In sensible gramma rs, these strings share some common c - - PDF document

SLIDE 1 Con text-F ree Grammars Notation for recursiv e description

• f

languages. Example: Rol l !< RO LL > C l ass S tuds < =RO LL > C l ass ! < C LAS S > T ext < =C LAS S > T ext ! C har T ext T ext ! C har C har ! a

• (other

c hars) S tuds ! S tud S tuds S tuds !

• S

tud !< S T U D > T ext < =S T U D >

• Generates

\do cumen ts" suc h as: <ROLL><CLASS>cs 154</ CLASS > <STUD>Sally</ST UD> <STUD>Fred</STU D>

• </ROLL>
• V

ariables (e.g., S tuds) represen t sets

• f

strings (i.e., languages).

✦

In sensible gramma rs, these strings share some common c haracteristic

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roll.

• T

erminals (e.g., a

• r

< RO LL >) = sym b

• ls
• f

whic h strings are comp

• sed.

✦

\T ags" lik e < RO LL > could b e considered either a single terminal

• r

the concatenation

• f

6 terminals.

• Pr
• ductions

= rules

• f

the form H ead ! B

• dy

.

✦

H ead is a v ariable.

✦

B

• dy

is a string

• f

zero

• r

more v ariables and/or terminals.

• Start

Symb

• l

= v ariable that represen ts \the language."

• Notation:

G = (V ; ; P ; S ) = (v ariables, terminals, pro ductions, start sym b

• l).

Example A simpler example generates strings

• f

0's and 1's suc h that eac h blo c k

• f

0's is follo w ed b y at least as man y 1's. S ! AS j

• A

! 0A1 j A1 j 01

• Note

v ertical bar separates dieren t b

• dies

for the same head. 1

SLIDE 2 Deriv ations

• A

)

• whenev

er there is a pro duction A !

• .

✦

Subscript with name

• f

grammar, e.g., ) G , if necessary .

✦

Example: 011AS ) 0110A1S .

• )

*

• means

string

• can

b ecome

• in

zero

• r

more deriv ation steps.

✦

Examples: 011AS ) * 011AS (zero steps); 011AS ) * 0110A1S (one step); 011AS ) * 0110011 (three steps). Language

• f

a CF G L(G) = set

• f

terminal strings w suc h that S ) * G w , where S is the start sym b

• l.

Aside: Notation

• a;

b; : : : are terminals; : : : ; y ; z are strings

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terminals.

• Greek

letters are strings

• f

v ariables and/or terminals,

• ften

called sentential forms.

• A;

B ; : : : are v ariables.

• :

: : ; Y ; Z are v ariables

• r

terminals.

• S

is t ypically the start sym b

• l.

Leftmost/Righ t most Deriv ations

• W

e ha v e a c hoice

• f

v ariable to replace at eac h step.

✦

Deriv ations ma y app ear dieren t

• nly

b ecause w e mak e the same replacemen ts in a dieren t

• rder.

✦

T

• a

v

• id

suc h dierences, w e ma y restrict the c hoice.

• A

leftmost deriv ation alw a ys replaces the leftmost v ariable in a sen ten tial form.

✦

Yields left-sentential forms.

• R

ightmost dened analogously .

• )

lm , ) rm , etc., used to indicate deriv ations are leftmost

• r

righ tmost. 2

SLIDE 3 Example

• S

) lm AS ) lm A1S ) lm 011S ) lm 011AS ) lm 0110A1S ) lm 0110011S ) lm 0110011

• S

) rm AS ) rm AAS ) rm AA ) rm A0A1 ) rm A0011 ) rm A10011 ) rm 0110011 Deriv ation T rees

• No

des = v ariables, terminals,

• r

.

✦

V ariables at in terior no des; terminals and

• at

lea v es.

✦

A leaf can b e

• nly

if it is the

• nly

c hild

• f

its paren t.

• A

no de and its c hildren from the left m ust form the head and b

• dy
• f

a pro duction. Example S A S A S

• A

1 1 A 1 1 Equiv alence

• f

P arse T rees, Leftmost, and Righ tmost Deriv ations The follo wing ab

• ut

a grammar G = (V ; ; P ; S ) and a terminal string w are all equiv alen t: 1. S ) * w (i.e., w is in L(G)). 2. S ) * lm w 3. S ) * rm w 4. There is a parse tree for G with ro

• t

S and yield (lab els

• f

lea v es, from the left) w .

• Ob

viously (2) and (3) eac h imply (1). 3

SLIDE 4 P arse T ree Implies LM/RM Deriv ations

• Generalize

all statemen ts to talk ab

• ut

an arbitrary v ariable A in place

• f

S .

✦

Except no w (1) no longer means w is in L(G).

• Induction
• n

the heigh t

• f

the parse tree. Basis : Heigh t 1: T ree is ro

• t

A and lea v es w = a 1 ; a 2 ; : : : ; a k .

• A

! w m ust b e a pro duction, so A ) lm w and A ) rm w . Induction : Heigh t > 1. T ree is ro

• t

A with c hildren X 1 ; X 2 ; : : : ; X k .

• Those

X i 's that are v ariables are ro

• ts
• f

shorter trees.

✦

Th us, the IH sa ys that they ha v e LM deriv ations

• f

their yields.

• Construct

a LM deriv ation

• f

w from A b y starting with A ) lm X 1 X 2

• X

k , then using LM deriv ations from eac h X i that is a v ariable, in

• rder

from the left.

• RM

deriv ation analogous. Deriv ations to P arse T rees Induction

• n

length

• f

the deriv ation. Basis : One step. There is an

• b

vious parse tree. Induction : More than

• ne

step.

• Let

the rst step b e A ) X 1 X 2

• X

k .

• Subsequen

t c hanges can b e reordered so that all c hanges to X 1 and the sen ten tial forms that replace it are done rst, then those for X 2 , and so

• n

(i.e., w e can rewrite the deriviation as a LM deriv ation).

• The

deriv ations from those X i 's that are v ariables are all shorter than the giv en deriviation, so the IH applies.

• By

the IH, there are parse trees for eac h

• f

these deriv ations.

• Mak

e the ro

• ts
• f

these trees b e c hildren

• f

a new ro

• t

lab eled A. Example Consider deriv ation S ) AS ) AAS ) AA ) A1A ) A10A1 ) 0110A1 ) 0110011 4

SLIDE 5

• Sub

deriv ation from A is: A ) A1 ) 011

• Sub

deriv ation from S is: S ) AS ) A ) 0A1 ) 0011

• Eac

h has a parse tree; put them together with new ro

• t

S . 5