CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 9 Mälardalen University
CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 6 Mälardalen University
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1 CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 6 Mälardalen University 2010
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CDT314
FABER
Formal Languages, Automata and Models of Computation
Lecture 6
Mälardalen University
2010
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Content
Midterm againContext-Free LanguagesPush-Down Automata, PDANPDA: Non-Deterministic PDAFormal Definitions for NPDAs NPDAs Accept Context-Free LanguagesConverting NPDA to Context-Free Grammar
Midterm
• 5 out of 13 passed…• Many failed on basic definitions
• Is something wrong with the course format?• Scheduled exercises• Don’t hesitate to ask for help!• Do labs, exercises etc.• Review lecture material before lectures• Strongly recommended to get a book if you
don’t have one• It doesn’t get easier…
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4
JFLAP demo
Concerning examination in the course:Exercises are voluntaryLabs are voluntaryMidterms are voluntaryLectures are voluntary…All of them are recommended!
http://www.cs.duke.edu/csed/jflap/movies
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Regular Languages
}{ nnba }{ Rww
Context-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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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 (certain kind of) information.
zxy
POP reading symbol PUSH writing symbol
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The States
Input
symbol
Pop old- reading
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 or a stack bottom symbol such as $)
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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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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
symbol)stack initial :($
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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
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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 symbol
current 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
38q0 q1 q2 q3
Input
Stack
a a a b b b$
aaTime 4:
Example Instantaneous Description
$),,( 1 aaabbbq
a
/, /, ab $/$,
/, abaa /,
),,( stackinputremainingstateurrentc
39q0 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 example
q0 q1 q2 q3 /, /, ab $/$,
/, abaa /,
),,( contentstackcurrentinputremainingstatecurrent
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
q0 q1 q2 q3 /, /, ab $/$,
/, abaa /,
),,( contentstackcurrentinputremainingstatecurrentA computation example
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
q0 q1 q2 q3 /, /, ab $/$,
/, abaa /,
),,( contentstackcurrentinputremainingstatecurrentA computation example
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
q0 q1 q2 q3 /, /, ab $/$,
/, abaa /,
),,( contentstackcurrentinputremainingstatecurrentA computation example
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
q0 q1 q2 q3 /, /, ab $/$,
/, abaa /,
),,( contentstackcurrentinputremainingstatecurrentA computation example
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
q0 q1 q2 q3 /, /, ab $/$,
/, abaa /,
),,( contentstackcurrentinputremainingstatecurrentA computation example
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
q0 q1 q2 q3 /, /, ab $/$,
/, abaa /,
),,( contentstackcurrentinputremainingstatecurrentA computation example
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$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
q0 q1 q2 q3 /, /, ab $/$,
/, abaa /,
),,( contentstackcurrentinputremainingstatecurrentA computation example
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For convenience we write
$),,($),,( 30 qaaabbbq
$),,($),,($),,($),,(
$),,($),,($),,(
$),,($),,(
3222
111
10
qqabqaabbq
aaabbbqaaabbbqaaabbbq
aaabbbqaaabbbq
),,( contentstackcurrentinputremainingstatecurrent
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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
)(MLba nn
q0 q1 q2 q3 /, /, ab $/$,
/, abaa /,NPDA M
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}0:{)( nbaML nnTherefore:
q0 q1 q2 q3 /, /, ab $/$,
/, abaa /,
NPDA M
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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 G to a NPDA M
with L(G) = L(M)
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Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Proof - Step 2:
Convert any NPDA M to a context-free grammar G with L(M) = L(G)
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Converting Context-Free Grammars
to NPDAs
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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/, $/$,
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The NPDA simulates
leftmost derivations of the grammar
L(Grammar) = L(NPDA)
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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
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q0 q1 2q
Input
Stack
a ab
Time 1
b
$S
S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
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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
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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
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q0 2q
Input
Stack
a ab
Time 7
b
$b
Ta
q1S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
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q0 2q
Input
Stack
a ab
Time 8
b
$ba
q1S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
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q0 2q
Input
Stack
a ab
Time 9
b
$b
q1S/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
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q0 q1 2q
Input
Stack
$
a ab
Time 10
b
acceptS/, $/$,
/,
/,
/,
/,
T
TaT
bS
aSTbS
/,
/,
bb
aa
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In general
Given any grammar Gwe can construct a NPDA M with
)()( MLGL
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q0 q1 2qS/, $/$,
wA /, /, aa
For any productionwA
For any terminala
Constructing NPDA M from grammar GTop-down parser
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Grammar G generates string w
if and only if
NPDA M accepts w
)()( MLGL
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For any context-free language
there is an NPDA
that accepts the same language
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Which means
Languages
Accepted by
NPDAs
Context-
FreeLanguages(Grammars)
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Converting NPDAsto
Context-Free Grammars
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For any NPDA M
we will construct
a context-free grammar G with
)()( GLML
81
in NPDA M
abcABCabcS
Input processed Stack contents
terminals variables
GA derivation in Grammar
The grammar simulates the machine
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Some Simplifications
First we modify the NPDA so that
• It has a single final state qf and
• It empties the stack
when it accepts the input
Original NPDA Empty Stack
fq
83
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.
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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.
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/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
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The Grammar Construction
)( jiBqq
In grammar G
Terminals:
Input symbols of NPDA
states
Stack symbol
Variables:
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iq jq/, BaFor each transition:
we add production: aBqq ji )(
88
For each transition:
we add production: ))(()( klljki DqqCqqaBqq
iq jqCDBa /,
for all states qk , ql
89
Start Variable )$( fo qq
Stack bottom symbol
Start state (Single) Final state
90
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
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)$)(1(|)$)(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 ,
92
Grammar production: )$( 0 fqq
0q fq
/1,
00/0,
$0/$,
a
a
a
/$,
/0,
11/1,
$1/$,
b
b
biq jq/, Ba
aBqq ji )(
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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()$(
00000
00000000
fffff
ff
qqqqbqqqqbqq
qqqqbqqqqbqq
)1)(1(|)1)(1()1(
)1)(1(|)1)(1()1(
00000
00000000
fffff
ff
qqqqbqqqqbqq
qqqqbqqqqbqq
}:{)( ba nnwML },{ ba
))(()( klljki DqqCqqaBqq
lk qqstatesallfor ,
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)$)(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
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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
96
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 )(
97
In general, in grammar:
wqq f
)$( 0
if and only if
w is accepted by the NPDA
98
Explanation
By construction of Grammar:
wAqq ji
)(
if and only if
in the NPDA going from qi to qj
the stack doesn’t change below
and A is removed from stack
iq jq/, Ba
aBqq ji )(
99
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
We have shown the procedure to convert
any NPDA M to a context-free
grammar G with L(M) = L(G)which means
100
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Therefore
END OF PROOF
101
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
