CSE 105: Introduction to the Theory of Comptuation
Fall 2010
Problem Set 6
Instructor: Daniele Micciancio
Due on:
Fri. Nov 19, 2010
Problem 1
Consider a context free grammar
G
over the alphabet
{
0
,
1
}
. Any such a grammar can be described as a
string of symbols (very much like computer programs can be stored in text files) over the alphabet
Σ =
{
0
,
1
, A,
[
,
]
, >,
;
}
, where grammar variables are represented as elements of an array
A
[
n
]
. Prove that the set
of all context free grammars as described above is regular by giving a DFA
M
for it.
Solution:
The set of context free grammars is accepted by the following DFA:
Alternatively, this set can be described by the regular expression “(A[(0+1)*]>(0+1+A[(0+1)*])*;)*”.
The given automaton was obtained from the regular expression, using jflap, by converting it first to an NFA,
then removing nondeterminism to get a DFA, and finally simplifying the result.
Problem 2
The language
L
=
L
(
M
)
described in problem 1 is a set of string over the 7 symbol alphabet
Σ =
{
0
,
1
, A,
[
,
]
, >,
;
}
. Consider the function
φ
(0) = 000
, φ
(1) = 001
, φ
(
A
) = 010
, φ
([) = 011
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 Fall '99
 Paturi
 Formal language, Halting problem, Contextfree grammar, Context free Grammars

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