Theory of Automata

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(CS402)
Theory of Automata
Lecture N0. 43
Introduction to Computer Theory
Chapter 16
Summary
Non-Context-Free-languages, Live Production, Dead Production, Theorem, self- embedded nonterminal,
Pumping lemma for CFLs, Examples
Non-Context-Free language
There arises a question, whether all languages are CFL? The answer is no.
Languages which are not Context-Free, are called Non-CFL.
To prove the claim that all languages are not Context-Free, the study of machines of word production from the
grammar is needed
Live production: A production of the form nonterminal Æ string of two nonterminals is called a live production.
Dead production: A production of the form nonterminal Æ terminal is called a dead production.
It may be noted that every CFG in CNF has only these types of productions.
Theorem
If a CFG is in CNF and if there is restriction to use the live production at most once each, then only the finite
many words can be generated.
It may be noted that every time a live production is applied during the derivation of a word it increases the
number of nonterminals by one.
Similarly applying dead production decreases the nonterminals by one. Which shows that to generate a word,
one more dead production are applied than the live productions e.g.
S fi XY
fiaY
fiaa
Here one live and two dead productions are used.
In general, if a CFG in CNF has p live and q dead productions then all words generated without repeating any
live production have at most (p+1) letters.
Theorem
If a CFG is in CNF with p live and q dead productions and if w is word generated by the CFG, having more than
2p letters then any derivation tree for w has a nonterminal z which is used twice, where the second z is the
descended from the first z.
It can be observed from the above theorem that generation tree of word w has more than p rows.
Self-embedded nonterminal
S
A nonterminal is said to be self-embedded, if in a given derivation
of a word, it ever occurs as a tree
X
A
descendant of itself, as shown in figure aside
A
S
a
a
X
A
A
S
a
a
b
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Here the nonterminal X is self-embedded.
Note
S
Consider the following CFG in CNF
S Æ AB
A Æ BC
B
A
C Æ AB
AÆa
BÆb
C
B
and the derivation tree of the word bbabbb
b
A
B
b
C
B
b
A
B
b
a
b
Note
The part of tree enclosed in upper triangle is identical to that enclosed in lower triangle, there is still another
option of replacing A by the same sequence of production shown in lower triangle.
The above fact provides the following pumping lemma for the CFLs.
Pumping lemma for CFLs
Theorem
If G is any CFG in CNF with p live productions, then every word w of length more than 2p can be partitioned
into five substrings as w = uvxyz, where x is not null string and v and y are not both null string.
Then all the words of the form uvnxynz, n = 1,2,3,... can also be generated by G.
Example
Consider the following CFG which is in CNF
S Æ PQ
Q Æ QS|b
PÆa
S
and a word abab generated by the above CFG with the
following derivation tree
Q
P
S
Q
a
u
Q
P
b
x
a
b
y
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Then w can be broken up as w = uvxyz where u = a, v = L, x = b, y = ab, z = L
Repeating the triangle from the second Q just as it descends from the first Q, the corresponding tree may be
expressed as follows
S
S
P
S
Q
a
u
S
Q
Q
P
Q
P
b
x
a
b
ab
y
y
Which shows that uvvxyyz=aLLbababL=ababab belongs to the language generated by the given CFG.
So, it can be generalized that words of the form uvnxynz, n=1,2,3,... belong to the language generated by the
given CFG.
Note
It may be noted that the pumping lemma is satisfied by all CFLs and the languages which don't hold this
pumping lemma, can't be Context Free languages. Such languages are non-CFLs.
Example
Consider the language
L={anbncn :n=1,2,3,...}, let the language L be Context Free language and let the word w=a200b200c200 of length
more than 2p, where p is the number of live productions of its CFG in CNF.
Note
It can be observed that no matter what choices are made for the substrings u,v,x,y and z, uv2xy2z can't belong to
L, as all the words in anbncn have
Only one substring ab
Only one substring bc
No substring ac
No substring ba
No substring ca
No substring cb
For any n=1,2,3,...
The above observations shows that if v or y is not single letter or L, then uv2xy2z may contain either two or
more substrings ab or bc or one or more substrings ac or ba or ca or cb i.e. these strings may be in the number
more than the number they are supposed to be.
Moreover, if v and y are either single letter or L, then one or two of letters a,b,c will be increased, where as the
other letter will not be increased in uv2xy2z, which shows uv2xy2z does not belong to L.
Thus pumping lemma is not satisfied. Hence L is non CFL.
