CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 8 Mälardalen University
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CD5560
FABER
Formal Languages, Automata and Models of Computation
Lecture 8
Mälardalen University
2006
2
Content
Context-Free LanguagesPush-Down Automata, PDANPDA: Non-Deterministic PDAFormal Definitions for NPDAs NPDAs Accept Context-Free LanguagesConverting NPDA to Context-Free Grammar
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Regular Languages
}{ nnba }{ RwwContext-Free Languages
Non-regular languages
}0:{ ! nan}0,:{ lncba lnln
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Context-Free Languages
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Context-Free Languages
Pushdown
Automata
Context-Free
Grammars
stack
automaton(CF grammars are
defined as generalized Regular Grammars)
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Definition: Context-Free Grammars
Grammar
Productions of the form:xA
x is string of variables and terminals
),,,( PSTVG
Variables Terminal
symbols
Start
variables
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Definition: Regular Grammars
xBA
Grammar
Right or Left Linear Grammars. Productions of the form:
x is string of terminals
),,,( PSTVG
Variables Terminal
symbols
Start
variables
BxA C xor
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Pushdown AutomataPDAs
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Pushdown Automaton - PDA
Input String
Stack
States
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The Stack
The stack allows pushdown automata to recognize some non-regular languages.
All access to the stack only on the top!(Stack top is written leftmost in the string, e.g. yxz)
A PDA can write symbols on stack and read them later on.
A stack is valuable as it can hold an unlimited amount of information.
zxy
POP reading symbol PUSH writing symbol
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The States
Input
symbol
Pop oldreading
stack symbol
Push new
writing
stack symbol
q1 q2cba /,
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a
b top
input
stack
a
Replace
eh
$eh
$
c
q1 q2cba /,
(An alternative is to start and finish with empty stack)
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a a
input
Pushb
eh
$eh
$
bc
top
stack
q1 q2ca /,
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a a input
Popb
eh
$eh
$
top
stack
q1 q2/,ba
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a a input
No Changeb
eh
$eh
$
btop
stack
q1 q2 /,a
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Input Stack
( )
Example 3.7 Salling: A PDA for simple nested parenthesis strings
sstart q(/),
( ( ) )
end
(/,( /(,)
Time 0
17
Input
Stack
( )
Example 3.7
sstart q(/),
(/,(
( ( ) )
/(,)
(
end
Time 1
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Input
Stack
( )
sstart q(/), end
( ( ) ) (
(
Example 3.7
(/,( /(,)
Time 2
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Input
( )
sstart q(/),
( ( ) )
Stack
(
((
end
Example 3.7
(/,( /(,)
Time 3
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Input
( )
sstart q(/),
( ( ) )
Stack
(
((
end
Example 3.7
(/,( /(,)
Time 4
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Input
( )
sstart (/),
( ( ) )
Stack
(
(
q
Example 3.7
end
(/,( /(,)
Time 5
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Input
Stack
( )
sstart q(/),
( ( ) )(
end
Example 3.7
(/,( /(,)
Time 6
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Input
( )
sstart q(/),
( ( ) )
Example 3.7
end(/),
(/,( /(,)
Time 7
Stack
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NPDAsNon-deterministic Push-Down Automata
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Non-Determinism
q1
q2cba /,
q3
q1 q2
transitioncba /,
cb /,
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A string is accepted if:
• All the input is consumed
• The last state is a final state
• Stack is in the initial condition(either: empty (when we started with empty stack), or: bottom symbol reached, or similar)
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}0:{ nbaL nn
is the language accepted by the NPDA:
Example NPDA
q0 q1 q2 q3 /, /,ab $/$,
/,abaa /,
},{ ba
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/,
/,
bb
aa
q1 q2 /,q0
NPDA M}{)( RwwML
bb
aa
/,
/,
$/$,
(Even-length palindromes)
},{ ba
Example NPDA
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Pushing Strings
q1 q2wba /,
Input
symbol
Pop
symbolPush
string
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q1 q2cdfba /,
a
b top
input
stack
a
Push
eh h
e
cdf
pushed
string
Example:
$ $
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Another NPDA example
$0/$,a00/0,a/1,a
$1/$,b11/1,b/0,b
$/$,q1 q2
NPDA M}:{)( ba nnwML
},{ ba
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Time 0
Input
a ab b ba
$
Stack
Current state
q1 q2
