Formal Languages

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Formal Languages PDAs accept CFGs Slides based on
RPI CSCI 2400 Thanks to Petros Drineas
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Theorem
Context-Free Languages (Grammars)
Languages Accepted by PDAs
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Proof - Step 1
Context-Free Languages (Grammars)
Languages Accepted by PDAs
Convert any context-free grammar to a PDA
with
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Proof - Step 2
Context-Free Languages (Grammars)
Languages Accepted by PDAs
Convert any PDA to a context-free
grammar with
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Converting Context-Free Grammarsto PDAs
Proof - step 1

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Context-Free Languages (Grammars)
Languages Accepted by PDAs
Convert any context-free grammar to a PDA
with
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We will convert grammar
to a PDA such that
simulates leftmost derivations of
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Convert grammar to PDA
Production in
Terminal in
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PDA computation
Grammar leftmost derivation
Simulates grammar leftmost derivations
Leftmost variable
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Example
Grammar
PDA
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Grammar derivation
PDA computation
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Derivation
Input
Time 0
Stack
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Derivation
Input
Time 0
Stack
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Derivation
Input
Time 1
Stack
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Derivation
Input
Time 2
Stack
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Derivation
Input
Time 3
Stack
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Derivation
Input
Time 4
Stack
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Derivation
Input
Time 5
Stack
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Derivation
Input
Time 6
Stack
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Derivation
Input
Time 7
Stack
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Derivation
Input
Time 8
Stack
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Derivation
Input
Time 9
Stack
accept
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In general, it can be shown that
Grammar generates string
If and Only if
PDA accepts
Therefore
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Therefore
For any context-free language there is a PDA
that accepts
Context-Free Languages (Grammars)
Languages Accepted by PDAs
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Converting PDAstoContext-Free Grammars
Proof - step 2

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Context-Free Languages (Grammars)
Languages Accepted by PDAs
Convert any PDA to a context-free
grammar with
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We will convert PDA to
a context-free grammar such that
simulates computations of with leftmost
derivations
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Some Necessary Modifications
If necessary, modify the PDA so that 1. The
stack is never empty during computation 2. It
has a single accept state and empties the
stack when it accepts a string 3. Has
transitions without popping
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1. Modify the PDA so that the stack is never
empty during computation
Stack
OK
OK
NOT OK
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Introduce the new symbol to mark the
bottom of the stack
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At the beginning insert into the stack
Original PDA
new initial state
original initial state
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Convert all transitions so that after popping
the automaton halts
pop
pop
halting state
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2. Modify the PDA so that at end it empties
stack and has a unique accept state
Empty stack
PDA
New accept state
Old accept states
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3. Modify the PDA so that it has no
transitions popping
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Example of a PDA in correct form (modifications
are not necessary)
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Grammar Construction
In grammar
PDA stack symbols
Variables
PDA input symbols
Terminals
Start Variable
Stack bottom symbol
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PDA transition
Grammar production
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PDA transition
Grammar production
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Grammar leftmost derivation
PDA computation
Leftmost variable
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Example PDA
Grammar
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Grammar Leftmost derivation
PDA Computation
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Derivation
Time 0
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Derivation
Time 1
Stack
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Derivation
Time 2
Stack
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Derivation
Time 3
Stack
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Derivation
Time 4
Stack
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Derivation
Time 5
empty
Stack
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However, this grammar conversion does not work
for all PDAs (AA??)
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Grammar
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Bad Derivation
Grammar
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The Correct Grammar Construction
In grammar
PDA stack symbol
Variables
PDA states
Terminals Input symbols of PDA
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PDA transition
Grammar production
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PDA transition
Grammar production
m1, not m
For all possible states in
PDA
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Stack bottom symbol
Start Variable
Start state
accept state
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Example
Grammar production
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Example
Grammar productions
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Example
Grammar production
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Resulting Grammar
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(No Transcript)
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Grammar Leftmost derivation
PDA computation
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Derivation
Time 0
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Derivation
Time 1
Stack
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Derivation
Time 2
Stack
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Derivation
Time 3
Stack
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Derivation
Time 4
Stack
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Derivation
Time 5
empty
Stack
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In general
If and Only if
Grammar
PDA
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Thus
Grammar generates
If and Only if
PDA accepts
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Therefore
For any PDA there is a context-free grammar that
accepts the same language
Context-Free Languages (Grammars)
Languages Accepted by PDAs