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Theory of Automata

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Theory of Automata
(CS402)
Theory of Automata
Lecture N0. 44
Reading Material
Introduction to Computer Theory
Chapter 18
Summary
Decidability, whether a CFG generates certain string (emptiness), examples, whether a nonterminal is used in
the derivation of some word (uselessness), examples, whether a CFL is finite (finiteness), example, whether the
given string is generated by the given CFG (membership), example, parsing techniques, top down parsing,
example
Decidablity
Following are the decidable problems w.r.t. CFG
Whether or not the given CFG generates any word? Problem of emptiness of CFL.
Whether or not the given CFG generates the finite language? Problem of finiteness.
Whether or not the given string w can be generated by the given CFG? Problem of membership.
Following are algorithms showing that the answers to the above three questions are yes.
Algorithm 1 (Emptiness)
If the given CFG contains a production of the form S∆L, then obviously the corresponding CFL is not empty. If
the CFG contains the production of the form St, where t is a terminal or string of terminal then t is a word of
the corresponding CFL and CFL is not empty.
If the CFG contains no such production then
For each nonterminal N with Nt, pick one production for N (if there are more than one) and replace N by t in
the right side of each production wherever it lies. Remove all such productions from the CFG. Doing so the
CFG will be changed, it will generate atleast one word of the old CFL.
Repeat the process until either it eliminates S or no new nonterminal is eliminated.
If S has been eliminated then CFG generates some words otherwise not.
Example
SAB, A BSB, BCC
CSS
Aa|b
C b|bb
Step (1). Picking Aa, C b, it can be written as
SaB
ABSB
Abb
BaaS
Bbb
CSS
Step (1). Picking Bbb and Abb, it can be written as
Sabb
AbbSbb
BaaS
CSS
Since Sabb has been obtained so, abb is a word in the corresponding CFL.
To determine whether the nonterminal X is ever used in the derivation of word from the given CFG, following
algorithm is used
Algorithm 2 (Uselessness)
Find all unproductive nonterminals (the nonterminal is unproductive if it cannot produce a string of terminals).
Eliminate all productions involving unproductive nonterminals.
Paint all X's blue.
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Theory of Automata
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If any nonterminal is in the left side of the production with any blue nonterminal in the right side, paint that
nonterminal blue and paint that nonterminal blue at all occurrences of it throughout the grammar.
Repeat step 4 until no new nonterminal is painted.
If S is blue then X is useful member of CFG, otherwise not.
Example
Consider the following CFG
SAba | bAZ | b
AXb | bZa
BbAA
XaZa|aaa
ZZAbA
To determine whether X is ever used to generate some words, unproductive nonterminals are determined. Z is
unproductive nonterminal, so eliminating the productions involving Z.
SAba|b
AXb
BbAA
Xaaa
X is blue, so A is blue. Thus B and S are also blue. Since S is blue so X can be used to generate certain word
from the given CFG.
Note:  It may be noted that a nonterminal is called useless if it cannot be used in a production of some word.
Following algorithm is used to determine whether the given CFG generate the finite language
Algorithm 3 (Finiteness)
Determine all useless nonterminals and eliminate all productions involving these nonterminals.
For each of the remaining nonterminals, determine whether they are self-embedded (using the following steps).
Stop if a self-embedded nonterminal is discovered.
To test whether X is self-embedded
Change all X's on the left side of the productions into a Greek letter Ψ and keep all X's on the right side as such.
Paint all X's blue.
If Y is any nonterminal on the left side of the production with X in the right side, then paint Y blue.
Repeat step (c) until no new nonterminal is painted.
If Ψ is painted, then the X is self-embedded, otherwise not.
If any nonterminal, left in the grammar, after step 1, is self-embedded then the language generated is infinite,
otherwise finite.
Example
Consider the CFG
SABa|bAZ|b
AXb|bZa
BbAA
XaZa|bA|aaa
ZZAbA
Here the nonterminal Z is useless, while all other are used in the derivation of some word. So eliminating the
productions involving Z
SABa|b
AXb
BbAA
XbA|aaa
Starting with nonterminal X. Replacing X on left side of the production by Ψ
SABa|b
AXb
BbAA
Ψ bA|aaa
X is blue so A is blue and so Ψ is blue. Since A is blue, so B is blue and so S is blue. Since Ψ is blue so X is
self-embedded and hence the CFG generates the infinite language.
To determine whether a string is generated by the given CFG, following algorithm is used
Algorithm 4 (The CYK algorithm)
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This algorithm was invented by John Cocke and later was published by Tandao Kasami and Daniel H. Younger.
Convert the given CFG in CNF.
