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cs466(Prasad) L8Norm 1
Normal Forms
Chomsky Normal Form
Griebach Normal Form
cs466(Prasad) L8Norm 2
• Language preserving transformations• Improve parsing efficiency
• Prove properties about languages and derivations
|SSS S
aSSS | aaSS |
Shorter derivations
cs466(Prasad) L8Norm 3
Elimination of rules
aaaS
abSL
bBL
bBB
aBSS
7
**)()(
*)(
|
|
Reduces the length of the derivation
aaaS
abSL
bBL
bbBB
aBSaSS
4
**)()(
)(
|
||
cs466(Prasad) L8Norm 4
• Aim: Restrict the grammar such that
• Approach:– Introduce S’
)'()(
)'},'{,},'{('
),,,(
GLGL
SSSPSVG
SPVG
L(G) iff Rules S
cs466(Prasad) L8Norm 5
– Add rules to capture the effect of -rules to be deleted.
–
(Ensures non-contracting rules)
bbBB
bbBB
bBB
|
||
|
.' add then)( If SGL
cs466(Prasad) L8Norm 6
Example
|
|
aAA
bBB
ABS
aaAA
bbBB
BAABS
SS
||
||
||
'
aaAA
bbBB
BAABS
SS
|
|
||
|'
baba
ba
**
cs466(Prasad) L8Norm 7
Determination of nullable non-terminals
AbCBcCC
CaAAACS
|
|
},,{},{}{ CASCAC
Bottom-up flow of information
cs466(Prasad) L8Norm 8
Algorithm Nullable Nonterminals
NULL := {A | A-> P};
repeat
PREV := NULL;
foreach A V do
if there is an A-rule A->w
and w PREV*
then NULL := NULL U {A}
until NULL = PREV;
cs466(Prasad) L8Norm 9
Proof of correctness
• Soundness– If A NULL(final) then A=>* .
• Induction on the number of iterations of the loop.
• Completeness– If A=>* then A NULL(final).
• Induction on the minimal derivation of the null string from a non-terminal.
• Termination• Bounded by the number of non-terminals.
cs466(Prasad) L8Norm 10
cC
cbbBB
cbbBaaAA
||
||||
Elimination of Chainrules
cC
CbbBB
BaaAA
||
||
Removing renaming rules: redundant procedure calls.
Top-down flow of information
cs466(Prasad) L8Norm 11
Construction of Chain(A)
Chain(A) := {A}; PREV := ;
repeat
NEW := Chain(A) - PREV;
PREV := Chain(A);
foreach B NEW do
if there is a rule B->C
then Chain(A) := Chain(A) U {C}
until Chain(A) = PREV;
cs466(Prasad) L8Norm 12
Examples
bB
BaaAA
BAABS
||
||
bB
baaAA
baaAABS
||
|||
SbBB
BSaA
AbaAS
|
|
||
SabaAbBB
baAbBSaA
bBSabaAS
|||
|||
|||
cs466(Prasad) L8Norm 13
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
GX
*)(
where*
Vu,v
VXuXvS G
cs466(Prasad) L8Norm 14
• 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.
cs466(Prasad) L8Norm 15
Examples
ccCC
aaAA
ABS
|
|
|
aaAA
S
|
B does not derive terminal string;C unreachable.
S
A unreachable.
DDBD
BBDB
BDS
|
|
Empty set of
productions
“Non-termination”
cs466(Prasad) L8Norm 16
• 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
cs466(Prasad) L8Norm 17
Construction
• Obtain an equivalent grammar that does not contain -rules, chain rules, and useless variables.
• Apply following conversion on rules of the form: bBcCA
cW
WCRBRQ
bPPQA
cs466(Prasad) L8Norm 18
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
cs466(Prasad) L8Norm 19
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 |
cs466(Prasad) L8Norm 20
)|...||( | )|...||( 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.
cs466(Prasad) L8Norm 21
bAaB
aBbA
|
|
baaBbB
aBbA
|)|(
|
(Cf. Gaussian Elimination)
baaBbaB
aBbA
||
|
babaZZ
baaZbaaB
aBbA
|
)|(|)|(
|
cs466(Prasad) L8Norm 22
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
cs466(Prasad) L8Norm 23
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.)
cs466(Prasad) L8Norm 24
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
cs466(Prasad) L8Norm 25
• 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”).
