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Theory of Automata

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Theory of Automata
(CS402)
Theory of Automata
Lecture N0. 33
Reading Material
Introduction to Computer Theory
Chapter 12
Summary
Example of trees, Polish Notation, examples, Ambiguous CFG, example
Example
Consider the following CFG
S S+S|S*S|number
where S and number are non-terminals and the operators behave like terminals.
The above CFG creates ambiguity as the expression 3+4*5 has two possibilities (3+4)*5=35 and 3+(4*5)=23
which can be expressed by the following production trees
S
S
S
S
S
S
(ii)
(i)
+
*
S
S
S
S
+
3
5
*
5
3
4
4
The expressions can be calculated starting from bottom to the top, replacing each nonterminal by the result of
calculation e.g.
S
S
fi
fi
23
(i) fi
S
3
3
20
+
+
5
4
*
S
S
Similarly
fi
fi 35
(ii) fi
7
5
5
S
*
*
4
3
+
The ambiguity that has been observed in this example can be removed with a change in the CFG as discussed in
the following example
Example
S (S+S)|(S*S)|number
where S and number are nonterminals, while (, *, +, ) and the numbers are terminals.
Here it can be observed that
S fi (S+S)
fi (S+(S*S))
fi (3+(4*5)) = 23
S fi (S*S)
fi ((S+S)*S)
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Theory of Automata
(CS402)
fi ((3+4)*5) = 35
Polish Notation (o-o-o)
There is another notation for arithmetic expressions for the CFG, SS+S|S*S|number. Consider the following
derivation trees
S
S
S
S
S
S
(i)
(ii)
+
*
S
S
S
S
3
5
+
*
4
3
5
4
The arithmetic expressions shown by the trees (i) and (ii) can be calculated from the following trees,
respectively
S
S
+
*
+
3
5
(i)
(ii)
*
3
4
5
4
Here most of the S's are eliminated.
The branches are connected directly with the operators. Moreover, the operators + and * are no longer terminals
as these are to be replaced by numbers (results).
To write the arithmetic expression, it is required to traverse from the left side of S and going onward around the
tree. The arithmetic expressions will be as under
(i) + 3 * 4 5
(ii) * +3 4 5
The above notation is called operator prefix notation.
To evaluate the strings of characters, the first substring (from the left) of the form operator-operand-operand
(o-o-o) is found and is replaced by its calculation e.g.
+3*4 5 = +3 20 = 23
*+3 4 5 = * 7 5 = 35
It may be noted that 4*5+3 is an infix arithmetic expression, while an arithmetic expression in (o-o-o) form is a
prefix arithmetic expression.
S
Example
To calculate the arithmetic expression of the following tree
*
+
6
5
*
+
+
1
3
4
2
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Theory of Automata
(CS402)
It can be written as *+*+1 2+3 4 5 6
The above arithmetic expression in (o-o-o) form can be calculated as
*+*+1 2+3 4 5 6 = *+*3+3 4 5 6
= *+*3 7 5 6 = *+21 5 6 = *26 6 = 156.
Note
The previous prefix arithmetic expression can be converted into the following infix arithmetic expression as
*+*+1 2+3 4 5 6
= *+*+1 2 (3+4) 5 6
= *+*(1+2) (3+4) 5 6
= *(((1+2)*(3+4)) + 5) 6
= (((1+2)*(3+4)) + 5)*6
Ambiguous CFG
The CFG is said to be ambiguous if there exists atleast one word of it's language that can be
generated by the different production trees.
Example: Consider the following CFG
SaS|Sa|a
The word aaa can be generated by the following three different trees
S
S
S
a
a
S
S
a
S
a
a
S
S
S
a
a
a
a
Thus the above CFG is ambiguous, while the CFG, SaS|a is not ambiguous as neither the word aaa nor any
other word can be derived from more than one production trees. The derivation tree for aaa is as follows
S
a
S
a
S
a
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