Let the string x under consideration has the form x=x1x2x3...xn where all xis may not be different. List all the
nonterminals in the given CFG, say, S, N1,N2, ...
List the nonterminals that generates single letter substrings of x i.e.
Substring
All producing nonterminals
x1
N...
x2
N...
x3
xn
N...
List the nonterminals that generates substrings of length 2 i.e.
Substring
All producing nonterminals
x1 x2
N...
x2 x3
N...
x3 x4
xn-1 xn
N...
Similarly, list of nonterminals generating substring of x of length 3
Substring
All producing nonterminals
x1 x2 x3
N...
x2 x3 x4
N...
x3 x4 x5
xn-2 xn-1 xn
N...
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Theory of Automata
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Continuing the process, the nonterminals that generate x1x2x3...xn can be determined as
Substring
All producing nonterminals
x1 x2 x3...xn
N...
If S is among the set of all producing nonterminals, then x can be generated by the CFG, otherwise not.
Example
Consider the following CFG in CNF
SAA
AAA
Aa
Let x = aaa. To determine whether x can be generated from the given CFG let x = x1x2x3 where x1 = x2 = x3 = a
According to CYK algorithm, the list of nonterminals producing single letter double letter substrings of x and
the string x itself, can be determined as follows
Substring
All producing nonterminals
x1 = a
A
x2 = a
A
x3 = a
A
x1 x2
S, A
x2 x3
S, A
x = x1 x2 x3
S, A
Since S is in the list of producing nonterminals, so aaa can be generated by the given CFG.
Parsing Techniques
Recall the CFG for arithmetic expression
SS+S|S*S|number
It was observed that the word 3+4*5 created ambiguity by considering its value either 23 or 35. To remove this
ambiguity, the CFG was modified to
S(S+S)|(S*S)|number
There arises a question that whether a new CFG can be defined without having parentheses with operator
hierarchy (i.e. * before +)? The answer is yes. Following is the required PLUS-TIMES grammar
SE, ET+E|T, TF*T|F, F(E)|i
Where i stands for any identifier i.e. number or of storage location name (variable). Following is the derivation
of i+i*i
S fiE
fiT+E
fiF+E
fii+E
fii+T
fii+F*T
fii+i*T
fii+i*F
fii+i*i
Parsing of word
Definition
The process of finding the derivation of word generated by particular grammar is called parsing.
There are different parsing techniques, containing the following three
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Top down parsing.
Bottom up parsing.
Parsing technique for particular grammar of arithmetic expression.
Top down parsing
Following is an example showing top down parsing technique
Example
Consider PLUS-TIMES grammar and a word i+i*i.
As can be observed from the name of top down parsing, the parsing starts from the nonterminal S and the
structure similar to that of total language tree is developed. The branches of the tree are extended till the
required word is found as a branch.
Some unwanted branches ( the branches that don't lead to the required word) are dropped. For the word i+i*i ,
the total language tree can be started as
S
E
Which can further be extended to
T
T+E
F+E F*T
F
F*T+E
S
E
T
T+E
F*T
F
F+E
F*T+E
(E)*T
i*T
(6)
(5)
i*T+E
(E)*T+E
(2)
(E)
i
(1)
i+E
(E)+E
(7)
(8)
(4)
(3)
Dropping the unwanted branches 1,3,5,7and 8
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S
E
T
T+E
F*T
F+E
F*T+E
i*T
(6)
i+E
i*T+E
(4)
(2)
Since first two letters in branches 2 and 6 are not that in i+i*i , so 2 and 6 can be dropped and using left most
derivation the nonterminal T is replaced as
S
E
T+E
F+E
i+E
i+T
i+T+E
i+F+E
i+F
i+F*T+E
i+F*T
(12)
(10)
(9)
(11)
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Theory of Automata
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since (9) gives more than five letters and (10) contains two + so (9) and (10) are dropped and left most
nonterminal F is replaced as
S
E
T+E
F+E
i+E
i+T
i+F
i+F*T
i+i*T
i+i
i+(E)*T
i+(E)
(16)
(14)
(13)
(15)
(13), (15) and (16) are again unwanted, so it can be written as
S
E
T+E
F+E
i+E
i+T
i+F*T
i+i*T
i+i*F
i+i*F*T
i+i*(E)
i+i*i
The above tree confirms the required derivation
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S fiE
fiT+E
fiF+E
fii+E
fii+T
fii+F*T
fii+i*T
fii+i*F
fii+i*i
Note
It can be noted that Bottom Up Parsing can be determined similar to that of Top Down Parsing with the change
that in this case, the process is started with the given string and the tree is extended till S is obtained.
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