Let convert be the function that on input an even

Info icon This preview shows page 2. Sign up to view the full content.

View Full Document Right Arrow Icon
simplicity, we will consider words of length just 2 bits. Let convert be the function that on input an even length string w , breaks the input into pairs of bits w 1 w 2 , w 3 w 4 , etc., and the replace each pair as follows: 01 is mapped to a , 10 is mapped to b , 00 is mapped to c and 11 is mapped to d . E.g., convert (10010111) = baad . If the input string has odd length, then convert appends a 0 to it before converting it. So, convert maps binary strings to strings over the alphabet { a, b, c, d } . Prove that regular languages are closed under conversion, i.e., if L is a regular language over the binary alphabet, then { convert ( w ): w L } is also a regular language (this time over the alphabet { a, b, c, d } . Then consider the regular expression (0+1)(01) * 1(101) * , and use your method to convert it into a regular expression R for the corresponding language over { a, b, c, d } . As part of this problem you should submit a brief explanation of your proof that regular languages are closed under this conversion, and a jflap file containing the regular expression R . In the solution of this problem you can use jflap as much as you want, to convert between automata, regular expression, etc., remove nondeterminism, simplify automata, etc. Problem 4 Let f be a function that on input a string w of odd length, removes the symbol in the middle of w . If the input string has even length, then f ( w ) = w leaves the input unchanged. E.g., f (10010) = 1010 , f (0011) = 0011 , f (1110111) = 111111 . Prove that regular languages are not closed under f , i.e., there is a regular language L such that { f ( w ): w L } is not regular. [In the solution to this problem you can use any closure property proved in class, from the textbook, previous homework assignment, or even problem 3 above. You can also use the fact that the language { a n b n : n 0 } is not regular, as proved in class. Hint: these are two big hints hidden in the list of things we told you can use.]
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern