dpda deterministic pda
DESCRIPTION
DPDA Deterministic PDA. Deterministic PDA: DPDA. Allowed transitions:. (deterministic choices). Allowed transitions:. (deterministic choices). Not allowed:. (non deterministic choices). DPDA example. Definition:. A language is deterministic context-free - PowerPoint PPT PresentationTRANSCRIPT
Fall 2003 Costas Busch - RPI 1
DPDA
Deterministic PDA
Fall 2003 Costas Busch - RPI 2
Deterministic PDA: DPDA
q1 q2wba ,
1q 2qwb,
Allowed transitions:
(deterministic choices)
Fall 2003 Costas Busch - RPI 3
Allowed transitions:
q1
q21, wb
q32, wc
q1
q21, wba
q32, wca
(deterministic choices)
Fall 2003 Costas Busch - RPI 4
q1
q21, wba
q32, wba
Not allowed:
(non deterministic choices)
q1
q21, wb
q32, wba
Fall 2003 Costas Busch - RPI 5
DPDA example
a, a
b, a q0 q1 q2 q3
b, a
, $ $
}0:{)( nbaML nn
a, a
Fall 2003 Costas Busch - RPI 6
}0:{)( nbaML nnThe language
is deterministic context-free
Definition:
A language is deterministic context-freeif there exists some DPDA that accepts it
L
Example:
Fall 2003 Costas Busch - RPI 7
Example of Non-DPDA (PDA)
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
}},{:{)( *bavvvML R
Fall 2003 Costas Busch - RPI 8
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
Not allowed in DPDAs
Fall 2003 Costas Busch - RPI 9
PDAs
Have More Power than
DPDAs
Fall 2003 Costas Busch - RPI 10
Deterministic
Context-FreeLanguages
(DPDA)
Context-Free
LanguagesPDAs
Since every DPDA is also a PDA
It holds that:
Fall 2003 Costas Busch - RPI 11
We will actually show:
We will show that there existsa context-free language which is notaccepted by any DPDA
L
LL
Deterministic
Context-FreeLanguages
(DPDA)
Context-Free
Languages(PDA)
Fall 2003 Costas Busch - RPI 12
The language is:
}{}{ 2nnnn babaL 0n
We will show:
L• is context-free
L• is not deterministic context-free
Fall 2003 Costas Busch - RPI 13
}{}{ 2nnnn babaL
Language is context-freeL
Context-free grammar for : L
21 | SSS
|11 baSS
|22 bbaSS
}{ nnba
}{ 2nnba
}{}{ 2nnnn baba
Fall 2003 Costas Busch - RPI 14
}{}{ 2nnnn babaL
is not deterministic context-free
Theorem:
The language
(there is no DPDA that accepts ) L
Fall 2003 Costas Busch - RPI 15
Proof: Assume for contradiction that
}{}{ 2nnnn babaL
is deterministic context free
Therefore:
there is a DPDA that accepts M L
Fall 2003 Costas Busch - RPI 16
DPDA with M
nnba nb
acceptsnnba 2
acceptsnnba
}{}{)( 2nnnn babaML
Fall 2003 Costas Busch - RPI 17
DPDA with M
nnba nb
}{}{)( 2nnnn babaML
Such a path exists due to determinism
M
Fall 2003 Costas Busch - RPI 18
The language is not context-free
}{ nnn cba
(we will prove this at a later class usingpumping lemma for context-free languages)
Fact 1:
Regular languages**ba
Context-free languagesnnba
Fall 2003 Costas Busch - RPI 19
The language is not context-free
}{ nnn cbaL
}){}{( 2nnnn babaL
Fact 2:
(we can prove this using pumping lemma for context-free languages)
Fall 2003 Costas Busch - RPI 20
We will construct a PDA that accepts:
}{ nnn cbaL
}){}{( 2nnnn babaL
which is a contradiction!
Fall 2003 Costas Busch - RPI 21
Modify M
M ncnnca
}{}{)( 2nnnn babaML
}{}{)( 2nnnn cacaML
Replacewith
bc
nnba nb
MDPDA
DPDA
Fall 2003 Costas Busch - RPI 22
A PDA that accepts }{ nnn cbaL
nnba nb
nc
M
M
nnca
Connect the final states of with the final states of M
M
Fall 2003 Costas Busch - RPI 23
Since is accepted by a PDA }{ nnn cbaL
it is context-free
Contradiction!
(since is not context-free)}{ nnn cbaL
Fall 2003 Costas Busch - RPI 24
Therefore:
}{}{ 2nnnn babaL
There is no DPDA that accepts it
End of Proof
Is not deterministic context free