CSE 401/M501 23au Homework 1 - Regular expressions / DFAs

Due: Thursday, October 5 by 11 pm. Please use Gradescope (gradescope.com, also linked from the CSE 401/M501 resources web page) to submit your homework online. Gradescope's web site has several instructional videos and help pages to outline the process, and this guide has specific information about scanning and uploading pdf files containing assignments (although this guide is now somewhat old, so you may have better methods of creating pdf files on your phone using more recent software).

We suggest you show your work to help us award partial credit if appropriate, and for TA sanity. You should do this assignment individually.

For questions that ask for regular expressions, you must restrict yourself to the basic operations and abbreviations outlined in the lecture slides: concatenation, |, *, +, ?, and character classes [...]. You may also specify character classes containing all characters except for specific characters, e.g., [^abc] means all characters in the entire alphabet except for a, b, and c. You can also use abbreviations and be somewhat informal when the meaning of the full regular expression it abbreviates is clear and unambiguous, e.g.,   allexceptx = a|b|c|...|w|y|z. Note that you cannot use something like "allexceptx" in a solution without first defining it as a name for a particular regular expression, as in the above example. You may not use additional "regular expression" operations found in libraries or languages like perl, python, java, javascript, ruby, etc., 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.

Sometimes a regular expression might need to include a literal character like * or +, which is also a regular expression operator. Use some simple convention to distinguish between the two. We suggest underlining literal characters, like * to distinguish them from operators like *. Please do not use backslash characters as escape characters, they are \v\e\r\y\h\a\r\d\t\o\r\e\a\d.

When drawing DFAs, you should not include an explicit "error state" or show transitions to this state for situations where the DFA rejects its input. We will assume that if there is no transition defined from a state on a particular input character then it corresponds to a transition to an error state, and the input is rejected. Diagrams are much simpler if we do not include this extra state and its many, many incoming transitions.

Feel free to submit clear, hand-written diagrams for DFAs. However, if you want to use computer diagraming tools, here are a couple of suggestions if you don't already have a favorite:

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 like "a 'c' followed by a 'o' followed by a 'w' followed by ..."). In other words, aim to describe the set of strings and not the details of how the regular expressions produce them.
    1. (a|xy)*
    2. b(oz)+o
    3. ((ε|0)1)*

  2. Give regular expressions that 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 any b's only appear in sequences of b's whose length is a multiple of 2 (a few examples: abba, bbbbabbaaa, a and ε are in this set; aba, b, abab, 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 in between the individual vowels.
       
  3. (Cooper & Torczon exercise 2 for section 2.2, parts (a) and (b) only. p. 80 2nd ed, p. 82 3rd ed). 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 }

    You do not need to go through the full subset construction to produce this DFA from a NFA, although you can use some of those ideas to help you produce your answer.

  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.)

    (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. Also, you may discover that the English description given above is not as precise as it might seem, so you may need to explore some edge cases more carefully.
     
  5. Comments can be included in html documents. Any text that starts with the 4-character sequence <!-- and ends with the separate 3-character sequence --> is treated as a comment, and a comment can span multiple lines. Examples: 
           <!-- this is a comment -->

          <!--  and
             so
    is
            this
          -->


    (a) Write a regular expression that generates html-style comments. You can restrict the alphabet to lower-case letters, digits, spaces, newlines (\n), carriage returns (\r) and the characters -, !, <, and >. Remember that these comments do not nest (i.e., the sequence --> marks the end of a comment no matter how many times <!-- appears before it). Any of the characters -, !, <, and > can appear in a comment in any combination, except that the three character sequence --> terminates the comment.

    (b) Draw a DFA that recognizes html comments as defined by your solution to part (a). As with the previous problems, you may draw this directly without tracing through the steps of the regexp->NFA->DFA algorithms.

    Hints: be careful about what is included between <!-- and -->. As with the previous problem, you may find it helpful to alternate between the regular expressions and DFA as you work on the problem.

    Note: the actual html standard does not allow the sequence <!-- to appear in the middle of a comment. We omitted that restriction here to simplify the problem.