It may be noted that the pumping lemma discussed for infinite regular language L, the word w can be
decomposed into three parts w=xyz, such that all words of the form xynz, n=1,2,3,..., belong to L.
Similarly, the pumping lemma discussed for CFLs can also stated as
134 Theory of Automata
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If w is a large enough word in a CF: then, w can be decomposed into w=uvxyz such that all words of the
form uvnxynz belong to L
It may be noted that proof of pumping lemma for regular languages needed that path of word w to be so large
enough so that it contains a circuit and circuit can be looped as many times as one can. The proof of the
pumping lemma for CFLs needs the derivation for w to be so large that it contains a sequence of productions
that can be repeated as many times as one can.
Moreover, the pumping lemma for regular languages does not hold for non regular language as that language
does not contain both xyz and xyyz.
Similarly pumping lemma for CFLs does not hold for non-CFL as that language does not contain both uvxyz
and uvvxyyz.
There is another difference between the pumping lemma for regular languages and that for CFLs that first one
acts on machines while other acts on algebraic structures i.e. grammar.
To achieve full power the pumping lemma for regular languages has modified by pumping lemma version II.
Similarly, full power for pumping lemma for CFLs is achieved by stating the following theorem
Theorem
If L is a CFL in CNF with p live productions then any word W in L of length more than 2p can be decomposed
as w=uvxyz s.t. length(vxy) 2p, length(x) > 0, length(v)+length(y) > 0
then the words of the form uvnxynz : n=1,2,3,... belong to L.
Example
Consider the language
L= {anbmanbm :m,n=1,2,3,...}
={abab,aabaab, abbabb, aabbaabb, aaabaaab,... }
The first version of pumping lemma for CFLs may be satisfied by L, but to apply the second version of pumping
lemma to L, let L be generated by CFG which is in CNF and has p live productions.
Consider the word decomposing w into uvxyz where length(vxy) < 2p which shows that v and y can't be single
letters separated by clumps of other letter because the separator letter is longer than the length of whole
substring vxy, which shows that uvvxyyz is not contained in L. Thus pumping lemma is not satisfied and L is
non CFL.
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Example
Consider the language EVENA i.e.
EVENA=(aa)n =a2n ={aa, aaaa, aaaaaa, ...}
The grammar for this language must be
S Æ SS|aa and its CNF will be
S Æ SS|AA, A Æ a, the PDA for this grammar will be as under
PUSH S
START
A
a
D
D
POP
ACCEPT
S
S
PUSH A
PUSH S
PUSH A
PUSH S
Its corresponding conversion form will be
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\$
START
PUSH \$
POP
A
\$
PUSH S
POP
POP
POP
S
a
a
a
PUSH S
A
PUSH A
POP
PUSH \$
\$
HERE
POP
PUSH \$
POP
POP
S
S
D
PUSH A
PUSH S
POP
PUSH S
PUSH A
\$
ACCEPT
The summary table corresponding to the above PDA in conversion form can be expressed as
FROM
TO
POP
PUSH
ROW
START
L
HERE
\$
S\$
1
L
HERE
HERE
S
SS
2
L
HERE
HERE
S
AA
3
L
HERE
A
--
4
HERE
a
S
S
5
RAED1
HERE
a
\$
\$
6
HERE
a
A
A
7
L
HERE
\$
\$
8
D
ACCEPT
\$
--
9
137 Theory of Automata
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Following are the productions defined from the summary table
S Æ Net(START, ACCEPT, \$)
Net(START, X, \$) Æ Row1 Net(HERE, Y, S)Net(Y, X, \$)
Net(HERE, X, S) Æ Row2 Net(HERE, Y, S)Net(Y, X, S)
Net(START, X, S) Æ Row3 Net(HERE, Y, A)Net(Y, X, A)
Net(READ1, X, S) Æ Row5 Net(HERE, X, S) gives four productions
Net(READ1, X, \$) Æ Row6 Net(HERE, X, \$) gives four productions
Net(READ1, X, A) Æ Row7 Net(HERE, X, A) gives four productions
Net(HERE, ACCEPT, \$) Æ Row8 Net(READ2, ACCEPT, \$)
Where X and Y are the corresponding joints
In addition to 44 productions following 9 productions complete the required CFG
Row1 Æ L
Row2 Æ L
Row3 Æ L
Row4 Æ L
Row5 Æ a
Row6 Æ a
Row7 Æ a
Row8 Æ L
Row9 Æ L
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