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Deterministic PDAs

Linz 6th, Section 7.3 DPDA’s and Deterministic Context-Free Languages

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To begin with we require:

A Non Context-Free Language

(We will prove it at the next class)

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Regular languages

**ba

Context-free languages

nnba

Non Context-free languages

nnn cba

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Deterministic PDAs

DPDAs

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DPDAs

Allowed: q1 q2wba →,

q1

q21, wba →

q32, wca →

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q1

q21, wba →

q32, wba →

Not allowed:

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Allowed:1q 2q

wb→,

q1

q21, wb→

q32, wc →

Something must be matched from the stack

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Not allowed:

q1

q21, wb→

q32, wba →

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DPDA example

a, → a

b, a→ q0 q1 q2 q3

b, a→

, $→ $

}0:{)( = nbaML nn

a, → a

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}0:{)( = nbaML nnThe language

is deterministic context-free

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Definition:

A language is deterministic context-free

if some DPDA accepts it

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Example of Non-DPDA (NPDA)

, $→ $q1

q2

bb

aa

→

→

,

,

, → q0

→

→

bb

aa

,

,

}{)( RwwML =

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, $→ $q1

q2

bb

aa

→

→

,

,

, → q0

→

→

bb

aa

,

,

Not allowed in DPDAs

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NPDAs

Have More Power than

DPDAs

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We will show:

there is a context-free language

which is not deterministic context-free

(accepted by a NPDA)

(not accepted by a DPDA)

L

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The language is:

}{}{ 2nnnn babaL =

0n

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}{}{ 2nnnn babaL =

The language is context-freeL

Context-free grammar for : L

21 | SSS →

|11 baSS →

|22 bbaSS →

there is an NPDA

that accepts L

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}{}{ 2nnnn babaL =

is not deterministic context-free

Theorem:

The language

(there is no DPDA that accepts ) L

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Proof: Assume for contradiction that

}{}{ 2nnnn babaL =

is deterministic context free

Therefore:

there is a DPDA that accepts M L

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DPDA with M

nnba nb

acceptsnnba 2

acceptsnnba

}{}{)( 2nnnn babaML =

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The language

is not context-free

}{ nnn cba

(we will prove it later)

Fact 1:

Example 8.1, page 217. The proof uses the context-free pumping lemma.

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The language

is not context-free

}{ nnn cbaL

}){}{( 2nnnn babaL =

Fact 2:

This is what we really need now. And, it can be proved by the pumping in the same manner as {𝑎𝑛𝑏𝑛𝑐𝑛}.

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We will construct a NPDA that accepts:

}{ nnn cbaL

}){}{( 2nnnn babaL =

Contradiction!

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We modify M

nc

Modified M

Replace with cb

nnca

}){}{( 2nnnn babaL =

}){}{( 2nnnn cacaL =

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The NPDA that accepts }{ nnn cbaL

nnba nb

nc

Modified M

Original M

nnca

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Since is accepted by a NPDA }{ nnn cbaL

it is context-free

Contradiction!

(since is not context-free)}{ nnn cbaL

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Therefore:

}{}{ 2nnnn babaL =

There is no DPDA that accepts

End of Proof

Not deterministic context free

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Positive Propertiesof

Context-Free languages

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Context-free languages

are closed under: Union

1L is context free

2L is context free

21 LL

is context-free

Union

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Example

|11 baSS →

|| 222 bbSaaSS →

Union

}{1nnbaL =

}{2RwwL =

21 | SSS →}{}{ Rnn wwbaL =

Language Grammar

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In general:

The grammar of the union

has new start variable

and additional production 21 | SSS →

For context-free languages

with context-free grammars

and start variables

21, LL

21, GG

21, SS

21 LL

S

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Context-free languages

are closed under: Concatenation

1L is context free

2L is context free

21LL

is context-free

Concatenation

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Example

|11 baSS →

|| 222 bbSaaSS →

Concatenation

}{1nnbaL =

}{2RwwL =

21SSS →}}{{ Rnn wwbaL =

Language Grammar

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In general:

The grammar of the concatenation

has new start variable

and additional production 21SSS →

For context-free languages

with context-free grammars

and start variables

21, LL

21, GG

21, SS

21LL

S

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Context-free languages

are closed under: Star-operation

L is context free *L is context-free

Star Operation

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|aSbS →}{ nnbaL =

|11 SSS →*}{ nnbaL =

Example

Language Grammar

Star Operation

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In general:

The grammar of the star operation

has new start variable

and additional production

For context-free language

with context-free grammar

and start variable

L

G

S

*L

1S

|11 SSS →

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Negative Propertiesof

Context-Free Languages

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Context-free languages

are not closed under: intersection

1L is context free

2L is context free

21 LL

not necessarily

context-free

Intersection

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Example

}{1mnn cbaL =

|

|

cCC

aAbA

ACS

→

→

→

Context-free:

}{2mmn cbaL =

|

|

bBcB

aAA

ABS

→

→

→

Context-free:

}{21nnn cbaLL = NOT context-free

Intersection

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Context-free languages

are not closed under: complement

L is context free L not necessarily

context-free

Complement

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}{2121nnn cbaLLLL ==

NOT context-free

Example

}{1mnn cbaL =

|

|

cCC

aAbA

ACS

→

→

→

Context-free:

}{2mmn cbaL =

|

|

bBcB

aAA

ABS

→

→

→

Context-free:

Complement

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Intersectionof

Context-free languagesand

Regular Languages

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The intersection of

a context-free language and

a regular language

is a context-free language

1L context free

2L regular

21 LL

context-free

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1Lfor for 2LNPDA

1M

DFA

2M

Construct a new NPDA machine

that accepts

Machine Machine

M

21 LL

context-free regular

M simulates in parallel and 1M 2M

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1M 2M

1q 2qcba →,

transition

1p 2pa

transition

NPDA DFA

11, pq cba →,

transition

MNPDA

22, pq

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1M 2M

0q

initial state

0p

initial state

NPDA DFA

Initial state

MNPDA

00, pq

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1M 2M

1q

final state

1p

final states

NPDA DFA

final states

MNPDA

11, pq

2p

21, pq

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M simulates in parallel and 1M 2M

M accepts string w if and only if

accepts string and w1M

accepts string w2M

)()()( 21 MLMLML =

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Therefore: )()( 21 MLML

(since is NPDA) M

is context-free

21 LL is context-free