Due: Monday April 07 by 11 pm. Please use the dropbox to submit your homework online. Please submit your work as some combination of PDF, plaintext, and PNG files. Please do not submit doc/docx files. You may submit a scanned copy of handwritten work, as long as it is legible and in one of the formats mentioned above. We suggest you show your work to help us award partial credit if appropriate, and for TA sanity.

For questions that ask for regular expressions, you should restrict yourself to the basic operations and abbreviations outlined in the slides: concatenation, |, *, +, ?, and character classes [...] or character classes containing all characters except for specific characters, e.g., [^abc]. You can also use abbreviations and be somewhat informal when the meaning is clear, e.g., allexceptx = a | b | c | ... | w | y | z. You should not use additional regular expression operators found in languages like perl, python, or ruby, or in unix tools like grep and sed. In particular, you cannot use not(re) as a a regular expression that matches all other regular expressions except for re.

You should do this homework individually.

1. For each of the following regular expressions, (i) give an example of two strings that can be generated by the regular expression and two that use the same alphabet but cannot be generated, and (ii) give an English description of the set of strings generated (for example, "all strings consisting of the word 'cow' followed by 1 or more 'x's and 'o's in any order"), not just a transliteration of the regular expression operations into English.
   1. (a|xy)*
   2. b(oz)+o
   3. ((ε|0)1)*
      
   
   
2. Give regular expressions that will generate the following sets of strings.
   1. All strings of a's and b's with at least 3 a's.
   2. All strings of a's and b's where b's only appear in sequences whose length is a multiple of 2 (a few examples: abba, bbbbabbaaa, and a are in this set; aba, b, ababa, and abbab are not).
   3. All strings of lower-case letters that contain the 5 vowels (aeiou) exactly once and in that order, with all other possible sequences of lower-case letters before, after, or between the individual vowels.
       
3. (Cooper & Torczon exercise 2 for section 2.2, parts (a) and (b) only). Construct a DFA accepting each of the following languages:
   1. {w in {a,b}* | w starts with 'a' and contains 'baba' as a substring}
   2. {w in {0,1}* | w contains '111' as a substring and does not contain '00' as a substring }
   
   
   
4. In The C Programming Language (Kernighan and Ritchie), an integer constant is defined as follows:

   An integer constant consisting of a sequence of digits is taken to be octal if it begins with 0 (digit zero), decimal otherwise. Octal constants do not contain the digits 8 or 9. A sequence of digits preceded by 0x or 0X (digit zero) is taken to be a hexadecimal integer. The hexadecimal digits include a or A through f or F with values 10 through 15.

   An integer constant may be suffixed with the letter u or U, to specify that it is unsigned. It may also be suffixed by the letter l or L to specify that it is long.

   (a) Write a regular expression that generates C integer constants as described above. (Of course you can break the solution down into a regular expression with several named parts if it makes things easier to write and read -- which it probably will.)
   
   (b) Draw a DFA that recognizes integer constants as defined by your solution to part (a). You may draw this directly; you don't need to formally trace through an algorithm for converting a regular expression to a NFA and then constructing a DFA from that. However, you might find it useful to do so at least partially.
   
   Hint: You might find it helpful to alternate between designing the DFA and writing the regular expressions as you work on your solution.
    
5. A commment in C is a sequence of characters /* ... */. Write a set of regular expressions that generate C-style comments. You can restrict the alphabet to lower-case letters, digits, spaces, newlines (\n), carriage returns (\r) and the characters * and /. Also, remember that in C comments do not nest (i.e., a */ marks the end of a comment no matter how many times /* appears before it.)
   
   Hint: be careful about what is included in the ... between /* and */.