COGS Q250, Fall 2008
Exam 2: Theory of Computation
11/04/08
Name: ____________________________
1. (10
pts.)
Define
an
edge
in
a
sequence
of
binary
digits
to
be
any
occurrence
of
a
1
following
one
or
more
consecutive
0’s,
or
any
occurrence
of
a
0
following
one
or
more
consecutive
1’s
(e.g.,
the
binary
string
000111100
has
two
edges,
indicated
by
bold
typeface).
Design
a
finite
state
transducer
that
outputs
whether
zero,
one,
or
more
than
one
edges
have
occurred
thus
far
in
a
sequence
of
binary
inputs.
The
output
alphabet
should
be
{0,
1,
more}.
2. (5
pts.)
Trace
the
computation
performed
by
your
finite
state
transducer
from
(1)
for
the
following
input
string.
Use
the
table
method
we
discussed
in
class
to
keep
track
of
the
state
visited
and
the
output
produced
at
each
step.
Input:
0001100
State q0 q1 q1
q1
q3 q3 q5 q5 Input 0 0 0 1 1 0 0 Output 0 0 0 1 1 more more 3. (10
pts.)
For
each
of
the
following,
decide
whether
the
statement
is
true
or
false,
and
BE
SURE
TO
EXPLAIN
YOUR
ANSWER,
as
you
will
not
receive
full
credit
unless
you
do
so.
a. If
the
transition
function
for
a
Turing
machine
is
a
total
function,
it
is
possible
for
that
machine
to
halt
for
some
input.
False. A Turing machine will only halt when it reaches a state/input pair for which no transition is specified. If the transition function is a total
function, that means that a transition is specified for every state/input pair. Thus, the Turing machine will never halt no matter what input it receives. b. For
any
set
A,
the
number
of
elements
in
A
is
always
less
than
the
number
of
elements
in
pow(A).
True. If a set A has n elements, then pow(A) has 2n elements. Since 2n > n for any natural number n, pow(A) will always have more elements than A. c. Any
language
that
can
be
recognized
by
a
Turing
machine
can
also
be
described
using
a
regular
expression.
False. Turing machines can recognize any of the recursively enumerable languages, while regular expressions can be used to describe only the regular languages. Since the regular languages are a proper subset of the recursively enumerable languages, there are languages that can be recognized by a Turing machine which cannot be described using a regular expression. 4. (10
pts.)
Suppose
I
design
a
new
kind
of
machine,
called
a
Williams
machine,
with
the
following
properties.
It
has
two
finite
state
controllers
with
separate
sets
of
states,
such
that
at
any
moment
the
overall
state
of
the
machine
is
described
by
the
states
of
each
of
its
finite
state
controllers.
At
each
time
step,
the
machine
receives
a
single
input
symbol
and,
depending
on
the
states
that
its
finite
state
controllers
are
in,
each
finite
state
controller
transitions
into
a
next
state.
The
machine
does
not
produce
any
output.
I’m
interested
in
developing
a
mathematical
description
for
the
Williams
machine.
So
far
I
have
the
sets
A
and
B,
which
are
the
sets
of
states
for
each
of
the
finite
state
controllers;
states
a0
∈
A
and
b0
∈
B,
which
are
the
starting
states
for
the
two
controllers;
and
the
input
alphabet
I.
The
one
remaining
component
that
I
need
to
specify
is
the
transition
function
f,
which
describes
the
transitions
made
by
both
finite
state
controllers
when
an
input
is
received.
Describe
f
using
the
function
notation
that
we
discussed
in
class.
(Hint:
your
answer
should
be
of
the
form
“f:
__________”,
where
you
will
need
to
fill
in
the
blank.)
f: A × B × I → A × B
5. (15
pts.)
Give
regular
expressions
to
describe
the
following
sets
of
strings.
a. All
strings
of
a’s,
b’s
and
c’s
with
at
most
one
a.
(b+c)*(P+a)(b+c)* b. All
binary
strings
that
have
exactly
two
1’s
in
the
first
three
symbols
and
contain
at
least
three
0’s
in
the
entire
string.
(110+101+011)(0+1)*0(0+1)*0(0+1)* 6. (15
pts.)
Give
regular
expressions
to
describe
the
sets
of
strings
recognized
by
the
following
finite
state
automata.
Assume
that
any
unspecified
transitions
go
immediately
to
a
trap
state.
a. (10+01)*00(11)* b. next
state
state
0
1
q0
‐‐
q1
q1
q0
q2
q2
‐‐
q3
q3
q4
q5
q4
q2
‐‐
q5
q2
‐‐
q0
and
q2
are
both
accepting
states.
(10)* + (10)*11(100+110)* 7. (15
pts.)
Design
finite
state
automata
to
recognize
the
following
regular
expressions.
You
are
free
to
assume
that
any
unspecified
transitions
go
immediately
to
a
trap
state,
or
you
may
specify
the
trap
state
explicitly.
a. (11+101)*00*
b. 10(0+1)*1(0+1)*1(0+1)*01
8. (20
pts.)
Describe
what
the
final
tape
will
look
like
when
the
following
Turing
machine
is
run
with
any
binary
string
as
input.
(Hint:
Although
it
is
not
required,
it
would
probably
be
a
good
idea
to
trace
the
machine’s
behavior
on
one
or
more
input
strings.
That
should
help
you
figure
out
what
the
machine
does,
as
well
as
provide
more
grounds
for
partial
credit.
Also,
note
that
short
input
sequences
will
probably
suffice
to
get
a
general
idea
of
what
the
machine
does.)
The final contents of the tape will be the reverse of the input string, with the head positioned over the leftmost non­blank cell of the tape.