meljun cortes automata theory 11

8/13/2019 MELJUN CORTES Automata Theory 11

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CSC 3130: Automata theory and formal languages

Limitations of context-free languages

Fall 2008ELJUN P. CORTES MBA MPA BSCS ACS

MELJUN CORTES



Non context-free languages

• Recall the pumping lemma for regular languagesallows us to show some languages are not

regular

Are these languages context-free?

L 1 = {an bn : n ≥ 0}

L 2 = {x : x has same number of as and bs}L 3 = {1n : n is prime}

L 4 = {an bn cn : n ≥ 0}

L 5 = {x #x R : x ∈ {0, 1}*}

L 6 = {x #x : x ∈ {0, 1}*}



Some intuition

• Let’s try to show this is context-free

L 4 = {an bn cn : n ≥ 0}

context-free grammar pushdown automaton

S → aBc

B → ??

read a / push 1

read c / pop 1

???



More intuition

• Suppose we could construct some CFG for L4,e.g.

• We do some derivations

of “long” strings

S BC

B CS | b

C SB | a

. . .

S BC CSC

aSC aBCC

abCC

abaC

abaSB

abaBCB ababCB

ababaB ababab



More intuition

• If derivation is long enough, some variable mustappear twice on same path in parse tree

S BC

CSC aSC

aBCC

abCC

abaC

abaSB

abaBCB

ababCB

ababaB

ababab

S

B C

BC S S

B C B C

a b a b a b



More intuition

• Then we can “cut and paste” part of parse tree S

B C

BC S S

B C B C

a b a b a b

BS

B C

a

b

b

B C

BC S S

B C Ba

b a b

bC

S

ababab ababbabb✗



More intuition

• We can repeat this many times

• Every sufficiently large derivation will have a part

that

can be repeated indefinitely – This is caused by cycles in the grammar

ababab✗ ababbabb✗ ababbbabbb

ababn abn bb



General picture

u u

v v

v

w

x

y y

u

v

v

v

w

x

y

uvwxy uv 3wx 3 y

A

A

A

A

A

A

A

A

A

x

x

x

x

wxvvwxxy



Example

• If L4 has a context-free grammar G, then

• What happens for an bn cn ?

• No matter how it is split, uv 2wx 2 y L 4!

If uvwxy can be derived in G, so can uv i wx i y for every i

L 4 = {an

bn

cn

: n ≥ 0}

wu yxva a a ... a a b b b ... b b c c c ... c c



Pumping lemma for context-free

languages

• Theorem: For every context-free language L

There exists a number n such that for every

string z in L , we can write z = uvwxy where

|vwx | ≤ n |vx | ≥ 1

For every i ≥ 0, the string uv i wx i y is in L .

wu yxv



Pumping lemma for context-free

languages

• So to prove L is not context-free, it is enough that

For every n there exists z in L , such that for

every way of writing z = uvwxy where

|vwx | ≤ n and |vx | ≥ 1, the string uv i wx i yisnot in L for some i ≥ 0.

wu yxv



Proving language is not context-free

• Just like for regular languages, need strategy that,

regardless of adversary, always wins you this

gameadversary

choose n

write z = uvwxy (|vwx | ≤ n ,|vx | ≥ 1)

you

choose z L

choose i

you win if uv i wx i y L

1

2



Example

adversary

choose n

write z = uvwxy (|vwx | ≤ n ,|vx | ≥ 1)

you

choose z L

choose i

you win if uv i wx i y L

1

2

adversary

n

write z = uvwxy

you

z = an bn cn

i = ? 1

2

L 4 = {an bn cn : n ≥ 0}

wu yxva a a ... a a b b b ... b b c c c ... c c



Example

• Case 1: v or x contains two kinds of symbols

Then uv 2wx 2 y not in L because pattern is wrong

• Case 2: v and x both contain one kind of symbol

Then uv 2wx 2 y does not have same number of as,

bs, cs

xva a a ... a a b b b ... b b c c c ... c c

xv

a a a ... a a b b b ... b b c c c ... c c



More examples

• Which of these is context-free?

L 1 = {an bn : n ≥ 0}

L 2 = {x : x has same number of as and bs}

L 3 = {1n

: n is prime} L 4 = {an bn cn : n ≥ 0}

L 5 = {x #x R : x {0, 1}*}

L 6 = {x #x : x {0, 1}*}

meljun cortes automata theory 11

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