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Solve Problems A-C on paper, and hand them in by noon on Friday. Solve Problems D-F in a single Python file Your code should include comments to explain any obscure or tricky bits. It should also include demonstration code. Hand in electronically, by dropping the file in your hand-in folder on the COURSES file server, by 11:59 PM on Friday.
A. Working over Σ = {a, b, c, d}, draw an NFA that recognizes the same language as the B. Our regular expressions conspicuously leave out the complementation operation, even though regular languages are closed under complementation. The reason is that complemen- tation can be implemented in terms of the other regular operations. Here’s a tiny example: Working over the alphabet Σ = {a, b}, find a regular expression to describe the complement of the language described by the regular expression a∗.
C. The Hamming distance H(w, x) between two bit strings w and x is defined as follows. If |w| = |x|, then H(w, x) = ∞. If |w| = |x|, then H(w, x) is the number of bits in which w and x differ. For example, H(00010, 10111) = 3. For any set A of bit strings, define N2(A) to be the set of bit strings within Hamming distance 2 of A: N2(A) = {w : ∃x ∈ A such that H(w, x) ≤ 2}.
Prove that if A ⊆ {0, 1}∗ is regular, then so is N2(A). (Hint: If A = L(M ), where M has states Q, then construct an NFA with states Q × {0, 1, 2}. Use the extra state information to track how many “errors” have occurred thus far.) D. Come up with a Python regular expression that describes mis-capitalized words. For the sake of this problem, a word is said to be mis-capitalized if it consists of two or more letters and any letter after the first one is upper-case. You may assume that only alphabetical characters appear in words. For example, when I feed your regular expression to re.findall(yourRegExp, ’This is okay tHis IS nOT Okay.’) I should get a result of [’tHis’, ’IS’, ’nOT’].
E. In this problem we learn how to harvest e-mail addresses from texts such as web pages.
You must promise never to use this power for evil.
An e-mail address such as consists of a local part, supersnake, and a hostname, The local part is a string made of one or more characters from this set: upper- and lower-case English letters, the digits 0 through 9, the characters !, #, $, %, &, ’, *, +, -, /, =, ?, ^, _, ‘, {, |, }, ~, and the period . The period is allowed to be neither the first nor the last character in the local part. The hostname is a string made of one or more characters from this set: lower-case English letters, the digits 0 through 9, the period ., and the hyphen -. The local part and the hostname are separated by a single @. (There are a few more rules to real e-mail addresses, but this is good enough for our purposes.) Design a Python regular expression that matches e-mail addresses as just specified. That is, when I call the matches function (from the Python REs tutorial) with your regular expression and a string w, it returns True if and only if w is a valid e-mail address. (In practice, you would probably call re.findall with a web page, to harvest all of the addresses. But I’m not asking F. I like to write my dates in the format yyyy/mm/dd, but sometimes I accidentally write mm/dd/yyyy because that’s how I was raised. Every date that I write is in one of these two formats. Having lived through the tumultuous Y2K apocalypse, I always write four-digit years.
When appropriate, I write months and days as single digits. I never write nonsense months or days such as month 13 or day 42, so don’t bother checking that.
Write a Python function fixdates that takes a string of text as input, uses a regular ex- pression to fix all dates in the string to my liking, and outputs the fixed string. (Hint: You need to use substitutions and multiple groups.)


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