6.045J/18.400J: Automata, Computability and Complexity
Prof. Nancy Lynch
Practice Quiz 1Solutions
Elena Grigorescu
Please write your name in the upper corner of each page.
PQ11
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentProblem 1
:
True or False (20 points)
Full credit will be given for correct answers. If you include justification for
your answers, you may obtain partial credit for incorrect answers.
In all parts of this question, the alphabet
Σ
is
{
0
,
1
}
.
1. True or False: A DFA with
n
states must accept at least one string of length greater than
n
.
False. The DFA might accept no strings at all.
2. True or False: A DFA with
n
states that accepts an infinite language must accept at least one string
x
such that
2
n <

x
 ≤
3
n
.
True. Follows by a pumping lemma argument. The DFA has to accept at least one string longer
than
3
n
. Now, pump this down. In each step, we pump down by at most
n
, and at least
1
. It follows
that if one does this sufficiently many times, one gets a string with length in the range [2n.
. . 3n].
3. If
R
is a regular language and
L
is some language, and
L
∪
R
is a regular language, then
L
must be a
regular language.
False. Let
R
= Σ
∗
and
L
be some nonregular language.
4. If
F
is a finite language and
L
is some language, and
L

F
is a regular language, then
L
must be a
regular language.
True.
L
= (
L

F
)
∪
F
′
, where
F
′
is some subset of
F
, and is therefore finite. The statement now
follows from the closure under union of regular languages.
5. True or False: Define
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 NancyLynch
 Formal language, Regular expression, Regular language, Nondeterministic finite state machine

Click to edit the document details