Languages and Finite Automata

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Formal Languages Deterministic PDAs Hinrich
Schütze IMS, Uni Stuttgart, WS 2007/08 Slides
based on RPI CSCI 2400 Thanks to Costas Busch
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Deterministic PDA DPDA
Allowed transitions
(deterministic choices)
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Allowed transitions
(deterministic choices)
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Not allowed
(non deterministic choices)
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DPDA example
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Definition
A language is deterministic context-free if
there exists some DPDA that accepts it
Example
The language
is deterministic context-free
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Example of Non-DPDA (PDA)
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Not allowed in DPDAs
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PDAs Have More Power thanDPDAs

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Trivially
Deterministic Context-Free Languages (DPDA)
Context-Free Languages PDAs
Since every DPDA is also a PDA
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We will actually show
Deterministic Context-Free Languages (DPDA)
Context-Free Languages (PDA)
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The language is
We will show

• is context-free

• is not deterministic context-free

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Proof?

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Language is context-free
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Theorem
The language
is not deterministic context-free
(there is no DPDA that accepts )
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Proof
Assume for contradiction that
is deterministic context free
Therefore
there is a DPDA that accepts
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DPDA with
accepts
accepts
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DPDA with
Such a path exists due to determinism
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Fact 1
The language is not
context-free
(we will prove this in a later class
using pumping lemma for context-free languages)
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Fact 2
The language is not
context-free
(we can prove this using pumping lemma for
context-free languages)
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We will construct a PDA that accepts
which is a contradiction!
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DPDA
Replace with
Modify
DPDA
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A PDA that accepts
Connect the final states of with the
final states of
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Since is accepted by a
PDA
it is context-free
Contradiction!
(since is not
context-free)
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Therefore
Is not deterministic context free
There is no DPDA that accepts it
End of Proof
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Positive Propertiesof Context-Free languages

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Union
Context-free languages are closed under
Union
is context free
is context free
is context-free
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Proof?

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Example
Language
Grammar
Union
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In general
For context-free languages with context-free
grammars and start variables
The grammar of the union has new start
variable and additional production
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Concatenation
Context-free languages are closed under
Concatenation
is context free
is context free
is context-free
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Proof?

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Example
Language
Grammar
Concatenation
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In general
For context-free languages with context-free
grammars and start variables
The grammar of the concatenation has new
start variable and additional production
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Star Operation
Context-free languages are closed under
Star-operation
is context free
is context-free
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Proof?

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Example
Language
Grammar
Star Operation
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In general
For context-free language with context-free
grammar and start variable
The grammar of the star operation has new
start variable and additional production
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Negative Propertiesof Context-Free Languages

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Intersection
Context-free languages are not closed under
intersection
is context free
is context free
not necessarily context-free
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Example
Context-free
Context-free
Intersection
NOT context-free
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Complement
Context-free languages are not closed under
complement
is context free
not necessarily context-free
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Example
Context-free
Context-free
Complement
NOT context-free
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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
context free
regular
context-free
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Machine
Machine
for
DFA
for
NPDA
regular
context-free
Construct a new NPDA machine that accepts
simulates in parallel and
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NPDA
DFA
transition
transition
NPDA
transition
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NPDA
DFA
transition
NPDA
transition
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NPDA
DFA
initial state
initial state
NPDA
Initial state
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NPDA
DFA
final state
final states
NPDA
final states
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Example
context-free
NPDA
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regular
DFA
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Intersection?

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context-free
Automaton for
NPDA
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In General
simulates in parallel and
accepts string
if and only if
accepts string and
accepts string
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Therefore
is NPDA
is context-free
is context-free
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Applications of Regular Closure
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The intersection of a context-free
language and a regular language is
a context-free language
Regular Closure
context free
regular
context-free
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An Application of Regular Closure
Prove that
is context-free
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We know
is context-free
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We also know
is regular
is regular
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regular
context-free
context-free
(regular closure)
is context-free
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Another Application of Regular Closure
Prove that
is not context-free
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Proof?

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is context-free
If
(regular closure)
Then
context-free
regular
context-free
Impossible!!!
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Decidable Propertiesof Context-Free Languages
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Membership Question
for context-free grammar find if string
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Membership Question
for context-free grammar find if string
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Empty Language Question
for context-free grammar find if
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Empty Language Question
for context-free grammar find if
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Infinite Language Question
for context-free grammar find if
is infinite
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Infinite Language Question
for context-free grammar find if
is infinite
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Example
Infinite language
Dependency graph
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