Siphon Languages that

are not context free

claim I -- hand"c" I neo 3 is not

context - free

Q : can we un pumpingof some kind

A : yes ,

but more complicated

Theorem2.atFor every CFL

L,thus exists p E

IN s . t

✓we L with Iwl Ep I uyxy.EE#s.tw--uvxyzand

① I uvixyiz EL Viso

② / Ivy I so

② I lvxylep

p : As L is aCFL then exists a

CFG

GIVE ,Ros ) in Chomsky normal form

such that ↳ L@ I

let p= 2Mt'( pumping length )

Soppon Iwl Epand look at a

parntree for a derivation of win

G

O

-

Gloi 6868

.

.

r -

e

r

- -

o O

O O l l

lo lo -

- - --

-

o O

-w

so I 2Mt ' leave, in T

Thu , the height ofT isat least I VHL

as the last step is of the formAi → a

Let HEI Se the height ofT

so h④ IZ IVI -12

look at a path of length htt

I⑤ -

} /repeated variable

③ so

1) alldifferent

,o

w → u v X yz

uvixyi2ELIvy 170Wxyt Ep

EXamph2.3.to B=3a" bncnlnzolisuotacfl

P : suppose L=L fora Chomsky grammar

G - LV ,-2 .RS/andCetp--zNHl

• We take w=aPsPcP

• adversary show , w - uvxyzwith

Ivy I > o and lvxylsp

e we claimthat uvoxyoz-uxz.GL

a a - - - ab b -- -bcc . -

- C

Ivxylsp ⇒cannot have both a. and c

in very

e can I very c-aPbP

Then uxz# L is it has mono's

than either adsorb

can 2-

Vxy C- bPcP

IfUx2dL (as in can I )

conclusion : no matter what partitionthe adversary tries we

find

that uxz EL I

so G cannotexist

,implying

that L is not context-free

Exampk2.rs?-C--haibick1oeisjskliascYIP:sopponC--L@1 for some Chomsky

grammar G -- ( V , -2 , Ras )and let

10=2141We Choon s=aPbPcP

adversary : S - uvxyz at

Ivy I > o and IVxylsp

Observation : neither v nor y cancontain two

different symbols

aa . . -abb . - - Scc -

- - c ⇒ uv2xy2z¢Lif

cant no a. inthe string cry

a a . . . abb - - - b cc - - - c ⇒ uxzctl

÷

Cant nob in vy

aa . - abt . - Scc . - c ⇒ uv2xy3&L-

vxy

So the string crycontains both an a

and as

aa -- a lob - - bcc - - c ⇒ uv2xy22 # L

TTY

Exainph-2.38D-hwwlwehodl.lySuppose D=L⑥

) for some Chomsky CFGG-fYE.RS)

we take s=OPlPoPgP when p=2WHl'

www

adversary gives us S=UVXy2 Sf lvyf > o and lvxytep

wi w

wi w e

tfO - -

- 01 . - - of :O . . -ol . - - I or o . - - ol - - - I :O - - -ol - - - l

l,W

✓ veryvery

e

.Thenwvhxg.kz/eL

So oo - r - ol - - - I :O- - ol - - - l

-very

Nowu×z= OPIIOIIP for some i ,j with

lsitjcrhp

Hence uxzotl I