Homework_13 - Theory of Computing CS 3810 Fall 2009 Homework 13 Due Friday Dec 4 General guidelines You may work with other people as long as you

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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 context-free 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 one-to-one correspondence between the elements of the sets. A set is countably infinite if its elements can be put in one-to-one correspondence with integers. For example, we can create a one-to-one 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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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).

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Homework_13 - Theory of Computing CS 3810 Fall 2009 Homework 13 Due Friday Dec 4 General guidelines You may work with other people as long as you

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