NPDAs continued

1
NPDAs continued

2
Pushing Strings
Pop symbol
Input symbol
Push string
3
Example
input
pushed string
stack
top
Push
4
Another NPDA example
NPDA
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Execution Example
Time 0
Input
Stack
current state
6
Time 1
Input
Stack
7
Time 3
Input
Stack
8
Time 4
Input
Stack
9
Time 5
Input
Stack
10
Time 6
Input
Stack
11
Time 7
Input
Stack
12
Time 8
Input
Stack
accept
13
Formalities for NPDAs

14
Transition function
15
Transition function
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Formal Definition
Non-Deterministic Pushdown Automaton
NPDA
Final states
States
Input alphabet
Stack start symbol
Transition function
Initial state
Stack alphabet
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Instantaneous Description
Current state
Current stack contents
Remaining input
18
Example
Instantaneous Description
Input
Time 4
Stack
19
Example
Instantaneous Description
Input
Time 5
Stack
20
We write
Time 4
Time 5
21
A computation
22
For convenience we write
23
Formal Definition
Language of NPDA
Initial state
Final state
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Example
NPDA
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NPDA
26
Therefore
NPDA
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NPDAs Accept Context-Free Languages
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Theorem
Context-Free Languages (Grammars)
Languages Accepted by NPDAs
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Proof - Step 1
Context-Free Languages (Grammars)
Languages Accepted by NPDAs
Convert any context-free grammar to a NPDA
with
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Proof - Step 2
Context-Free Languages (Grammars)
Languages Accepted by NPDAs
Convert any NPDA to a context-free
grammar with
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Converting Context-Free Grammarsto NPDAs

32
An example grammar
What is the equivalent NPDA?
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Grammar
NPDA
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The NPDA simulates leftmost derivations of the
grammar
L(Grammar) L(NPDA)
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Grammar
A leftmost derivation
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NPDA execution
Time 0
Input
Stack
current state
37
Time 1
Input
Stack
38
Time 2
Input
Stack
39
Time 3
Input
Stack
40
Time 4
Input
Stack
41
Time 5
Input
Stack
42
Time 6
Input
Stack
43
Time 7
Input
Stack
44
Time 8
Input
Stack
45
Time 9
Input
Stack
46
Time 10
Input
Stack
accept
47
In general
Given any grammar
We can construct a NPDA
With
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Constructing NPDA from grammar
For any production
For any terminal
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Grammar generates string
if and only if
NPDA accepts
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Therefore
For any context-free language there is an
NPDA that accepts the same language
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Converting NPDAstoContext-Free Grammars

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For any NPDA
we will construct a context-free grammar
with
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Intuition
The grammar simulates the machine
A derivation in Grammar
Current configuration in NPDA
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A derivation in Grammar
terminals
variables
Input processed
Stack contents
in NPDA
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Some Necessary Modifications

• First, we modify the NPDA
• It has a single final state
• It empties the stack
• when it accepts the input

Empty Stack
Original NPDA
56
Second, we modify the NPDA transitions all
transitions will have form
or
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Example of a NPDA in correct form
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The Grammar Construction
In grammar
Stack symbol
Variables
states
Terminals Input symbols of NPDA
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For each transition
We add production
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For each transition
We add production
For all states
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Stack bottom symbol
Start Variable
Start state
final 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)
67
Derivation of string
68
In general, in Grammar
if and only if
is accepted by the NPDA
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Explanation
By construction of Grammar
if and only if
in the NPDA going from to the stack
doesnt change below and is removed from
stack