pushdown automata pdas
DESCRIPTION
Pushdown Automata PDAs. Pushdown Automaton -- PDA. Input String. Stack. States. Initial Stack Symbol. Stack. Stack. stack head. top. bottom. special symbol. The States. Pop symbol. Input symbol. Push symbol. input. stack. top. Replace. input. stack. top. Push. input. - PowerPoint PPT PresentationTRANSCRIPT
Fall 2004 COMP 335 1
Pushdown AutomataPDAs
Fall 2004 COMP 335 2
Pushdown Automaton -- PDA
Input String
Stack
States
Fall 2004 COMP 335 3
Initial Stack Symbol
Stack
$
Stack
z
bottomspecial symbol
stackhead
top
Fall 2004 COMP 335 4
The States
q1 q2a, b c
Inputsymbol
Popsymbol
Pushsymbol
Fall 2004 COMP 335 5
q1 q2a, b c
a
b top
input
stack
a
Replaceeh
$eh
$
c
Fall 2004 COMP 335 6
q1 q2a, c
a a
Pushb
eh
$eh
$
bc
top
input
stack
Fall 2004 COMP 335 7
q1 q2a, b
a a
Popb
eh
$eh
$
top
input
stack
Fall 2004 COMP 335 8
q1 q2a,
a a
No Changeb
eh
$eh
$
btop
input
stack
Fall 2004 COMP 335 9
q1 q2$,a
a a
Pop$ top
input
stack
A Possible Transition
empty
Fall 2004 COMP 335 10
q1 q2cba ,
a input
A Bad Transition
The automaton Halts in state and Rejects the input string
Empty stack
q1
HALT
Fall 2004 COMP 335 11
q1 q2ca ,
a input
A Bad Transition
The automaton Halts in state and Rejects the input string
Empty stack
q1
HALT
Fall 2004 COMP 335 12
q1 q2zyx ,
No transition is allowed to be followedwhen the stack is empty
Empty stack
Fall 2004 COMP 335 13
q1 q2ba $,
a a
Pop$ top
input
stack
A Good Transition
b
Fall 2004 COMP 335 14
Non-Determinism
q1
q2
q3dba ,
q1 q2, b c
transition
These are allowed transitions in a Non-deterministic PDA (NPDA)
cba ,
Fall 2004 COMP 335 15
NPDA: Non-Deterministic PDA
Example:
,
a, a
b, a q0 q1 q2 q3
b, a
, $ $
Fall 2004 COMP 335 16
a, a
b, a 0q q1 q2 q3
Execution Example:
Input
a a a b b b
currentstate
b, a
Time 0
, , $ $
Stack
$
Fall 2004 COMP 335 17
a, a
b, a q0 q1 q2 q3
Input
a a a b b b
b, a
Time 1
, , $ $
Stack
$
Fall 2004 COMP 335 18
a, a
b, a q0 q1 q2 q3
Input
Stack
a a a b b b
$
a
b, a
Time 2
, , $ $
Fall 2004 COMP 335 19
a, a
b, a q0 q1 q2 q3
Input
Stack
a a a b b b$
aa
b, a
Time 3
, , $ $
Fall 2004 COMP 335 20
a, a
b, a q0 q1 q2 q3
Input
Stack
a a a b b b
$
aaa
b, a
Time 4
, , $ $
Fall 2004 COMP 335 21
a, a
b, a q0 q1 q2 q3
Input
a a a b b b
Stack
$
aaa
b, a
Time 5
, , $ $
Fall 2004 COMP 335 22
a, a
b, a q0 q1 q2 q3
Input
a a a b b b$
a
Stack
b, a
Time 6
, , $ $
a
Fall 2004 COMP 335 23
a, a
b, a q0 q1 q2 q3
Input
a a a b b b$
Stack
b, a
Time 7
, , $ $
a
Fall 2004 COMP 335 24
a, a
b, a q0 q1 q2 q3
Input
a a a b b b
b, a
Time 8
accept, , $ $
$
Stack
Fall 2004 COMP 335 25
A string is accepted if there is a 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
Fall 2004 COMP 335 26
The input stringis accepted by the NPDA:
aaabbb
a, a
b, a q0 q1 q2 q3
b, a
, , $ $
Fall 2004 COMP 335 27
}0:{ nbaL nn
is the language accepted by the NPDA:
a, a
b, a q0 q1 q2 q3
b, a
In general,
, , $ $
Fall 2004 COMP 335 28
NPDA -- Another example
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
NPDA M
}{)( RwwML
Fall 2004 COMP 335 29
Execution Example:
Input
Time 0
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
a ab b
Fall 2004 COMP 335 30
Input
a ab
Time 1
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
ab
Fall 2004 COMP 335 31
Input
Time 2
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
aa ab b
b
Fall 2004 COMP 335 32
Input
Time 3
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
aa ab b
b
Guess the middle of string
Fall 2004 COMP 335 33
Input
Time 4
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
aa ab b
b
Fall 2004 COMP 335 34
Input
Time 5
Stack
$
, $ $1q q2
bb
aa
,
,
, q0
bb
aa
,
,
a ab b a
Fall 2004 COMP 335 35
Input
Time 6
Stack
$
, $ $q1
bb
aa
,
,
, q0
bb
aa
,
,
a ab b
accept
q2
Fall 2004 COMP 335 36
Rejection Example:
Input
Time 0
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
a b b b
Fall 2004 COMP 335 37
Input
Time 1
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
aa b b b
Fall 2004 COMP 335 38
Input
Time 2
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
ab
a b b b
Fall 2004 COMP 335 39
Input
Time 3
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
ab
Guess the middle of string
a b b b
Fall 2004 COMP 335 40
Input
Time 4
