down flow of information

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Construction of Chain(A)

Chain(A) := {A}; PREV := φ;

repeat

NEW := Chain(A) - PREV;

PREV := Chain(A);

foreach B ε NEW do

if there is a rule B->C

then Chain(A) := Chain(A) U {C}

until Chain(A) = PREV;

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Examples

bB

BaaAA

BAABS

→→→

||

||

bB

baaAA

baaAABS

→→→

||

|||

SbBB

BSaA

AbaAS

|

|

||

→→→

SabaAbBB

baAbBSaA

bBSabaAS

|||

|||

|||

→→→

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Elimination of useless symbols

• A variable is usefuluseful if it occurs in a derivation that begins with the start symbol and generates a terminal string.

• Reachable from S

• Derives terminal string*

*

where Σ∈

⇒

ωωGX

*)(

where*

Σ∪∈∈⇒

Vu,v

VXuXvS G

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• Construction of the set of variables that derive terminal string.– Bottom-up flow of information

• Similar to the computation of nullable variables.

• Construction of the set of variables that are reachable– Top-down flow of information

• Similar to the computation of chained variables.

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Examples

ccCC

aaAA

ABS

|

|

|

→→→ λ

aaAA

S

|→→ λ

B does not derive terminal string;C unreachable.

λ→S

A unreachable.

DDBD

BBDB

BDS

|

|

→→→ Empty set of

productions

“Non-termination”

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• A CFG is in Chomsky Normal Form if each rule is of the form:

• Theorem: There is an algorithm to construct a grammar G’ in CNF that is equivalent to a CFG G.

Chomsky Normal Form

}{ where SVB,C

S

aA

BCA

−∈→→→

λ

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Construction

• Obtain an equivalent grammar that does not contain λ-rules, chain rules, and useless variables.

• Apply following conversion on rules of the form: bBcCA →

cW

WCRBRQ

bPPQA

→→→→→

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Significance of CNF

• Length of derivation of a string of length n in CNF = (2n-1)(Cf. Number of nodes of a strictly binary tree with n-leaves)

• Maximum depth of a parse tree = n• Minimum depth of a parse tree = 1log2 +n

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Removal of direct left recursion

• Causes infinite loop in top-down (depth-first) parsers.

• Approach: Generate string from left to right.

aaZZ

bbZA

|

|

→→

+==

aZL

baAL

)(

*)(

*)( baAL =bAaA |→

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)|...||( | )|...||( 2121 kj vvvuuuAA →

* )|...||)(|...||( : 2121 jk uuuvvvRΕ

ZuuuuuuZ

ZvvvvvvA

jj

kk

)|...||( | )|...||(

)|...||( | )|...||(

2121

2121

→→

Note that absence of direct left recursion does not imply absence of left recursion.

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bAaB

aBbA

|

|

→→

baaBbB

aBbA

|)|(

|

→→

(Cf. Gaussian Elimination)

baaBbaB

aBbA

||

|

→→

babaZZ

baaZbaaB

aBbA

|

)|(|)|(

|

→→→

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Griebach Normal Form (* Constructs terminal prefixes that facilitates discovery of dead-ends *)

• A CFG is in Griebach Normal Form if each rule is of the form

• Theorem: There is an algorithm to construct a grammar G’ in GNF that is equivalent to a CFG G.

}{ where

...21

SVA

S

aA

AAaAA

i

n

−∈→→→

λ

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Analogy: solving linear simultaneous equations

62

12

3

++−=++=+=

zxz

zyy

yx

6231

62 )3(

1

3

++−+=+++−=

−−=+=

zz

zyz

zy

yx

What are the values of x,y, and z?

4

1

2

==

−=

x

y

z

(Solving for z andthen back substituiting.)

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Example: conversion to GNF

aABC

bCAB

BCA

|

|

→→→

CBA

aBBCC

bCAB

BCA

|

|

→→

→

aBCbBCCAC ||→

CBARCBAR

a|bCB |

RabCBC

|

)|(

→

→

Elim

inat

ing

left

recu

rsio

n Introducing terminals as first element on RHS

)|()|...|(

|||

|

|||

|

|||

CBCBRbCbCBRACR

bCaACbCBAC

aRACbcBRACA

baAbCBA

aRAbcBRAB

abCBaRbCBRC

→

→

→→

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• The size of the equivalent GNF can be large compared to the original grammar.

• Example CFG has 5 rules, but the corresponding GNF has 24 rules!!

• Length of the derivation in GNF = Length of the string.• GNF is useful in relating CFGs

(“generators”) to pushdown automata (“recognizers”/”acceptors”).

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