Execution Example:
$0/$,a00/0,a/1,a
$1/$,b11/1,b/0,b
$/$,
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Input
a ab b ba
Stack
$0
q1 q2
$0/$,a00/0,a/1,a
$1/$,b11/1,b/0,b
$/$,
Time 1
34
Input
a bb b a
Stack
a
q1 q2
$0
$0/$,a00/0,a/1,a
$1/$,b11/1,b/0,b
$/$,
Time 2
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Input
a bb b a
Stack
a
$1
q1 q2
$0/$,a00/0,a/1,a
$1/$,b11/1,b/0,b
$/$,
Time 3
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Input
a bb b a
Stack
a
$11
q1 q2
$0/$,a00/0,a/1,a
$1/$,b11/1,b/0,b
$/$,
Time 4
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Input
a bb b a
Stack
a
q1 q2
$11
$0/$,a00/0,a/1,a
$1/$,b11/1,b/0,b
$/$,
Time 5
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Input
a bb b a
Stack
a
q1 q2
$1
$0/$,a00/0,a/1,a
$1/$,b11/1,b/0,b
$/$,
Time 6
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Input
a bb b a a
$
Stack
q1 q2
accept
$0/$,a00/0,a/1,a
$1/$,b11/1,b/0,b
$/$,
Time 7
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Formal Definitions for NPDAs
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q1 q2wba /,
)},{(),,( 21 wqbaq
Transition function:
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Transition function:
q1
q2wba /,
q3
wba /,
current state
)},(),,{(),,( 321 wqwqbaq
current input symbolcurrent stack top
new state
new stack top
An unspecified transition function is to the null set and represents a dead configuration for the NPDA.
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Formal Definition
Non-Deterministic Pushdown Automaton NPDA
),,,,,( FzQM States
Input
alphabetStack
alphabet
Transition
function
Final
states
Stack
start
symbol
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Instantaneous Description
),,( suq
Current
state Remaining
input
Current
stack
contents
45q0 q1 q2 q3
Input
Stack
a a a b b b$
aaTime 4:
Example Instantaneous Description
$),,( 1 aaabbbq
a
/, /,ab $/$,
/,abaa /,
),,( stackinputremainingstateurrentc
46q0 q1 q2 q3
Input
Stack
a a a b b b$
aaTime 5:
Example Instantaneous Description
$),,( 2 aabbq
a
/, /,ab $/$,
/,abaa /,
),,( stackinputremainingstateurrentc
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We write
$),,($),,( 21 aabbqaaabbbq Time 4 Time 5
),,( contentstackcurrentinputremainingstatecurrent
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
A computation
q0 q1 q2 q3 /, /,ab $/$,
/,abaa /,
),,( contentstackcurrentinputremainingstatecurrent
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
A computation
q0 q1 q2 q3 /, /,ab $/$,
/,abaa /,
),,( contentstackcurrentinputremainingstatecurrent
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
A computation
q0 q1 q2 q3 /, /,ab $/$,
/,abaa /,
),,( contentstackcurrentinputremainingstatecurrent
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
A computation
q0 q1 q2 q3 /, /,ab $/$,
/,abaa /,
),,( contentstackcurrentinputremainingstatecurrent
52
$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
A computation
q0 q1 q2 q3 /, /,ab $/$,
/,abaa /,
),,( contentstackcurrentinputremainingstatecurrent
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
A computation
q0 q1 q2 q3 /, /,ab $/$,
/,abaa /,
),,( contentstackcurrentinputremainingstatecurrent
54
$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
A computation
q0 q1 q2 q3 /, /,ab $/$,
/,abaa /,
),,( contentstackcurrentinputremainingstatecurrent
55
$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
A computation
q0 q1 q2 q3 /, /,ab $/$,
/,abaa /,
),,( contentstackcurrentinputremainingstatecurrent
56
For convenience we write
$),,($),,( 30 qaaabbbq
$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
),,( contentstackcurrentinputremainingstatecurrent
57
Formal Definition
Language of NPDA M
)}',,(),,(:{)( 0 sqswqwML f
Initial state Final state
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Example:,$),(,$),( 30 qaaabbbq
q0 q1 q2 q3
NPDA :M
)(MLaaabbb
/, /,ab $/$,
/,abaa /,
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,$),(,$),( 30 qbaq nn
NPDA :M
)(MLba nn
q0 q1 q2 q3 /, /,ab $/$,
/,abaa /,
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NPDA :M
}0:{)( nbaML nnTherefore:
q0 q1 q2 q3 /, /,ab $/$,
/,abaa /,
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NPDAs Accept Context-Free Languages
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Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Theorem
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Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Proof - Step 1:
Convert any context-free grammar
to a NPDA with
GM )()( MLGL
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Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Proof - Step 2:
Convert any NPDA to a context-free
grammar with: GM
)()( GLML
65
Converting Context-Free Grammars
to NPDAs
66
An example grammar:
T
TaT
bS
aSTbS
What is the equivalent NPDA?