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
ab
a b b b
Fall 2004 COMP 335 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
Fall 2004 COMP 335 42
Another computation on the same string:
Input Time 0
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
a b b b
Fall 2004 COMP 335 43
Input
Time 1
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
aa b b b
Fall 2004 COMP 335 44
Input
Time 2
Stack
$
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
ab
a b b b
Fall 2004 COMP 335 45
Input
Time 3
Stack
$ab
a b b b
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
b
Fall 2004 COMP 335 46
Input
Time 4
Stack
a b b b
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
$abbb
Fall 2004 COMP 335 47
Input
Time 5
Stack
a b b b
$abbb
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
No final state is reached
Fall 2004 COMP 335 48
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
There is no computation that accepts string abbb
)(MLabbb
Fall 2004 COMP 335 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
Fall 2004 COMP 335 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
Fall 2004 COMP 335 51
q0
aa ,
1q
ab,
$,bab,
NPDA – Another Example
$,b
}1:{)( mnbaML mn
}0,:{}1:{ 1 kmbamba mkmmm
Fall 2004 COMP 335 52
Pushing Strings
q1 q2a, b w
Inputsymbol
Popsymbol
Pushstring
Fall 2004 COMP 335 53
q1 q2a, b cdf
a
b top
input
stack
a
Push
eh h
e
cdf
pushedstring
Example:
$ $
Fall 2004 COMP 335 54
Another NPDA example
$$, q1 q2
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
NPDA M
})()(:{)( wnwnwML ba
Fall 2004 COMP 335 55
Time 0
Input
a ab b b
currentstate
a
$
Stack
$$, q1 q2
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
Execution Example:
Fall 2004 COMP 335 56
Time 1
Input
a ab b ba
$
Stack
0
$$, q1 q2
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
Fall 2004 COMP 335 57
Time 3
Input
a bb b a
$
Stack
a
$
$$, q1 q2
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
0
Fall 2004 COMP 335 58
Time 4
Input
a bb b a
$
Stack
a
1
$$, q1 q2
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
Fall 2004 COMP 335 59
Time 5
Input
a bb b a
$
Stack
a
11
$$, q1 q2
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
Fall 2004 COMP 335 60
Time 6
Input
a bb b a
$
Stack
a
1
$$, q1 q2
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
1
Fall 2004 COMP 335 61
Time 7
Input
a bb b a
$
Stack
a
1
$$, q1 q2
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
Fall 2004 COMP 335 62
Time 8
Input
a bb b a a
$
Stack
$$, q1 q2
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
accept
Fall 2004 COMP 335 63
Formalities for NPDAs
Fall 2004 COMP 335 64
q1 q2a, b w
)},{(),,( 21 wqbaq
Transition function:
Fall 2004 COMP 335 65
q1
q2wba ,
q3uba ,
)},(),,{(),,( 321 uqwqbaq
Transition function:
Fall 2004 COMP 335 66
Formal Definition
Non-Deterministic Pushdown Automaton NPDA
),,,δ,Γ,Σ,( 0 FzqQM
States
Inputalphabet
Stackalphabet
Transitionfunction
Finalstates
Stackendsymbol
Initialstate
Fall 2004 COMP 335 67
Instantaneous Description (ID)
),,( uvq
Currentstate Remaining
unread input
Currentstackcontents
Fall 2004 COMP 335 68
aa /,
/,abq0 q1 q2 q3
Input
Stack
a a a b b b$
aaTime 4:
/, $/$,
Example: Instantaneous Description (ID)
$),,( 1 aaabbbqa
/,ab
Fall 2004 COMP 335 69
a, a
b, a q0 q1 q2 q3
Input
Stack
a a a b b b$
aa
b, a
Time 5:
, , $ $
Example: Instantaneous Description (ID)
$),,( 2 aabbq
a
Fall 2004 COMP 335 70
We write:
Time 4 Time 5
$),,($),,( 21 aabbqaaabbbq
Fall 2004 COMP 335 71
a, a
b, a q0 q1 q2 q3
b, a
, , $ $
,$),(,$),($),,($),,(
$),,($),,($),,(
,$),(,$),(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
A computation:
Fall 2004 COMP 335 72
,$),(,$),($),,($),,(
$),,($),,($),,(
,$),(,$),(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
For convenience we write:
,$),(,$),( 30 qaaabbbq
Fall 2004 COMP 335 73
Formal Definition
Language of NPDA :M
}),,,(),,(,:{)( **
0* uuqzwqwwML fM
Initial ID Final ID
)(ML
Fall 2004 COMP 335 74
Example:,$),(,$),( 30 qaaabbbq
a, a
b, a q0 q1 q2 q3
b, a
, , $ $
NPDA :M
)(MLaaabbb
Fall 2004 COMP 335 75
,$),(,$),( 30 qbaq nn
a, a
b, a q0 q1 q2 q3
b, a
, , $ $
NPDA :M
)(MLba nn
Fall 2004 COMP 335 76
a, a
b, a q0 q1 q2 q3
b, a
, , $ $
NPDA :M
}0:{)( nbaML nnTherefore: