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Theory of Computing
Homework 13
CS 3810, Fall 2009
Due Friday, Dec 4
General guidelines: You may work with other people, as long as you write up your
solution in your own words and understand everything you turn in.
Make sure to justify
your answers—they should be clear and concise, with no irrelevant information.
Please put your name and net id on your homework!
1.
Select a topic from the list of topics on the course website beyond finite automata.
You may select any topic after the heading contextfree languages.
Write a half page
explanation of your topic that should be understandable to any student in the class.
Then
create two problems on the topic and work out solutions.
These might be used by
someone teaching the course as examples.
Then create eight more homework exercises
that would help a student understand the topic.
To illustrate what is intended the following are three write ups by students.
Diagonalization
If we want to compare the cardinalities of two finite sets, we can simply count the
elements in each set and compare the number of elements.
For infinite sets, such as the set of integers or the set of real numbers, we can compare the
cardinalities of two sets by finding a onetoone correspondence between the elements of
the sets.
A set is countably infinite
if its elements can be put in onetoone correspondence with
integers.
For example, we can create a onetoone correspondence between the set of all even
numbers and the set of all integers.
Therefore, the set of all even numbers is countably
infinite.
Even
numbers
:
02468
…
Integers:
0
1234
…
If a set is countable, then we can list the elements of the set or, equivalently, we can put
the elements of the set in a table.
We can show that certain sets are not countably infinite by diagonalization
.
For example, diagonalization can be used to show that the set of all real numbers between
0 and 1 is not countably infinite.
To do this, try to put all the elements in the following
table:
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View Full Document Theory of Computing
Homework 13
CS 3810, Fall 2009
Due Friday, Dec 4
0.
1
01010
0.
0
1
0101
0.
1
1
1
111
0.
0
0
1
0
01
0.
1
1
0
1
1
0
0.
0
0
0
0
0
1
By selecting all the elements along the diagonal beginning with the digit in the tenths
place, namely 111011, we have selected an element that has not already been listed in the
table.
This shows that all the real numbers cannot be listed in a table.
Thus, the set of
real numbers is not countably infinite.
There exist a countably infinite number of computer programs, or names, but there are
too many sets, or functions to be counted.
Thus, there exist functions that cannot be
computed by a computer program.
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This note was uploaded on 11/07/2009 for the course CS 3810 taught by Professor Hopcroft during the Fall '07 term at Cornell University (Engineering School).
 Fall '07
 HOPCROFT

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