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Grammar NPDA
/,
/,
/,
/,
T
TaT
bS
aSTbS
T
TaT
bS
aSTbS
/,
/,
bb
aa
q0 q1 2qS/, $/$,
68
The NPDA simulates
leftmost derivations of the grammar
L(Grammar) = L(NPDA)
69
Grammar:
T
TaT
bS
aSTbS
A leftmost derivation:
abababTabbTabTbSaS
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NPDA execution:
0q q1 2qS/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
Input
Stack
$
a ab
Time 0
b
Start
71
q0 q1 2q
Input
Stack
a ab
Time 1
b
$S
S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
72
q0 2q
Input
Stack
a ab
Time 2
b
$
a
b
ST
q1S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
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q0 2q
Input
Stack
a ab
Time 3
b
$
a
b
ST
q1S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
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q0 2q
Input
Stack
a ab
Time 4
b
$bTb
q1S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
75
q0 2q
Input
Stack
a ab
Time 5
b
$bTb
q1S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
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q0 2qS, , $ $
bb
aa
,
,
Input
Stack
a ab
Time 6
b
T
TaT
bS
aSTbS
,
,
,
, $b
Ta
q1
77
q0 2q
Input
Stack
a ab
Time 7
b
$b
Ta
q1S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
78
q0 2q
Input
Stack
a ab
Time 8
b
$ba
q1S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
79
q0 2q
Input
Stack
a ab
Time 9
b
$b
q1S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
80
q0 q1 2q
Input
Stack
$
a ab
Time 10
b
acceptS/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
81
In general
Given any grammar G
We can construct a NPDA M
With )()( MLGL
82
q0 q1 2qS/, $/$,
wA /, /, aa
For any productionwA
For any terminala
Constructing NPDA from grammarM GTop-down parser
83
Grammar generates string G w
if and only if
MNPDA accepts w
)()( MLGL
84
Therefore:
For any context-free language
there is an NPDA
that accepts the same language
85
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Which means
86
Converting NPDAsto
Context-Free Grammars
87
For any NPDA M
we will construct
a context-free grammar withG
)()( GLML
88
in NPDA M
abcABCabcS
Input processed Stack contents
terminals variables
GA derivation in Grammar
The grammar simulates the machine
89
Some Simplifications
First we modify the NPDA so that
• It has a single final state
• It empties the stack
when it accepts the input
Original NPDA Empty Stack
fq
fq
90
Second we modify the NPDA transitions.
All transitions will have form:
iq jq/, Ba
or
iq jqCDBa /,
which means that each move
increases/decreases stack by a single symbol.
91
Those simplifications do not affect generality of our argument.
It can be shown that for any NPDA there exists an equivalent one having above two properties
i.e.the equivalent NPDA with a single final state
which empties its stack when it accepts the input, and which for each move increases/decreases stack by a single symbol.
