costas busch - rpi1 pushdown automata pdas. costas busch - rpi2 pushdown automaton -- pda input...
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Costas Busch - RPI 1
Pushdown AutomataPDAs
Costas Busch - RPI 2
Pushdown Automaton -- PDA
Input String
Stack
States
Costas Busch - RPI 3
Initial Stack Symbol
Stack
$
Stack
z
bottomspecial symbol
stackhead
top
Costas Busch - RPI 4
The States
q1 q2a, b c
Inputsymbol
Popsymbol
Pushsymbol
Costas Busch - RPI 5
q1 q2a, b c
a
b top
input
stack
a
Replaceeh
$eh
$
c
Costas Busch - RPI 6
q1 q2a, c
a a
Pushb
eh
$eh
$
bc
top
input
stack
Costas Busch - RPI 7
q1 q2a, b
a a
Popb
eh
$eh
$
top
input
stack
Costas Busch - RPI 8
q1 q2a,
a a
No Changeb
eh
$eh
$
btop
input
stack
Costas Busch - RPI 9
q1 q2$,a
a a
Pop$ top
input
stack
A Possible Transition
empty
Costas Busch - RPI 10
q1 q2cba ,
a input
A Bad Transition
The automaton Halts in state and Rejects the input string
Empty stack
q1
HALT
Costas Busch - RPI 11
q1 q2ca ,
a input
A Bad Transition
The automaton Halts in state and Rejects the input string
Empty stack
q1
HALT
Costas Busch - RPI 12
q1 q2zyx ,
No transition is allowed to be followedWhen the stack is empty
Empty stack
Costas Busch - RPI 13
q1 q2ba $,
a a
Pop$ top
input
stack
A Good Transition
b
Costas Busch - RPI 14
Non-Determinism
q1
q2a, b c
q3a, b c
q1 q2, b c
transition
These are allowed transitions in a Non-deterministic PDA (NPDA)
Costas Busch - RPI 26
The input stringis accepted by the NPDA:
aaabbb
a, a
b, a q0 q1 q2 q3
b, a
, , $ $
Costas Busch - RPI 28
NPDA example
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
NPDA M
}{)( RwwML
Costas Busch - RPI 29
Execution Example:
Input
Time 0
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
a ab b
Costas Busch - RPI 30
Input
a ab
Time 1
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
ab
Costas Busch - RPI 31
Input
Time 2
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
aa ab b
b
Costas Busch - RPI 32
Input
Time 3
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
aa ab b
b
Guess the middle of string
Costas Busch - RPI 33
Input
Time 4
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
aa ab b
b
Costas Busch - RPI 34
Input
Time 5
Stack
$
, $ $1q q2
bb
aa
,
,
, q0
bb
aa
,
,
a ab b a
Costas Busch - RPI 35
Input
Time 6
Stack
$
, $ $q1
bb
aa
,
,
, q0
bb
aa
,
,
a ab b
accept
q2
Costas Busch - RPI 36
Rejection Example:
Input
Time 0
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
a b b b
Costas Busch - RPI 37
Input
Time 1
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
aa b b b
Costas Busch - RPI 38
Input
Time 2
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
ab
a b b b
Costas Busch - RPI 39
Input
Time 3
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
ab
Guess the middle of string
a b b b
Costas Busch - RPI 40
Input
Time 4
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
ab
a b b b
Costas Busch - RPI 41
Input
Time 5
Stack
$
, $ $1q q2
bb
aa
,
,
, q0
bb
aa
,
,
aa b b b
There is no possible transition.
Input is not consumed
Costas Busch - RPI 42
Another computation on same string:
Input Time 0
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
a b b b
Costas Busch - RPI 43
Input
Time 1
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
aa b b b
Costas Busch - RPI 44
Input
Time 2
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
ab
a b b b
Costas Busch - RPI 45
Input
Time 3
Stack
$ab
a b b b
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
b
Costas Busch - RPI 46
Input
Time 4
Stack
a b b b
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
$abbb
Costas Busch - RPI 47
Input
Time 5
Stack
a b b b
$abbb
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
No final state is reached
Costas Busch - RPI 48
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
There is no computation that accepts string abbb
)(MLabbb
Costas Busch - RPI 49
A string is rejected if there is no computation such that:
All the input is consumed AND The last state is a final state
At the end of the computation,we do not care about the stack contents
Costas Busch - RPI 50
In other words, a string is rejected if in every computation with this string:
The input cannot be consumed OR
The input is consumed and the last state is not a final state
ORThe stack head moves below the bottom of the stack
Costas Busch - RPI 61
Pushing Strings
q1 q2a, b w
Inputsymbol
Popsymbol
Pushstring
Costas Busch - RPI 62
q1 q2a, b cdf
a
b top
input
stack
a
Push
eh h
e
cdf
pushedstring
Example:
$ $