92
/1,
00/0,
$0/$,
a
a
a
/$,0q fq
}:{)( ba nnwML
Example of a NPDA in an appropriate form
symbol)stack initial :($/0,
11/1,
$1/$,
b
b
b},{ ba
93
The Grammar Construction
)( jiBqq
In grammar G
Terminals:
Input symbols of NPDA
states
Stack symbol
Variables:
94
iq jq/, BaFor each transition:
we add production: aBqq ji )(
95
For each transition:
we add production: ))(()( klliki DqqCqqaBqq
iq jqCDBa /,
for all states lk qq ,
96
Start Variable )$( fo qq
Stack bottom symbol
Start state (Single) Final state
97
Example
aqq )1( 00Grammar production:
0q fq
/1,
00/0,
$0/$,
a
a
a
/$,
/0,
11/1,
$1/$,
b
b
biq jq/, Ba
aBqq ji )(
}:{)( ba nnwML },{ ba
98
)$)(1(|)$)(1()$(
)$)(1(|)$)(1()$(
00000
00000000
fffff
ff
qqqqbqqqqbqq
qqqqbqqqqbqq
Grammar productions:
0q fq
/1,
00/0,
$0/$,
a
a
a
/$,
/0,
11/1,
$1/$,
b
b
b iq jqCDBa /,
))(()( klljki DqqCqqaBqq
lk qqstatesallfor ,
99
Grammar production: )$( 0 fqq
0q fq
/1,
00/0,
$0/$,
a
a
a
/$,
/0,
11/1,
$1/$,
b
b
biq jq/, Ba
aBqq ji )(
100
0q fq
/1,
00/0,
$0/$,
a
a
a
/$,
/0,
11/1,
$1/$,
b
b
b
Resulting Grammar
ablestart vari:)$( 0 fqq
)$)(1(|)$)(1()$(
)$)(1(|)$)(1()$(
00000
00000000
fffff
ff
qqqqbqqqqbqq
qqqqbqqqqbqq
)1)(1(|)1)(1()1(
)1)(1(|)1)(1()1(
00000
00000000
fffff
ff
qqqqbqqqqbqq
qqqqbqqqqbqq
}:{)( ba nnwML },{ ba
))(()( klljki DqqCqqaBqq
lk qqstatesallfor ,
101
)$)(0(|)$)(0()$(
)$)(0(|)$)(0()$(
00000
00000000
fffff
ff
qqqqaqqqqaqq
qqqqaqqqqaqq
Resulting Grammar, cont.
)0)(0(|)0)(0()0(
)0)(0(|)0)(0()0(
00000
00000000
fffff
ff
qqqqaqqqqaqq
qqqqaqqqqaqq
0q fq
/1,
00/0,
$0/$,
a
a
a
/$,
/0,
11/1,
$1/$,
b
b
b }:{)( ba nnwML
102
bqq
aqq
)0(
)1(
00
00
)$( 0 fqq0q fq
/1,
00/0,
$0/$,
a
a
a
/$,
/0,
11/1,
$1/$,
b
b
b
Resulting Grammar, cont.
iq jq/, Ba
aBqq ji )(
}:{)( ba nnwML
103
Derivation of string abba
)$( 0 fqq )$)(0( 000 fqqqqa
)$( 0 fqqab
)$)(1( 000 fqqqqabb
)$( 0 fqqabba
abba
0q fq
/1,
00/0,
$0/$,
a
a
a
/$,
/0,
11/1,
$1/$,
b
b
b
iq jq/, Ba
aBqq ji )(
104
In general, in grammar:
wqq f
)$( 0
if and only if
w is accepted by the NPDA
105
Explanation
By construction of Grammar:
wAqq ji
)(
if and only if
in the NPDA going from to
the stack doesn’t change below
and is removed from stack
iq jq
A
iq jq/, Ba
aBqq ji )(
106
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
We have shown the procedure to convert
any NPDA to a context-free
grammar with:
which means
GM
)()( GLML
107
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Therefore
END OF PROOF
108
q0 q1
Example (Sudkamp 8.1.2)
/, Ab
/,
/, AbAa /,
}0:{}0:{)( ibaiaML iii
Language consisting solely of a’s or an equal number of a´s and b´s.
q2 /, A
